Loading...

A thesis presented to the faculty of the Russ College of Engineering and Technology of Ohio University

In partial fulfillment of the requirements for the degree Master of Science

Kiran R. Suravaram August 2007

This thesis titled MODELING THE INTERARRIVAL TIMES FOR NON-SIGNALIZED FREEWAY ENTRANCE RAMPS

by KIRAN R. SURAVARAM

has been approved for the Department of Civil Engineering and the Russ College of Engineering and Technology by

Helmut T. Zwahlen Russ Professor Emeritus of Industrial and Systems Engineering

Dennis Irwin Dean, College of Engineering and Technology

Abstract SURAVARAM, KIRAN R, M.S., August 2007, Civil Engineering MODELING THE INTERARRIVAL TIMES FOR NON-SIGNALIZED FREEWAY ENTRANCE RAMPS (270 pp.) Director of Thesis: Helmut T. Zwahlen The nature of the headways or the interarrival times (IATs) on non-signalized freeway entrance ramps was studied. Cumulative interarrival time distributions for three different entrance ramps with different configurations were developed which can be used in digital computer traffic simulation models. Data collected in the study “Improved Work Zone Design Guidelines and Enhanced Model for Travel Delays in Work Zones” for the Ohio Department of Transportation were used in this study. The IAT distributions for the entrance ramps were compared with the IAT distributions of the lanes for freeways and were found to be close so that the IATs for the selected freeway lanes can be used to approximate the IATs for the entrance ramps. The cumulative IAT distributions obtained for freeways were extrapolated using the hyperbolic relationship to cover low and high traffic volumes outside the empirical observed data range. The IAT distributions were further investigated whether they could be described accurately by a negative exponential, an Erlang with K=2, 3 or a Pearson type III distribution, but no close matches were found. Approved: _____________________________________________________________ Helmut T. Zwahlen Russ Professor Emeritus of Industrial and Systems Engineering

Acknowledgments First and the foremost I would like to thank Dr. Helmut T. Zwahlen for being my advisor and helping me out to get to this point. This effort would not have been possible with out his help, support and constant motivation for the whole period of my graduate study. I thank all my thesis committee members Dr. Gayle Mitchell, Dr. Lloyd Herman and Dr. Nick Pavel for their time, help and support. Special thanks to Dr.Mitchell for her constant academic support and help in my graduate study. Many thanks to Rebecca Gill for her affection and administrative support. I would like to thank my colleagues Erdinc Oner, Housila Tiwari, Amey Gowaikar and David Allard for their help. More thanks to my friends Raghu Gorthy and Avinash Patnala for their timely help and encouragement.

5 Table of Contents Page Abstract ............................................................................................................................... 3 Acknowledgments............................................................................................................... 4 List of Tables ...................................................................................................................... 6 List of Figures ................................................................................................................... 10 1 Introduction............................................................................................................... 15 2 Statement of the problem .......................................................................................... 18 3 Objective of the study ............................................................................................... 19 4 Scope of work ........................................................................................................... 20 5 Literature review....................................................................................................... 21 5.1 Headway distributions ...................................................................................... 21 5.2 Headway distributions for urban traffic............................................................ 29 5.3 Headway distributions for low and high traffic volumes.................................. 32 6

Methodology ............................................................................................................. 34 6.1 Modeling the interarrival (IATs) times for the entrance ramps........................ 34 6.1.1 Data collection .............................................................................................. 34 6.1.2 Data analysis ................................................................................................. 36 6.1.3 I-71 South to I-270 West entrance ramp....................................................... 39 6.1.4 I-71 to I-270 West entrance ramp ................................................................. 49 6.1.5 SR-2 to I-90 East entrance ramp................................................................... 59 6.1.5.1 SR-2 to I-90 East entrance ramp driving lane....................................... 61 6.1.5.2 SR-2 to I-90 East entrance ramp passing lane ...................................... 69 6.2 Comparison of Cumulative IATs of the entrance ramps with the freeways..... 77 6.2.1 I-71 South to I-270 West entrance ramp....................................................... 78 6.2.2 I-71 to I-270 west entrance ramp .................................................................. 86 6.2.3 SR-2 to I-90 East entrance ramp................................................................... 90 6.2.3.1 Driving lane .......................................................................................... 90 6.2.3.2 Passing lane........................................................................................... 94 6.3 Extrapolation of the Cumulative IAT distributions for low/high traffic volumes …………………………………………………………………………………98

6 6.4 7

Analysis of the shape of the cumulative IAT distributions............................. 111

Results and discussion ............................................................................................ 133 7.1 Modeling the IATs for the entrance ramps and comparison with the IATs for selected lanes of the freeways ..................................................................................... 133 7.2 Extrapolation of cumulative IAT distributions for freeways.......................... 141 7.3 Analysis of the shape of the PDF for cumulative IAT distributions for freeways ……………………………………………………………………………….141

8

Conclusions............................................................................................................. 142 8.1 Modeling the IATs for the entrance ramps and comparison with the IATs for selected lanes of the freeways ..................................................................................... 142 8.2 Extrapolation of cumulative IAT distributions for freeways.......................... 142 8.3 Analysis of the shape of the PDF for cumulative IAT distributions for freeways ………………………………………………………………………………………..142

9

List of References ................................................................................................... 144

Appendix A: Hyperbolic plots obtained from the observed data for 1%, 2%, 5%, 10%, 30%, 40%, 60%, 80%, 90%, 95%, 98%, 100% points. .................................................. 149 Appendix B: Comparison of entrance ramp cumulative IATs with the freeways for different hourly traffic volumes. ..................................................................................... 174 Appendix C: Extrapolated cumulative IAT distribution tables for 2-lane lane1, 3-lane lane2, 3-lane lane3, 4-lane lane1, 4-lane lane2 and 4-lane lane4. .................................. 258

7 List of Tables Page Table 1: Summary table of the data used to develop IAT distributions for all the entrance ramp sites. ......................................................................................................................... 36 Table 2 : Hyperbolic formulae and R2 values obtained from the fits to the data for the I-71 south to I-270 west entrance ramp. ................................................................................... 42 Table 3: Cumulative interarrival time (in sec) distribution table obtained for I-71 south to I270 west entrance ramp for traffic volume range of 200-800 vehicles/hour based on the hyperbolic fits. .................................................................................................................. 46 Table 4: Adjustment factors obtained from the average IAT of the fit and actual IATs for I-71 south to I-270 west entrance ramp for traffic volume range of 200-800 vehicles/hour. .. 47 Table 5: Adjusted cumulative interarrival time (in sec) distribution table obtained for I-71 south to I-270 west entrance ramp for traffic volume range of 200-800 vehicles/hour. .. 48 Table 6: Hyperbolic fits and the R2 values obtained for the I-71 to I-270 west entrance ramp. ........................................................................................................................................... 52 Table 7: Cumulative interarrival time (in sec) distribution table obtained for I-71 to I-270 west entrance ramp for traffic volume range of 300-1600 vehicles/hour based on the hyperbolic fits. .................................................................................................................. 56 Table 8: Adjustment factors obtained from the average IAT of the fit and actual IATs for I-71 to I-270 west entrance ramp for traffic volume range of 300-1600 vehicles/hour. .......... 57 Table 9: : Adjusted cumulative interarrival time (in sec) distribution table obtained for I-71 to I-270 west entrance ramp for traffic volume range of 300-1600 vehicles/hour. .............. 58 Table 10: Hyperbolic fits and the R2 values obtained for the SR-2 to I-90 east entrance ramp driving lane. ...................................................................................................................... 62 Table 11: Cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp driving lane for traffic volume range of 200-800 vehicles/hour based on the hyperbolic fits. ....................................................................................................... 66

8 Table 12: Adjustment factors obtained from the average IAT of the fit and actual IATs for SR-2 to I-90 East entrance ramp driving lane for traffic volume range of 200-800 vehicles/hour. .................................................................................................................... 67 Table 13: Adjusted cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp driving lane for traffic volume range of 200-800 vehicles/hour. 68 Table 14: Hyperbolic fits and the R2 values obtained for the SR-2 to I-90 east entrance ramp passing lane....................................................................................................................... 70 Table 15: Cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp passing lane for traffic volume range of 200-1300 vehicles/hour based on the hyperbolic fits. ....................................................................................................... 74 Table 16: Adjustment factors obtained from the average IAT of the fit and actual IATs for SR-2 to I-90 East entrance ramp passing lane for traffic volume range of 200-1300 vehicles/hour. .................................................................................................................... 75 Table 17: Adjusted cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp passing lane for traffic volume range of 200-1300 vehicles/hour. ........................................................................................................................................... 76 Table 18: Cumulative interarrival times (in sec) for I-71 south to I-270 west entrance ramp, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 200 vehicles/hour b) 800 vehicles/hour used for comparison. .................................................................................. 79 Table 19: Cumulative interarrival times (in sec) for I-71 to I-270 west entrance ramp, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 300 vehicles/hour b) 1100 vehicles/hour used for comparison. .................................................................................. 87 Table 20: Cumulative interarrival times (in sec) for SR-2 to I-90 east entrance ramp driving lane, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 200 vehicles/hour b) 800 vehicles/hour used for comparison. ........................................................................... 91 Table 21: Cumulative interarrival times (in sec) for SR-2 to I-90 east entrance ramp passing lane, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 300 vehicles/hour b) 1100 vehicles/hour used for comparison. ......................................................................... 95

9 Table 22: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 2 of 2-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13]........................................................................................... 103 Table 23: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 1 of 3-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13]........................................................................................... 105 Table 24: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 3 of 4-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13]........................................................................................... 107 Table 25: Cumulative IATs (in sec) obtained for traffic volumes of 4 vehicles/hour, 958 vehicles/hour and 2478 vehicles/hour for lane 2 of a 3-lane freeway. ........................... 109 Table 26: Values obtained for the average IAT of the fit, variance, standard deviation and coefficient of variation for lane 1 of 2-lane freeway [13] for hourly traffic volume range of 200-1600 vehicles/hour. ............................................................................................. 112 Table 27: Values of coefficient of variation obtained for all lanes of 2-lane, 3-lane and 4-lane freeways for the observed hourly traffic volume ranges. ............................................... 118 Table 28: Summary of the coefficient of variation analysis for all the cumulative interarrival time distributions in [13]................................................................................................. 119 Table 29: Parameters used to calculate the PDF for a negative exponential, an Erlang K = 2, Erlang K = 3 and a Pearson type III distributions for a traffic volume of 1025 vehicles/hour from lane 1 of 3-lane freeway. ................................................................. 121 Table 30: Probability densities obtained for negative exponential, Erlang with K = 2, 3 and Pearson type III distributions for a traffic volume of 1025 vehicles/hour from lane 1 of 3lane freeway. ................................................................................................................... 122 Table 31: Discrete probabilities obtained for OU, negative exponential, Erlang with K = 2, 3 and Pearson type III distributions for a traffic volume of 1025 vehicles/hour from lane 1 of 3-lane freeway. ........................................................................................................... 124 Table 32: Summary table obtained after comparisons showing the best approximation of the selected lane IATs of the freeways for the entrance ramp IATs with average of the maximum and maximum absolute differences. .............................................................. 133

10

List of Figures Page Figure 1: Image showing signalized Martin Luther King drive to I-90 east entrance ramp in Cleveland, Ohio. ............................................................................................................... 16 Figure 2: Image showing non-signalized I-71 north to I-270 west entrance ramp in Columbus, Ohio................................................................................................................................... 17 Figure 3: Measured time headway distributions for different traffic volumes adopted from [1], [11].................................................................................................................................... 26 Figure 4: Typical configuration of the microwave trailer and ORITE equipment used for validation for a two lane freeway. Adapted from Zwahlen et al. [21].............................. 35 Figure 5: Map of the work zone construction site on I-270 adapted from [3]. ........................ 39 Figure 6: Image showing the I-71 south to I-270 west entrance ramp and the trailer location. ........................................................................................................................................... 40 Figure 7: Hourly Traffic volumes observed for 15-minute intervals from 23/6/2006 3:00 PM to 26/6/2006 3:00 PM for I-71 south to I-270 west entrance ramp. ................................. 41 Figure 8: Typical hyperbolic plots obtained for I-71 south to I-270 west entrance ramp for a) 20% value b) 50% value. .................................................................................................. 43 Figure 9: Typical hyperbolic plots obtained for I-71 south to I-270 west entrance ramp for a) 70% value b) 99% value. .................................................................................................. 44 Figure 10: Image showing the I-71 south to I-270 west entrance ramp and the trailer location. ........................................................................................................................................... 49 Figure 11: Hourly Traffic volumes observed for 15-minute intervals from 28/8/2004 0:00 AM to 31/8/2004 midnight I-71 to I-270 west entrance ramp. ................................................ 50 Figure 12: Typical hyperbolic plots obtained for I-71 to I-270 west entrance ramp for a) 20% value b) 50% value............................................................................................................ 53 Figure 13: Typical hyperbolic plots obtained for I-71 to I-270 west entrance ramp for a) 70% value b) 99% value............................................................................................................ 54 Figure 14: Map adapted from [3] showing the I-90 work zone construction in Cleveland. .... 59

11 Figure 15: Image showing the SR-2 to I-90 east entrance ramp and the trailer location......... 60 Figure 16: Hourly Traffic volumes observed for 15-minute intervals from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM for SR-2 to I-90 east entrance ramp driving lane. ........................ 61 Figure 17: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp driving lane for a) 20% value b) 50% value. ........................................................................................ 63 Figure 18: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp driving lane for a) 20% value b) 50% value. ........................................................................................ 64 Figure 19: Hourly Traffic volumes observed for 15-minute intervals from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM for SR-2 to I-90 east entrance ramp passing lane. ........................ 69 Figure 20: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp passing lane for a) 20% value b) 50% value. ........................................................................................ 71 Figure 21: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp passing lane for a) 70% value b) 99% value. ........................................................................................ 72 Figure 22: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane1 b) lane2 of 2lane freeway. ..................................................................................................................... 80 Figure 23: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane1 b) lane2 c) lane3 of 3 lane freeway. .................................................................................................... 81 Figure 24: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane1 b) lane2 of 4lane freeway. ..................................................................................................................... 82 Figure 25: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane3 b) lane4 of 4lane freeway. ..................................................................................................................... 83 Figure 26: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) 2-lane b)................. 84 Figure 27: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with a) 2-lane b) 3-lane c) 4lane freeway. ..................................................................................................................... 85

12 Figure 28: Comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.............................................................................................................................. 88 Figure 29: Comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.............................................................................................................................. 89 Figure 30: Comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with a) 2-lane b) 3lane c) 4-lane freeway....................................................................................................... 92 Figure 31: Comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with a) 2-lane b) 3lane c) 4-lane freeway....................................................................................................... 93 Figure 32: Comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with a) 2-lane b) 3lane c) 4-lane freeway....................................................................................................... 96 Figure 33: Comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with a) 2-lane b) 3lane c) 4-lane freeway....................................................................................................... 97 Figure 34: Adjusted cumulative interarrival time plot for 1% for lane 1 of 3-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic fits used in [13]. ......................................................................................................................................... 102 Figure 35: Cumulative IAT distribution plots obtained for traffic volume of a) 4 vehicles/hour b) 958 vehicles/hour c) 2478 vehicles/hour for lane2 of 3-lane freeway. ...................... 110 Figure 36: Average IAT and standard deviation as a function of traffic volume for a) lane 1 with 200-1600 vehicles/hour b) lane 2 with 150-1450 vehicles/hour for the 2-lane freeway............................................................................................................................ 114 Figure 37: Average IAT and standard deviation as a function of traffic volume for a) lane 1 with 200-1800 vehicles/hour b) lane 2 with 200-1550 vehicles/hour c) lane 3 with 1501650 vehicles/hour for the 3-lane freeway...................................................................... 115

13 Figure 38: Average IAT and standard deviation as a function of traffic volume for a) lane 1 with 100-1150 vehicles/hour b) lane 2 with 150-1400 vehicles/hour for the 4-lane freeway............................................................................................................................ 116 Figure 39: Average IAT and standard deviation as a function of traffic volume for a) lane 3 with 100-1400 vehicles/hour b) lane 4 with 100-1300 vehicles/hour for the 4-lane freeway............................................................................................................................ 117 Figure 40: Comparison of discrete OU PDF for an hourly traffic volume of 1025 vehicles/hour with a) Negative exponential PDF b) Erlang K = 2 PDF......................... 125 Figure 41: Comparison of discrete OU PDF for an hourly traffic volume of 1025 vehicles/hour with a) Erlang K = 3 PDF b) Pearson type III PDF. ................................ 126 Figure 42: Comparison of discrete OU PDF for an hourly traffic volume of 225 vehicles/hour with a) Negative exponential PDF b) Erlang K = 2 PDF. .............................................. 127 Figure 43: Comparison of discrete OU PDF for an hourly traffic volume of 225 vehicles/hour with a) Erlang K = 3 PDF b) Pearson type III PDF. ....................................................... 128 Figure 44: Comparison of discrete OU PDF for an hourly traffic volume of 1775 vehicles/hour with a) Negative exponential PDF b) Erlang K = 2 PDF......................... 129 Figure 45: Comparison of discrete OU PDF for an hourly traffic volume of 1775 vehicles/hour with a) Erlang K = 3 PDF b) Pearson type III PDF. ................................ 130 Figure 46: Discrete OU PDFs for lane1 of 3-lane freeway for traffic volumes of 225, 1025 and 1775 vehicles/hour on a normalized abscissa scale. ....................................................... 131 Figure 47: Cumulative IAT distributions for lane1 of 3-lane freeway for traffic volumes of 75,175,225,575, 1025, 1275, 1425 and 1775 vehicles/hour. .......................................... 132 Figure 48: Comparison of the cumulative IAT distributions for a) I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour b) I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with lane 2 of 2-lane freeway. ......................... 135 Figure 49: Comparison of the cumulative IAT distributions for a)SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour b) SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with average of lane1, lane2 of the 2-lane freeway........................................................................................................... 136

14 Figure 50: Comparison of the cumulative IAT distributions for a) I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour b) I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with lane 3 of 3-lane freeway. ......................... 137 Figure 51: Comparison of the cumulative IAT distributions for a) SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour b) SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane3 of the 3-lane freeway............................................................................................................................ 138 Figure 52: Comparison of the cumulative IAT distributions for a) I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with lane 4 of 4-lane freeway b) I71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with lane 1 of 4lane freeway. ................................................................................................................... 139 Figure 53: Comparison of the cumulative IAT distributions for a) SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour b) SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane1 of the 4-lane freeway............................................................................................................................ 140

15 1

Introduction Time headway or the interarrival time (IAT) is one of the important flow

characteristics which affect the safety, level of service, driver behavior and capacity of the transportation system [1]. Time headway distributions are required to determine the opportunity for passing, merging and crossing on a highway. The minimum time headway and the headway distributions under capacity-flow conditions can be used to determine the capacity of a highway system and therefore it is important to model the headways or interarrival times [1]. Mathematical forms of headway distributions give more insight into the behavior of the traffic. The distributions can also be used as an input to a traffic simulation model. Traffic situations which are hard to observe in real world can be investigated with the use of the simulation models which use the mathematical headway distribution models. Further, digital computer simulation can be used to simulate the flow of traffic in work zone bottlenecks. Simulation of the bottlenecks for work zones on freeways involves the use of headway distributions. Queues and travel time delays generated in a work zone can be effectively simulated if all the inputs required for a complex simulation model are known and defined. The behavior of the traffic thorough the work zones, merging traffic from the entrance ramps and exiting traffic thorough the exit ramps if known can be effectively used for developing an accurate simulation model for the work zones. Mathematical headway distribution models for free flowing traffic on the freeways have been studied and established in the past and there is need for the headway distributions for the traffic on the entrance and the exit ramps.

16 Traffic entering the freeways from the entrance ramps can be divided into two: traffic entering from signalized freeway entrance ramps and traffic entering from nonsignalized freeway entrance ramps. Typical examples for the signalized freeway entrance ramp and non-signalized freeway entrance ramp are shown in Figure 1 and Figure 2.

Figure 1: Image showing signalized Martin Luther King drive to I-90 east entrance ramp in Cleveland, Ohio.

It can be seen that traffic goes thorough an intersection with a traffic signal before the traffic enters the I-90 east entrance ramp from Martin Luther King drive.

17

Figure 2: Image showing non-signalized I-71 north to I-270 west entrance ramp in Columbus, Ohio.

It can be seen in the image that the traffic from I-71 north directly enters the I-270 west entrance ramp with going thorough any intersection or a traffic signal.

18 2

Statement of the problem Work zones on freeways create problems to motorists by increasing the risk of

accidents and travel delays. The reduction of lanes in work zones is often the cause for the slower movement of the traffic and the associated delays in the work zone. The delays become much more aggravated by the merging traffic from the entrance ramps on the freeways creating bottlenecks at either upstream or downstream locations on the freeways with respect to the merge points. For traffic simulation of bottlenecks in work zones which contains a number of entrance ramps, IAT distributions for entrance ramps are needed in addition to the IAT distributions of the mainlines. Therefore it is necessary to model the traffic on the freeway entrance ramps which could be used as an input in complex work zone traffic simulation models to see the effects of merging traffic. This could provide more insight into the change in the behavior of traffic in the work zones due to the merging traffic. Modeling the inter arrival time distributions for the traffic flow on the freeway entrance ramps would enhance the accuracy of the simulation models used to calculate the travel delay times in the work zones. The headway distributions obtained could also be used to develop a set of design guidelines and criteria for closure of the ramps to provide more safety and efficient traffic flow operations with in a work zone.

19 3

Objective of the study The major objectives of the study are to investigate the nature of IATs, model the

IAT distributions for free flowing traffic on non-signalized entrance ramps for freeways and see whether the IAT distributions obtained are similar to the IAT distributions for the freeways. This study is a part of the study “Phase II: Improved work zone design guidelines and enhanced model of travel delays in work zones” [2] which is a continuation of the study “Phase I: Improved work zone design guidelines and enhanced model of travel delays in work zones: Portability and scalability of interarrival and service time probability distribution functions for different locations in Ohio and the establishment of improved work zone design guidelines” [3] which were conducted by Ohio Research Institute for Transportation and Environment (ORITE) at Ohio University sponsored by Ohio Department of Transportation. The minor objectives of the study are to extrapolate the IAT distributions obtained for the freeways in [3] for low, high traffic volumes beyond the observed range of the traffic and investigate the shape of the probability distribution function for the interarrival time distributions obtained in [3] to see whether these distributions can be modeled closely with a negative exponential, an Erlang with K=2, 3 or a Pearson type III distribution.

20 4

Scope of work The work to be performed in this study is described in the following steps: 1. Perform a literature review on existing headway distribution models available in the literature for modeling the headways in a traffic flow. 2. Analyze the data collected in all phases of the study “Improved Work Zone Design Guidelines and Enhanced Model for Travel Delays in Work Zones” [2] [3] for the non-signalized entrance ramps on freeways and model the interarrival time distributions. 3. Compare the IAT distributions obtained for the entrance ramps with the interarrival time distributions for selected lanes of all freeways and see which lane IAT distribution can be used for best approximation of entrance ramp IATs. 4. Extrapolate the IAT distributions obtained for the freeways in [3] for low and high traffic volumes beyond the observed range using the hyperbolic relationships obtained in [3]. 5. Investigate the shape of the probability distribution function of the IAT distributions obtained in [3] and see whether they can be modeled using a negative exponential, an Erlang with K=2, 3 or a Pearson type III distribution.

21 5 5.1

Literature review Headway distributions A semi Poisson model for traffic flow was postulated by Buckley [4] in his study.

The model was based on the assumption that the reason for the traffic process to deviate from the Poisson process is the existence of a zone of emptiness in front of each vehicle. The experimental validation of the model was carried out using the data collected on General Holmes Drive in Sydney, Australia, which is a six lane freeway. Twenty one hours of data were collected when the traffic was free flowing during day hours. The data were grouped into seven flow levels with more than 1000 headways in each group. The model was compared with four other distributions namely the exponential distribution, the displaced exponential distribution, the gamma distribution and the displaced gamma distribution to further validate its use. The author concludes that the model provided some more insight into the behavior of road traffic even though it involves complex mathematical and statistical methods. A log normal headway distribution was developed by Tolle [5] based on normal law of probability. Two and three parameter log normal distributions were discussed. The author says that reliable estimates of mean and variance are needed to have a good fit for the distributions. The models were tested with the data collected on I-71 corridor near downtown Columbus, Ohio using photogrammetric techniques developed by Transportation Research Center at Ohio State University. The data were collected from a helicopter by taking photographs of vehicles for one second intervals. The data obtained were compared with the theoretical distributions using graphical and statistical techniques

22 for traffic volume ranges of 900 vehicles/hour/lane to 1900 vehicles/hour/lane. The three parameter model, i.e., the shifted log normal distribution with a shift of 0.3 seconds to the right gave a better fit than the two parameter model. The author concluded that the model gave a good fit for the data with high traffic volumes and chi-square method was hard to use to get statistically good fits for the data. Cowan [6] proposed four headway models in his paper “Useful headway models” in 1975. The models proposed by Cowan [6] were based on “the value of the models for use as arrival processes in stochastic model building, some traffic situations where it is known from theoretical considerations that the models are appropriate and empirical evidence to support the models”. The uses and theoretical derivations of the four models, namely M1, M2, M3 and M4, were presented. Model M1 and M2 were subsets of model M3, i.e., changes in the parameters of M3 model yielded M1 and M2 models. Model M4 was more generalized form of M3 model which was considered more realistic by the author but M3 was adequate enough to model most of the traffic flow situations. The use of these models was supported empirically using the data collected on westbound lane of Mona Vale road, Sydney, Australia, which was a two lane road. The 1,324 headways obtained were plotted on a histogram which was compared with M4 model probability density function. The author hoped that the models proposed would be more useful in analyzing various traffic situations even though they were validated using a small data sample. Branston [7] presented models of time headway distributions for single lane traffic in his study. He categorized the headway distributions into two classes: simple

23 headway distribution models having properties of observed headways which are easy to estimate and mixed headway models consisting of two distributions for modeling the following and non following vehicle headways. The author says that it is worthwhile to use mixed models even though the estimation of parameters is difficult, if a good fit for the data can be obtained [7]. A lognormal distribution with a shift of 0.3 seconds and the queuing model, the semi Poisson model proposed by Buckley [4], were used for simple and mixed models respectively. Both the models were tested using the data collected on M4 motorway in West London and two-lane roads near Bloomington, Indiana. The data were sampled into “short term” counts (vehicles per minute) and “long term” counts (tens of minutes) to produce unbiased estimates of the parameters for the models. The author concluded that the mean and variance of the log normal distribution can be held constant without loosing the goodness of fit for the observed data. Tolle [8] examined the mathematical models that can be used to describe the headway distributions for two lane and urban freeways. The models were tested using real time data with graphical and statistical procedures. Six headway distribution models namely Poisson, negative exponential, shifted exponential, pearson type III, composite and log normal distributions were tested and the results were presented. The 22 hours of data were collected from I-71 freeway in Ohio which was grouped into hourly and 5minute intervals. Three distributions namely composite, pearson type III and lognormal distributions were compared in the results section and pearson type III gave a good fit to the data at high traffic volumes but it was not statistically sound. The lognormal model gave statistically good fits based on the plots obtained in the study for the observed data.

24 A semi Poisson distribution model was used by Wasielewski [9] for modeling the car following headways on freeways. All the cars in the observed data were divided into leaders and followers. The study aimed at modeling the time headways for the followers as a function of traffic volume. The data used in this study were collected on the center westbound lane of I-94, a six lane divided highway at the Whittier street overpass in Detroit, Michigan, for 32 hours to obtain 42,000 headways. The headways collected had an hourly traffic volume range of 900-2000 vehicles/hour/lane. The main results of this study were that the follower headway distribution was independent of the traffic flow with a mean, standard deviation of 1.32 seconds, 0.52 seconds respectively and the semi poisson model fits the 42,000 headway data quite well. Headway distribution models for two-lane rural highways based on data from North Carolina were described by Khasnabis and Heimbach [10]. The model developed was a part of simulation technique feasibility study for modeling the traffic on two lane rural highways. The field data collected at nine sites (eighteen sets of data, two for each site) were tested with different models with increasing the complexity of the model until there was a match between the field data and output of the model. Three models, namely negative exponential, pearson type III and schuhl models, were tested for the traffic volume range of 80-632 vehicles/hour/lane, and none of the models fitted the data well with statistical soundness over the entire range of traffic. Even the combination of two headway models did not improve the predictive certainty but most of the models gave a good fit for the data with low traffic volumes (approximately 150 vehicles/hour/lane).

25 A classification scheme of theoretical probabilistic distributions that can be applied to headways or interarrival time distributions (IATs) was described by May [1]. The headways are dependant on the traffic flow rate and distributions that fit to different flow rates have been defined and the mathematical formulations of these distributions were also given. Distributions that can be used to model random, constant and intermediate headways were proposed. For a low traffic flow level, the headways were considered to be random as the vehicles traveling were independent of each other. Even though minimum time headway is required from safety point of view, the headways for low traffic levels were classified as random headways and a random time headway distribution could be used. According to May [1] “The negative exponential distribution is the mathematical distribution that represents the distribution of random intervals such as time headways”. It was found that the negative exponential distribution was best if used for low traffic volumes when compared with the actual measured distributions and the distributions were different at high traffic volumes. The normal distribution was used for high traffic volumes where a constant headway state exists because the traffic flow level is at the capacity flow level. May [1] said that “The normal distribution is a mathematical distribution that can be used when either the time headways are all constant or when drivers attempt to drive at a constant headway state”. In this state, there is more interaction between vehicles and the driver tends to maintain a constant headway which can vary due to driver errors. This distribution was best fit with the actual measured distributions at high traffic flow levels

26 and shift of 0.5-1 sec was observed from the measured distributions. The theoretical and the measured distributions were quite different at low traffic volumes. For intermediate headway state, Pearson type III distribution has been used. Most of the real time traffic has intermediate headways which is difficult to analyze using a single distribution. A generalized form of Pearson type III model has been defined which can be modified from a random to constant distribution by modifying the parameters of the distribution. The Pearson type III distribution yields to a negative exponential, a gamma, an Erlang and a shifted negative exponential distribution by modifying the parameters of the distribution. As different distributions fit for different traffic flow levels, sensitivity analysis of the distribution parameters has to be performed to find the best fit of the Pearson type III distribution.

Figure 3: Measured time headway distributions for different traffic volumes adopted from [1], [11].

27 Figure 3 shows the measured time headway distributions reported by May [11] in 1965 for different traffic volumes based on minute flow rates. For low traffic volumes, May used negative exponential distribution which is not reflected in Figure 1. From Figure 1, the ratio of standard deviation and the mean is 0.78 for the traffic volume of 600 vehicles/hour. For a negative exponential distribution, the ratio of the standard deviation to the mean is one [12]. A composite model approach was discussed for intermediate headway state by May [1] as a single distribution was not sufficient to model the headways for a wide range of traffic. The composite model used had a combination of normal distribution for car-following state and shifted negative exponential distribution for non-interacting vehicles i.e. random headway state. The probabilities obtained from the theoretical distributions were greater than measured distributions for headways greater than 4 seconds and less than measured distributions for headways between 2.5-4 seconds. Zwahlen et al. [13] developed approximated headway distributions for the free flowing traffic on Ohio freeways. Cumulative interarrival time distributions as a function of traffic volume were developed for 2-lane, 3-lane and 4-lane freeways in Ohio which are available at http://webce.ent.ohiou.edu/orite/cumulativeIATdistributions.html. The distributions developed were also investigated for portability so that universal time headway distributions could be obtained. Data collected from six different work-zone sites was used to model the interarrival times observed. The six sites selected for data collection were; I-76 Westbound near Akron (2 lanes), I-270 Westbound in Columbus (2 lanes), I-270 Eastbound in Columbus (3 lanes), I-90 Eastbound in Cleveland (3 lanes), I-

28 90 Westbound in Cleveland (4-lanes), and I-75 Southbound in Dayton (3 lanes). The data were collected using microwave radar trailers which collected time-stamped data for the arrival of vehicles in 2.5 millisecond increments at the point of measurement for each lane of the freeway. Most of the measurements were made at free flowing speeds near the speed limit at the work zones [3]. The successive points of measurement on each freeway site were more than 2500 feet apart to get uninterrupted traffic measurements. All the traffic volume data collected were grouped into 15-minute intervals and cumulative interarrival times were obtained for 16 percentile points for the observed range of traffic. The interarrival times obtained for the 16 percentile points were fitted using hyperbolic fits using least squares fitting method. The cumulative interarrival time distributions obtained were validated using data from I-70 Westbound near SR-256 which were collected by Schnell et al. [14]. A total of nine approximated cumulative interarrival time distribution tables were developed in an easy to use excel spreadsheet for observed traffic volume ranges. The portability analysis in the study showed that approximated cumulative interarrival time distributions for 2-lane, 3-lane and 4-lane were similar for a given hourly traffic volume and lane. The methodology used for generating the cumulative interarrival time distribution for non-signalized freeway entrance ramps in this study was adapted from [13]. A more detailed description of the sites used, data collection equipment and the procedure is presented in the methodology section.

29 5.2

Headway distributions for urban traffic Griffiths and Hunt [15] studied headways in busy urban areas and the study was

carried out primarily for the estimation of delays experienced by the pedestrians while crossing busy streets at points where there was no pedestrian crossing. This study also helped the Transport and Road Research Laboratory to simulate the arrival patterns of vehicles in urban area. Headway data were collected from 45 sites in United Kingdom for single lanes of traffic, and a total of 82,388 headways were collected generating 86 data sets with traffic volume ranges of 172-1351 vehicles/hour/lane. The requirements that were used to select the distribution were that the distribution should represent the qualities of the data sets, should be easy to use for traffic engineers for estimation of the distribution parameters from traffic counts and generate vehicles arrival times that could be used in traffic simulations. The authors investigated a double displaced negative exponential distribution with four parameters to fit the data which were tested using a chi-square test. The fit obtained using parameters which yielded small chi-square values at 5% significance level was considered as a good fit and 78 out of the 86 data sets collected had a good fit. A easy to use ‘look up’ table for estimating the parameters of the distribution based on traffic volume was developed which was considered less precise as the model gave good fits to the data collected in the study and its transferability to other sites was not known for sure. A comparison study of Cowan M3 model and double displaced negative exponential distributions (DDNED) to model the urban traffic flow was performed by Sullivan and Troutbeck [16]. For a given headway set of data both models provided good

30 fit for headways greater than 4 seconds but Cowan M3 model was easy to use and accurate. The Cowan M3 model was preferred over the DDNED for its ease to estimate the parameters of the distribution using simple relationships based on traffic flow rate which reduced the need for extensive data collection effort. Ali S. Al-Ghamdi [17] analyzed the time headways on urban roads in Riyadh, Saudi Arabia in a case study. The objective of his study was to model the time headways and determine boundaries for the traffic flow levels (low, medium and high) based on the data collected in Riyadh. Data were collected for 20 urban sites (13 freeways, 7 arterials) for single lanes of traffic at sections which were level, straight and free of traffic signals, intersections and median openings. The data used for fitting the distributions were selected on the criteria that it was statistical enough for fitting distributions and any variations in the data behavior for fitting distributions with respect to flow level increments could be detected [17]. The chi-square technique was used to test the goodness of fit of the data because the author believed there was enough data to use the chi-square test. The boundaries for different flow levels were determined partially based on grouped distributions according to the flow levels. The flow ranges for low, medium and high were described as 0-400 vehicles/hour, 400-1200 vehicles/hour and greater than 1200 vehicles/hour respectively. Negative exponential, shifted exponential, gamma and Erlang distributions were observed to fit the traffic flow levels described above respectively for the collected data. The author concludes that very little research was done on modeling headways for high traffic flows and despite of the large data sample used in the study, the modeling of headways for high traffic flows was not clear. The

31 author said that the flow boundaries and the headway distributions used for fitting the data are valid for traffic data in Riyadh only and does not represent traffic conditions elsewhere. Arasan and Koshy [18] modeled the headway distributions for heterogeneous traffic on urban roads in Chennai, India. The models developed for homogenous traffic could not be applied for heterogeneous traffic conditions because of the presence of more two-wheelers and a zero headway state was experienced as the two wheelers arrived in parallel even in a single lane of traffic. The mixed traffic conditions ruled out the lanewise observation of headways as the two wheelers had more choice to move within a traffic stream. The data used in the study were collected using a video camera mounted on a roadside post on a four lane divided highway (Sardar Patel road in Chennai, India) during peak and non-peak hours. The data collected had traffic volume ranges of 2506000 vehicles/hour and had the following mix of vehicles: two-wheelers (49%), cars (25%), auto-rickshaws (10%), buses (5%), bicycles (5%), trucks (3%) and light commercial vehicles (3%). The headways from the data collected were extracted using a computer program using keystrokes during the playback of the video. The video was converted into mpeg format and played at 0.25 times the actual speed to facilitate the extraction of vehicles arriving at a very fast rate. Negative exponential distribution was used to model the headways, and it was found that it gave a good fit for the wide range of traffic volumes observed because of the zero or near-zero headways. The author opined that headway analysis for heterogeneous traffic should be done measuring the headways

32 over the entire road width and the use of video camera for measuring heterogeneous traffic headways was satisfactory. 5.3

Headway distributions for low and high traffic volumes Mei and Bullen [19] studied the headway distributions for high traffic flows both

theoretically and empirically in their study “Lognormal distributions for high traffic flows”. According to the authors lognormal distribution should have been used for high traffic flows based on theoretical analysis with car following state but it was found that a shifted lognormal distribution gave the best fit for the data. The model was tested using the data collected at Milroy overpass in downtown Pittsburgh from two south bound lanes of I-279 four lane freeway. The average traffic volumes observed were in a range of 2500-2900 vehicles/hour and a total of 10000 headways were collected out of which only 1375 headways were used as 8 data sets for testing the model. It was found that a lognormal distribution with a shift of 0.3 to 0.4 seconds gave an excellent fit for the data which were tested using chi-square and Kolmogorov-Smirnov goodness of fit tests. Pearson type III distribution gave good results for the data but the authors believed that the calibration of the model is difficult and lacked explanation for the traffic process. Normal distribution did not give any acceptable fit for any of the 8 data sets in this study. Sadeghhosseini and Benekohal [20] studied the headway characteristics for wide range (low to high) of traffic volumes obtained from the interstate highways. The study also developed relationships for the traffic volume and parameters of the model. Basic freeways sections with two lanes without horizontal and vertical curves were selected to collect the headway data for wide range of traffic volumes. Two sites on I-74, one on I-55

33 and two sites on I-57 in Illinois were selected to collect the headway data. The data were collected by videotaping the traffic stream from an overpass between 7:00 am to 10:00 am and from 3:00 pm to 6:00 pm to obtain wide range of traffic volumes. The data were reduced manually to obtain 10,000 headways out of which 5000 headways obtained from the median lanes were used for analysis. The headway data were grouped into 20 data sets with an hourly traffic volume range of 140-1704 vehicles. The effect of volume on headway mean, headway standard deviation, median, mode and speed were studied and relationships were established which were all found to be power series. Log normal and Pearson type III models (shifted gamma) were examined because of their ease in estimation of the parameters. The model parameters were estimated then tested with the headway data collected, and it was found that the log normal distribution with a shift of 0.36 seconds gave the best fit for 15 sets of data. The relationships of the traffic volume with the mean and variance of the shifted log normal model were established in order to compute the headway distributions for any traffic volume. Headway distributions for traffic volumes of 200 to 2400 vehicles/hour were generated using the model which showed that the percentage of larger headways decreased with the increase in traffic volume and vice versa.

34 6

Methodology

6.1 6.1.1

Modeling the interarrival (IATs) times for the entrance ramps Data collection The data used for modeling the interarrival times for the entrance ramps were

collected as a part of the data collection effort in [3] “Improved Work Zone Design Guidelines and Enhanced Model of Travel Delays in Work Zones Phase I” and in [2] which was the continuation of [3]. Data were collected in six freeway work zone construction sites in [3] and at two of the same sites after construction in [2]. Data collected for three non-signalized freeway entrance ramp sites, one from I-270, one from I-90 in [3] and one from I-270 in [2] were selected for analysis. A more detailed description of the entrance ramp sites and the data used is provided in subsequent sections. The data were collected using microwave radar trailer units [13] which were specially designed for data collection on freeways using non-intrusive methods. They record the data from the reflections of the radar beam from the passing traffic. The trailer consists of two microwave radar units which can be mounted on two collapsible poles and can be used in forward fire or side fire mode to detect the traffic from the point of observation. The data used in this study were collected by using the trailer in side fire mode, and a typical configuration of the trailer equipment is shown in Figure 4. The data collected from the trailer was validated independently using a video camera by recording the traffic for an hour [13]. The speeds were also measured using Kustom Signals TR-6 radar unit and were recorded by hand for the time the video was recorded. The data

35 collected on the trailers were stored in text files which could be downloaded using a laptop.

Figure 4: Typical configuration of the microwave trailer and ORITE equipment used for validation for a two lane freeway. Adapted from Zwahlen et al. [21].

Table 1 gives a brief summary of the data used to develop the IAT distributions for the non-signalized freeway entrance ramps. The table shows the dates the data was collected, observed traffic volumes and number of IATs or headways used for data analysis.

36 Table 1: Summary table of the data used to develop IAT distributions for all the entrance ramp sites.

6.1.2

23/6/2006-26/6/2006 28/8/2004-31/8/2004

Observed traffic volumes (vehicles/hour) 8 - 836 44 -1616

IATs or headways collected 12954 39224

13/9/2004-16/9/2004

4 - 692

15443

13/9/2004-16/9/2005

24 -1320

35130

Entrance ramp

Dates data collected

I-71 S to I-270 W I-71 to I-270 W SR-2 to I-90 E Driving lane SR-2 to I-90 E Passing lane

Data analysis As mentioned earlier the method used for developing the interarrival time

distributions for traffic on non-signalized freeway entrance ramps was adapted from Zwahlen et al. [13]. The headway or IAT distributions for three entrance ramps with different configurations were developed using the same procedure described below. Three days of data collected using the microwave radar trailers was analyzed for each site [13]. The data collection period for each freeway site was about a week which included the time for moving the trailers to the locations, calibration of the trailers and taking them off from the field after data collection. Due to these time restrictions generally three days of data was collected for each trailer location. The text file downloaded from the trailer was imported into excel and the data were grouped according to the lane of travel. Traffic volume data for 15 minute intervals for three days were obtained and tabulated. The data were grouped into 15-minute intervals to further reduce the variations observed in traffic behavior when the data were grouped into hourly intervals. The time stamps recorded for the arrivals of vehicles in 2.5 millisecond

37 increments were all converted into seconds to calculate the interarrival times (IATs) or the headways which was the difference between two successive vehicle arrival time stamps. The data collected for validation of the trailer measurements were entered into excel and compared with the data collected from the trailer. As the trailers were not 100% accurate in measuring the traffic data [21], correction factors were calculated for each lane of each site so that adjusted traffic volumes could be obtained. All the 15-minute traffic volumes obtained in the previous step were multiplied by the correction factors to obtain the adjusted counts. Hourly traffic volumes for each 15-minute interval were then obtained by multiplying the adjusted 15-minute counts by four [13]. Microsoft Excel Data analysis/Histogram tool was used to compute the cumulative percentages for all the 15-minute interval data. An IAT of 0.1 seconds was assigned to 0% value and bins were set up from this value to the maximum IAT observed in a 15-minute interval. IATs for sixteen percentile values: 1%, 2%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99%, and 100% were extracted after calculating the cumulative percentage for all the 15-minute interval data sets [13]. A excel sheet was set up with the 15 minute time period, adjusted hourly vehicle count and IATs for the sixteen percentile values according to the site, date and lane of travel. Cumulative percentages were used because cumulative percentage values can be easily used with Kolmogorov-Smirnov (KS) goodness of fit test [13]. For each percentile value, a hyperbolic relationship of the form y= (a/x) + b was used to generate the relationship between the IAT values and the observed traffic volume

38 [13]. The coefficients “a” and “b” were calculated using the least squares fitting method. Hyperbolic fits were used because the average IAT for an hourly traffic volume is inversely proportional to the hourly traffic volume. After determining the hyperbolic fit relationship for all the 16 percentile values, cumulative interarrival times for 16 percentile values based on observed traffic volumes in increments of 50 were generated using the hyperbolic fits. The average IAT of the fit for the distribution for each traffic volume was calculated using the formula given in equation 1 [13] to compare it with the average IAT obtained for a given traffic volume. The average IAT of the fit is given by: 16

Average IAT of the fit =

∑ [( p i =1

i +1

− pi ) *

( y i +1 + y i ) ] 2

(1)

where pi = cumulative percentage value from the hyperbolic fit table yi = corresponding IAT for pi The average IAT for a given traffic volume was computed by using the formula (3600/traffic volume). An adjustment factor was calculated for each traffic volume by dividing the average IAT for a given traffic volume divided by the average IAT of fit obtained from the cumulative interarrival time distributions [13]. All the cumulative interarrival times computed using the hyperbolic fits for a given traffic volume were then multiplied by the respective adjustment factor in order to obtain the corrected cumulative IAT values. Then all the adjusted cumulative interarrival times were tabulated for the observed traffic volumes in increments of 50 vehicles/hour.

39 6.1.3

I-71 South to I-270 West entrance ramp

I-71 south to I-270 west entrance ramp which is a single lane ramp was located near the work zone site identified as FRA-270-0.60 on construction drawings [3]. The entrance ramp site was located just before the start of the work zone. The data used for this entrance ramp was collected as a part of data collection effort in [2] after the construction was finished. Figure 5 shows the map of the work zone site during construction where as Figure 6 shows the entrance ramp and the place where the trailer was set up to collect the traffic data.

Figure 5: Map of the work zone construction site on I-270 adapted from [3].

40

Figure 6: Image showing the I-71 south to I-270 west entrance ramp and the trailer location.

The data for this site were collected from 21/6/2006 to 26/6/2006 but three days of data from 23/6/2006 3:00 PM to 26/6/2006 3:00 PM were used for analysis. A total of 12,954 interarrival times or headways were obtained for three days of data with hourly traffic volume ranging from 8-836 vehicles/hour with a average of 176 vehicles/hour. Figure 7 shows the plot of hourly traffic volumes observed for the 15-minute intervals from 23/6/2006 3:00 PM to 26/6/2006 3:00 PM. As the data collected by the trailer were not accurate [19], a correction factor of 0.95 determined with validation data obtained by ORITE [12] was multiplied for all the 15-minute traffic volumes to obtain the adjusted hourly traffic counts which had a range of 8-794 vehicles/hour after they were rounded off to the nearest integer.

41 900 800

Maximum = 836 vehicles/hour Minimum = 8 vehicles/hour Average = 176 vehicles/hour

Traffic volume (vph)

700

23/6/2006-24/6/2006 24/6/2006-25/6/2006 25/6/2006-26/6/2006

600 500 400 300 200 100

15 .

00 15 -15 .4 .1 5 5 16 -16 .3 .0 0- 0 17 16 .1 .4 5 5 18 -17 .0 .3 0- 0 18 18 .4 .1 5 5 19 -19 .3 .0 0- 0 20 19 .1 .4 5 5 21 -20 .0 .3 0- 0 21 21 .4 .1 5 5 22 -22 .3 .0 0 0 23 -22 .1 .4 5- 5 2 0. 3.3 00 0 0. 0.1 45 5 -1 1. .0 30 0 2. 1.4 15 5 -2 3. .3 00 0 3. 3.1 45 5 -4 4. .0 30 0 5. 4.4 15 5 6. 5.3 00 0 -6 6. .1 45 5 7. 7.0 30 0 8. 7.4 15 5 -8 9. .3 00 0 9. -9. 45 15 10 -10 .3 . 0 0 0 11 -10 .1 .4 5- 5 12 11 .0 .3 0 0 12 -12 .4 .1 5- 5 13 13 .3 .0 0 0 14 -13 .1 .4 5- 5 14 .3 0

0

15 minute interval

Figure 7: Hourly Traffic volumes observed for 15-minute intervals from 23/6/2006 3:00 PM to 26/6/2006 3:00 PM for I-71 south to I-270 west entrance ramp.

Cumulative interarrival times (IATs) for the sixteen percentile values mentioned in the methodology were computed for all the 15 minute interval data sets. A total of 80 out of 288 15-minute intervals with traffic volumes ranging from 200-794 vehicles/hour had IAT values for all the sixteen percentile values i.e. the 15-minute intervals with low hourly traffic volumes did not have IAT values for some percentile values. The IAT data from the 15-minute intervals with low traffic volumes from 4-200 vehicles/hour were not used to increase the accuracy of the hyperbolic fits. The equations of the hyperbolic fits

42 and the R-squares values obtained for this site are tabulated in Table 2. As expected the R2 value for the percentile points around 50% is more where as it decreases at the ends because of the variability in the data at the tails of the distribution which can be seen in Table 2. This can be also due to the variability in traffic speeds at low traffic volumes.

Table 2 : Hyperbolic formulae and R2 values obtained from the fits to the data for the I-71 south to I-270 west entrance ramp. Cumulative percentage (%) 0 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 100

Model 0.1 y = 37.26/x + 0.571 y = 61.175/x + 0.608 y = 71.48/x + 0.716 y = 131.189/x + 0.789 y = 375.02/x + 0.659 y = 902.68/x + 0.179 y = 1506.30/x - 0.126 y = 2264.84/x - 0.461 y = 3282.53/x - 0.875 y = 4278.85/x - 0.855 y = 5247.62/x + 0.691 y = 8220.31/x + 0.068 y = 9738.12/x + 2.130 y = 9442.25/x + 8.915 y = 11258.78/x + 9.153 y = 13518.21/x + 15.321

R2

0.072 0.156 0.220 0.282 0.328 0.485 0.514 0.607 0.777 0.868 0.878 0.802 0.756 0.643 0.548 0.346

Figure 8 and Figure 9 show the typical hyperbolic plots obtained for the observed data for 20%, 50% and 70%, 99% percentile values respectively. The rest of the plots for other percentile points are provided in Appendix A.

43 4.5

4 N=80 R-Square = 0.328

Interarrival time (sec)

3.5 3

20% Hyperbolic 2.5

2

1.5 1

0.5

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

a) 14

12

N=80 R-Square = 0.607

Interarrival time (sec)

10 50% Hyperbolic 8

6

4

2

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

b) Figure 8: Typical hyperbolic plots obtained for I-71 south to I-270 west entrance ramp for a) 20% value b) 50% value.

44 25 N=80 R-Square = 0.868

Interarrival time (sec)

20

15 70% Hyperbolic 10

5

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

a) 90 80

N=80 R-Square = 0.548

Interarrival time (sec)

70 60 50 99% Hyperbolic

40 30 20 10 0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

b) Figure 9: Typical hyperbolic plots obtained for I-71 south to I-270 west entrance ramp for a) 70% value b) 99% value.

45 The hyperbolic relationships in Table 2 were used to obtain the cumulative interarrival times for the sixteen percentile values for traffic volumes ranging from 200800 vehicles/hour in increments of 50 vehicles/hour. The cumulative interarrival times obtained for the sixteen percentile points using the hyperbolic relationships are tabulated in Table 3. The average IAT value of the fit for the interarrival time distribution was calculated using equation 1 for each hourly traffic volume which was compared with the average IAT calculated directly from the hourly traffic volume. The adjustment factors for all hourly traffic volumes were then determined which are given in Table 4. The adjusted cumulative interarrival times were obtained after multiplying the adjustment factors with the interarrival times obtained in Table 3 for each hourly traffic volume which are tabulated in Table 5.

46 Table 3: Cumulative interarrival time (in sec) distribution table obtained for I-71 south to I-270 west entrance ramp for traffic volume range of 200-800 vehicles/hour based on the hyperbolic fits. Number of Vehicles per hour 200 250 300 350 400 450 500 550 600 650 700 750 800

Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.76 0.72 0.70 0.68 0.66 0.65 0.65 0.64 0.63 0.63 0.62 0.62 0.62

0.91 0.85 0.81 0.78 0.76 0.74 0.73 0.72 0.71 0.70 0.70 0.69 0.68

1.07 1.00 0.95 0.92 0.89 0.87 0.86 0.85 0.84 0.83 0.82 0.81 0.81

1.44 1.31 1.23 1.16 1.12 1.08 1.05 1.03 1.01 0.99 0.98 0.96 0.95

2.53 2.16 1.91 1.73 1.60 1.49 1.41 1.34 1.28 1.24 1.19 1.16 1.13

4.69 3.79 3.19 2.76 2.44 2.18 1.98 1.82 1.68 1.57 1.47 1.38 1.31

7.41 5.90 4.90 4.18 3.64 3.22 2.89 2.61 2.38 2.19 2.03 1.88 1.76

10.86 8.60 7.09 6.01 5.20 4.57 4.07 3.66 3.31 3.02 2.77 2.56 2.37

15.54 12.26 10.07 8.50 7.33 6.42 5.69 5.09 4.60 4.18 3.81 3.50 3.23

20.54 16.26 13.41 11.37 9.84 8.65 7.70 6.92 6.28 5.73 5.26 4.85 4.49

26.93 21.68 18.18 15.68 13.81 12.35 11.19 10.23 9.44 8.76 8.19 7.69 7.25

41.17 32.95 27.47 23.55 20.62 18.34 16.51 15.01 13.77 12.71 11.81 11.03 10.34

50.82 41.08 34.59 29.95 26.48 23.77 21.61 19.84 18.36 17.11 16.04 15.11 14.30

56.13 46.68 40.39 35.89 32.52 29.90 27.80 26.08 24.65 23.44 22.40 21.50 20.72

65.45 54.19 46.68 41.32 37.30 34.17 31.67 29.62 27.92 26.47 25.24 24.16 23.23

82.91 69.39 60.38 53.94 49.12 45.36 42.36 39.90 37.85 36.12 34.63 33.35 32.22

47 Table 4: Adjustment factors obtained from the average IAT of the fit and actual IATs for I-71 south to I-270 west entrance ramp for traffic volume range of 200-800 vehicles/hour. Number of vehicles per hour

Average IAT of the fit (sec)

200 250 300 350 400 450 500 550 600 650 700 750 800

16.340 13.165 11.048 9.536 8.402 7.520 6.814 6.237 5.756 5.349 5.000 4.698 4.433

Actual IAT based on traffic volume (sec) 18.000 14.400 12.000 10.286 9.000 8.000 7.200 6.545 6.000 5.538 5.143 4.800 4.500

Adjustment factor

1.102 1.094 1.086 1.079 1.071 1.064 1.057 1.049 1.042 1.035 1.029 1.022 1.015

48 Table 5: Adjusted cumulative interarrival time (in sec) distribution table obtained for I-71 south to I-270 west entrance ramp for traffic volume range of 200-800 vehicles/hour. Number of Vehicles per hour 200 250 300 350 400 450 500 550 600 650 700 750 800

Adjusted Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.11 0.11 0.11 0.11 0.11 0.11 0.11 0.10 0.10 0.10 0.10 0.10 0.10

0.83 0.79 0.76 0.73 0.71 0.70 0.68 0.67 0.66 0.65 0.64 0.63 0.63

1.01 0.93 0.88 0.84 0.82 0.79 0.77 0.75 0.74 0.73 0.72 0.70 0.69

1.18 1.10 1.04 0.99 0.96 0.93 0.91 0.89 0.87 0.86 0.84 0.83 0.82

1.59 1.44 1.33 1.26 1.20 1.15 1.11 1.08 1.05 1.03 1.00 0.98 0.97

2.79 2.36 2.07 1.87 1.71 1.59 1.49 1.41 1.34 1.28 1.23 1.18 1.14

5.17 4.15 3.46 2.97 2.61 2.32 2.10 1.91 1.75 1.62 1.51 1.41 1.33

8.16 6.45 5.32 4.51 3.90 3.43 3.05 2.74 2.49 2.27 2.08 1.92 1.78

11.97 9.41 7.70 6.48 5.57 4.86 4.30 3.84 3.45 3.13 2.85 2.61 2.41

17.12 13.41 10.93 9.17 7.85 6.83 6.01 5.35 4.79 4.32 3.92 3.58 3.28

22.63 17.79 14.56 12.26 10.54 9.21 8.14 7.27 6.54 5.93 5.41 4.96 4.56

29.67 23.72 19.75 16.92 14.79 13.14 11.82 10.74 9.84 9.07 8.42 7.86 7.36

45.35 36.04 29.84 25.41 22.09 19.51 17.44 15.76 14.35 13.17 12.15 11.27 10.50

55.98 44.94 37.57 32.31 28.36 25.29 22.83 20.82 19.14 17.72 16.50 15.44 14.52

61.83 51.06 43.87 38.71 34.84 31.81 29.37 27.37 25.70 24.27 23.04 21.97 21.03

72.10 59.27 50.71 44.57 39.95 36.35 33.46 31.09 29.10 27.41 25.96 24.69 23.58

91.34 75.91 65.59 58.19 52.61 48.26 44.75 41.87 39.46 37.40 35.62 34.07 32.71

49 6.1.4

I-71 to I-270 West entrance ramp

This entrance ramp site is the same as the I-71 south to I-270 west site which is discussed earlier but the only difference was that the ramp was configured in a way that all the traffic entering from I-71, i.e., I 71-south, I-71 north onto I-270 west was made to merge using a single entrance ramp. The data for this site was collected as a part of data collection effort in [3] when the construction was under progress in the work zone. The location of the site and trailer where the data were collected is shown in Figure 10.

Figure 10: Image showing the I-71 south to I-270 west entrance ramp and the trailer location.

50 When the data was collected for this location in [2] it was used only for the traffic merging from I-71 south to I-270 west for which the interarrival time distributions have been established in the previous section. The data for this site was collected from 27/8/2004 to 1/9/2004 but three days of data from 29/8/2004 to 31/8/2004 were used for the analysis. A total of 39224 headways were obtained with observed hourly traffic volumes ranging from 44-1616 vehicles/hour with a average of 544 vehicles/hour. This site had the maximum traffic volume of all the three sites used in the analysis. Figure 11 shows the plot of hourly traffic volumes observed based on 15 minute interval counts.

1800 1600

29/8/2004 30/8/2004 31/8/2004

Traffic volume (vph)

1400 1200

Maximum = 1616 vehicles/hour Minimum = 44 vehicles/hour Average = 544 vehicles/hour

1000 800 600 400 200

0. 00 0. -0.1 45 5 1. -1.0 30 0 2. -1.4 15 5 3. -2.3 00 0 3. -3.1 45 5 4. -4.0 30 0 5. -4.4 15 5 6. -5.3 00 0 6. -6.1 45 5 7. -7.0 30 0 8. -7.4 15 5 9. -8.3 0 0 9. 0-9 4 .1 10 5-1 5 . 3 0 .0 11 0-1 0 .1 0.4 12 5-1 5 .0 1.3 12 0-1 0 .4 2.1 13 5-1 5 .3 3.0 14 0-1 0 .1 3.4 15 5-1 5 .0 4.3 15 0-1 0 .4 5.1 16 5-1 5 .3 6.0 17 0-1 0 .1 6.4 18 5-1 5 .0 7.3 18 0-1 0 .4 8.1 19 5-1 5 .3 9.0 20 0-1 0 .1 9.4 21 5-2 5 .0 0.3 21 0-2 0 .4 1.1 22 5-2 5 .3 2.0 23 0-2 0 .1 2.4 5- 5 23 .3 0

0

15 minute time interval

Figure 11: Hourly Traffic volumes observed for 15-minute intervals from 28/8/2004 0:00 AM to 31/8/2004 midnight I-71 to I-270 west entrance ramp.

51 No adjustment was made to the 15-minute interval traffic counts as the correction factor obtained for this site using the ORITE validation procedure [12] was equal to 1.00. Cumulative interarrival times were extracted for all the 15-minute interval data sets based on the procedure described in the methodology section. A total of 195 data sets out of 288 data sets were used to generate the cumulative interarrival times for traffic volumes ranging from 300-1616 vehicles/hour. Fifteen minute interval data sets with traffic volumes less than 300 vehicles/hour did not have cumulative IATs for all the 16 percentile values. The equations of the hyperbolic fits and the R2 values obtained for the fits are given in Table 6.

52 Table 6: Hyperbolic fits and the R2 values obtained for the I-71 to I-270 west entrance ramp. Cumulative percentage (%) 0 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 100

Model 0.1 y = 29.951/x + 0.541 y = 62.239/x + 0.569 y = 99.663/x + 0.654 y = 174.086\x+0.709 y = 359.735\x + 0.740 y = 659.698\x + 0.788 y = 1167\x + 0.387 y = 2005.903\x - 0.118 y = 3082.805\x - 0.609 y = 4503.669\x - 0.927 y = 6365\x-1.127 y = 9022\x - 0.318 y = 11192.16\x + 0.969 y = 13325.22\x + 3.231 y = 15034.26/x + 4.155 y = 16532/x + 11.094

R2

0.068 0.238 0.349 0.505 0.728 0.788 0.828 0.898 0.942 0.955 0.973 0.962 0.933 0.889 0.842 0.550

Figure 12 and Figure 13 show the hyperbolic plots obtained for the observed data for 20%, 50% and 70%, 99% percentile values, respectively. The hyperbolic plots obtained for the rest of the percentile points are provided in the Appendix A.

53 2.5 N = 195 R-square = 0.728 2

Interarrival time (sec)

20% Hyperbolic

1.5

1

0.5

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

a) 9

8

N = 195 R-square = 0.898

7

50% Hyperbolic Interarrival time (sec)

6

5

4

3

2

1

0

0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

b) Figure 12: Typical hyperbolic plots obtained for I-71 to I-270 west entrance ramp for a) 20% value b) 50% value.

54 16

14

N = 195 R-square = 0.955

Interarrival time (sec)

12

70% Hyperbolic

10

8

6

4

2

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

a) 80

70

N = 195 R-square = 0.842

Interarrival time (sec)

60

99% Hyperbolic

50

40

30

20

10

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

b) Figure 13: Typical hyperbolic plots obtained for I-71 to I-270 west entrance ramp for a) 70% value b) 99% value.

55 The hyperbolic relationships in Table 6 were used to obtain the cumulative interarrival times for the sixteen percentile values for traffic volumes ranging from 3001600 vehicles/hour at increments of 50 vehicles/hour. The cumulative interarrival times obtained for the sixteen percentile points using the hyperbolic relationships are tabulated in Table 7. The average IAT value of the fit for the interarrival time distribution was calculated using equation 1 for each hourly traffic volume which was compared with the average IAT calculated directly from the hourly traffic volume. The adjustment factors for all hourly traffic volumes were then determined which are given in Table 8. The adjusted cumulative interarrival times were obtained after multiplying the adjustment factors with the interarrival times obtained in Table 7 for each hourly traffic volume and are tabulated in Table 9.

56 Table 7: Cumulative interarrival time (in sec) distribution table obtained for I-71 to I-270 west entrance ramp for traffic volume range of 300-1600 vehicles/hour based on the hyperbolic fits. Number of Vehicles per hour 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600

Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.64 0.63 0.62 0.61 0.60 0.60 0.59 0.59 0.58 0.58 0.58 0.58 0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.56 0.56 0.56 0.56 0.56 0.56 0.56 0.56

0.78 0.75 0.72 0.71 0.69 0.68 0.67 0.66 0.66 0.65 0.65 0.64 0.64 0.63 0.63 0.63 0.63 0.62 0.62 0.62 0.62 0.62 0.61 0.61 0.61 0.61 0.61

0.99 0.94 0.90 0.88 0.85 0.84 0.82 0.81 0.80 0.79 0.78 0.77 0.76 0.76 0.75 0.75 0.74 0.74 0.74 0.73 0.73 0.73 0.73 0.72 0.72 0.72 0.72

1.29 1.21 1.14 1.10 1.06 1.03 1.00 0.98 0.96 0.94 0.93 0.91 0.90 0.89 0.88 0.87 0.87 0.86 0.85 0.85 0.84 0.84 0.83 0.83 0.83 0.82 0.82

1.94 1.77 1.64 1.54 1.46 1.39 1.34 1.29 1.25 1.22 1.19 1.16 1.14 1.12 1.10 1.08 1.07 1.05 1.04 1.03 1.02 1.01 1.00 0.99 0.98 0.97 0.96

2.99 2.67 2.44 2.25 2.11 1.99 1.89 1.80 1.73 1.67 1.61 1.56 1.52 1.48 1.45 1.42 1.39 1.36 1.34 1.32 1.30 1.28 1.26 1.24 1.23 1.21 1.20

4.28 3.72 3.30 2.98 2.72 2.51 2.33 2.18 2.05 1.94 1.85 1.76 1.68 1.62 1.55 1.50 1.45 1.40 1.36 1.32 1.28 1.25 1.22 1.19 1.17 1.14 1.12

6.57 5.61 4.90 4.34 3.89 3.53 3.23 2.97 2.75 2.56 2.39 2.24 2.11 1.99 1.89 1.79 1.71 1.63 1.55 1.49 1.43 1.37 1.31 1.27 1.22 1.18 1.14

9.67 8.20 7.10 6.24 5.56 5.00 4.53 4.13 3.80 3.50 3.24 3.02 2.82 2.64 2.47 2.33 2.19 2.07 1.96 1.86 1.76 1.67 1.59 1.52 1.45 1.38 1.32

14.09 11.94 10.33 9.08 8.08 7.26 6.58 6.00 5.51 5.08 4.70 4.37 4.08 3.81 3.58 3.36 3.17 2.99 2.83 2.68 2.54 2.41 2.29 2.18 2.08 1.98 1.89

20.09 17.06 14.79 13.02 11.60 10.45 9.48 8.67 7.97 7.36 6.83 6.36 5.95 5.57 5.24 4.93 4.66 4.41 4.18 3.97 3.77 3.59 3.42 3.26 3.12 2.98 2.85

29.76 25.46 22.24 19.73 17.73 16.09 14.72 13.56 12.57 11.71 10.96 10.30 9.71 9.18 8.70 8.27 7.88 7.53 7.20 6.90 6.62 6.36 6.13 5.90 5.70 5.50 5.32

38.28 32.95 28.95 25.84 23.35 21.32 19.62 18.19 16.96 15.89 14.96 14.14 13.40 12.75 12.16 11.63 11.14 10.70 10.30 9.92 9.58 9.26 8.96 8.69 8.43 8.19 7.96

47.65 41.30 36.54 32.84 29.88 27.46 25.44 23.73 22.27 21.00 19.89 18.91 18.04 17.26 16.56 15.92 15.34 14.82 14.34 13.89 13.48 13.10 12.75 12.42 12.11 11.83 11.56

54.27 47.11 41.74 37.56 34.22 31.49 29.21 27.28 25.63 24.20 22.95 21.84 20.86 19.98 19.19 18.47 17.82 17.23 16.68 16.18 15.72 15.29 14.89 14.52 14.18 13.85 13.55

66.20 58.33 52.42 47.83 44.16 41.15 38.65 36.53 34.71 33.14 31.76 30.54 29.46 28.50 27.63 26.84 26.12 25.47 24.87 24.32 23.81 23.34 22.90 22.50 22.12 21.76 21.43

57 Table 8: Adjustment factors obtained from the average IAT of the fit and actual IATs for I-71 to I-270 west entrance ramp for traffic volume range of 300-1600 vehicles/hour. Number of vehicles per hour

Average IAT of the fit (sec)

Actual IAT based on traffic volume (sec)

Adjustment factor

300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600

11.709 10.063 8.829 7.868 7.100 6.472 5.948 5.505 5.125 4.796 4.508 4.253 4.027 3.825 3.643 3.479 3.329 3.192 3.067 2.952 2.846 2.747 2.656 2.571 2.491 2.417 2.347

12.00 10.29 9.00 8.00 7.20 6.55 6.00 5.54 5.14 4.80 4.50 4.24 4.00 3.79 3.60 3.43 3.27 3.13 3.00 2.88 2.77 2.67 2.57 2.48 2.40 2.32 2.25

1.025 1.022 1.019 1.017 1.014 1.011 1.009 1.006 1.004 1.001 0.998 0.996 0.993 0.991 0.988 0.986 0.983 0.981 0.978 0.976 0.973 0.971 0.968 0.966 0.963 0.961 0.959

58 Table 9: : Adjusted cumulative interarrival time (in sec) distribution table obtained for I-71 to I-270 west entrance ramp for traffic volume range of 300-1600 vehicles/hour. Number of Vehicles per hour 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600

Adjusted Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.66 0.64 0.63 0.62 0.61 0.60 0.60 0.59 0.59 0.58 0.58 0.57 0.57 0.57 0.56 0.56 0.56 0.56 0.55 0.55 0.55 0.55 0.54 0.54 0.54 0.54 0.54

2 0.80 0.76 0.74 0.72 0.70 0.69 0.68 0.67 0.66 0.65 0.65 0.64 0.63 0.63 0.62 0.62 0.61 0.61 0.61 0.60 0.60 0.60 0.59 0.59 0.59 0.59 0.58

5 1.01 0.96 0.92 0.89 0.87 0.84 0.83 0.81 0.80 0.79 0.78 0.77 0.76 0.75 0.74 0.74 0.73 0.73 0.72 0.72 0.71 0.71 0.70 0.70 0.69 0.69 0.69

10 1.32 1.23 1.17 1.11 1.07 1.04 1.01 0.98 0.96 0.94 0.93 0.91 0.90 0.88 0.87 0.86 0.85 0.84 0.84 0.83 0.82 0.81 0.81 0.80 0.79 0.79 0.78

20 1.99 1.81 1.67 1.57 1.48 1.41 1.35 1.30 1.26 1.22 1.19 1.16 1.13 1.11 1.09 1.07 1.05 1.03 1.02 1.00 0.99 0.98 0.97 0.95 0.94 0.93 0.92

30 3.06 2.73 2.48 2.29 2.14 2.01 1.90 1.81 1.74 1.67 1.61 1.56 1.51 1.47 1.43 1.40 1.36 1.34 1.31 1.28 1.26 1.24 1.22 1.20 1.18 1.17 1.15

40 4.38 3.80 3.37 3.03 2.76 2.54 2.35 2.20 2.06 1.94 1.84 1.75 1.67 1.60 1.54 1.48 1.42 1.37 1.33 1.29 1.25 1.21 1.18 1.15 1.12 1.10 1.07

50 6.73 5.74 4.99 4.41 3.95 3.57 3.25 2.99 2.76 2.56 2.39 2.23 2.10 1.97 1.87 1.77 1.68 1.59 1.52 1.45 1.39 1.33 1.27 1.22 1.17 1.13 1.09

60 9.91 8.38 7.24 6.35 5.63 5.05 4.57 4.16 3.81 3.50 3.24 3.00 2.80 2.61 2.44 2.29 2.16 2.03 1.92 1.81 1.72 1.63 1.54 1.47 1.39 1.33 1.26

70 14.43 12.20 10.53 9.23 8.19 7.34 6.64 6.04 5.53 5.08 4.69 4.35 4.05 3.78 3.53 3.31 3.11 2.93 2.76 2.61 2.47 2.34 2.22 2.10 2.00 1.90 1.81

80 20.59 17.44 15.07 13.24 11.77 10.56 9.56 8.72 7.99 7.37 6.82 6.33 5.90 5.52 5.18 4.86 4.58 4.32 4.09 3.87 3.67 3.48 3.31 3.15 3.00 2.86 2.73

90 30.49 26.02 22.67 20.06 17.98 16.27 14.85 13.65 12.61 11.72 10.94 10.25 9.64 9.09 8.60 8.15 7.75 7.38 7.04 6.73 6.44 6.18 5.93 5.70 5.49 5.29 5.10

95 39.23 33.68 29.51 26.27 23.68 21.56 19.79 18.30 17.02 15.91 14.93 14.08 13.31 12.63 12.02 11.46 10.95 10.49 10.07 9.68 9.32 8.99 8.68 8.39 8.12 7.87 7.63

98 48.83 42.22 37.25 33.39 30.30 27.77 25.66 23.88 22.35 21.02 19.85 18.83 17.91 17.10 16.36 15.69 15.08 14.53 14.02 13.55 13.12 12.72 12.34 12.00 11.67 11.37 11.08

99 55.62 48.15 42.55 38.19 34.70 31.85 29.47 27.45 25.72 24.22 22.91 21.75 20.72 19.79 18.96 18.21 17.52 16.89 16.32 15.79 15.30 14.84 14.42 14.03 13.66 13.31 12.99

100 67.84 59.62 53.44 48.63 44.78 41.62 38.99 36.75 34.83 33.17 31.71 30.41 29.26 28.23 27.30 26.45 25.68 24.97 24.33 23.73 23.17 22.66 22.18 21.73 21.31 20.91 20.54

59 6.1.5

SR-2 to I-90 East entrance ramp

SR-2 to I-90 east entrance ramp site was part of the work zone construction site near Cleveland which is shown in Figure 14. Data collected as a part of data collection effort in [3] were used for analysis of this two lane entrance ramp. Figure 15 shows the location of the entrance ramp and the trailer used for data collection.

Figure 14: Map adapted from [3] showing the I-90 work zone construction in Cleveland.

60

Figure 15: Image showing the SR-2 to I-90 east entrance ramp and the trailer location.

The data for the site were collected from 13/9/2004 to 18/9/2004 for both the lanes. Three days of data from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM were used for analysis. The data collected for this entrance ramp site was segregated according to lane of travel and interarrival time distributions were developed for both the driving and the passing lane separately.

61 6.1.5.1 SR-2 to I-90 East entrance ramp driving lane

A total of 15,443 interarrival times or headways generated from three days of data from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM with an hourly traffic volume range of 4692 vehicles/hour with an average of 215 vehicles/hour. A plot of the observed hourly traffic volumes based on the 15-minute interval traffic counts is shown in Figure 16.

800 Maximum = 692 vehiles/hour Minimum = 4 vehicles/hour Average = 215 vehicles/hour

700

13/9/2004-14/9/2004 14/9/2004-15/9/2004 15/9/2004-16/9/2004

Traffic volume (vph)

600

500

400

300

200

100

15 .

00 15 -15 .4 .1 5 5 16 -16 .3 .0 0- 0 17 16 .1 .4 5 5 18 -17 .0 .3 0- 0 18 18 .4 .1 5 5 19 -19 .3 .0 0 0 20 -19 .1 .4 5- 5 21 20 .0 .3 0 0 21 -21 .4 .1 5- 5 22 22 .3 .0 0 0 23 -22 .1 .4 5- 5 23 0. .3 00 0 0. 0.1 45 5 1. 1.0 30 0 -1 2. .4 15 5 3. 2.3 00 0 3. 3.1 45 5 4. 4.0 30 0 5. 4.4 15 5 6. 5.3 00 0 -6 6. .1 45 5 7. 7.0 30 0 8. 7.4 15 5 9. 8.3 00 0 9. -9. 45 15 10 -10 .3 . 0 0 0 11 -10 .1 .4 5 5 12 -11 .0 .3 0 0 12 -12 .4 .1 5- 5 13 13 .3 .0 0 0 14 -13 .1 .4 5- 5 14 .3 0

0

15 minute interval

Figure 16: Hourly Traffic volumes observed for 15-minute intervals from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM for SR-2 to I-90 east entrance ramp driving lane.

62 A correction factor of 1.08 determined by the ORITE validation procedure [13] was used to adjust the 15-minute traffic volume counts to generate the adjusted hourly traffic counts. A total of 118 15-minute interval data sets were used out of 288 intervals to generate the cumulative interarrival times for hourly traffic volumes ranging from 200800 vehicles/hour. Cumulative interarrival times for the 118 15-minute data sets were extracted and the hyperbolic fits were used to fit the data. The hyperbolic fits obtained and the R2 values obtained for the fits are given in Table 10. Figure 17 and Figure 18 show the hyperbolic plots obtained for the observed data for 20%, 50% and 70%, 99% percentile values respectively for the SR-2 to I-90 east entrance ramp driving lane. Rest of the plots for other percentile points are provided in Appendix A.

Table 10: Hyperbolic fits and the R2 values obtained for the SR-2 to I-90 east entrance ramp driving lane. Cumulative percentage (%) 0 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 100

Model 0.1 y = 10.385/x + 0.567 y = 74.660/x + 0.562 y = 140.226/x + 0.633 y = 261.099/x + 0.691 y = 574.038/x + 0.720 y = 1024.641/x + 0.597 y = 1536.756/x + 0.611 y = 2295.759/x + 0.393 y = 3195.259/x + 0.167 y = 4526.68/x - 0.315 y = 6173.49/x - 0.425 y = 9762.123/x - 1.543 y = 11942.63/x - 0.190 y = 12594.39/x + 5.725 y = 14197.76 + 8.097 y = 14478.99/x + 15.975

R2

0.001 0.083 0.154 0.259 0.457 0.541 0.678 0.751 0.814 0.878 0.873 0.844 0.775 0.710 0.653 0.457

63 5 4.5

N = 118 R-square = 0.457

4

20% Hyperbolic

Interarrival time (sec)

3.5 3 2.5 2 1.5 1 0.5 0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

a) 14 N = 118 R-square = 0.751 12

50% Hyperbolic

Interarrival time (sec)

10

8

6

4

2

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

b) Figure 17: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp driving lane for a) 20% value b) 50% value.

64 25

N = 118 R-square = 0.878 20

Interarrival time (sec)

70% Hyperbolic

15

10

5

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

a) 100 90

N = 118 R-square = 0.653

80

99% Hyperbolic

Interarrival time (sec)

70 60 50 40 30 20 10 0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

b) Figure 18: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp driving lane for a) 20% value b) 50% value.

65 The cumulative interarrival times for the 16 percentile values were calculated using the hyperbolic relationships obtained in Table 10 for traffic volumes ranging from 200-800 vehicles/hour which are tabulated in Table 11. The average IAT of the interarrival time distribution for each hourly traffic volume was then determined using equation 1 and was compared with the average IAT calculated from the hourly traffic volume directly. The adjustment factors for each hourly traffic volume were then calculated which are tabulated in Table 12. The adjusted cumulative interarrival time distribution table was then generated by multiplying the adjustment factors obtained in Table 12 with the cumulative interarrival times calculated using the hyperbolic fits in Table 11. The adjusted cumulative interarrival time distribution table for SR-2 to I-90 east entrance ramp driving lane is given in Table 13.

66 Table 11: Cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp driving lane for traffic volume range of 200-800 vehicles/hour based on the hyperbolic fits. Number of Vehicles per hour 200 250 300 350 400 450 500 550 600 650 700 750 800

Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.62 0.61 0.60 0.60 0.59 0.59 0.59 0.59 0.58 0.58 0.58 0.58 0.58

0.94 0.86 0.81 0.78 0.75 0.73 0.71 0.70 0.69 0.68 0.67 0.66 0.66

1.33 1.19 1.10 1.03 0.98 0.94 0.91 0.89 0.87 0.85 0.83 0.82 0.81

2.00 1.74 1.56 1.44 1.34 1.27 1.21 1.17 1.13 1.09 1.06 1.04 1.02

3.59 3.02 2.63 2.36 2.16 2.00 1.87 1.76 1.68 1.60 1.54 1.49 1.44

5.72 4.70 4.01 3.52 3.16 2.87 2.65 2.46 2.30 2.17 2.06 1.96 1.88

8.29 6.76 5.73 5.00 4.45 4.03 3.68 3.41 3.17 2.98 2.81 2.66 2.53

11.87 9.58 8.05 6.95 6.13 5.49 4.98 4.57 4.22 3.92 3.67 3.45 3.26

16.14 12.95 10.82 9.30 8.16 7.27 6.56 5.98 5.49 5.08 4.73 4.43 4.16

22.32 17.79 14.77 12.62 11.00 9.74 8.74 7.91 7.23 6.65 6.15 5.72 5.34

30.44 24.27 20.15 17.21 15.01 13.29 11.92 10.80 9.86 9.07 8.39 7.81 7.29

47.27 37.51 31.00 26.35 22.87 20.15 17.98 16.21 14.73 13.48 12.41 11.48 10.66

59.52 47.58 39.62 33.93 29.67 26.35 23.70 21.52 19.71 18.18 16.87 15.73 14.74

68.70 56.10 47.71 41.71 37.21 33.71 30.91 28.62 26.72 25.10 23.72 22.52 21.47

79.09 64.89 55.42 48.66 43.59 39.65 36.49 33.91 31.76 29.94 28.38 27.03 25.84

88.37 73.89 64.24 57.34 52.17 48.15 44.93 42.30 40.11 38.25 36.66 35.28 34.07

67 Table 12: Adjustment factors obtained from the average IAT of the fit and actual IATs for SR-2 to I-90 East entrance ramp driving lane for traffic volume range of 200-800 vehicles/hour. Number of vehicles per hour

Average IAT of the fit (sec)

Actual IAT based on traffic volume (sec)

Adjustment factor

200 250 300 350 400 450 500 550 600 650 700 750 800

18.599 14.964 12.540 10.809 9.510 8.501 7.693 7.032 6.481 6.015 5.615 5.269 4.966

18.00 14.40 12.00 10.29 9.00 8.00 7.20 6.55 6.00 5.54 5.14 4.80 4.50

0.968 0.962 0.957 0.952 0.946 0.941 0.936 0.931 0.926 0.921 0.916 0.911 0.906

68 Table 13: Adjusted cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp driving lane for traffic volume range of 200-800 vehicles/hour. Number of Vehicles per hour 200 250 300 350 400 450 500 550 600 650 700 750 800

Adjusted Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.10 0.10 0.10 0.10 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09 0.09

0.60 0.59 0.58 0.57 0.56 0.56 0.55 0.55 0.54 0.54 0.53 0.53 0.53

0.90 0.83 0.78 0.74 0.71 0.68 0.67 0.65 0.64 0.62 0.61 0.60 0.59

1.29 1.15 1.05 0.98 0.93 0.89 0.85 0.83 0.80 0.78 0.76 0.75 0.73

1.93 1.67 1.49 1.37 1.27 1.20 1.14 1.09 1.04 1.01 0.97 0.95 0.92

3.47 2.90 2.52 2.25 2.04 1.88 1.75 1.64 1.55 1.48 1.41 1.35 1.30

5.54 4.52 3.84 3.35 2.99 2.70 2.48 2.29 2.13 2.00 1.89 1.79 1.70

8.03 6.50 5.49 4.76 4.21 3.79 3.45 3.17 2.94 2.74 2.57 2.42 2.29

11.49 9.22 7.70 6.62 5.80 5.17 4.67 4.25 3.91 3.61 3.36 3.15 2.96

15.62 12.46 10.35 8.85 7.72 6.84 6.14 5.56 5.08 4.68 4.33 4.03 3.77

21.60 17.12 14.14 12.01 10.41 9.17 8.18 7.37 6.69 6.12 5.63 5.21 4.84

29.46 23.35 19.29 16.38 14.20 12.51 11.16 10.05 9.13 8.35 7.69 7.11 6.61

45.75 36.10 29.67 25.08 21.64 18.97 16.83 15.09 13.64 12.41 11.36 10.45 9.66

57.61 45.79 37.91 32.29 28.07 24.80 22.18 20.04 18.25 16.74 15.45 14.33 13.36

66.48 53.99 45.65 39.69 35.21 31.73 28.93 26.64 24.73 23.11 21.72 20.51 19.45

76.54 62.44 53.04 46.31 41.25 37.31 34.16 31.57 29.40 27.57 25.99 24.62 23.42

85.52 71.11 61.47 54.57 49.37 45.32 42.06 39.38 37.13 35.22 33.58 32.14 30.88

69 6.1.5.2 SR-2 to I-90 East entrance ramp passing lane

A total of 35,130 interarrival times or headways were obtained from the three days of data analyzed from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM with an hourly traffic volume range of 24-1320 vehicles/hour with an average hourly traffic volume of 488 vehicles/hour. Figure 19 shows the plot of the observed hourly traffic volumes based on 15-minute interval traffic counts for the SR-2 to I-90 east entrance ramp passing lane.

1400

1200

Traffic volume (vph)

1000

Maximum = 1320 vehicles/hour Minimum = 24 vehicles/hour Average = 488 vehicles/hour 13/9/2004-14/9/2004 14/9/2004-15/9/2004 15/9/2004-16/9/2004

800

600

400

200

15 .0 0 15 -15 .4 .1 5 5 16 -16 .3 .0 0- 0 17 16 .1 .4 5 5 18 -17 .0 .3 0 0 18 -18 .4 .1 5 5 19 -19 .3 .0 0- 0 20 19 .1 .4 5 5 21 -20 .0 .3 0- 0 21 21 .4 .1 5 5 22 -22 .3 .0 0 0 23 -22 .1 .4 5- 5 2 0. 3.3 00 0 0 . 0 .1 45 5 1 . 1 .0 30 0 2 . 1 .4 15 5 3. 2 .3 00 0 -3 3. .1 45 5 4 . 4 .0 30 0 5 . 4 .4 15 5 6 . 5 .3 00 0 6. 6 .1 45 5 -7 7. .0 30 0 8 . 7 .4 15 5 9 . 8 .3 00 0 9. -9. 45 15 10 -10 .3 .0 0 0 11 -10 .1 .4 5 5 12 -11 .0 .3 0 0 12 -12 .4 .1 5- 5 13 13 .3 .0 0 0 14 -13 .1 .4 5- 5 14 .3 0

0

15 minute interval

Figure 19: Hourly Traffic volumes observed for 15-minute intervals from 13/9/2004 3:00 PM to 16/9/2004 3:00 PM for SR-2 to I-90 east entrance ramp passing lane.

70 A correction factor of 0.998 was applied to the 15-minute interval traffic counts to get the adjusted traffic volumes. 215 out of 288 15-minute intervals were used to generate the cumulative interarrival time distributions for hourly traffic volume ranging from 2001300 vehicles/hour. The hyperbolic fits obtained and the R2 values obtained for the fits are given in Table 14. Figure 20 and Figure 21 show the hyperbolic plots obtained for the observed data for 20%, 50% and 70%, 99% percentile values respectively for the SR-2 to I-90 east entrance ramp passing lane whereas the rest are provided in Appendix A.

Table 14: Hyperbolic fits and the R2 values obtained for the SR-2 to I-90 east entrance ramp passing lane. Cumulative percentage (%) 0 1 2 5 10 20 30 40 50 60 70 80 90 95 98 99 100

Model 0.1 y = 22.119/x + 0.555 y = 86.562/x + 0.546 y = 138.500/x + 0.619 y = 256.117/x + 0.644 y = 582.660/x + 0.552 y = 951.067/x + 0.479 y = 1594.577/x + 0.125 y = 2230.929/x - 0.030 y = 3207.518/x - 0.525 y = 4394.864/x - 0.689 y = 6023.856/x - 0.894 y = 8362.026/x - 0.174 y = 10337.76/x + 1.201 y = 12754.07/x + 2.882 y = 14433.08/x + 4.539 y = 14899.86/x + 12.211

R2

0.023 0.279 0.417 0.562 0.707 0.795 0.860 0.908 0.946 0.959 0.969 0.950 0.928 0.867 0.808 0.620

71 7

6 N = 215 R-square = 0.707

Interarrival time (sec)

5

20% Hyperbolic

4

3

2

1

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

a) 16

14 N = 215 R-square = 0.908

Interarrival time (sec)

12

50% Hyperbolic

10

8

6

4

2

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

b) Figure 20: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp passing lane for a) 20% value b) 50% value.

72 25

Interarrival time (sec)

20

N = 215 R-square = 0.959 15

70% Hyperbolic

10

5

0

0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

a) 100 90 80

N = 215 R-square = 0.808

Interarrival time (sec)

70

99% Hyperbolic

60 50 40 30 20 10 0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

b) Figure 21: Typical hyperbolic plots obtained for SR-2 to I-90 east entrance ramp passing lane for a) 70% value b) 99% value.

73 Cumulative interarrival times were generated from hourly traffic volumes ranging from 200-1300 vehicles/hour in increments of 50 vehicles/hour for the 16 percentile points based on the hyperbolic models given in Table 14. The cumulative interarrival time distribution table generated using the hyperbolic fits are tabulated in Table 15. The adjustment factors based on the average IAT of the distribution and average IAT calculated based on hourly traffic volume were calculated which are tabulated in Table 16. Then the cumulative interarrival times calculated in Table 15 were adjusted using the adjustment factors by multiplying the corresponding adjustment factor with the respective hourly traffic volumes. Table 17 shows the adjusted cumulative interarrival time distribution table calculated for SR-2 to I-90 east entrance ramp passing lane.

74 Table 15: Cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp passing lane for traffic volume range of 200-1300 vehicles/hour based on the hyperbolic fits. Number of Vehicles per hour 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300

Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.67 0.64 0.63 0.62 0.61 0.60 0.60 0.60 0.59 0.59 0.59 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.58 0.57 0.57 0.57 0.57

0.98 0.89 0.83 0.79 0.76 0.74 0.72 0.70 0.69 0.68 0.67 0.66 0.65 0.65 0.64 0.64 0.63 0.63 0.62 0.62 0.62 0.62 0.61

1.31 1.17 1.08 1.01 0.97 0.93 0.90 0.87 0.85 0.83 0.82 0.80 0.79 0.78 0.77 0.76 0.76 0.75 0.74 0.74 0.73 0.73 0.73

1.92 1.67 1.50 1.38 1.28 1.21 1.16 1.11 1.07 1.04 1.01 0.99 0.96 0.95 0.93 0.91 0.90 0.89 0.88 0.87 0.86 0.85 0.84

3.47 2.88 2.49 2.22 2.01 1.85 1.72 1.61 1.52 1.45 1.38 1.33 1.28 1.24 1.20 1.17 1.13 1.11 1.08 1.06 1.04 1.02 1.00

5.23 4.28 3.65 3.20 2.86 2.59 2.38 2.21 2.06 1.94 1.84 1.75 1.67 1.60 1.54 1.48 1.43 1.38 1.34 1.31 1.27 1.24 1.21

8.09 6.50 5.44 4.68 4.11 3.67 3.31 3.02 2.78 2.58 2.40 2.25 2.12 2.00 1.90 1.80 1.72 1.64 1.57 1.51 1.45 1.40 1.35

11.12 8.89 7.41 6.34 5.55 4.93 4.43 4.03 3.69 3.40 3.16 2.94 2.76 2.59 2.45 2.32 2.20 2.09 2.00 1.91 1.83 1.75 1.69

15.81 12.55 10.37 8.81 7.64 6.74 6.01 5.42 4.92 4.50 4.14 3.83 3.56 3.32 3.11 2.91 2.74 2.59 2.45 2.32 2.20 2.09 1.99

21.29 16.89 13.96 11.87 10.30 9.08 8.10 7.30 6.64 6.07 5.59 5.17 4.80 4.48 4.19 3.94 3.71 3.50 3.31 3.13 2.97 2.83 2.69

29.22 23.20 19.18 16.31 14.16 12.49 11.15 10.06 9.14 8.37 7.71 7.14 6.63 6.19 5.80 5.45 5.13 4.84 4.58 4.34 4.13 3.92 3.74

41.64 33.27 27.70 23.72 20.73 18.41 16.55 15.03 13.76 12.69 11.77 10.98 10.28 9.66 9.12 8.63 8.19 7.79 7.43 7.10 6.79 6.52 6.26

52.89 42.55 35.66 30.74 27.05 24.17 21.88 20.00 18.43 17.11 15.97 14.98 14.12 13.36 12.69 12.08 11.54 11.05 10.60 10.19 9.82 9.47 9.15

66.65 53.90 45.40 39.32 34.77 31.22 28.39 26.07 24.14 22.50 21.10 19.89 18.82 17.89 17.05 16.31 15.64 15.03 14.48 13.97 13.51 13.09 12.69

76.70 62.27 52.65 45.78 40.62 36.61 33.41 30.78 28.59 26.74 25.16 23.78 22.58 21.52 20.58 19.73 18.97 18.28 17.66 17.09 16.57 16.09 15.64

86.71 71.81 61.88 54.78 49.46 45.32 42.01 39.30 37.04 35.13 33.50 32.08 30.84 29.74 28.77 27.90 27.11 26.40 25.76 25.17 24.63 24.13 23.67

75 Table 16: Adjustment factors obtained from the average IAT of the fit and actual IATs for SR-2 to I-90 East entrance ramp passing lane for traffic volume range of 200-1300 vehicles/hour. Number of vehicles per hour

Average IAT of the fit (sec)

Actual IAT based on traffic volume (sec)

Adjustment factor

200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300

17.419 13.974 11.676 10.036 8.805 7.848 7.082 6.456 5.934 5.492 5.113 4.785 4.498 4.245 4.019 3.818 3.636 3.472 3.323 3.187 3.062 2.947 2.841

18.00 14.40 12.00 10.29 9.00 8.00 7.20 6.55 6.00 5.54 5.14 4.80 4.50 4.24 4.00 3.79 3.60 3.43 3.27 3.13 3.00 2.88 2.77

1.033 1.031 1.028 1.025 1.022 1.019 1.017 1.014 1.011 1.008 1.006 1.003 1.000 0.998 0.995 0.993 0.990 0.987 0.985 0.982 0.980 0.977 0.975

76 Table 17: Adjusted cumulative interarrival time (in sec) distribution table obtained for SR-2 to I-90 east entrance ramp passing lane for traffic volume range of 200-1300 vehicles/hour. Number of Vehicles per hour 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300

Adjusted Cumulative Interarrival Time (sec) Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

0.69 0.66 0.65 0.63 0.62 0.62 0.61 0.60 0.60 0.59 0.59 0.59 0.58 0.58 0.58 0.57 0.57 0.57 0.57 0.56 0.56 0.56 0.56

1.01 0.92 0.86 0.81 0.78 0.75 0.73 0.71 0.70 0.68 0.67 0.66 0.65 0.65 0.64 0.63 0.63 0.62 0.62 0.61 0.61 0.60 0.60

1.36 1.21 1.11 1.04 0.99 0.94 0.91 0.88 0.86 0.84 0.82 0.81 0.79 0.78 0.77 0.76 0.75 0.74 0.73 0.73 0.72 0.71 0.71

1.99 1.72 1.54 1.41 1.31 1.24 1.18 1.13 1.08 1.05 1.02 0.99 0.96 0.94 0.92 0.91 0.89 0.88 0.86 0.85 0.84 0.83 0.82

3.58 2.97 2.56 2.27 2.05 1.88 1.75 1.63 1.54 1.46 1.39 1.33 1.28 1.23 1.19 1.16 1.12 1.09 1.07 1.04 1.02 0.99 0.97

5.41 4.41 3.75 3.28 2.92 2.64 2.42 2.24 2.09 1.96 1.85 1.75 1.67 1.59 1.53 1.47 1.42 1.37 1.32 1.28 1.25 1.21 1.18

8.36 6.70 5.59 4.79 4.20 3.74 3.37 3.06 2.81 2.60 2.42 2.26 2.12 1.99 1.89 1.79 1.70 1.62 1.55 1.48 1.42 1.37 1.32

11.50 9.17 7.61 6.50 5.67 5.02 4.51 4.08 3.73 3.43 3.18 2.95 2.76 2.59 2.44 2.30 2.18 2.07 1.97 1.88 1.79 1.71 1.64

16.34 12.93 10.65 9.03 7.81 6.87 6.11 5.49 4.98 4.54 4.17 3.84 3.56 3.31 3.09 2.89 2.71 2.55 2.41 2.28 2.15 2.04 1.94

22.00 17.41 14.35 12.16 10.53 9.25 8.24 7.40 6.71 6.12 5.62 5.19 4.81 4.47 4.17 3.91 3.67 3.45 3.26 3.08 2.91 2.76 2.62

30.20 23.91 19.71 16.72 14.48 12.73 11.34 10.20 9.25 8.44 7.76 7.16 6.64 6.18 5.77 5.41 5.08 4.78 4.51 4.27 4.04 3.83 3.64

43.02 34.29 28.47 24.31 21.19 18.77 16.83 15.24 13.92 12.80 11.84 11.01 10.28 9.64 9.07 8.56 8.11 7.69 7.31 6.97 6.66 6.37 6.10

54.65 43.85 36.65 31.50 27.64 24.64 22.24 20.28 18.64 17.25 16.06 15.03 14.13 13.33 12.63 11.99 11.42 10.91 10.44 10.01 9.62 9.25 8.92

68.87 55.54 46.65 40.30 35.54 31.83 28.86 26.43 24.41 22.69 21.22 19.95 18.83 17.85 16.97 16.19 15.48 14.84 14.26 13.72 13.24 12.79 12.37

79.26 64.17 54.11 46.92 41.52 37.32 33.96 31.21 28.91 26.97 25.30 23.86 22.59 21.47 20.48 19.59 18.78 18.05 17.39 16.79 16.23 15.72 15.24

89.60 74.00 63.59 56.15 50.56 46.20 42.71 39.85 37.46 35.43 33.69 32.18 30.85 29.68 28.63 27.69 26.84 26.07 25.36 24.72 24.13 23.58 23.07

77 6.2

Comparison of Cumulative IATs of the entrance ramps with the freeways

Cumulative interarrival time distributions for the entrance ramps having different configurations and hourly traffic volume ranges have been established. The next step was to check whether the cumulative interarrival time distributions obtained for the entrance ramps are similar to the cumulative interarrival time distributions obtained for the freeways in [13]. The comparisons were done using the graphical plots (visual comparison) which are shown in the next section. The graphical comparisons were made by plotting the cumulative interarrival times for both the entrance ramps and the freeways for similar hourly traffic volumes. The entrance ramp cumulative interarrival times were compared with cumulative interarrival times for all lanes of 2-lane, 3-lane and 4-lane freeways generated in [13]. Comparisons were done for hourly traffic volumes selected over the whole range of observed hourly traffic volumes for each entrance ramp to see whether the interarrival time distributions were different from distributions for the freeways for different traffic volumes. For each traffic volume, a total of nine cumulative interarrival time distribution plots were generated for all lanes of 2-lane, 3-lane and 4-lane freeways to compare with the entrance ramp cumulative interarrival times.

78 6.2.1

I-71 South to I-270 West entrance ramp

The range of hourly traffic volumes for I-71 south to I-270 west entrance ramp cumulative interarrival time distributions was 200-800 vehicles/hour. Cumulative interarrival time plots for the entrance ramp were generated for the following hourly traffic volumes: 200 vehicles/hour, 400 vehicles/hour, 600 vehicles/hour and 800 vehicles/hour. The cumulative interarrival times for all lanes of the 2-lane, 3-lane and 4lane freeway sites were generated from the tables in [13]. The cumulative interarrival times used for comparison for traffic volumes of 200 vehicles/hour and 800 vehicles/hour are tabulated in Table 18. Figure 22, Figure 23 and Figure 24, Figure 25 show the comparisons of the cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp with selected lanes of the 2-lane, 3-lane, 4-lane freeways for traffic volume of 200 vehicles/hour respectively. The comparisons with selected lanes of all freeways for rest of the traffic volumes are provided in Appendix B. Figure 26 and Figure 27 show the comparison of cumulative IAT distributions of the I-71 south to I-270 west entrance ramp with 2-lane, 3-lane and 4-lane freeways for traffic volumes of 200 and 800 vehicles/hour respectively. The comparisons for 400 vehicles/hour and 600 vehicles/hour are provided in Appendix B.

79 Table 18: Cumulative interarrival times (in sec) for I-71 south to I-270 west entrance ramp, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 200 vehicles/hour b) 800 vehicles/hour used for comparison.

a) Traffic volume 200 vph Site I-71 S to I-270 W 2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.11 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.83 1.12 0.58 0.80 1.03 0.51 0.74 0.92 0.93 0.51

2 1.01 1.28 0.69 0.99 1.23 0.70 1.04 1.08 1.37 0.82

5 1.18 2.12 1.01 1.72 1.88 1.04 1.42 1.85 1.85 1.08

10 1.59 2.99 1.48 2.65 2.69 1.54 2.18 2.71 2.87 1.84

20 2.79 4.90 2.62 4.70 4.36 3.13 3.86 4.82 4.90 3.45

30 5.17 7.17 4.22 7.16 6.35 5.29 5.57 7.14 7.33 6.22

40 8.16 9.83 6.51 10.09 9.00 8.06 8.56 10.14 10.42 9.11

50 11.97 13.14 9.97 12.97 12.21 11.93 12.39 13.42 13.69 12.75

60 17.12 17.34 15.09 17.06 16.40 16.83 18.18 17.63 18.16 17.24

70 22.63 22.91 21.99 22.12 21.60 23.04 24.27 22.82 23.32 23.17

80 29.67 30.46 32.26 29.55 29.84 31.85 32.30 30.60 30.18 32.63

90 45.35 42.26 49.77 43.48 44.33 46.40 44.23 43.61 41.59 44.50

95 55.98 53.37 66.27 55.37 59.19 60.09 56.41 52.99 52.91 56.42

98 61.83 70.21 84.53 74.00 79.49 74.64 63.85 63.12 63.33 66.42

99 72.10 73.01 98.23 77.67 84.83 87.63 77.37 76.30 74.09 83.40

100 91.34 212.47 113.76 94.65 111.67 99.77 85.93 81.56 80.46 92.69

70 4.56 5.30 4.95 5.42 5.33 5.11 4.79 5.50 5.45 5.38

80 7.36 7.00 7.37 7.18 7.06 7.36 7.14 7.21 7.19 7.45

90 10.50 10.05 11.80 10.07 10.17 11.30 11.21 9.95 10.01 11.63

95 14.52 13.22 16.79 13.25 13.42 15.37 15.48 12.88 12.52 15.69

98 21.03 16.78 23.67 16.97 17.67 20.77 21.60 16.77 16.42 20.93

99 23.58 19.89 26.42 20.81 20.48 23.96 24.37 18.42 18.33 23.63

100 32.71 39.08 38.28 27.42 28.45 34.03 38.69 29.44 28.84 38.10

b) Traffic volume 800 vph Site I-71 S to I-270 W 2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.63 0.65 0.48 0.64 0.62 0.57 0.60 0.58 0.62 0.49

2 0.69 0.79 0.56 0.76 0.72 0.64 0.67 0.68 0.70 0.54

5 0.82 1.01 0.67 0.93 0.90 0.78 0.83 0.87 0.89 0.68

10 0.97 1.27 0.80 1.16 1.14 0.94 1.01 1.11 1.14 0.78

20 1.14 1.69 1.02 1.56 1.57 1.22 1.34 1.57 1.59 1.04

30 1.33 2.14 1.34 2.01 2.04 1.59 1.79 2.08 2.12 1.33

40 1.78 2.66 1.81 2.57 2.59 2.09 2.23 2.67 2.73 1.87

50 2.41 3.29 2.46 3.31 3.27 2.74 2.79 3.40 3.45 2.68

60 3.28 4.15 3.39 4.21 4.16 3.69 3.50 4.30 4.33 3.87

80 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

b) Figure 22: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane1 b) lane2 of 2-lane freeway.

81 120

a)

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane1

60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec) 120

b)

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure 23: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane1 b) lane2 c) lane3 of 3 lane freeway.

82 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

b) Figure 24: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane1 b) lane2 of 4-lane freeway.

83 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

b) Figure 25: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) lane3 b) lane4 of 4-lane freeway.

84 120

a)

Cumulative percentage (%)

100

80

60

Ent ramp

40

2-Lane (lane1) Lane 2 20

0 0.10

1.00

10.00 100.00 Cumulative interarrival time (sec)

1000.00

120

b)

Cumulative percentage (%)

100

80 Ent ramp 3-Lane (lane1) Lane 2 Lane 3

60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60 Ent ramp 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00 100.00 Cumulative interarrival time (sec)

1000.00

Figure 26: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 200 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

85 120

a)

Cumulative percentage (%)

100

80

60 Ent ramp 2-Lane (lane1) Lane 2

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60 Ent ramp 3-Lane (lane1) Lane 2 Lane 3

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec) 120

c)

Cumulative percentage (%)

100

80

60 Ent ramp 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure 27: Comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

86 6.2.2

I-71 to I-270 west entrance ramp

Cumulative interarrival time distributions were developed for hourly traffic volumes from 300-1600 vehicles/hour. Cumulative interarrival time plots were generated for 300 vehicles/hour, 600 vehicles/hour, 900 vehicles/hour and 1100 vehicles/hour which were compared with the cumulative interarrival times for all lanes of the 2-lane, 3lane and 4-lane freeway sites from [13]. Cumulative interarrival times for the I-71 to I270 west entrance ramp and 2-lane, 3-lane, 4-lane freeways for hourly traffic volumes of 300 vehicles/hour, 1100 vehicles/hour used for comparison are tabulated in Table 19. The comparisons for the remaining hourly traffic volumes are provided in the Appendix B. Figure 28 and Figure 29 show the comparisons of the cumulative interarrival time distributions for I-71 to I-270 west entrance ramp with 2-lane, 3-lane, 4-lane freeways for hourly traffic volumes of 300 vehicles/hour and 1100 vehicles/hour respectively.

87 Table 19: Cumulative interarrival times (in sec) for I-71 to I-270 west entrance ramp, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 300 vehicles/hour b) 1100 vehicles/hour used for comparison. a) Traffic volume 300 vph Site I-71 to I-270 W 2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.66 0.92 0.55 0.74 0.86 0.54 0.68 0.78 0.80 0.51

2 0.80 1.08 0.65 0.90 1.01 0.68 0.88 0.91 1.07 0.70

5 1.01 1.63 0.87 1.37 1.46 0.91 1.16 1.43 1.43 0.91

10 1.32 2.24 1.19 1.99 2.01 1.22 1.67 2.01 2.11 1.37

20 1.99 3.49 1.92 3.31 3.13 2.12 2.75 3.38 3.44 2.38

30 3.06 4.94 2.95 4.87 4.44 3.31 3.90 4.90 5.02 4.04

40 4.38 6.65 4.43 6.74 6.16 4.87 5.75 6.83 7.01 5.89

50 6.73 8.76 6.63 8.67 8.25 7.02 8.12 8.97 9.14 8.27

60 9.91 11.47 9.88 11.34 10.97 9.81 11.64 11.70 12.01 11.29

70 14.43 15.08 14.40 14.69 14.38 13.46 15.59 15.11 15.37 15.26

80 20.59 20.02 21.18 19.60 19.72 18.77 21.10 20.19 19.96 21.43

90 30.49 27.93 32.88 28.60 29.15 27.66 29.56 28.62 27.55 29.90

95 39.23 35.52 44.30 36.62 38.85 36.21 38.24 35.15 34.95 38.34

98 48.83 46.45 57.58 48.60 52.01 45.89 45.16 42.55 42.49 46.27

99 100 55.62 67.84 49.44 135.09 66.40 80.57 52.40 64.81 56.25 74.73 53.65 64.74 53.90 65.21 50.55 58.66 49.30 57.61 56.88 68.67

70 3.11 3.72 3.43 3.90 3.85 3.49 3.05 3.93 3.83 3.78

80 4.58 4.89 5.14 5.15 4.99 5.14 4.87 5.10 5.11 5.18

90 7.75 7.14 8.38 7.06 7.09 8.12 8.21 6.93 7.14 8.63

95 10.95 9.57 12.28 9.44 9.29 11.31 11.72 9.25 8.86 11.95

98 15.08 11.95 18.02 11.83 12.10 15.86 17.63 12.51 12.15 16.70

99 17.52 15.02 19.81 15.61 14.64 18.16 19.44 13.18 13.26 18.13

b) Traffic volume 1100 vph Site I-71 to I-270 W 2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.56 0.60 0.45 0.61 0.58 0.57 0.57 0.53 0.59 0.48

2 0.61 0.74 0.53 0.72 0.66 0.63 0.62 0.62 0.63 0.51

5 0.73 0.89 0.62 0.84 0.81 0.75 0.77 0.77 0.80 0.63

10 0.85 1.10 0.72 1.01 0.99 0.88 0.89 0.95 0.97 0.67

20 1.05 1.39 0.86 1.26 1.30 1.04 1.10 1.26 1.28 0.82

30 1.36 1.67 1.07 1.53 1.64 1.25 1.44 1.61 1.65 0.89

40 1.42 2.00 1.37 1.89 2.00 1.55 1.65 1.99 2.03 1.22

50 1.68 2.40 1.78 2.43 2.45 1.91 1.92 2.48 2.51 1.78

60 2.16 2.96 2.35 3.04 3.05 2.51 2.18 3.09 3.08 2.66

100 25.68 23.75 30.97 21.21 20.87 27.96 34.01 24.34 24.02 32.80

88 120

a)

Cumulative percentage (%)

100

80 Ent ramp Right lane (lane1) Lane 2

60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative Interarrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp Right lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec) 120

c)

Cumulative percentage (%)

100

80

60

Ent ramp Right lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00 10.00 Cumulative Interarrival time (sec)

100.00

Figure 28: Comparison of cumulative interarrival time distributions for I-71 to I270 west entrance ramp for traffic volume of 300 vehicles/hour with a) 2-lane b) 3lane c) 4-lane freeway.

89 120

a)

Cumulative percentage (%)

100

80

60

40 Ent ramp Right lane(lane 1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec) 120

b)

Cumulative percentage (%)

100

80

60

Ent ramp Right lane (lane1) Lane 2 Lane 3

40

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80 Ent ramp 4-Lane (lane1) Lane 2 Lane 3 Lane 4

60

40

20

0 0.10

1.00 10.00 Cumulative interarrival time (sec)

100.00

Figure 29: Comparison of cumulative interarrival time distributions for I-71 to I270 west entrance ramp for traffic volume of 1100 vehicles/hour with a) 2-lane b) 3lane c) 4-lane freeway.

90 6.2.3

SR-2 to I-90 East entrance ramp

6.2.3.1 Driving lane

Interarrival time distributions were developed for hourly traffic volumes range of 200-800 vehicles/hour for the driving lane of SR-2 to I-90 east entrance ramp. They were compared with the cumulative interarrival time distributions for all lanes of 2-lane, 3-lane and 4-lane freeways obtained in [13] for the following hourly traffic volumes: 200 vehicles/hour, 400 vehicles/hour, 600 vehicles/hour and 800 vehicles/hour. The cumulative interarrival times used for comparison for the entrance ramp and the freeways are tabulated in Table 20 where as the comparison plots for the hourly traffic volumes of 200 vehicles/hour and 800 vehicles/hour are shown in Figure 30 and Figure 31. The comparison plots for 400 vehicles/hour and 600 vehicles/hour are provided in Appendix B.

91 Table 20: Cumulative interarrival times (in sec) for SR-2 to I-90 east entrance ramp driving lane, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 200 vehicles/hour b) 800 vehicles/hour used for comparison. a) Traffic volume 200 vph Site SR-2 to I-90 E Driving

2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.60 1.12 0.58 0.80 1.03 0.51 0.74 0.92 0.93 0.51

2 0.90 1.28 0.69 0.99 1.23 0.70 1.04 1.08 1.37 0.82

5 1.29 2.12 1.01 1.72 1.88 1.04 1.42 1.85 1.85 1.08

10 1.93 2.99 1.48 2.65 2.69 1.54 2.18 2.71 2.87 1.84

20 3.47 4.90 2.62 4.70 4.36 3.13 3.86 4.82 4.90 3.45

30 5.54 7.17 4.22 7.16 6.35 5.29 5.57 7.14 7.33 6.22

40 8.03 9.83 6.51 10.09 9.00 8.06 8.56 10.14 10.42 9.11

50 11.49 13.14 9.97 12.97 12.21 11.93 12.39 13.42 13.69 12.75

60 15.62 17.34 15.09 17.06 16.40 16.83 18.18 17.63 18.16 17.24

70 21.60 22.91 21.99 22.12 21.60 23.04 24.27 22.82 23.32 23.17

80 29.46 30.46 32.26 29.55 29.84 31.85 32.30 30.60 30.18 32.63

90 45.75 42.26 49.77 43.48 44.33 46.40 44.23 43.61 41.59 44.50

95 57.61 53.37 66.27 55.37 59.19 60.09 56.41 52.99 52.91 56.42

98 66.48 70.21 84.53 74.00 79.49 74.64 63.85 63.12 63.33 66.42

99 100 76.54 85.52 73.01 212.47 98.23 113.76 77.67 94.65 84.83 111.67 87.63 99.77 77.37 85.93 76.30 81.56 74.09 80.46 83.40 92.69

70 4.84 5.30 4.95 5.42 5.33 5.11 4.79 5.50 5.45 5.38

80 6.61 7.00 7.37 7.18 7.06 7.36 7.14 7.21 7.19 7.45

90 9.66 10.05 11.80 10.07 10.17 11.30 11.21 9.95 10.01 11.63

95 13.36 13.22 16.79 13.25 13.42 15.37 15.48 12.88 12.52 15.69

98 19.45 16.78 23.67 16.97 17.67 20.77 21.60 16.77 16.42 20.93

99 23.42 19.89 26.42 20.81 20.48 23.96 24.37 18.42 18.33 23.63

b) Traffic volume 800 vph Site SR-2 to I-90 E Driving

2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.09 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.53 0.65 0.48 0.64 0.62 0.57 0.60 0.58 0.62 0.49

2 0.59 0.79 0.56 0.76 0.72 0.64 0.67 0.68 0.70 0.54

5 0.73 1.01 0.67 0.93 0.90 0.78 0.83 0.87 0.89 0.68

10 0.92 1.27 0.80 1.16 1.14 0.94 1.01 1.11 1.14 0.78

20 1.30 1.69 1.02 1.56 1.57 1.22 1.34 1.57 1.59 1.04

30 1.70 2.14 1.34 2.01 2.04 1.59 1.79 2.08 2.12 1.33

40 2.29 2.66 1.81 2.57 2.59 2.09 2.23 2.67 2.73 1.87

50 2.96 3.29 2.46 3.31 3.27 2.74 2.79 3.40 3.45 2.68

60 3.77 4.15 3.39 4.21 4.16 3.69 3.50 4.30 4.33 3.87

100 30.88 39.08 38.28 27.42 28.45 34.03 38.69 29.44 28.84 38.10

92 120

a)

Cumulative percentage (%)

100

80

60 Ent ramp-driving 2-Lane (lane1) Lane 2

40

20

0 0.10

1.00

10.00 100.00 Cumulative interarrival time (sec)

1000.00

120

b)

Cumulative percentage (sec)

100

80

60

40

Ent ramp-driving 3-Lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

Ent ramp-driving 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00 100.00 Cumulative interarrival time (sec)

1000.00

Figure 30: Comparison of cumulative interarrival time distributions for SR-2 to I90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with a) 2lane b) 3-lane c) 4-lane freeway.

93 120

a)

Cumulative percentage (%)

100

80

60

40

Ent ramp-driving 2-Lane (lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec) 120

b)

Cumulative percentage (%)

100

80

60

Ent ramp-driving 3-Lane (lane1) Lane 2 Lane 3

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60 Ent ramp-driving 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure 31: Comparison of cumulative interarrival time distributions for SR-2 to I90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with a) 2lane b) 3-lane c) 4-lane freeway.

94 6.2.3.2 Passing lane

Cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane were developed for hourly traffic volume range of 200-1300 vehicles/hour. These distributions were compared with the cumulative interarrival time distributions for all lanes of 2-lane, 3-lane and 4-lane freeways from [13] for hourly traffic volumes of 300 vehicles/hour, 600 vehicles/hour, 900 vehicles/hour and 1100 vehicles/hour. The cumulative interarrival times used for comparison are tabulated in Table 21 and the comparison plots for traffic volumes of 300 vehicles/hour, 1100 hour vehicles/hour are shown in Figure 32 and Figure 33 respectively.

95 Table 21: Cumulative interarrival times (in sec) for SR-2 to I-90 east entrance ramp passing lane, 2-lane, 3-lane and 4-lane freeways for traffic volumes of a) 300 vehicles/hour b) 1100 vehicles/hour used for comparison. a) Traffic volume 300 vph Site SR-2 to I-90 E Passing

2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.65 0.92 0.55 0.74 0.86 0.54 0.68 0.78 0.80 0.51

2 0.86 1.08 0.65 0.90 1.01 0.68 0.88 0.91 1.07 0.70

5 1.11 1.63 0.87 1.37 1.46 0.93 1.16 1.43 1.43 0.91

10 1.54 2.24 1.19 1.99 2.01 1.27 1.67 2.01 2.11 1.37

20 2.56 3.49 1.92 3.31 3.13 2.29 2.75 3.38 3.44 2.38

30 3.75 4.94 2.95 4.87 4.44 3.64 3.90 4.90 5.02 4.04

40 5.59 6.65 4.43 6.74 6.16 5.41 5.75 6.83 7.01 5.89

50 7.61 8.76 6.63 8.67 8.25 7.84 8.12 8.97 9.14 8.27

60 10.65 11.47 9.88 11.34 10.97 10.98 11.64 11.70 12.01 11.29

70 14.35 15.08 14.40 14.69 14.38 15.07 15.59 15.11 15.37 15.26

80 19.71 20.02 21.18 19.60 19.72 20.96 21.10 20.19 19.96 21.43

90 28.47 27.93 32.88 28.60 29.15 30.80 29.56 28.62 27.55 29.90

95 36.65 35.52 44.30 36.62 38.85 40.22 38.24 35.15 34.95 38.34

98 46.65 46.45 57.58 48.60 52.01 50.71 45.16 42.55 42.49 46.27

99 100 54.11 63.59 49.44 135.09 66.40 80.57 52.40 64.81 56.25 74.73 59.35 70.62 53.90 65.21 50.55 58.66 49.30 57.61 56.88 68.67

70 3.26 3.72 3.43 3.90 3.85 3.49 3.05 3.93 3.83 3.78

80 4.51 4.89 5.14 5.15 4.99 5.14 4.87 5.10 5.11 5.18

90 7.31 7.14 8.38 7.06 7.09 8.12 8.21 6.93 7.14 8.63

95 10.44 9.57 12.28 9.44 9.29 11.31 11.72 9.25 8.86 11.95

98 14.26 11.95 18.02 11.83 12.10 15.86 17.63 12.51 12.15 16.70

99 17.39 15.02 19.81 15.61 14.64 18.16 19.44 13.18 13.26 18.13

b) Traffic volume 1100 vph Site SR-2 to I-90 E Passing

2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Cumulative Percentage(%) 0 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

1 0.57 0.60 0.45 0.61 0.58 0.57 0.57 0.53 0.59 0.48

2 0.62 0.74 0.53 0.72 0.66 0.63 0.62 0.62 0.63 0.51

5 0.73 0.89 0.62 0.84 0.81 0.75 0.77 0.77 0.80 0.63

10 0.86 1.10 0.72 1.01 0.99 0.88 0.89 0.95 0.97 0.67

20 1.07 1.39 0.86 1.26 1.30 1.04 1.10 1.26 1.28 0.82

30 1.32 1.67 1.07 1.53 1.64 1.25 1.44 1.61 1.65 0.89

40 1.55 2.00 1.37 1.89 2.00 1.55 1.65 1.99 2.03 1.22

50 1.97 2.40 1.78 2.43 2.45 1.91 1.92 2.48 2.51 1.78

60 2.41 2.96 2.35 3.04 3.05 2.51 2.18 3.09 3.08 2.66

100 25.36 23.75 30.97 21.21 20.87 27.96 34.01 24.34 24.02 32.80

96 120

a)

Cumulative percentage (%)

100

80

60

40 Ent ramp passing Right Lane (lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative Interarrival time (sec) 120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp passing Right lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

40

Ent ramp passing Right lane (lane1) Lane 2 Lane 3 Lane 4

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

Figure 32: Comparison of cumulative interarrival time distributions for SR-2 to I90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with a) 2lane b) 3-lane c) 4-lane freeway.

97 120

Cumulative percentage (%)

100

a)

80

60

40 Ent ramp passing Right lane (lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec) 120

Cumulative percentage (%)

100

b)

80

60

Ent ramp passing Right lane (lane1) Lane 2 Lane 3

40

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

Ent ramp passing 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure 33: Comparison of cumulative interarrival time distributions for SR-2 to I90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

98 6.3

Extrapolation of the Cumulative IAT distributions for low/high traffic volumes

A total of nine cumulative interarrival time distributions were developed for all lanes of 2-lane, 3-lane and 4-lane highways in [13] with different traffic volume ranges specific to each lane for all the freeways. The maximum traffic volume range for which the cumulative IAT distributions were developed was for lane 1 (right lane) of the 3-lane freeway with a traffic volume range of 200-1800 vehicles/hour [13]. This section discusses how the cumulative interarrival time distributions obtained in [13] were extrapolated for low and high traffic volumes beyond the observed range of traffic volume data. The extrapolated cumulative interarrival time distributions can be used for work zone traffic simulations where the hourly traffic volumes are too low or high to model the traffic flow. The first step in this process was to determine the limits for the low and high hourly traffic volumes for extrapolation which are practical and close to real-world traffic situations observed on the freeways. The uncertainty in the results obtained by the extrapolation methods in general are limited by the assumptions made for the function used for extrapolation [22], [24]. To approximate limits for infinite sequences with good accuracy, a large amount of data and extensive calculations are required [23]. According to Sidi [23] “These limits can be approximated economically and with high accuracy by applying suitable extrapolation methods to small number of terms” which makes the use of extrapolation viable if the underlying process of the method used for extrapolation and the nature of the process is

99 well known. Extrapolation methods have been used as convergence acceleration methods for many years and give solutions to problems which are otherwise unsolvable [24]. Considering extrapolation for low traffic volumes, traffic flow at low volumes is relatively stable with independent random arrivals and no traffic congestion and time delays are experienced by the motorists. Traffic simulation models can be used without any problems for low hourly traffic volumes. The low observed traffic volume was 150 vehicles/hour [13] and the cumulative interarrival time distributions were extrapolated to 2 vehicle/hour on the lower side of the observed traffic range in [13] using the hyperbolic fits. The maximum hourly traffic volume for which cumulative interarrival time distributions were developed in [13] was 1800 vehicles/hour for lane 1 of 3-lane freeway. Therefore this case was used to determine the limit for the high traffic volume to which the distributions would be extrapolated in order to reduce the variability due to extrapolation. A trade off between the safety of the motorists and traffic volumes that could be measured in real-traffic situations had to be made to obtain the limit for the high traffic volume to be used for cumulative IAT distribution extrapolation. The interarrival times or the headways for traffic flow with large traffic volumes are generally low i.e. the drivers would be in car following state. In order for the driver to be safe, a minimum time headway which is greater than the driver reaction time should be maintained to avoid an accident. The minimum reaction times required for the driver to react to a situation in traffic based on the braking method used are given in the driver performance data book [25]. The reaction times using left foot poised on the brake is 0.42

100 seconds for males and 0.54 seconds for females with an average of 0.48 seconds [25]. A minimum time headway approaching this value is not desirable to ensure the safety of the motorist. The traffic engineering handbook [26] discusses the speed flow relationships and the saturation flow rates observed on freeways. The saturation capacity for a two lane freeway occurs between 2000-2800 vehicles/hour whereas for multilane freeway occurs at 2200 vehicles/hour/lane [26]. Based on the literature review performed it is observed that traffic volumes greater than 2500 vehicles/hour/lane are hard to observe in real world situations in a free flowing state without generating any traffic queues. To apply the aspects of safety, driver reaction times and the saturation flows observed on freeways to extrapolate the cumulative interarrival time distributions, the cumulative interarrival times for 1 percentile point were extrapolated so that only one in hundred would be in a situation with an IAT close to the minimum reaction time specified in the driver performance data handbook [25]. The cumulative IATs for 1 % of 3-lane lane1 were extrapolated from 1800 vehicles/hour/lane to 2500 vehicles/hour/lane in intervals of 50 vehicles/hour using the same hyperbolic plots used in [13] to develop the cumulative IAT distributions for the observed range of the traffic volume of 200-1800 vehicles/hour. The interarrival times obtained were adjusted based on the average IAT of the fit and the actual average IAT obtained based on hourly traffic volume. The adjusted cumulative IAT for a traffic volume of 2500 vehicles/hour was 0.50 second which was close to the average of the minimum reaction time required for the driver to react to any situation based on driver performance data handbook [25]. It was found that the adjusted

101 cumulative interarrival time for one percentile varied from 0.532 seconds for traffic volume of 1775 vehicles/hour to 0.500 seconds for 2475 vehicles/hour with only an approximate change of three hundredth’s of a second which further justified the traffic volume of 2500 vehicles/hour as the maximum limit for extrapolation. The hyperbolic plot obtained for cumulative interarrival time for one percentile for lane 1 of 3-lane freeway after extrapolation is shown in Figure 34. Once the limits for the low and high hourly traffic volumes were established to be 2 vehicle/hour and 2500 vehicles/hour respectively, all the nine cumulative interarrival time distributions obtained for all lanes of 2-lane, 3-lane and 4-lane freeways were extrapolated for a range of 2-2500 vehicles/hour in increments of 50 vehicles/hour using the hyperbolic fits obtained for each cumulative IAT distribution in [13]. The cumulative interarrival times obtained were adjusted based on the adjustment factors obtained from the average IAT of the fit and the actual average IAT based on traffic volume. Cumulative IAT distributions for any hourly traffic volume were calculated by linearly interpolating between the cumulative IAT distributions of the traffic volumes defining the interval into which desired hourly traffic volume falls into [13]. For example IAT distribution for a traffic volume of 263 vehicles/hour could be calculated by interpolating the values of the IAT distributions for the traffic volumes of 250 and 300 vehicles/hour. The extrapolated adjusted cumulative interarrival time distribution tables for 2-lane lane 2, 3-lane lane 1 and 4-lane lane 3 are shown in Table 22, Table 23 and Table 24 respectively. The rest are provided in the Appendix C.

102 2.50

N = 50

Interarrival time (sec)

2.00 Extrapolated traffic volume range with low traffic volumes volumes

1.50

1.00

Observed traffic volume range

Extrapolated Observed

Extrapolated traffic volume range with high traffic volumes

0.50

0.00 0

0 10

0 20

0 30

0 40

0 50

0 60

0 70

0 80

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 250

Traffic volume (vph)

Figure 34: Adjusted cumulative interarrival time plot for 1% for lane 1 of 3-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic fits used in [13].

To illustrate how the extrapolated distribution tables in the excel spreadsheets given at http://webce.ent.ohiou.edu/orite/cumulativeIATdistributions.html can be used to generate cumulative IAT distributions for traffic volumes from 2-2500 vehicles/hour, three traffic volumes of 4 vehicles/hour, 958 vehicles/hour and 2478 vehicles/hour were selected for lane2 of 3-lane freeway. The cumulative IATs for these traffic volumes are given in Table 25 and where as the cumulative IAT distribution plots are give in Figure 35.

103 Table 22: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 2 of 2-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13]. Hourly traffic volume range 2-50

Cumulative Percentage (%)

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100 764.82

25

0.10

1.08

1.44

3.50

6.99

16.22

29.10

47.64

75.90

118.19

172.20

251.57

383.87

500.39

615.85

726.51

50-100

75

0.10

0.68

0.85

1.57

2.80

5.75

9.97

16.03

25.24

38.98

56.80

83.08

127.18

166.80

207.48

243.66

264.13

100-150

125

0.10

0.60

0.73

1.18

1.95

3.66

6.14

9.71

15.10

23.15

33.73

49.39

75.85

100.09

125.78

147.08

163.89

150-200

175

0.10

0.56

0.67

1.01

1.59

2.76

4.49

7.00

10.76

16.36

23.85

34.96

53.86

71.49

90.75

105.67

120.86

200-250

225

0.10

0.54

0.64

0.91

1.38

2.25

3.58

5.49

8.35

12.60

18.36

26.95

41.64

55.61

71.27

82.66

96.90

250-300

275

0.10

0.52

0.62

0.85

1.25

1.93

3.00

4.53

6.82

10.20

14.87

21.85

33.87

45.50

58.87

68.01

81.61

300-350

325

0.10

0.51

0.60

0.80

1.15

1.71

2.59

3.86

5.75

8.54

12.46

18.33

28.50

38.50

50.27

57.86

70.99

350-400

375

0.10

0.50

0.59

0.77

1.08

1.54

2.29

3.38

4.97

7.33

10.69

15.75

24.56

33.37

43.96

50.42

63.17

400-450

425

0.10

0.49

0.57

0.74

1.03

1.42

2.06

3.00

4.38

6.40

9.34

13.77

21.55

29.45

39.13

44.72

57.17

450-500

475

0.10

0.48

0.56

0.72

0.98

1.31

1.88

2.71

3.91

5.67

8.27

12.22

19.17

26.35

35.31

40.22

52.41

500-550

525

0.10

0.48

0.56

0.70

0.95

1.23

1.74

2.47

3.53

5.08

7.41

10.96

17.25

23.84

32.21

36.56

48.53

550-600

575

0.10

0.47

0.55

0.68

0.91

1.16

1.61

2.27

3.21

4.60

6.70

9.92

15.67

21.76

29.65

33.55

45.31

600-650

625

0.10

0.46

0.54

0.67

0.89

1.10

1.51

2.10

2.95

4.19

6.11

9.05

14.34

20.02

27.50

31.01

42.60

650-700

675

0.10

0.46

0.53

0.65

0.86

1.05

1.42

1.96

2.73

3.84

5.60

8.31

13.20

18.54

25.66

28.85

40.27

700-750

725

0.10

0.45

0.53

0.64

0.84

1.01

1.35

1.84

2.53

3.54

5.17

7.67

12.23

17.26

24.07

26.98

38.25

750-800

775

0.10

0.45

0.52

0.63

0.82

0.97

1.28

1.73

2.36

3.28

4.79

7.12

11.38

16.15

22.68

25.35

36.47

800-850

825

0.10

0.44

0.51

0.62

0.81

0.94

1.22

1.64

2.22

3.05

4.46

6.64

10.64

15.17

21.46

23.92

34.91

850-900

875

0.10

0.44

0.51

0.61

0.79

0.90

1.17

1.55

2.09

2.85

4.16

6.21

9.98

14.31

20.38

22.65

33.51

900-950

925

0.10

0.43

0.50

0.60

0.77

0.88

1.12

1.48

1.97

2.67

3.90

5.82

9.39

13.53

19.41

21.52

32.25

950-1000

975

0.10

0.43

0.50

0.59

0.76

0.85

1.08

1.41

1.86

2.51

3.67

5.48

8.87

12.84

18.54

20.50

31.12

1000-1050

1025

0.10

0.43

0.49

0.59

0.75

0.83

1.04

1.35

1.77

2.37

3.46

5.17

8.39

12.21

17.75

19.58

30.09

1050-1100

1075

0.10

0.42

0.49

0.58

0.74

0.82

1.03

1.34

1.76

2.35

3.43

5.13

8.33

12.12

17.61

19.43

29.85

1100-1150

1125

0.10

0.42

0.49

0.57

0.73

0.79

0.97

1.24

1.61

2.12

3.09

4.64

7.57

11.13

16.38

17.98

28.27

1150-1200

1175

0.10

0.42

0.48

0.56

0.72

0.77

0.94

1.20

1.54

2.01

2.94

4.41

7.22

10.66

15.78

17.28

27.48

Note: Cells shaded in grey show the extrapolated traffic volume range.

104 Table 22: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 2 of 2-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13] (continued). Cumulative Percentage (%)

Hourly traffic volume range

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1250-1300

1275

0.10

0.41

0.47

0.55

0.70

0.74

0.89

1.12

1.41

1.82

2.66

4.00

6.59

9.82

14.71

16.04

26.05

1300-1350

1325

0.10

0.40

0.47

0.55

0.69

0.72

0.87

1.08

1.35

1.74

2.54

3.82

6.31

9.45

14.24

15.49

25.41

1350-1400

1375

0.10

0.40

0.46

0.54

0.68

0.71

0.85

1.05

1.30

1.66

2.42

3.66

6.05

9.11

13.80

14.98

24.81

1400-1450

1425

0.10

0.40

0.46

0.54

0.67

0.69

0.82

1.01

1.26

1.59

2.32

3.50

5.82

8.79

13.39

14.51

24.25

1450-1500

1475

0.10

0.39

0.46

0.53

0.66

0.68

0.81

0.98

1.21

1.52

2.22

3.36

5.59

8.49

13.00

14.06

23.72

1500-1550

1525

0.10

0.39

0.45

0.52

0.65

0.67

0.79

0.96

1.17

1.46

2.13

3.22

5.39

8.21

12.64

13.65

23.22

1550-1600

1575

0.10

0.39

0.45

0.52

0.65

0.66

0.77

0.93

1.13

1.40

2.04

3.10

5.19

7.95

12.31

13.26

22.75

1600-1650

1625

0.10

0.39

0.45

0.51

0.64

0.65

0.75

0.91

1.09

1.34

1.96

2.98

5.01

7.71

11.99

12.89

22.31

1650-1700

1675

0.10

0.38

0.44

0.51

0.63

0.64

0.74

0.88

1.06

1.29

1.89

2.87

4.84

7.48

11.69

12.55

21.88

1700-1750

1725

0.10

0.38

0.44

0.51

0.63

0.63

0.72

0.86

1.03

1.24

1.82

2.77

4.68

7.26

11.41

12.22

21.48

1750-1800

1775

0.10

0.38

0.44

0.50

0.62

0.62

0.71

0.84

1.00

1.20

1.75

2.67

4.53

7.06

11.14

11.91

21.10

1800-1850

1825

0.10

0.37

0.43

0.50

0.61

0.61

0.70

0.82

0.97

1.16

1.69

2.58

4.39

6.86

10.88

11.62

20.74

1850-1900

1875

0.10

0.37

0.43

0.49

0.61

0.60

0.68

0.80

0.94

1.12

1.63

2.50

4.25

6.68

10.64

11.35

20.39

1900-1950

1925

0.10

0.37

0.43

0.49

0.60

0.59

0.67

0.79

0.91

1.08

1.58

2.41

4.13

6.51

10.41

11.08

20.06

1950-2000

1975

0.10

0.37

0.42

0.48

0.60

0.59

0.66

0.77

0.89

1.04

1.53

2.34

4.01

6.34

10.20

10.83

19.74

2000-2050

2025

0.10

0.36

0.42

0.48

0.59

0.58

0.65

0.75

0.87

1.01

1.48

2.26

3.89

6.19

9.99

10.60

19.44

2050-2100

2075

0.10

0.36

0.42

0.48

0.59

0.57

0.64

0.74

0.84

0.98

1.43

2.19

3.78

6.04

9.79

10.37

19.15

2100-2150

2125

0.10

0.36

0.41

0.47

0.58

0.56

0.63

0.72

0.82

0.95

1.38

2.13

3.68

5.89

9.60

10.15

18.87

2150-2200

2175

0.10

0.36

0.41

0.47

0.58

0.56

0.62

0.71

0.80

0.92

1.34

2.07

3.58

5.76

9.42

9.95

18.60

2200-2250

2225

0.10

0.35

0.41

0.47

0.57

0.55

0.61

0.70

0.78

0.89

1.30

2.01

3.49

5.63

9.24

9.75

18.34

2250-2300

2275

0.10

0.35

0.40

0.46

0.57

0.54

0.60

0.68

0.77

0.86

1.26

1.95

3.40

5.51

9.08

9.56

18.08

2300-2350

2325

0.10

0.35

0.40

0.46

0.56

0.54

0.59

0.67

0.75

0.84

1.23

1.90

3.31

5.39

8.92

9.38

17.84

2350-2400

2375

0.10

0.35

0.40

0.46

0.56

0.53

0.58

0.66

0.73

0.81

1.19

1.84

3.23

5.27

8.76

9.20

17.61

2400-2450

2425

0.10

0.34

0.40

0.45

0.55

0.53

0.58

0.65

0.72

0.79

1.16

1.79

3.15

5.17

8.62

9.04

17.38

2450-2500

2475

0.10

0.34

0.39

0.45

0.55

0.52

0.57

0.64

0.70

0.77

1.13

1.75

3.08

5.06

8.47

8.87

17.17

Note: Cells shaded in grey show the extrapolated traffic volume range.

105 Table 23: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 1 of 3-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13]. Hourly traffic volume range 2-50

Cumulative Percentage (%)

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100 687.70

25

0.11

1.97

2.73

8.38

15.51

32.31

52.63

76.64

98.49

130.98

170.18

227.93

340.04

428.95

580.32

580.47

50-100

75

0.10

1.06

1.38

3.25

5.62

11.12

17.74

25.58

32.87

43.59

56.60

75.75

112.56

142.37

191.95

194.68

232.58

100-150

125

0.10

0.87

1.10

2.22

3.64

6.88

10.76

15.37

19.75

26.11

33.88

45.31

67.07

85.06

114.28

117.52

141.54

150-200

175

0.10

0.79

0.98

1.78

2.79

5.06

7.76

10.99

14.13

18.62

24.15

32.27

47.58

60.50

81.00

84.44

102.51

200-250

225

0.10

0.75

0.92

1.53

2.32

4.04

6.10

8.56

11.00

14.46

18.74

25.03

36.75

46.86

62.52

66.06

80.81

250-300

275

0.10

0.72

0.87

1.37

2.01

3.40

5.04

7.01

9.01

11.81

15.30

20.42

29.86

38.18

50.76

54.36

67.00

300-350

325

0.10

0.69

0.84

1.26

1.80

2.95

4.31

5.94

7.64

9.98

12.92

17.23

25.09

32.17

42.62

46.26

57.42

350-400

375

0.10

0.68

0.82

1.18

1.65

2.62

3.77

5.15

6.63

8.63

11.17

14.89

21.60

27.77

36.66

40.32

50.40

400-450

425

0.10

0.66

0.80

1.11

1.53

2.37

3.36

4.55

5.85

7.61

9.84

13.10

18.93

24.40

32.10

35.77

45.02

450-500

475

0.10

0.65

0.78

1.06

1.43

2.17

3.03

4.08

5.25

6.79

8.78

11.69

16.82

21.74

28.50

32.18

40.77

500-550

525

0.10

0.64

0.77

1.02

1.36

2.01

2.77

3.69

4.75

6.14

7.93

10.54

15.12

19.59

25.59

29.27

37.32

550-600

575

0.10

0.63

0.76

0.99

1.29

1.88

2.55

3.38

4.34

5.60

7.22

9.60

13.71

17.82

23.19

26.86

34.47

600-650

625

0.10

0.63

0.75

0.96

1.24

1.76

2.37

3.11

4.00

5.14

6.63

8.81

12.53

16.33

21.17

24.84

32.08

650-700

675

0.10

0.62

0.74

0.93

1.19

1.67

2.21

2.88

3.71

4.75

6.13

8.13

11.52

15.06

19.46

23.12

30.03

700-750

725

0.10

0.61

0.73

0.91

1.15

1.58

2.08

2.69

3.46

4.42

5.69

7.55

10.66

13.96

17.98

21.63

28.27

750-800

775

0.10

0.61

0.72

0.89

1.11

1.51

1.96

2.52

3.24

4.13

5.32

7.04

9.90

13.01

16.69

20.34

26.72

800-850

825

0.10

0.60

0.71

0.87

1.08

1.45

1.86

2.37

3.05

3.87

4.98

6.60

9.24

12.17

15.56

19.20

25.37

850-900

875

0.10

0.60

0.71

0.85

1.05

1.39

1.76

2.23

2.87

3.64

4.69

6.21

8.65

11.43

14.56

18.19

24.16

900-950

925

0.10

0.59

0.70

0.84

1.03

1.34

1.68

2.12

2.72

3.44

4.43

5.85

8.13

10.77

13.67

17.29

23.09

950-1000

975

0.10

0.59

0.70

0.82

1.00

1.29

1.61

2.01

2.59

3.26

4.19

5.54

7.66

10.18

12.88

16.48

22.12

1000-1050

1025

0.10

0.58

0.69

0.81

0.98

1.25

1.54

1.91

2.46

3.10

3.98

5.26

7.24

9.65

12.16

15.75

21.24

1050-1100

1075

0.10

0.58

0.69

0.80

0.96

1.21

1.48

1.83

2.35

2.95

3.79

5.00

6.86

9.16

11.51

15.08

20.45

1100-1150

1125

0.09

0.58

0.68

0.79

0.94

1.18

1.43

1.75

2.25

2.81

3.61

4.76

6.51

8.72

10.91

14.48

19.72

1150-1200

1175

0.09

0.57

0.68

0.78

0.93

1.14

1.37

1.67

2.16

2.69

3.45

4.55

6.20

8.32

10.37

13.92

19.06

1200-1250

1225

0.09

0.57

0.67

0.77

0.91

1.11

1.33

1.61

2.07

2.58

3.30

4.35

5.90

7.95

9.88

13.41

18.44

Note: Cells shaded in grey show the extrapolated traffic volume range.

106 Table 23: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 1 of 3-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13] (continued). Cumulative Percentage (%)

Hourly traffic volume range

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1250-1300

1275

0.09

0.57

0.67

0.76

0.90

1.09

1.29

1.55

1.99

2.47

3.17

4.17

5.64

7.61

9.42

12.94

17.88

1300-1350

1325

0.09

0.56

0.66

0.75

0.88

1.06

1.25

1.49

1.92

2.38

3.04

4.00

5.39

7.30

9.00

12.51

17.35

1350-1400

1375

0.09

0.56

0.66

0.74

0.87

1.04

1.21

1.44

1.85

2.29

2.93

3.85

5.16

7.01

8.61

12.10

16.86

1400-1450

1425

0.09

0.56

0.65

0.74

0.86

1.02

1.17

1.39

1.79

2.20

2.82

3.70

4.95

6.74

8.24

11.73

16.41

1450-1500

1475

0.09

0.55

0.65

0.73

0.85

0.99

1.14

1.34

1.73

2.13

2.72

3.57

4.75

6.48

7.90

11.38

15.98

1500-1550

1525

0.09

0.55

0.65

0.72

0.84

0.98

1.11

1.30

1.67

2.05

2.62

3.44

4.56

6.25

7.59

11.05

15.58

1550-1600

1575

0.09

0.55

0.64

0.72

0.83

0.96

1.08

1.26

1.62

1.99

2.54

3.33

4.39

6.03

7.30

10.74

15.21

1600-1650

1625

0.09

0.54

0.64

0.71

0.82

0.94

1.06

1.22

1.57

1.92

2.45

3.22

4.23

5.82

7.02

10.45

14.86

1650-1700

1675

0.09

0.54

0.64

0.70

0.81

0.92

1.03

1.19

1.53

1.86

2.38

3.11

4.07

5.63

6.76

10.18

14.52

1700-1750

1725

0.09

0.54

0.63

0.70

0.80

0.91

1.01

1.15

1.48

1.81

2.30

3.02

3.93

5.44

6.52

9.92

14.21

1750-1800

1775

0.09

0.54

0.63

0.69

0.79

0.89

0.98

1.12

1.44

1.75

2.23

2.92

3.80

5.27

6.29

9.68

13.91

1800-1850

1825

0.09

0.532

0.62

0.69

0.78

0.88

0.96

1.09

1.41

1.70

2.17

2.84

3.67

5.11

6.07

9.45

13.63

1850-1900

1875

0.09

0.530

0.62

0.68

0.77

0.86

0.94

1.06

1.37

1.66

2.11

2.75

3.55

4.96

5.87

9.23

13.36

1900-1950

1925

0.09

0.527

0.62

0.68

0.77

0.85

0.92

1.04

1.34

1.61

2.05

2.68

3.43

4.81

5.67

9.02

13.11

1950-2000

1975

0.09

0.524

0.62

0.67

0.76

0.84

0.91

1.01

1.30

1.57

1.99

2.60

3.33

4.67

5.49

8.83

12.86

2000-2050

2025

0.09

0.522

0.61

0.67

0.75

0.83

0.89

0.99

1.27

1.53

1.94

2.53

3.22

4.54

5.31

8.64

12.63

2050-2100

2075

0.09

0.519

0.61

0.66

0.74

0.82

0.87

0.96

1.24

1.49

1.89

2.47

3.13

4.42

5.15

8.46

12.41

2100-2150

2125

0.09

0.517

0.61

0.66

0.74

0.80

0.86

0.94

1.21

1.45

1.84

2.40

3.03

4.30

4.99

8.29

12.20

2150-2200

2175

0.09

0.514

0.60

0.65

0.73

0.79

0.84

0.92

1.19

1.42

1.80

2.34

2.95

4.19

4.84

8.13

12.00

2200-2250

2225

0.09

0.512

0.60

0.65

0.73

0.78

0.83

0.90

1.16

1.38

1.76

2.28

2.86

4.08

4.70

7.97

11.81

2250-2300

2275

0.09

0.509

0.60

0.64

0.72

0.77

0.81

0.88

1.14

1.35

1.71

2.23

2.78

3.97

4.56

7.82

11.62

2300-2350

2325

0.09

0.507

0.59

0.64

0.71

0.77

0.80

0.86

1.11

1.32

1.67

2.18

2.71

3.88

4.43

7.68

11.44

2350-2400

2375

0.09

0.505

0.59

0.64

0.71

0.76

0.79

0.85

1.09

1.29

1.64

2.13

2.63

3.78

4.31

7.54

11.27

2400-2450

2425

0.08

0.502

0.59

0.63

0.70

0.75

0.77

0.83

1.07

1.26

1.60

2.08

2.56

3.69

4.19

7.41

11.11

2450-2500

2475

0.08

0.500

0.59

0.63

0.70

0.74

0.76

0.81

1.05

1.24

1.57

2.03

2.49

3.61

4.08

7.29

10.95

Note: Cells shaded in grey show the extrapolated traffic volume range.

107 Table 24: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 3 of 4-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13]. Hourly traffic volume range 2-50

Cumulative Percentage (%)

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100 537.58

25

0.10

3.48

7.17

10.19

18.19

34.33

53.66

79.03

105.11

141.63

182.89

235.42

323.50

413.54

481.70

571.64

50-100

75

0.10

1.51

2.71

3.78

6.43

11.76

18.13

26.43

35.03

46.99

60.58

78.10

107.40

137.11

160.96

190.22

186.83

100-150

125

0.10

1.11

1.81

2.49

4.08

7.24

11.03

15.91

21.01

28.06

36.11

46.64

64.18

81.83

96.81

113.94

116.65

150-200

175

0.10

0.94

1.43

1.94

3.07

5.31

7.98

11.40

15.01

19.95

25.63

33.15

45.66

58.14

69.32

81.25

86.56

200-250

225

0.10

0.85

1.22

1.64

2.51

4.23

6.29

8.90

11.67

15.44

19.81

25.66

35.37

44.98

54.04

63.09

69.82

250-300

275

0.10

0.79

1.08

1.44

2.15

3.55

5.21

7.30

9.55

12.57

16.10

20.90

28.83

36.60

44.32

51.53

59.15

300-350

325

0.10

0.74

0.98

1.30

1.90

3.07

4.46

6.20

8.08

10.59

13.54

17.60

24.29

30.81

37.59

43.53

51.76

350-400

375

0.10

0.71

0.91

1.20

1.72

2.72

3.92

5.39

7.00

9.14

11.66

15.18

20.97

26.56

32.66

37.67

46.33

400-450

425

0.10

0.69

0.86

1.13

1.58

2.46

3.50

4.77

6.17

8.02

10.22

13.33

18.43

23.31

28.88

33.18

42.17

450-500

475

0.10

0.67

0.82

1.07

1.47

2.24

3.17

4.28

5.52

7.14

9.09

11.87

16.43

20.75

25.90

29.64

38.88

500-550

525

0.10

0.65

0.78

1.02

1.38

2.07

2.90

3.89

5.00

6.43

8.17

10.69

14.80

18.67

23.49

26.77

36.20

550-600

575

0.10

0.64

0.75

0.97

1.30

1.93

2.68

3.56

4.56

5.85

7.41

9.72

13.46

16.96

21.49

24.41

33.99

600-650

625

0.10

0.62

0.73

0.94

1.24

1.81

2.49

3.29

4.19

5.35

6.78

8.90

12.34

15.52

19.82

22.42

32.13

650-700

675

0.10

0.61

0.71

0.91

1.19

1.71

2.33

3.05

3.88

4.93

6.23

8.20

11.38

14.29

18.39

20.72

30.53

700-750

725

0.10

0.60

0.69

0.88

1.14

1.62

2.19

2.85

3.61

4.57

5.77

7.60

10.55

13.24

17.16

19.26

29.16

750-800

775

0.10

0.60

0.67

0.86

1.10

1.55

2.07

2.67

3.38

4.26

5.36

7.07

9.83

12.32

16.09

17.99

27.95

800-850

825

0.10

0.59

0.66

0.84

1.06

1.48

1.97

2.52

3.18

3.98

5.00

6.61

9.20

11.51

15.15

16.88

26.89

850-900

875

0.10

0.58

0.64

0.82

1.03

1.42

1.87

2.38

2.99

3.74

4.69

6.21

8.64

10.80

14.31

15.89

25.95

900-950

925

0.10

0.58

0.63

0.80

1.00

1.36

1.79

2.26

2.83

3.52

4.41

5.84

8.14

10.16

13.57

15.00

25.10

950-1000

975

0.10

0.57

0.62

0.79

0.98

1.32

1.72

2.15

2.69

3.32

4.15

5.52

7.70

9.59

12.90

14.21

24.34

1000-1050

1025

0.10

0.57

0.61

0.78

0.95

1.27

1.65

2.05

2.55

3.15

3.93

5.23

7.29

9.07

12.29

13.50

23.65

1050-1100

1075

0.10

0.56

0.60

0.76

0.93

1.23

1.59

1.96

2.43

2.99

3.72

4.96

6.93

8.60

11.75

12.85

23.02

1100-1150

1125

0.10

0.56

0.60

0.75

0.91

1.20

1.53

1.88

2.33

2.84

3.53

4.72

6.59

8.18

11.25

12.26

22.44

1150-1200

1175

0.10

0.55

0.59

0.74

0.89

1.16

1.48

1.80

2.23

2.71

3.36

4.50

6.29

7.79

10.79

11.72

21.91

1200-1250

1225

0.10

0.55

0.58

0.73

0.88

1.13

1.43

1.74

2.14

2.58

3.20

4.29

6.01

7.43

10.37

11.23

21.42

Note: Cells shaded in grey show the extrapolated traffic volume range.

108 Table 24: Extrapolated adjusted cumulative interarrival time (in sec) distribution table for lane 3 of 4-lane freeway for traffic volume range of 2-2500 vehicles/hour based on the hyperbolic plots used in [13] (continued). Cumulative Percentage (%)

Hourly traffic volume range

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1250-1300

1275

0.10

0.54

0.57

0.72

0.86

1.10

1.39

1.67

2.05

2.47

3.06

4.10

5.75

7.10

9.99

10.77

20.97

1300-1350

1325

0.10

0.54

0.57

0.71

0.85

1.08

1.35

1.61

1.97

2.37

2.92

3.93

5.51

6.80

9.63

10.35

20.55

1350-1400

1375

0.10

0.54

0.56

0.70

0.83

1.05

1.31

1.56

1.90

2.27

2.80

3.77

5.29

6.52

9.30

9.96

20.16

1400-1450

1425

0.10

0.53

0.56

0.70

0.82

1.03

1.28

1.51

1.83

2.18

2.68

3.62

5.09

6.26

8.99

9.59

19.79

1450-1500

1475

0.10

0.53

0.55

0.69

0.81

1.01

1.24

1.46

1.77

2.10

2.58

3.48

4.90

6.01

8.70

9.25

19.45

1500-1550

1525

0.10

0.53

0.55

0.68

0.80

0.99

1.21

1.42

1.71

2.02

2.47

3.35

4.72

5.79

8.43

8.94

19.12

1550-1600

1575

0.10

0.52

0.54

0.68

0.79

0.97

1.18

1.37

1.66

1.95

2.38

3.23

4.55

5.57

8.18

8.64

18.82

1600-1650

1625

0.10

0.52

0.54

0.67

0.78

0.95

1.16

1.34

1.61

1.88

2.29

3.12

4.40

5.37

7.94

8.36

18.53

1650-1700

1675

0.10

0.52

0.53

0.66

0.77

0.94

1.13

1.30

1.56

1.81

2.21

3.01

4.25

5.19

7.72

8.10

18.26

1700-1750

1725

0.10

0.52

0.53

0.66

0.76

0.92

1.11

1.26

1.51

1.75

2.13

2.91

4.11

5.01

7.51

7.85

18.00

1750-1800

1775

0.10

0.51

0.52

0.65

0.75

0.90

1.08

1.23

1.47

1.70

2.06

2.82

3.98

4.84

7.31

7.62

17.76

1800-1850

1825

0.10

0.51

0.52

0.65

0.74

0.89

1.06

1.20

1.43

1.64

1.99

2.73

3.86

4.69

7.12

7.40

17.52

1850-1900

1875

0.10

0.51

0.52

0.64

0.73

0.88

1.04

1.17

1.39

1.59

1.92

2.64

3.74

4.54

6.95

7.20

17.30

1900-1950

1925

0.10

0.51

0.51

0.64

0.73

0.86

1.02

1.14

1.36

1.54

1.86

2.56

3.63

4.40

6.78

7.00

17.09

1950-2000

1975

0.10

0.50

0.51

0.63

0.72

0.85

1.00

1.12

1.32

1.50

1.80

2.49

3.52

4.26

6.62

6.81

16.89

2000-2050

2025

0.10

0.50

0.51

0.63

0.71

0.84

0.99

1.09

1.29

1.45

1.75

2.41

3.42

4.14

6.47

6.63

16.69

2050-2100

2075

0.10

0.50

0.50

0.62

0.70

0.83

0.97

1.07

1.26

1.41

1.69

2.34

3.33

4.02

6.32

6.46

16.51

2100-2150

2125

0.10

0.50

0.50

0.62

0.70

0.82

0.95

1.05

1.23

1.37

1.64

2.28

3.24

3.90

6.19

6.30

16.33

2150-2200

2175

0.10

0.50

0.50

0.62

0.69

0.81

0.94

1.03

1.20

1.34

1.60

2.22

3.15

3.79

6.05

6.15

16.16

2200-2250

2225

0.10

0.49

0.49

0.61

0.69

0.80

0.92

1.00

1.17

1.30

1.55

2.16

3.07

3.69

5.93

6.00

16.00

2250-2300

2275

0.10

0.49

0.49

0.61

0.68

0.79

0.91

0.98

1.15

1.27

1.51

2.10

2.99

3.59

5.81

5.86

15.84

2300-2350

2325

0.10

0.49

0.49

0.61

0.67

0.78

0.90

0.97

1.12

1.23

1.46

2.05

2.92

3.49

5.69

5.73

15.69

2350-2400

2375

0.10

0.49

0.49

0.60

0.67

0.77

0.88

0.95

1.10

1.20

1.42

1.99

2.85

3.40

5.58

5.60

15.54

2400-2450

2425

0.10

0.49

0.48

0.60

0.66

0.76

0.87

0.93

1.07

1.17

1.39

1.94

2.78

3.31

5.48

5.48

15.40

2450-2500

2475

0.10

0.48

0.48

0.59

0.66

0.75

0.86

0.91

1.05

1.14

1.35

1.90

2.71

3.23

5.37

5.36

15.26

Note: Cells shaded in grey show the extrapolated traffic volume range.

109 Table 25: Cumulative IATs (in sec) obtained for traffic volumes of 4 vehicles/hour, 958 vehicles/hour and 2478 vehicles/hour for lane 2 of a 3-lane freeway. Vehicles per hour

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

4

0.10

25.04

31.57

61.05

97.19

175.65

271.28

404.70

565.15

773.93

1029.11

1441.24

2161.73

2896.90

3913.34

4070.41

5260.66

958

0.10

0.57

0.65

0.80

1.00

1.34

1.71

2.12

2.63

3.31

4.21

5.51

7.88

10.36

13.56

16.08

22.64

2478

0.09

0.46

0.53

0.60

0.70

0.83

0.96

1.04

1.16

1.32

1.58

1.86

2.44

3.09

3.78

5.73

9.11

110 120

a)

Cumulative percentage (%)

100

80

60

40

20

0 0.10

1.00

10.00

100.00

1000.00

10000.00

Cumulative interarrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

40

20

0 0.10

1.00 Cumulative interarrival time (sec)

10.00

Figure 35: Cumulative IAT distribution plots obtained for traffic volume of a) 4 vehicles/hour b) 958 vehicles/hour c) 2478 vehicles/hour for lane2 of 3-lane freeway.

111 6.4

Analysis of the shape of the cumulative IAT distributions

There was an interest to investigate the shape of the probability distribution function (PDF) and see whether the shape of the PDF for the cumulative interarrival time distribution obtained in [13] for the freeways were similar to other headway distributions models like negative exponential, Erlang and Pearson type III distribution. It was also not known how the shape of the PDF for the cumulative interarrival time distributions obtained in [13] changed with respect to the hourly traffic volume i.e. whether the shape of the PDF was similar for the whole range of hourly traffic volume data. For the analysis of the shape of the cumulative interarrival time distribution, the average IAT of the fit and the variance of the fit were calculated using equation 1 [13] and equation 2 respectively for all the nine cumulative interarrival time distributions from [13] for all hourly traffic volumes observed. 16

Average IAT of the fit =

∑ [( p i =1

Variance =

16

∑ [( i =1

i +1

− pi ) *

( y i +1 + y i ) ] 2

_ y i +1 + y i ) − t ] 2 * ( pi +1 − pi ) 2 _

Coefficient of variation (CV) = S / t

(1)

(2)

(3)

where pi = cumulative percentage value from the hyperbolic fit table _

yi = corresponding IAT for pi , t = Average IAT of the fit S= Variance The standard deviation of the fit was then estimated by taking the square root of the variance and the coefficient of variation (CV) was also calculated by dividing the

112 standard deviation by the mean. All these values obtained for lane 1 of 2-lane freeway for the whole range of observed hourly traffic volumes for the site are shown in Table 26.

Table 26: Values obtained for the average IAT of the fit, variance, standard deviation and coefficient of variation for lane 1 of 2-lane freeway [13] for hourly traffic volume range of 200-1600 vehicles/hour. Hourly traffic volume range

Hourly traffic volume

Average IAT of fit (sec)

Variance

Standard deviation (sec)

Coefficient of variation

200-250 250-300 300-350 350-400 400-450 450-500 500-550 550-600 600-650 650-700 700-750 750-800 800-850 850-900 900-950 950-1000 1000-1050 1050-1100 1100-1150 1150-1200 1200-1250 1250-1300 1300-1350 1350-1400 1400-1450 1450-1500 1500-1550 1550-1600

225 275 325 375 425 475 525 575 625 675 725 775 825 875 925 975 1025 1075 1125 1175 1225 1275 1325 1375 1425 1475 1525 1575

16.00 13.09 11.08 9.60 8.47 7.58 6.86 6.26 5.76 5.33 4.97 4.65 4.36 4.11 3.89 3.69 3.51 3.35 3.20 3.06 2.94 2.82 2.72 2.62 2.53 2.44 2.36 2.29

268.69 176.41 123.89 91.28 69.72 54.76 43.98 35.98 29.88 25.14 21.39 18.38 15.92 13.90 12.21 10.79 9.59 8.56 7.68 6.92 6.25 5.67 5.16 4.71 4.31 3.96 3.64 3.36

16.39 13.28 11.13 9.55 8.35 7.40 6.63 6.00 5.47 5.01 4.63 4.29 3.99 3.73 3.49 3.28 3.10 2.93 2.77 2.63 2.50 2.38 2.27 2.17 2.08 1.99 1.91 1.83

1.02 1.01 1.00 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.91 0.90 0.89 0.88 0.87 0.87 0.86 0.85 0.84 0.84 0.83 0.82 0.82 0.81 0.80

113 It can be seen from Table 26 that the coefficient of variation is close to one for low traffic volumes and decreases as the hourly traffic volume increases which implies that the average IAT of the fit is close to standard deviation of the distribution. All the nine cumulative interarrival time distributions showed similar phenomenon with the average IAT of the distribution close to standard deviation of the distribution. Figure 36, Figure 37, Figure 38 and Figure 39 show the plot of average IATs, standard deviations versus hourly traffic volume for 2-lane, 3-lane and 4-lane freeways respectively for all lanes. The variability of the coefficient of variation with respect to hourly traffic volume was not the same as observed for lane 1 of 2-lane freeway for all the nine cumulative interarrival time distributions in [13]. Some times the CV decreased with the increase in the hourly traffic volume and where as some times it increased. Out of the nine cases investigated, the CV increased in 4 cases where as it decreased in 5 cases with the increase in the traffic volume. The only common trend that was observed for most of them was that the CV’s obtained were always close to one which is shown in Table 27. This implied that the cumulative interarrival time distributions obtained in [13] were close to negative exponential for the entire range of observed traffic volumes which led to the investigation of the shape of the probability distribution function for the cumulative IAT distributions obtained in [13]. A summary table for the coefficient of variation analysis performed is given in Table 28 with mean of average IATs, mean of the standard deviations and the traffic volume where the average IAT curve intersected the standard deviation curve.

114 18 N = 28 16

Average IAT Standard deviation

14

Interarrival time (sec)

12

10

8

6

4

2

0 0

200

400

600

800 1000 Traffic volume (vph)

1200

1400

1600

1800

a) 25

N = 26 Average IAT

20

Interarrival time (sec)

Standard deviation

15

10

5

0 0

200

400

600

800

1000

1200

1400

1600

Traffic volume (vph)

b) Figure 36: Average IAT and standard deviation as a function of traffic volume for a) lane 1 with 200-1600 vehicles/hour b) lane 2 with 150-1450 vehicles/hour for the 2lane freeway.

115 18

N = 32

16

Average IAT

Interarrival time (sec)

14

a)

Standard deviation

12

10

8

6

4

2

0 0

200

400

600

800

1000

1200

1400

1600

1800

2000

Traffic volume (vph)

18

N = 27 16 Average IAT Standard deviation

b)

Interarrival time (sec)

14

12

10

8

6

4

2

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph) 25

N = 30 20 Average IAT

c)

Interarrival time (sec)

Standard deviation 15

10

5

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure 37: Average IAT and standard deviation as a function of traffic volume for a) lane 1 with 200-1800 vehicles/hour b) lane 2 with 200-1550 vehicles/hour c) lane 3 with 150-1650 vehicles/hour for the 3-lane freeway.

116 35

N = 21

30

Average IAT Standard deviation

Interarrival time (sec)

25

20

15

10

5

0 0

200

400

600

800

1000

1200

Traffic volume (vph)

a) 25

N = 25

20

Average IAT

Interarrival time (sec)

Standard deviation

15

10

5

0 0

200

400

600

800

1000

1200

1400

1600

Traffic volume (vph)

b) Figure 38: Average IAT and standard deviation as a function of traffic volume for a) lane 1 with 100-1150 vehicles/hour b) lane 2 with 150-1400 vehicles/hour for the 4lane freeway.

117 35

30

N = 26 Average IAT Standard deviation

Interarrival time (sec)

25

20

15

10

5

0 0

200

400

600

800

1000

1200

1400

1600

Traffic volume (vph)

a) 35

N = 24

30

Average IAT Standard deviation

Interarrival time (sec)

25

20

15

10

5

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

b) Figure 39: Average IAT and standard deviation as a function of traffic volume for a) lane 3 with 100-1400 vehicles/hour b) lane 4 with 100-1300 vehicles/hour for the 4lane freeway.

118 Table 27: Values of coefficient of variation obtained for all lanes of 2-lane, 3-lane and 4-lane freeways for the observed hourly traffic volume ranges. Hourly traffic volume range 100-150 150-200 200-250 250-300 300-350 350-400 400-450 450-500 500-550 550-600 600-650 650-700 700-750 750-800 800-850 850-900 900-950 950-1000 1000-1050 1050-1100 1100-1150 1150-1200 1200-1250 1250-1300 1300-1350 1350-1400 1400-1450 1450-1500 1500-1550 1550-1600 1600-1650 1650-1700 1700-1750 1750-1800

Hourly traffic volume 125 175 225 275 325 375 425 475 525 575 625 675 725 775 825 875 925 975 1025 1075 1125 1175 1225 1275 1325 1375 1425 1475 1525 1575 1625 1675 1725 1775

2-lane freeway Lane1

1.02 1.01 1.00 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.91 0.90 0.89 0.88 0.87 0.87 0.86 0.85 0.84 0.84 0.83 0.82 0.82 0.81 0.80

Lane2

1.14 1.14 1.15 1.15 1.15 1.15 1.16 1.16 1.16 1.17 1.17 1.17 1.18 1.18 1.19 1.19 1.19 1.20 1.25 1.20 1.21 1.21 1.22 1.22 1.22 1.23

3-lane freeway Lane1

Lane2

0.93 0.93 0.92 0.92

1.00 0.99 0.98 0.97

0.91 0.91 0.91 0.90 0.90 0.90 0.89 0.89 0.88 0.88 0.88 0.88 0.87 0.87 0.87 0.86 0.86 0.86 0.86 0.85 0.85 0.85 0.85 0.84 0.84 0.84 0.84 0.84

0.96 0.96 0.95 0.94 0.93 0.92 0.91 0.90 0.89 0.89 0.88 0.87 0.86 0.85 0.85 0.84 0.83 0.82 0.82 0.81 0.80 0.79 0.79

4-lane freeway

Lane3

Lane1

1.03 1.03 1.03 1.03 1.04

0.92 0.93 0.94 0.95 0.96 0.98

1.04 1.04 1.05 1.05 1.05 1.06 1.06 1.06 1.07 1.07 1.08 1.08 1.08 1.09 1.09 1.10 1.10 1.11 1.11 1.12 1.12 1.13 1.13 1.14 1.14

0.99 1.00 1.01 1.03 1.04 1.05 1.07 1.08 1.09 1.11 1.12 1.14 1.15 1.17 1.18

Lane2

Lane3

Lane4

0.89 0.88 0.88 0.88 0.88

0.87 0.87 0.86 0.86 0.86 0.86

0.94 0.95 0.97 0.98 0.99 1.00

0.88 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.87 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86 0.86

0.86 0.86 0.86 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85 0.85

1.01 1.02 1.04 1.05 1.06 1.07 1.09 1.10 1.11 1.12 1.14 1.15 1.16 1.17 1.19 1.20 1.21 1.22

Note: Empty cells indicate that no traffic volumes were observed in the lane for that hourly traffic volume range.

119 Table 28: Summary of the coefficient of variation analysis for all the cumulative interarrival time distributions in [13]. Mean of the Average IATs (AM) (sec) 5.341 6.270 4.938 5.454 5.744 8.340 6.420 7.281 7.856

Case 2-Lane lane1 2-Lane lane2 3-Lane lane1 3-Lane lane2 3-Lane lane3 4-Lane lane1 4-Lane lane2 4-Lane lane3 4-Lane lane4

Mean of the Std dev (AS) (sec) 5.042 7.316 4.419 5.044 6.068 8.302 5.610 6.255 7.665

Ratio (AS/AM) 0.944 1.167 0.895 0.925 1.056 0.995 0.874 0.859 0.976

Point of intersection (Vehicles/hour) 375

475

375

Note: Empty cells indicate that the average IAT curve did not intersect the standard deviation curve. AM = Mean of the average IATs; AS = Mean of the standard deviations.

To compare the shape of the PDF of the cumulative interarrival time distribution over the entire range, the analysis was performed for low, medium and high hourly traffic volumes. The PDF of the cumulative interarrival time distributions obtained in [13] were compared with negative exponential, Erlang and Pearson type III distributions to see whether these distributions can be used to model the cumulative IAT distributions obtained for the freeways in [13]. The probability density functions for different distributions and their respective parameters used are defined as follows according to [1]: For negative exponential distribution − λt P (t) = λ * e _

where λ = 1/ t ,

(4)

120 _

t = Average IAT of the fit for an hourly traffic volume (sec)

t = time for which probability is calculated (sec), (0≤ t≤ ∞) For Erlang distribution [1] P (t) =

λ ( K − 1)!

* [λt ] K −1 * e −λt

(5)

_

where λ = k / t _

t = Average IAT of the fit for an hourly traffic volume (sec)

t = time for which probability is calculated (sec), (0≤ t≤ ∞) K varies from 2 to ∞ For Pearson type III distribution [1] [27] P (t) =

λ Γ( K )

* [λ (t − α )]K −1 * e −λ ( t −α )

(6)

_

where K = ( t - α)/ s _

t = Average IAT of the fit for an hourly traffic volume (sec)

S= standard deviation of the fit for an hourly traffic volume (sec) α = minimum IAT for which the probability is calculated = 0.1 seconds _

λ = K/ ( t - α) which affects the shape of the distribution Γ(K ) = gamma function = (K-1)! t = time for which probability is calculated (sec), (α≤ t≤ ∞) Cumulative interarrival time distributions obtained for lane 1 of the 3-lane freeway in [13] were used to compare with other distributions because it had the maximum range of hourly traffic volume observed from 200-1800 vehicles/hour. Sample calculations for

121 traffic volume of 1025 vehicles/hour are provided here. The average IAT, standard deviation, maximum IAT observed for an hourly traffic volume of 1025 vehicles/hour was 3.512 seconds, 3.063 seconds and 21.24 seconds respectively. The probabilities for all the distributions were calculated for every 0.7 seconds (maximum IAT/√N, N = 967 for lane 1 of 3-lane freeway) starting from 0.1 seconds as the minimum IAT for which the probability is calculated. The parameters used to calculate the PDFs for different distributions are provided in Table 29. The probabilities were calculated for regular IAT time intervals so that the shape of the distribution is not lost. The probabilities calculated for intervals of 0.7 seconds till the maximum IAT for all the distributions using the parameters in Table 29 are shown in Table 30.

Table 29: Parameters used to calculate the PDF for a negative exponential, an Erlang K = 2, Erlang K = 3 and a Pearson type III distributions for a traffic volume of 1025 vehicles/hour from lane 1 of 3-lane freeway.

Negative exponential λ = 0.284

Erlang K=2

Erlang K=3

λ = 0.569 (K-1)! = 1

λ = 0.854 (K-1)! = 2

Pearson type III α = 0.1 K=1.113 λ = 0.326 Γ(K ) = 0.946

Note: The value of the gamma function Γ(K ) was calculated from [28].

122 Table 30: Probability densities obtained for negative exponential, Erlang with K = 2, 3 and Pearson type III distributions for a traffic volume of 1025 vehicles/hour from lane 1 of 3-lane freeway. Probabilities obtained in % IAT (In sec)

Negative exponential

Erlang K=2

Erlang K=3

Pearson type III

0.1 0.7 1.4 2.1 2.8 3.5 4.2 4.9 5.6 6.3 7 7.7 8.4 9.1 9.8 10.5 11.2 11.9 12.6 13.3 14 14.7 15.4 16.1 16.8 17.5 18.2 18.9 19.6 20.3 21

27.673 23.327 19.112 15.658 12.829 10.511 8.611 7.055 5.780 4.736 3.880 3.179 2.605 2.134 1.748 1.432 1.174 0.961 0.788 0.645 0.529 0.433 0.355 0.291 0.238 0.195 0.160 0.131 0.107 0.088 0.072

3.063 15.237 20.455 20.596 18.433 15.467 12.458 9.757 7.485 5.652 4.216 3.113 2.279 1.658 1.198 0.862 0.617 0.440 0.313 0.222 0.157 0.110 0.078 0.054 0.038 0.027 0.019 0.013 0.009 0.006 0.004

0.286 8.397 18.472 22.857 22.347 19.203 15.208 11.384 8.177 5.692 3.864 2.572 1.683 1.086 0.693 0.437 0.274 0.170 0.105 0.064 0.039 0.024 0.014 0.009 0.005 0.003 0.002 0.001 0.001 0.000 0.000

0.000 23.589 20.497 17.129 14.104 11.521 9.365 7.587 6.132 4.946 3.984 3.205 2.576 2.069 1.660 1.332 1.068 0.855 0.685 0.549 0.439 0.351 0.281 0.225 0.180 0.144 0.115 0.092 0.073 0.059 0.047

123 Once the probabilities were obtained using the probability density functions for all the distributions they were discretised using the trapezoidal rule to obtain discrete probabilities for all the distributions which are given in Table 31. All the discrete probabilities obtained were then plotted to compare the shapes of the discretised probability distribution functions obtained for all distributions. Even though the probabilities were discretised they were joined to get a better idea of the shape of the distribution. Figure 40, Figure 41, Figure 42,Figure 43, Figure 44 and Figure 45 show the comparisons of discrete OU (Ohio University) probability distribution function with negative exponential, Erlang K=2, Erlang K=3, Pearson type III for traffic volumes of 1025 vehicles/hour, 225 vehicles/hour and 1775 vehicles/hour respectively. Figure 46 shows the discretised PDF plots obtained for traffic volumes of 225 vehicles/hour, 1025 vehicles/hour and 1775 vehicles/hour for lane 1 of 3-lane freeways on a normalized abscissa scale. The abscissa scale was normalized by the dividing the IAT by the average IAT for a given traffic volume. Figure 47 shows the cumulative IAT distributions obtained for lane 1of 3-lane freeway for the entire traffic volume range observed which show that the shape of the distribution remains fairly the same over the entire range.

124 Table 31: Discrete probabilities obtained for OU, negative exponential, Erlang with K = 2, 3 and Pearson type III distributions for a traffic volume of 1025 vehicles/hour from lane 1 of 3-lane freeway. Discrete probabilities obtained in % IAT (In sec) 0.4 1.05 1.75 2.45 3.15 3.85 4.55 5.25 5.95 6.65 7.35 8.05 8.75 9.45 10.15 10.85 11.55 12.25 12.95 13.65 14.35 15.05 15.75 16.45 17.15 17.85 18.55 19.25 19.95 20.65 Sum of probabilities

OU

Negative exponential

Erlang K=2

Erlang K=3

Pearson type III

2.000 23.144 18.261 11.907 9.253 7.168 5.483 4.518 3.522 3.522 2.172 1.455 1.455 1.321 0.837 0.837 0.837 0.431 0.195 0.195 0.195 0.195 0.161 0.127 0.127 0.127 0.127 0.127 0.127 0.172

17.850 14.854 12.170 9.971 8.169 6.693 5.483 4.493 3.681 3.016 2.471 2.024 1.658 1.359 1.113 0.912 0.747 0.612 0.502 0.411 0.337 0.276 0.226 0.185 0.152 0.124 0.102 0.083 0.068 0.056

5.490 12.492 14.368 13.660 11.865 9.774 7.775 6.034 4.598 3.454 2.565 1.887 1.378 0.999 0.721 0.518 0.370 0.263 0.187 0.132 0.093 0.066 0.046 0.032 0.023 0.016 0.011 0.008 0.005 0.004

2.605 9.404 14.465 15.822 14.543 12.044 9.307 6.846 4.854 3.345 2.253 1.489 0.969 0.623 0.396 0.249 0.155 0.096 0.059 0.036 0.022 0.013 0.008 0.005 0.003 0.002 0.001 0.001 0.000 0.000

7.077 15.430 13.169 10.932 8.969 7.310 5.933 4.802 3.877 3.126 2.516 2.024 1.626 1.305 1.047 0.840 0.673 0.539 0.432 0.346 0.277 0.221 0.177 0.142 0.113 0.091 0.072 0.058 0.046 0.037

100.0

99.8

98.8

99.6

93.2*

Note: OU- Empirical cumulative IAT distributions obtained for the freeways in [13]. *The discrete probabilities will add up to 99.0 for Pearson Type III distribution if an interval of 0.1 seconds is used for IATs.

125 30 28 26 24 N = 30

Discrete probability (%)

22 20

OU Negative exponential

18 16 14 12 10 8 6 4 2 0 0

2

4

6

8

10

12

14

16

18

20

22

Interarrival time (sec)

a) 30 28 26 24 N = 30

Discrete probability (%)

22 20

OU Erlang K=2

18 16 14 12 10 8 6 4 2 0 0

2

4

6

8

10

12

14

16

18

20

22

Interarrival time (sec)

b) Figure 40: Comparison of discrete OU PDF for an hourly traffic volume of 1025 vehicles/hour with a) Negative exponential PDF b) Erlang K = 2 PDF.

126 30 28 26 24 N = 30

Discrete probability (%)

22 20

OU Erlang K = 3

18 16 14 12 10 8 6 4 2 0 0

2

4

6

8

10

12

14

16

18

20

22

Interarrival time (sec)

a) 30 28 26 24 N = 30

Discrete probability (%)

22 20

OU Pearson type III

18 16 14 12 10 8 6 4 2 0 0

2

4

6

8

10

12

14

16

18

20

22

Interarrival time (sec)

b) Figure 41: Comparison of discrete OU PDF for an hourly traffic volume of 1025 vehicles/hour with a) Erlang K = 3 PDF b) Pearson type III PDF.

127 16

14

Discrete probability (%)

12 N = 30

10

OU Negative Exponential

8

6

4

2

0 0

10

20

30

40

50

60

70

80

90

Interarrival time (sec)

a) 16

14

Discrete probability (%)

12 N = 30

10 OU Erlang K=2

8

6

4

2

0 0

10

20

30

40

50

60

70

80

90

Interarrival time (sec)

b) Figure 42: Comparison of discrete OU PDF for an hourly traffic volume of 225 vehicles/hour with a) Negative exponential PDF b) Erlang K = 2 PDF.

128 16

14

Discrete probability (%)

12 N = 30

10

OU Erlang K=3 8

6

4

2

0 0

10

20

30

40

50

60

70

80

90

Interarrival time (sec)

a) 16

14

Discrete probability (%)

12 N = 30

10

OU Pearson type III

8

6

4

2

0 0

10

20

30

40

50

60

70

80

90

Interarrival time (sec)

b) Figure 43: Comparison of discrete OU PDF for an hourly traffic volume of 225 vehicles/hour with a) Erlang K = 3 PDF b) Pearson type III PDF.

129 30

25

Discrete probability (%)

N = 30

20 OU Negative exponential 15

10

5

0 0

2

4

6

8

10

12

14

Interarrival time (sec)

a) 30

25

Discrete probability (%)

N = 30

20

OU Erlang K=2

15

10

5

0 0

2

4

6

8

10

12

14

Interarrival time (sec)

b) Figure 44: Comparison of discrete OU PDF for an hourly traffic volume of 1775 vehicles/hour with a) Negative exponential PDF b) Erlang K = 2 PDF.

130 30

25

Discrete probability (%)

N = 30

20

OU Erlang K=3

15

10

5

0 0

2

4

6

8

10

12

14

Interarrival time (sec)

a) 30

25

Discrete probability (%)

N = 30

20

OU Pearson type III

15

10

5

0 0

2

4

6

8

10

12

14

Interarrival time (sec)

b) Figure 45: Comparison of discrete OU PDF for an hourly traffic volume of 1775 vehicles/hour with a) Erlang K = 3 PDF b) Pearson type III PDF.

131

30

Discrete probability (%)

25

20

225 vehicles/hour 1025 vehicles/hour 1775 vehicles/hour

15

10

5

0 0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

5.50

6.00

Normalized scale (IAT/Mean)

Figure 46: Discrete OU PDFs for lane1 of 3-lane freeway for traffic volumes of 225, 1025 and 1775 vehicles/hour on a normalized abscissa scale.

132 120

Cumulative percentage (%)

100

80

75 vehicles/hour 175 vehicles/hour 225 vehicles/hour 575 vehicles/hour 1025 vehicles/hour 1275 vehicles/hour 1425 vehicles/hour 1775 vehicles/hour

60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure 47: Cumulative IAT distributions for lane1 of 3-lane freeway for traffic volumes of 75,175,225,575, 1025, 1275, 1425 and 1775 vehicles/hour.

133 7 7.1

Results and discussion Modeling the IATs for the entrance ramps and comparison with the IATs for selected lanes of the freeways

The interarrival time distributions for non-signalized freeway entrance ramps were modeled using hyperbolic fits. This provided some more insight into the nature of headways or interarrival times observed on the entrance ramps. The cumulative interarrival time distributions obtained for the entrance ramps were compared with the cumulative interarrival time distributions obtained for all lanes of 2-lane, 3-lane and 4lane freeways in [13]. Table 32 given below shows the results of the comparisons done in this study which shows the best approximation of the selected lane IATs of the freeways for the IATs of the entrance ramps.

Table 32: Summary table obtained after comparisons showing the best approximation of the selected lane IATs of the freeways for the entrance ramp IATs with average of the maximum and maximum absolute differences. Freeway

Entrance ramp 2-lane I-71 S to I-270 W Lane 2 (4.75%,8%) (200-800 veh/hr) I-71 to I-270 W Lane 2 (5.50%,10%) (300-1600 veh/hr) SR-2 to I-90 E Average of lane1 and Driving lane lane 2 (2.38%,3%) (200-800 veh/hr) SR-2 to I-90 E Average of lane1 and Passing lane lane 2 (2.50%,3%) (200-1300 veh/hr)

3-lane

4-lane

Lane 3 (2.75%,5%)

Lane4 (3.5%,5%)

Lane 3 (3.50%,4%)

Lane1 (7.50%,10%)

Lane 3 (2%,3%) *

Lane1 (2.75%,4%)

Lane 3 (2.13%,3%)

Lane1 (2.25%,3%)

Note: * Best approximation case

134 The values in each bracket in Table 32 shows the average of the maximum absolute differences and the maximum absolute differences obtained between the distributions of the freeways and the entrance ramps when compared. Based on all the comparisons it was found that the cumulative interarrival time distributions for the entrance ramp were similar to the interarrival time distributions for different lanes for different freeways. The IAT distributions for the first two entrance ramps were similar to lane 2 when compared with 2-lane freeway. But the results were found to be little different for SR-2 to I-90 east entrance ramp. The driving and the passing lane for this entrance ramp were close to the average of IAT distributions for lane 1 and 2 when compared with a 2-lane freeway. The results consistently showed that when the entrance ramp IAT distributions for all entrance ramps were compared with a 3-lane freeway, they were similar to the IAT distributions of lane 3 of 3-lane freeway. The comparisons with the 4-lane freeway did not yield any unique results with respect to a single lane of the 4lane freeway but the IAT distributions for all the entrance ramps were closer to either lane 1 or lane 4 of 4-lane freeway. The entrance ramp IATs were close to lane 1 of 4-lane freeway for all cases except the I-71 south to I-270 west entrance ramp which was close to lane 4 of 4-lane freeway when compared. Figure 48 thorough Figure 53 show the comparisons which graphically show the results obtained in Table 31 with the best approximated selected lanes IAT distributions of the freeways compared with the entrance ramp IAT distributions for typical hourly traffic volumes.

135 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane2 60 Maximum absolute difference = 3% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane2 60 Maximum absolute difference = 3% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

b) Figure 48: Comparison of the cumulative IAT distributions for a) I-71 south to I270 west entrance ramp for traffic volume of 400 vehicles/hour b) I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with lane 2 of 2-lane freeway.

136 120

Cumulative percentage (%)

100

80

Ent ramp-driving

60

Avg of lane1,lane2 Maximum absolute difference = 3%

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp-passing Avg of lane1,lane2 60 Maximum absolute difference = 2% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

b) Figure 49: Comparison of the cumulative IAT distributions for a)SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour b) SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with average of lane1, lane2 of the 2-lane freeway.

137 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane3 60 Maximum absolute difference = 1.5% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane3 60 Maximum absolute difference = 4% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

b) Figure 50: Comparison of the cumulative IAT distributions for a) I-71 south to I270 west entrance ramp for traffic volume of 400 vehicles/hour b) I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with lane 3 of 3-lane freeway.

138 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane3 60 Maximum absolute difference = 1.5% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane3 60 Maximum absolute difference = 2% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

b) Figure 51: Comparison of the cumulative IAT distributions for a) SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour b) SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane3 of the 3-lane freeway.

139 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane4 60 Maximum absolute difference = 3% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane1 60

Maximum absolute difference = 5%

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

b) Figure 52: Comparison of the cumulative IAT distributions for a) I-71 south to I270 west entrance ramp for traffic volume of 400 vehicles/hour with lane 4 of 4-lane freeway b) I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with lane 1 of 4-lane freeway.

140 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane1 60 Maximum absolute difference = 2% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

a) 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane1 60 Maximum absolute difference = 1.5% 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

b) Figure 53: Comparison of the cumulative IAT distributions for a) SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour b) SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane1 of the 4-lane freeway.

141 7.2

Extrapolation of cumulative IAT distributions for freeways

The cumulative IAT distributions obtained in [13] were extrapolated using the hyperbolic fits beyond the range of observed traffic volumes. All nine cumulative IAT distributions were extrapolated for hourly traffic volume range of 2-2500 vehicles/hour taking into consideration that traffic flow process is stable for low traffic volumes where as aspects of driver safety and saturation flows that can be observed for free flowing traffic on freeways were used for high traffic volumes. 7.3

Analysis of the shape of the PDF for cumulative IAT distributions for freeways

The shape of the probability distribution function for the cumulative interarrival time distributions obtained in [13] were compared with four other distributions: negative exponential, Erlang K=2, Erlang K=3 and Pearson type III in the analysis of the shape of the cumulative IAT distributions section. The probability distribution functions for all the distributions were calculated for regular intervals of IATs, and the discretised probabilities were plotted and compared. None of the four distributions matched the cumulative interarrival time distributions obtained for the freeways in [13] but the negative exponential distribution was close to IAT distributions obtained for freeways for low volumes where the average IAT of the fit was always close to the standard deviation.

142 8 8.1

Conclusions Modeling the IATs for the entrance ramps and comparison with the IATs for selected lanes of the freeways

Based on this study of the interarrival time distributions for non-signalized freeway entrance ramps it can be concluded that the IAT distributions for the nonsignalized entrance ramps are similar to the IAT distributions of the freeways for the same hourly traffic volumes. It is recommended that lane 3 of 3-lane freeway can be used as the best approximation as it had the lowest average of maximum absolute difference and also the least maximum absolute difference when the entrance ramps IATs were compared with the freeway IATs. These approximations are judged to be accurate enough for the purpose of computer simulation of bottlenecks in the work zones. It should be noted that the effect of geometrics (alignment, grade, etc.) was not taken into consideration for the modeling effort. 8.2

Extrapolation of cumulative IAT distributions for freeways

The extrapolated cumulative interarrival time distributions for low and high traffic volumes are judged to be accurate enough for the purpose of computer simulation of the bottlenecks in work zones where large range of traffic volumes are observed. 8.3

Analysis of the shape of the PDF for cumulative IAT distributions for freeways

The analysis of the shape of the IAT probability distribution functions showed that the distributions obtained for the freeways in [13] are different from the distributions provided in the literature for modeling the interarrival times and no close matches were

143 found. The shape of the distributions remained the same over the whole range of observed traffic volumes which suggests that the classification scheme using different mathematical IAT distributions provided by May [1] for different headway states is doubtful and appears to be questionable based on this study. May [1] used three mathematical IAT distributions namely the negative exponential, the Pearson type III and the normal distributions for low, medium and high traffic volumes respectively. Studies performed by [10], [17] also used different headway distribution for different traffic volume ranges which appears not to be applicable for IAT distributions developed in [13]. It would appear that the cumulative IAT distributions defined by 17 percentile points (0%, 1%, 2%, 5%, 10%, 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%, 95%, 98%, 99% and 100%) given in an excel spreadsheet for any lane of 2-lane, 3-lane and 4-lane freeway for any selected hourly traffic volume given by [13] are the most accurate and efficient way to characterize headway or IAT distributions. Further research is needed to model the entrance ramp IATs for signalized freeway entrance ramps and to determine the effects of ramp metering on the mainline traffic.

144 9

List of References

[1]

May, Adolf D., Traffic Flow Fundamentals, Prentice Hall Inc., 1990, pp.11-51.

[2]

Zwahlen, Helmut T., Improved Work Zone Design Guidelines and Enhanced Model of Travel Delays in Work Zones Phase II, State job number 134270, September 2005, Ohio Department of Transportation.

[3]

Zwahlen, Helmut T. and Oner, Erdinc., Improved Work Zone Design Guidelines and Enhanced Model of Travel Delays in Work Zones Phase I: Portability and Scalability of Interarrival and Service Time Probability Distribution Functions for Different Locations in Ohio and the Establishment of Improved Work Zone Design Guidelines. Report number FHWA-OH-2006/1, Ohio Department of Transportation, U.S Department of Transportation and Federal Highway Administration, January 2006, 224 pages.

[4]

Buckley, D. J., A Semi-Poisson Model of Traffic Flow, Transportation Science, Volume 2, Number 2, 1968, pp. 107-133.

[5]

Tolle, J. E., The Lognormal Headway Distribution Model, Traffic Engineering and Control, Volume 13, Number 1, 1971, pp. 22-24.

[6]

Cowan, R. J., Useful Headway Models, Transportation Research, Vol.9, 1975, pp. 371-375.

[7]

Branston,

D.,

Models

of

Single

Lane

Time

Headway

Distributions,

Transportation Science, Volume 10, Number 2, 1976, pp. 125-148. [8]

Tolle,

J. E.,

Vehicular

Headway

Distributions:

Testing

Transportation Research Record, Number 567, 1976, pp. 56-64.

and Results,

145 [9]

Wasielewski, P., Car Following Headways on Freeways Interpreted by SemiPoisson Headway Distribution Model, Transportation Science, Volume 13, Number 1, pp. 36-55.

[10]

Khasnabis, S. and C. L. Heimbach, Headway-Distribution Models for Two-Lane Rural Highways, In Transportation Research Record, Number 772, 1980, pp. 4450.

[11]

May, Adolf D., Gap Availability Studies, In Highway Research Board Record, Number 72, 1965 pp. 105-136.

[12]

Walpole, Ronald E., Myers, Raymond H. and Myers, Sharon L., Probability and Statistics for Engineers and Scientists, Prentice Hall, 1998, p. 169.

[13]

Zwahlen, Helmut T., Oner, Erdinc. and Suravaram, Kiran R., Approximated Headway Distributions of Free-Flowing Traffic on Ohio Freeways for Work Zone Traffic Simulations, Paper Number: 07-1511, 86th Annual Transportation Research Board meeting CD-ROM, January 2007, National Academy of Sciences, Washington D.C.

[14]

Schnell T., J. Mohror and F. Aktan. Evaluation Of Traffic Flow Analysis Tools Applied To Work Zones Based On Flow Data Collected In The Field. In Transportation Research Record, Number 1811, 2002, pp. 57-66.

[15]

Griffiths, J.D. and Hunt, J.G., Vehicle Headways in Urban Areas, Traffic Engineering and Control, Volume 32, Number 10, 1991, pp. 458-462.

[16]

Sullivan, D. and Troutbeck, R., The Use of Cowan's M3 Headway Distribution for Modeling Urban Traffic Flow, Volume 35, Number 7, 1994, pp. 445-450.

146 [17]

Al-Ghamdi, Ali S., Analysis of Time Headways on Urban Roads: Case Study from Riyadh. Journal of Transportation Engineering, July-August 2001, pp. 289294.

[18]

Arasan, V.T. and Koshy, R.Z., Headway Distribution of Heterogeneous Traffic on Urban Arterials, Institution of Engineers journal (India), Volume 84, November 2003, pp. 210-215.

[19]

Mei, M. and Bullen, A. G., Lognormal Distribution for High Traffic Flows, Transportation Research Record, Number 1398, 1993, pp. 125-128.

[20]

Sadeghhosseini, S., and Benekohal, R. F., Headway Models for Low to High Volume Highway Traffic, CD-ROM, Transportation Research Board, National Research Council, Washington, D.C., 2002.

[21]

Zwahlen, Helmut T., Russ, Andrew, Oner, Erdinc, and Parthasarathy, Meghna, Evaluation of Microwave Radar Trailers for Nonintrusive Traffic Measurements, Transportation Research Record, Number 1917, 2005, pp. 127-140.

[22]

Wikipedia

webpage

on

Extrapolation.

Accessed

online

at

http://en.wikipedia.org/wiki/Extrapolation. [23]

Sidi, A., Practical Extrapolation Methods: Theory and Application, Cambridge University Press, 2003, Preface.

[24]

Brezenski, C. and Redivo Zaglia, M., Extrapolation Methods: Theory and Practice, Elsevier Science Publishers B.V, 1991, Preface.

[25]

Driver performance data book. U.S Department of Transportation and Federal Highway Administration, June 1987, p. 2-10.

147 [26]

Pline, James L., Traffic Engineering Handbook, Fifth Edition, Institute of Transportation Engineers, 1999, pp.62-63.

[27]

Wikipedia

webpage

on

Pearson

distribution.

Accessed

online

at

Accessed

online

at

http://en.wikipedia.org/wiki/Pearson_distribution. [28]

Efunda

webpage

on

Gamma

function

finder.

http://www.efunda.com/math/gamma/findgamma.cfm?search_string=factorial%2 0calculator.

148 List of Appendices Appendix A: Hyperbolic plots obtained from the observed data for 1%, 2%, 5%, 10%, 30%, 40%, 60%, 80%, 90%, 95%, 98%, 100% points for the entrance ramps. I-71 South to I-270 West entrance ramp I-71 to I-270 West entrance ramp SR-2 to I-90 East entrance ramp Appendix B: Comparison of entrance ramp cumulative IATs with the freeways for different hourly traffic volumes. I-71 South to I-270 West entrance ramp comparisons with selected lanes of all freeways for 400 vehicles/hour, 600 vehicles/hour, 800 vehicles/hour I-71 South to I-270 West entrance ramp comparisons for 400 vehicles/hour, 600 vehicles/hour I-71 to I-270 West entrance ramp comparisons with selected lanes of all freeways for 300 vehicles/hour, 600 vehicles/hour, 900 vehicles/hour, 1100 vehicles/hour I-71 to I-270 West entrance ramp comparisons for 600 vehicles/hour, 900 vehicles/hour SR-2 to I-90 East entrance ramp driving lane comparisons with selected lanes of all freeways for 200 vehicles/hour, 400 vehicles/hour, 600 vehicles/hour, 800 vehicles/hour SR-2 to I-90 East entrance ramp driving lane comparisons for 400 vehicles/hour, 600 vehicles/hour SR-2 to I-90 East entrance ramp driving lane comparisons with selected lanes of all freeways for 300 vehicles/hour, 600 vehicles/hour, 900 vehicles/hour, 1100 vehicles/hour SR-2 to I-90 East entrance ramp passing lane comparisons for 600 vehicles/hour, 900 vehicles/hour Appendix C: Extrapolated cumulative IAT distribution tables for 2-lane lane1, 3-lane lane2, 3-lane lane3, 4-lane lane1, 4-lane lane2 and 4-lane lane4. Extrapolated tables are available at http://webce.ent.ohiou.edu/orite/cumulativeIATdistributions.html.

149 Appendix A: Hyperbolic plots obtained from the observed data for 1%, 2%, 5%, 10%, 30%, 40%, 60%, 80%, 90%, 95%, 98%, 100% points. I-71 South to I-270 West entrance ramp I-71 to I-270 West entrance ramp SR-2 to I-90 East entrance ramp

150 Appendix A: Hyperbolic plots for I-71 South to I-270 West entrance ramp. 1.2

1

Interarrival time (sec)

0.8

0.6

0.4

N=80 R-Square = 0.072 1% Hyperbolic

0.2

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 1% value. 1.4

1.2

Interarrival time (sec)

1

0.8

0.6

N=80 R-Square = 0.156

0.4

2% Hyperbolic

0.2

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 2% value.

151 Appendix A: Hyperbolic plots for I-71 South to I-270 West entrance ramp continued. 1.6

1.4

Interarrival time (sec)

1.2

1

0.8

0.6

N=80 R-Square = 0.220

0.4 5% Hyperbolic 0.2

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 5% value. 2.5

N=80 R-Square = 0.282

2

Interarrival time (sec)

10% Hyperbolic 1.5

1

0.5

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 10% value.

152 Appendix A: Hyperbolic plots for I-71 South to I-270 West entrance ramp continued. 7

6

Interarrival time (sec)

5

4

N=80 R-Square = 0.485 3 30% Hyperbolic 2

1

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 30% value. 12

10

Interarrival time (sec)

8

N=80 R-Square = 0.514 6 40% Hyperbolic 4

2

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 40% value.

153 Appendix A: Hyperbolic plots for I-71 South to I-270 West entrance ramp continued. 20

18

16

Interarrival time (sec)

14

12

N=80 R-Square = 0.777

10

8

60% Hyperbolic

6

4

2

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 60% value. 30

25

Interarrival time (sec)

20

N=80 R-Square = 0.878 15 80% Hyperbolic 10

5

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 80% value.

154 Appendix A: Hyperbolic plots for I-71 South to I-270 West entrance ramp continued. 50

45

40

Interarrival time (sec)

35

30

N=80 R-Square = 0.802

25

20 90% Hyperbolic

15

10

5

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 90% value. 70

60

Interarrival time (sec)

50

40

N=80 R-Square = 0.756 30 95% Hyperbolic 20

10

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 95% value.

155 Appendix A: Hyperbolic plots for I-71 South to I-270 West entrance ramp continued. 80

70

Interarrival time (sec)

60

50

N=80 R-Square = 0.643

40

98% Hyperbolic

30

20

10

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 98% value. 140

120

Interarrival time (sec)

100

N=80 R-Square = 0.346

80

100% Hyperbolic

60

40

20

0 0

100

200

300

400

500

600

700

800

900

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 South to I-270 West entrance ramp for 100% value.

156 Appendix A: Hyperbolic plots for I-71 to I-270 West entrance ramp continued. 0.9

0.8

0.7

Interarrival time (sec)

0.6

0.5

0.4

N = 195 R-square = 0.068

0.3 1% Hyperbolic 0.2

0.1

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 1% value. 1

0.9

0.8

Interarrival time (sec)

0.7

0.6

0.5

N = 195 R-square = 0.238

0.4

0.3

2% Hyperbolic

0.2

0.1

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 2% value.

157 Appendix A: Hyperbolic plots for I-71 to I-270 West entrance ramp continued. 1.6

1.4

Interarrival time (sec)

1.2

1

0.8

0.6

N = 195 R-square = 0.349

0.4

5% Hyperbolic

0.2

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 5% value. 1.8

1.6

1.4

Interarrival time (sec)

1.2

1

0.8

0.6

N = 195 R-square = 0.505

0.4 10% Hyperbolic 0.2

0 0

200

400

600

800

1000

1200

1400

1600

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 10% value.

1800

158 Appendix A: Hyperbolic plots for I-71 to I-270 West entrance ramp continued. 4

3.5

Interarrival time (sec)

3

2.5

N = 195 R-square = 0.788

2

30% Hyperbolic

1.5

1

0.5

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 30% value. 6

5

Interarrival time (sec)

4

N = 195 R-square = 0.828

3

40% Hyperbolic

2

1

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 40% value.

159 Appendix A: Hyperbolic plots for I-71 to I-270 West entrance ramp continued. 12

10

Interarrival time (sec)

8

N = 195 R-square = 0.942

6

60% Hyperbolic 4

2

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 60% value. 25

Interarrival time (sec)

20

15

N = 195 R-square = 0.973

10

80% Hyperbolic 5

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 80% value.

160 Appendix A: Hyperbolic plots for I-71 to I-270 West entrance ramp continued. 35

30

Interarrival time (sec)

25

20

N = 195 R-square = 0.962 15 90% Hyperbolic 10

5

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 90% value. 45

40

35

Interarrival time (sec)

30

25

N = 195 R-square = 0.933

20

15

95% Hyperbolic

10

5

0 0

200

400

600

800

1000

1200

1400

1600

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 95% value.

1800

161 Appendix A: Hyperbolic plots for I-71 to I-270 West entrance ramp continued. 60

50

Interarrival time (sec)

40

N = 195 R-square = 0.889

30

98% Hyperbolic 20

10

0 0

200

400

600

800

1000

1200

1400

1600

1800

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 98% value. 140

120

Interarrival time (sec)

100

80

N = 195 R-square = 0.550 60 100% Hyperbolic 40

20

0 0

200

400

600

800

1000

1200

1400

1600

Traffic volume (vph)

Figure showing the hyperbolic plot for I-71 to I-270 West entrance for 100% value.

1800

162 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp driving lane continued. 1.4

1.2

N = 118 R-square = 0.001 1

1%

Interarrival time (sec)

Hyperbolic 0.8

0.6

0.4

0.2

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 1% value. 1.6

1.4

N = 118 R-square = 0.083 1.2 2% Interarrival time (sec)

Hyperbolic 1

0.8

0.6

0.4

0.2

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 2% value.

163 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp driving lane continued. 2.5

N = 118 R-square = 0.154

2

5% Interarrival time (sec)

Hyperbolic 1.5

1

0.5

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 5% value. 3.5

3

N = 118 R-square = 0.259

2.5

Interarrival time (sec)

10% Hyperbolic 2

1.5

1

0.5

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 10% value.

164 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp driving lane continued. 7

6

N = 118 R-square = 0.541 5

Interarrival time (sec)

30% Hyperbolic 4

3

2

1

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 30% value. 9

8

7

N = 118 R-square = 0.678 Interarrival time (sec)

6 40% Hyperbolic 5

4

3

2

1

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 40% value.

165 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp driving lane continued. 16

14

12

Interarrival time (sec)

N = 118 R-square = 0.814 10 60% Hyperbolic 8

6

4

2

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 60% value. 35

30

Interarrival time (sec)

25

N = 118 R-square = 0.873 20 80% Hyperbolic 15

10

5

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 80% value.

166 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp driving lane continued. 50

45

40

Interarrival time (sec)

35

N = 118 R-square = 0.844

30 90% Hyperbolic

25

20

15

10

5

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 90% value. 60

50

N = 118 R-square = 0.775

Interarrival time (sec)

40

95% Hyperbolic 30

20

10

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 95% value.

167 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp driving lane continued. 70

60

N = 118 R-square = 0.710 50

Interarrival time (sec)

98% Hyperbolic 40

30

20

10

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 98% value. 120

100

N = 118 R-square = 0.457

Interarrival time (sec)

80

100% Hyperbolic 60

40

20

0 0

100

200

300

400

500

600

700

800

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp driving lane for 100% value.

168 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp passing lane continued. 1.2

1

Interarrival time(sec)

0.8

0.6

0.4

N = 215 R-square = 0.023 0.2 1% Hyperbolic

0 0

200

400

600

800

1000

1200

1400

Traffic volume(vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 1% value. 1.6

1.4

Interarrival time (sec)

1.2

1

0.8

0.6

0.4

N = 215 R-square = 0.279 0.2 2% Hyperbolic 0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 2% value.

169 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp passing lane continued. 2.5

Interarrival time (sec)

2

N = 215 R-square = 0.417

1.5

5% Hyperbolic 1

0.5

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 5% value. 3

2.5

N = 215 R-square = 0.562

Interarrival time (sec)

2

10% Hyperbolic 1.5

1

0.5

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 10% value.

170 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp passing lane continued. 8

7

Interarrival time (sec)

6

5

N = 215 R-square = 0.795

4 30% Hyperbolic 3

2

1

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 30% value. 12

10

Interarrival tiime (sec)

8

N = 215 R-square = 0.860

6

40% Hyperbolic 4

2

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 40% value.

171 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp passing lane continued. 18

16

14

Interarrival time (sec)

12

10

8

N = 215 R-square = 0.946

6 60% Hyperbolic 4

2

0 0

200

400

600

800

1000

1200

1400

Trafffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 60% value. 35

30

Interarrival time (sec)

25

20

N = 215 R-square = 0.969 15 80% Hyperbolic 10

5

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 80% value.

172 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp passing lane continued. 50

45

40

Interarrival time (sec)

35

30

N = 215 R-square = 0.950

25

20 90% Hyperbolic

15

10

5

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 90% value. 70

60

Interarrival time (sec)

50

40

N = 215 R-square = 0.928 30 95% Hyperbolic 20

10

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 95% value.

173 Appendix A: Hyperbolic plots for SR-2 to I-90 East entrance ramp passing lane continued. 90

80

70

Interarrival time (sec)

60

50

40

N = 215 R-square = 0.867

30 98% Hyperbolic 20

10

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 98% value. 140

120

Interarrival time (sec)

100

N = 215 R-square = 0.620 80 100% Hyperbolic 60

40

20

0 0

200

400

600

800

1000

1200

1400

Traffic volume (vph)

Figure showing the hyperbolic plot for SR-2 to I-90 East entrance ramp passing lane for 100% value.

174 Appendix B: Comparison of entrance ramp cumulative IATs with the freeways for different hourly traffic volumes.

I-71 South to I-270 West entrance ramp comparisons with selected lanes of all freeways for 400 vehicles/hour, 600 vehicles/hour, 800 vehicles/hour I-71 South to I-270 West entrance ramp comparisons for 400 vehicles/hour, 600 vehicles/hour

I-71 to I-270 West entrance ramp comparisons with selected lanes of all freeways for 300 vehicles/hour, 600 vehicles/hour, 900 vehicles/hour, 1100 vehicles/hour I-71 to I-270 West entrance ramp comparisons for 600 vehicles/hour, 900 vehicles/hour

SR-2 to I-90 East entrance ramp driving lane comparisons with selected lanes of all freeways for 200 vehicles/hour, 400 vehicles/hour, 600 vehicles/hour, 800 vehicles/hour SR-2 to I-90 East entrance ramp driving lane comparisons for 400 vehicles/hour, 600 vehicles/hour

SR-2 to I-90 East entrance ramp driving lane comparisons with selected lanes of all freeways for 300 vehicles/hour, 600 vehicles/hour, 900 vehicles/hour, 1100 vehicles/hour SR-2 to I-90 East entrance ramp passing lane comparisons for 600 vehicles/hour, 900 vehicles/hour

175 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp. 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 2-lane lane2 freeway.

176 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 3-lane lane2 freeway.

177 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 4-lane lane1 freeway.

178 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 4-lane lane3 freeway.

179 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with 4-lane lane4 freeway.

180 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80

Ent ramp 2-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 2-lane lane2 freeway.

181 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 3-lane lane2 freeway.

182 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane1 freeway.

183 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane3 freeway.

184 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-Lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane4 freeway.

185 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp . 120

Cumulative percentage (%)

100

80

Ent ramp 2-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 2-lane lane2 freeway.

186 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 3-lane lane2 freeway.

187 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80

Ent ramp 3-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80

Ent ramp 4-Lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 4-lane lane1 freeway.

188 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80

Ent ramp 4-Lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80

Ent ramp 4-Lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 4-lane lane3 freeway.

189 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80

Ent ramp 4-Lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 800 vehicles/hour with 4-lane lane4 freeway.

190 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

a)

Cumulative percentage (%)

100

80

60

Ent ramp 2-Lane (lane1) Lane 2

40

20

0 0.10

1.00

10.00

100.00

Cumulative interrrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp 3-Lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

Ent ramp 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 400 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

191 Appendix B: Comparison of cumulative IATs for I-71 South to I-270 West entrance ramp continued. 120

a)

Cumulative percentage (%)

100

80

60 Ent ramp 2-Lane (lane1) Lane 2

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp 3-Lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

40 Ent ramp 4-Lane (lane1) Lane 2 Lane 3 Lane 4

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 south to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

192 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 2-lane lane2 freeway.

193 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 3-lane lane2 freeway.

194 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80

60

Ent ramp 4-lane lane1

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 4-lane lane1 freeway.

195 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 4-lane lane3 freeway.

196 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 300 vehicles/hour with 4-lane lane4 freeway.

197 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 2-lane lane2 freeway.

198 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 3-lane lane2 freeway.

199 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane1 freeway.

200 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane3 freeway.

201 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with 4-lane lane4 freeway.

202 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 2-lane lane2 freeway.

203 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 3-lane lane2 freeway.

204 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 4-lane lane1 freeway.

205 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 4-lane lane3 freeway.

206 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with 4-lane lane4 freeway.

207 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 2-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 2-lane lane2 freeway.

208 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 3-lane lane1 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 3-lane lane2 freeway.

209 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 3-lane lane3 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 4-lane lane1 freeway.

210 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 4-lane lane2 freeway. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 4-lane lane3 freeway.

211 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

Cumulative percentage (%)

100

80 Ent ramp 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 1100 vehicles/hour with 4-lane lane4 freeway.

212 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

a)

Cumulative percentage (%)

100

80

60

40 Ent ramp Right lane(lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec) 120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp Right lane(lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

40

Ent ramp Right lane(lane1) Lane 2 Lane 3 Lane 4

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 600 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

213 Appendix B: Comparison of cumulative IATs for I-71 to I-270 West entrance ramp continued. 120

a)

Cumulative percentage (%)

100

80

60

40 Ent ramp Right lane(lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60 Ent ramp Right lane(lane1) Lane 2 Lane 3

40

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec) 120

c)

Cumulative percentage (%)

100

80

60

40

Ent ramp Right lane(lane1) Lane 2 Lane 3 Lane 4

20

0 0.10

1.00 10.00 Cumulative Interarrival time (sec)

100.00

Figure showing the comparison of cumulative interarrival time distributions for I-71 to I-270 west entrance ramp for traffic volume of 900 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

214 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane2 of 2-lane freeway.

215 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80

60 Ent ramp-driving 3-lane lane1 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane2 of 3-lane freeway.

216 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80

60 Ent ramp-driving 3-lane lane3 40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80

60 Ent ramp-driving 4-lane lane1 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane1 of 4-lane freeway.

217 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80

60 Ent ramp-driving 4-lane lane2 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80

60 Ent ramp-driving 4-lane lane3 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane3 of 4-lane freeway.

218 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80

60 Ent ramp-driving 4-lane lane4 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 200 vehicles/hour with lane4 of 4-lane freeway.

219 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane2 of 2-lane freeway.

220 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane2 of 3-lane freeway.

221 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane1 of 4-lane freeway.

222 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane3 of 4-lane freeway.

223 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 400 vehicles/hour with lane4 of 4-lane freeway.

224 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane2 of 2-lane freeway.

225 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane2 of 3-lane freeway.

226 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane1 of 4-lane freeway.

227 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane3 of 4-lane freeway.

228 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 600 vehicles/hour with lane4 of 4-lane freeway.

229 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane2 of 2-lane freeway.

230 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane2 of 3-lane freeway.

231 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane1 of 4-lane freeway.

232 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane3 of 4-lane freeway.

233 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-driving 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp driving lane for traffic volume of 800 vehicles/hour with lane4 of 4-lane freeway.

234 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

a)

Cumulative percentage (%)

100

80

60

40

Ent ramp-driving 2-Lane (lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative interrrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp-driving 3-Lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec) 120

c)

Cumulative percentage (%)

100

80

60

Ent ramp-driving 4-Lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 East entrance ramp driving lane for traffic volume of 400 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

235 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp driving lane continued. 120

a)

Cumulative percentage (%)

100

80

60 Ent ramp-driving 40

2-Lane (lane1) Lane 2

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp-driving 3-Lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

120

c)

Cumulative percentage (%)

100

80

60

40 Ent ramp-driving 4-Lane (lane1) Lane 2 Lane 3 Lane 4

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 East entrance ramp driving lane for traffic volume of 600 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

236 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

1000.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80

60

Ent ramp-passing 2-lane lane2

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane2 of 2-lane freeway.

237 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80

60

Ent ramp-passing 3-lane lane1

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80

60

Ent ramp-passing 3-lane lane2

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane2 of 3-lane freeway.

238 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80

60 Ent ramp-passing 3-lane lane3 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80

60 Ent ramp-passing 4-lane lane1 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane1 of 4-lane freeway.

239 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80

60

Ent ramp-passing 4-lane lane2

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80

60 Ent ramp-passing 4-lane lane3 40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane3 of 4-lane freeway.

240 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80

60

Ent ramp-passing 4-lane lane4

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 300 vehicles/hour with lane4 of 4-lane freeway.

241 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane2 of 2-lane freeway.

242 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane2 of 3-lane freeway.

243 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane1 of 4-lane freeway.

244 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane3 of 4-lane freeway.

245 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 600 vehicles/hour with lane4 of 4-lane freeway.

246 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane2 of 2-lane freeway.

247 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane2 of 3-lane freeway.

248 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane1 of 4-lane freeway.

249 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane3 of 4-lane freeway.

250 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 900 vehicles/hour with lane4 of 4-lane freeway.

251 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane1 of 2-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 2-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane2 of 2-lane freeway.

252 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane1 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane2 of 3-lane freeway.

253 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 3-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane3 of 3-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane1 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane1 of 4-lane freeway.

254 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane2 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane2 of 4-lane freeway. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane3 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane3 of 4-lane freeway.

255 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

Cumulative percentage (%)

100

80 Ent ramp-passing 4-lane lane4 60

40

20

0 0.10

1.00

10.00

100.00

Cumulative interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 east entrance ramp passing lane for traffic volume of 1100 vehicles/hour with lane4 of 4-lane freeway.

256 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

a)

Cumulative percentage (%)

100

80

60

40 Ent ramp passing Right lane (lane1) Lane 2

20

0 0.10

1.00 10.00 Cumulative Interarrival time (sec)

100.00

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp passing Right lane (lane 1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec) 120

c)

Cumulative percentage (%)

100

80

60

40

Ent ramp passing Right lane (lane1) Lane 2 Lane 3 Lane 4

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 East entrance ramp passing lane for traffic volume of 600 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

257 Appendix B: Comparison of cumulative IATs for SR-2 to I-90 East entrance ramp passing lane continued. 120

a)

Cumulative percentage (%)

100

80

60

40

Ent ramp passing Right lane (lane1) Lane 2

20

0 0.10

1.00 10.00 Cumulative Interarrival time (sec)

100.00

120

b)

Cumulative percentage (%)

100

80

60

40

Ent ramp passing Right lane (lane1) Lane 2 Lane 3

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec) 120

c)

Cumulative percentage (%)

100

80

60

Ent ramp passing Right lane (lane1) Lane 2 Lane 3 Lane 4

40

20

0 0.10

1.00

10.00

100.00

Cumulative Interarrival time (sec)

Figure showing the comparison of cumulative interarrival time distributions for SR-2 to I-90 East entrance ramp passing lane for traffic volume of 900 vehicles/hour with a) 2-lane b) 3-lane c) 4-lane freeway.

258 Appendix C: Extrapolated cumulative IAT distribution tables for 2-lane lane1, 3-lane lane2, 3-lane lane3, 4-lane lane1, 4-lane lane2 and 4-lane lane4.

259 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 1 of 2-lane freeway. Hourly traffic volume range

Vehicles per hour

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100 1699.73

2-50

25

0.10

4.87

5.07

11.22

17.24

31.88

49.77

70.68

96.96

129.74

173.13

230.65

316.87

395.25

525.74

523.85

50-100

75

0.10

1.96

2.12

4.17

6.22

11.06

16.92

23.77

32.36

43.13

57.40

76.42

105.28

131.79

174.75

176.29

554.60

100-150

125

0.10

1.37

1.53

2.76

4.02

6.89

10.35

14.39

19.45

25.81

34.25

45.57

62.96

79.10

104.55

106.77

325.64

150-200

175

0.10

1.12

1.27

2.15

3.07

5.10

7.53

10.36

13.91

18.39

24.33

32.36

44.83

56.51

74.47

76.97

227.55

200-250

225

0.10

0.98

1.13

1.82

2.54

4.11

5.96

8.13

10.83

14.26

18.82

25.02

34.76

43.97

57.76

60.41

173.09

250-300

275

0.10

0.89

1.03

1.60

2.21

3.48

4.97

6.71

8.88

11.64

15.32

20.34

28.35

35.98

47.13

49.87

138.45

300-350

325

0.10

0.83

0.97

1.45

1.97

3.04

4.28

5.72

7.52

9.82

12.89

17.11

23.91

30.46

39.77

42.57

114.50

350-400

375

0.10

0.78

0.92

1.34

1.80

2.71

3.77

5.00

6.53

8.49

11.11

14.74

20.66

26.40

34.37

37.21

96.95

400-450

425

0.10

0.74

0.89

1.25

1.67

2.47

3.38

4.45

5.77

7.47

9.75

12.93

18.17

23.30

30.24

33.12

83.54

450-500

475

0.10

0.72

0.86

1.19

1.56

2.27

3.07

4.01

5.17

6.67

8.68

11.50

16.21

20.86

26.99

29.88

72.98

500-550

525

0.10

0.69

0.83

1.13

1.48

2.11

2.82

3.65

4.68

6.02

7.81

10.34

14.62

18.87

24.35

27.26

64.43

550-600

575

0.10

0.67

0.81

1.08

1.41

1.98

2.62

3.36

4.28

5.48

7.09

9.39

13.30

17.24

22.17

25.09

57.39

600-650

625

0.10

0.65

0.79

1.04

1.35

1.87

2.44

3.12

3.94

5.03

6.49

8.58

12.20

15.86

20.34

23.27

51.48

650-700

675

0.10

0.64

0.78

1.01

1.29

1.77

2.30

2.91

3.65

4.65

5.98

7.90

11.26

14.69

18.79

21.71

46.46

700-750

725

0.10

0.63

0.76

0.98

1.25

1.69

2.17

2.73

3.41

4.31

5.54

7.31

10.45

13.68

17.45

20.37

42.13

750-800

775

0.10

0.62

0.75

0.96

1.21

1.62

2.06

2.57

3.19

4.03

5.15

6.80

9.75

12.80

16.28

19.20

38.38

800-850

825

0.10

0.60

0.74

0.93

1.17

1.56

1.96

2.43

3.00

3.77

4.82

6.35

9.13

12.03

15.25

18.17

35.08

850-900

875

0.10

0.60

0.73

0.91

1.14

1.50

1.87

2.31

2.83

3.55

4.52

5.95

8.58

11.35

14.34

17.26

32.17

900-950

925

0.09

0.59

0.72

0.89

1.12

1.45

1.79

2.20

2.68

3.35

4.25

5.60

8.10

10.74

13.53

16.45

29.58

950-1000

975

0.09

0.58

0.71

0.88

1.09

1.40

1.72

2.10

2.55

3.17

4.01

5.28

7.66

10.19

12.80

15.72

27.27

1000-1050

1025

0.09

0.57

0.71

0.86

1.07

1.36

1.66

2.01

2.43

3.01

3.80

4.99

7.26

9.69

12.15

15.06

25.18

1050-1100

1075

0.09

0.57

0.70

0.85

1.05

1.33

1.60

1.93

2.32

2.86

3.60

4.73

6.90

9.24

11.55

14.46

23.29

1100-1150

1125

0.09

0.56

0.69

0.83

1.03

1.29

1.55

1.85

2.22

2.73

3.42

4.50

6.58

8.84

11.01

13.91

21.58

1150-1200

1175

0.09

0.55

0.68

0.82

1.01

1.26

1.50

1.79

2.12

2.60

3.26

4.28

6.28

8.46

10.52

13.41

20.02

1200-1250

1225

0.09

0.55

0.68

0.81

0.99

1.23

1.46

1.72

2.04

2.49

3.11

4.08

6.01

8.12

10.06

12.94

18.59

Note: Cells shaded in grey show the extrapolated traffic volume range.

260 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 1 of 2-lane freeway continued. Hourly traffic volume range 1250-1300

Cumulative Percentage (%)

Vehicles per hour

0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1275

0.09

0.54

0.67

0.80

0.98

1.20

1.42

1.67

1.96

2.39

2.97

3.90

5.75

7.80

9.64

12.52

17.27

1300-1350

1325

0.09

0.54

0.67

0.79

0.96

1.18

1.38

1.61

1.89

2.29

2.85

3.73

5.52

7.51

9.26

12.12

16.06

1350-1400

1375

0.09

0.53

0.66

0.78

0.95

1.15

1.34

1.56

1.82

2.20

2.73

3.57

5.30

7.24

8.90

11.76

14.94

1400-1450

1425

0.09

0.53

0.66

0.77

0.93

1.13

1.31

1.52

1.76

2.12

2.62

3.43

5.10

6.99

8.56

11.42

13.91

1450-1500

1475

0.09

0.52

0.65

0.76

0.92

1.11

1.28

1.48

1.70

2.05

2.52

3.29

4.92

6.75

8.25

11.10

12.94

1500-1550

1525

0.09

0.52

0.65

0.75

0.91

1.09

1.25

1.44

1.65

1.97

2.42

3.17

4.74

6.53

7.96

10.80

12.05

1550-1600

1575

0.09

0.52

0.64

0.75

0.90

1.07

1.22

1.40

1.60

1.91

2.33

3.05

4.58

6.32

7.69

10.52

11.21

1600-1650

1625

0.09

0.51

0.64

0.74

0.89

1.06

1.20

1.36

1.55

1.84

2.25

2.94

4.42

6.13

7.44

10.26

10.43

1650-1700

1675

0.09

0.51

0.64

0.73

0.88

1.04

1.17

1.33

1.51

1.78

2.17

2.83

4.28

5.95

7.20

10.01

9.70

1700-1750

1725

0.09

0.51

0.63

0.73

0.87

1.02

1.15

1.30

1.46

1.73

2.10

2.74

4.14

5.78

6.97

9.78

9.01

1750-1800

1775

0.09

0.50

0.63

0.72

0.86

1.01

1.13

1.27

1.42

1.68

2.03

2.64

4.02

5.62

6.76

9.56

8.37

1800-1850

1825

0.09

0.50

0.62

0.71

0.85

1.00

1.11

1.24

1.39

1.63

1.96

2.56

3.90

5.46

6.56

9.35

7.76

1850-1900

1875

0.09

0.50

0.62

0.71

0.84

0.98

1.09

1.21

1.35

1.58

1.90

2.47

3.78

5.32

6.37

9.15

7.19

1900-1950

1925

0.09

0.49

0.62

0.70

0.84

0.97

1.07

1.19

1.32

1.54

1.84

2.40

3.67

5.18

6.19

8.96

6.65

1950-2000

1975

0.09

0.49

0.61

0.70

0.83

0.96

1.05

1.16

1.29

1.49

1.79

2.32

3.57

5.05

6.02

8.78

6.14

2000-2050

2025

0.09

0.49

0.61

0.69

0.82

0.95

1.04

1.14

1.26

1.45

1.73

2.25

3.47

4.93

5.86

8.61

5.65

2050-2100

2075

0.09

0.48

0.61

0.69

0.81

0.93

1.02

1.12

1.23

1.42

1.68

2.19

3.38

4.81

5.70

8.45

5.20

2100-2150

2125

0.09

0.48

0.60

0.68

0.81

0.92

1.00

1.10

1.20

1.38

1.64

2.12

3.29

4.70

5.55

8.30

4.76

2150-2200

2175

0.09

0.48

0.60

0.68

0.80

0.91

0.99

1.08

1.17

1.34

1.59

2.06

3.21

4.59

5.41

8.15

4.35

2200-2250

2225

0.09

0.48

0.60

0.67

0.79

0.90

0.98

1.06

1.15

1.31

1.55

2.00

3.13

4.49

5.28

8.00

3.96

2250-2300

2275

0.09

0.47

0.59

0.67

0.79

0.89

0.96

1.04

1.12

1.28

1.51

1.95

3.05

4.39

5.15

7.87

3.59

2300-2350

2325

0.09

0.47

0.59

0.66

0.78

0.88

0.95

1.02

1.10

1.25

1.47

1.90

2.98

4.30

5.03

7.74

3.24

2350-2400

2375

0.09

0.47

0.59

0.66

0.78

0.88

0.94

1.01

1.08

1.22

1.43

1.85

2.91

4.21

4.91

7.61

2.90

2400-2450

2425

0.09

0.47

0.59

0.66

0.77

0.87

0.92

0.99

1.06

1.19

1.39

1.80

2.84

4.12

4.80

7.49

2.58

2450-2500

2475

0.08

0.46

0.58

0.65

0.77

0.86

0.91

0.98

1.04

1.17

1.36

1.75

2.77

4.04

4.69

7.38

2.27

Note: Cells shaded in grey show the extrapolated traffic volume range.

261 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 2 of 3-lane freeway continued. Hourly traffic volume range

Vehicles per hour

2-50 50-100

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

25

0.10

4.42

5.52

10.26

16.08

28.65

43.92

65.14

90.67

123.90

164.58

230.14

344.84

461.94

623.64

650.44

842.31

75

0.10

1.80

2.21

3.81

5.78

9.98

15.05

22.01

30.42

41.35

54.79

76.35

114.13

152.75

205.91

216.16

281.25

100-150

125

0.10

1.28

1.54

2.52

3.72

6.24

9.27

13.39

18.36

24.84

32.84

45.60

67.99

90.92

122.37

129.31

169.03

150-200

175

0.10

1.05

1.26

1.96

2.83

4.64

6.79

9.69

13.20

17.77

23.43

32.42

48.22

64.42

86.58

92.09

120.94

200-250

225

0.10

0.92

1.10

1.65

2.34

3.75

5.42

7.64

10.33

13.84

18.20

25.10

37.24

49.70

66.69

71.41

94.22

250-300

275

0.10

0.84

1.00

1.45

2.02

3.18

4.54

6.33

8.50

11.34

14.87

20.44

30.25

40.34

54.05

58.25

77.22

300-350

325

0.10

0.79

0.93

1.32

1.81

2.79

3.93

5.42

7.24

9.60

12.57

17.21

25.42

33.86

45.29

49.15

65.44

350-400

375

0.10

0.74

0.87

1.22

1.65

2.50

3.49

4.76

6.31

8.33

10.88

14.85

21.87

29.11

38.87

42.47

56.81

400-450

425

0.10

0.71

0.83

1.14

1.52

2.28

3.14

4.25

5.60

7.36

9.59

13.04

19.16

25.47

33.97

37.36

50.21

450-500

475

0.10

0.69

0.80

1.08

1.42

2.10

2.87

3.85

5.04

6.60

8.57

11.62

17.02

22.61

30.10

33.33

44.99

500-550

525

0.10

0.66

0.77

1.03

1.35

1.96

2.66

3.52

4.59

5.98

7.75

10.46

15.29

20.29

26.97

30.07

40.77

550-600

575

0.10

0.65

0.75

0.98

1.28

1.84

2.47

3.25

4.21

5.46

7.07

9.51

13.86

18.38

24.38

27.38

37.29

600-650

625

0.10

0.63

0.73

0.95

1.22

1.74

2.32

3.03

3.89

5.03

6.49

8.71

12.66

16.77

22.21

25.11

34.36

650-700

675

0.10

0.62

0.72

0.92

1.18

1.65

2.19

2.83

3.63

4.66

6.01

8.02

11.64

15.40

20.37

23.19

31.86

700-750

725

0.10

0.61

0.70

0.89

1.13

1.58

2.08

2.67

3.39

4.35

5.59

7.44

10.76

14.22

18.77

21.53

29.71

750-800

775

0.10

0.60

0.69

0.87

1.10

1.52

1.98

2.52

3.19

4.07

5.22

6.93

10.00

13.19

17.39

20.08

27.83

800-850

825

0.10

0.59

0.68

0.85

1.07

1.46

1.89

2.39

3.01

3.83

4.90

6.48

9.32

12.29

16.17

18.81

26.19

850-900

875

0.10

0.58

0.67

0.83

1.04

1.41

1.82

2.28

2.86

3.62

4.61

6.08

8.73

11.50

15.10

17.68

24.73

900-950

925

0.10

0.57

0.66

0.81

1.01

1.36

1.75

2.18

2.72

3.42

4.36

5.72

8.20

10.79

14.14

16.68

23.42

950-1000

975

0.10

0.56

0.65

0.80

0.99

1.32

1.69

2.09

2.59

3.25

4.13

5.41

7.72

10.15

13.28

15.78

22.26

1000-1050

1025

0.10

0.56

0.64

0.78

0.97

1.29

1.63

2.00

2.48

3.10

3.93

5.12

7.29

9.57

12.50

14.97

21.20

1050-1100

1075

0.10

0.55

0.63

0.77

0.95

1.25

1.58

1.93

2.37

2.96

3.74

4.86

6.90

9.05

11.80

14.23

20.24

1100-1150

1125

0.10

0.55

0.63

0.76

0.93

1.22

1.53

1.86

2.28

2.83

3.57

4.62

6.55

8.58

11.16

13.56

19.37

1150-1200

1175

0.10

0.54

0.62

0.75

0.91

1.19

1.49

1.80

2.19

2.71

3.41

4.41

6.22

8.15

10.58

12.95

18.57

1200-1250

1225

0.10

0.54

0.61

0.74

0.90

1.17

1.45

1.74

2.11

2.60

3.27

4.21

5.93

7.75

10.05

12.39

17.84

Note: Cells shaded in grey show the extrapolated traffic volume range.

262 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 2 of 3-lane freeway continued. Hourly traffic volume range

Vehicles per hour

1250-1300 1300-1350

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1275

0.10

0.53

0.61

0.73

0.88

1.14

1.41

1.69

2.04

2.50

3.14

4.02

5.65

7.38

9.55

11.87

17.17

1325

0.10

0.53

0.60

0.72

0.87

1.12

1.38

1.64

1.97

2.41

3.02

3.85

5.40

7.05

9.10

11.39

16.54

1350-1400

1375

0.09

0.52

0.60

0.71

0.86

1.10

1.35

1.59

1.91

2.33

2.90

3.70

5.17

6.73

8.68

10.95

15.96

1400-1450

1425

0.09

0.52

0.59

0.70

0.85

1.08

1.32

1.55

1.85

2.25

2.80

3.55

4.95

6.44

8.28

10.53

15.42

1450-1500

1475

0.09

0.52

0.59

0.70

0.84

1.06

1.29

1.51

1.80

2.17

2.70

3.42

4.75

6.17

7.92

10.15

14.92

1500-1550

1525

0.09

0.51

0.58

0.69

0.82

1.04

1.27

1.47

1.75

2.10

2.61

3.29

4.56

5.92

7.58

9.79

14.45

1550-1600

1575

0.09

0.51

0.58

0.68

0.82

1.03

1.24

1.44

1.70

2.04

2.53

3.17

4.38

5.68

7.26

9.45

14.01

1600-1650

1625

0.09

0.51

0.58

0.68

0.81

1.01

1.22

1.41

1.65

1.98

2.45

3.06

4.22

5.46

6.97

9.14

13.60

1650-1700

1675

0.09

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

0.00

1700-1750

1725

0.09

0.50

0.57

0.66

0.79

0.98

1.18

1.35

1.57

1.87

2.30

2.86

3.92

5.06

6.42

8.56

12.84

1750-1800

1775

0.09

0.50

0.57

0.66

0.78

0.97

1.16

1.32

1.53

1.82

2.23

2.77

3.78

4.87

6.18

8.30

12.50

1800-1850

1825

0.09

0.49

0.56

0.65

0.77

0.96

1.14

1.29

1.50

1.77

2.17

2.68

3.65

4.70

5.94

8.05

12.17

1850-1900

1875

0.09

0.49

0.56

0.65

0.77

0.95

1.12

1.27

1.46

1.72

2.11

2.59

3.52

4.54

5.72

7.81

11.86

1900-1950

1925

0.09

0.49

0.56

0.64

0.76

0.93

1.10

1.24

1.43

1.68

2.05

2.52

3.41

4.38

5.51

7.59

11.57

1950-2000

1975

0.09

0.49

0.55

0.64

0.75

0.92

1.09

1.22

1.40

1.64

2.00

2.44

3.30

4.23

5.31

7.38

11.29

2000-2050

2025

0.09

0.48

0.55

0.63

0.75

0.91

1.07

1.20

1.37

1.60

1.95

2.37

3.19

4.09

5.13

7.18

11.02

2050-2100

2075

0.09

0.48

0.55

0.63

0.74

0.90

1.06

1.18

1.34

1.56

1.90

2.30

3.09

3.96

4.95

6.99

10.77

2100-2150

2125

0.09

0.48

0.54

0.62

0.73

0.89

1.05

1.16

1.32

1.53

1.85

2.24

3.00

3.83

4.78

6.81

10.53

2150-2200

2175

0.09

0.48

0.54

0.62

0.73

0.88

1.03

1.14

1.29

1.49

1.81

2.18

2.91

3.71

4.62

6.63

10.30

2200-2250

2225

0.09

0.47

0.54

0.62

0.72

0.88

1.02

1.12

1.27

1.46

1.77

2.12

2.82

3.60

4.46

6.47

10.08

2250-2300

2275

0.09

0.47

0.54

0.61

0.72

0.87

1.01

1.11

1.25

1.43

1.73

2.07

2.74

3.49

4.31

6.31

9.87

2300-2350

2325

0.09

0.47

0.53

0.61

0.71

0.86

0.99

1.09

1.22

1.40

1.69

2.01

2.66

3.38

4.17

6.16

9.67

2350-2400

2375

0.09

0.47

0.53

0.60

0.71

0.85

0.98

1.07

1.20

1.37

1.65

1.96

2.59

3.28

4.04

6.01

9.48

2400-2450

2425

0.09

0.46

0.53

0.60

0.70

0.84

0.97

1.06

1.18

1.35

1.61

1.91

2.51

3.19

3.91

5.87

9.29

2450-2500

2475

0.09

0.46

0.53

0.60

0.70

0.84

0.96

1.04

1.16

1.32

1.58

1.87

2.44

3.10

3.79

5.74

9.12

Note: Cells shaded in grey show the extrapolated traffic volume range.

263 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 3 of 3-lane freeway continued. Hourly traffic volume range

Vehicles per hour

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

2-50

25

0.10

0.19

0.85

1.95

3.70

10.22

19.11

30.40

46.43

66.15

90.41

123.77

178.08

227.74

276.31

326.07

344.85

50-100

75

0.10

0.32

0.77

1.53

2.71

7.00

12.85

20.29

30.83

43.85

59.96

82.21

118.53

151.91

185.04

218.17

233.75

100-150

125

0.10

0.42

0.71

1.19

1.91

4.43

7.84

12.20

18.35

26.02

35.60

48.97

70.89

91.24

112.02

131.84

144.85

150-200

175

0.10

0.47

0.68

1.04

1.57

3.32

5.69

8.74

13.01

18.38

25.16

34.72

50.48

65.23

80.72

94.84

106.73

200-250

225

0.10

0.49

0.66

0.96

1.38

2.71

4.50

6.81

10.04

14.13

19.36

26.80

39.14

50.79

63.32

74.29

85.54

250-300

275

0.10

0.51

0.65

0.90

1.26

2.32

3.74

5.59

8.15

11.43

15.68

21.77

31.92

41.60

52.25

61.20

72.05

300-350

325

0.10

0.51

0.64

0.87

1.17

2.04

3.22

4.74

6.84

9.56

13.12

18.28

26.93

35.23

44.59

52.14

62.69

350-400

375

0.10

0.52

0.64

0.84

1.11

1.84

2.83

4.12

5.88

8.19

11.25

15.73

23.26

30.56

38.96

45.49

55.83

400-450

425

0.10

0.53

0.63

0.82

1.06

1.69

2.53

3.64

5.15

7.15

9.82

13.77

20.46

27.00

34.66

40.41

50.57

450-500

475

0.10

0.53

0.63

0.80

1.02

1.57

2.30

3.27

4.57

6.32

8.69

12.23

18.25

24.18

31.27

36.40

46.41

500-550

525

0.10

0.53

0.62

0.78

0.99

1.47

2.11

2.96

4.10

5.65

7.78

10.99

16.46

21.90

28.51

33.15

43.04

550-600

575

0.10

0.53

0.62

0.77

0.96

1.39

1.96

2.71

3.71

5.10

7.02

9.96

14.98

20.01

26.24

30.46

40.25

600-650

625

0.10

0.54

0.61

0.76

0.94

1.32

1.82

2.50

3.39

4.64

6.39

9.09

13.74

18.43

24.33

28.20

37.91

650-700

675

0.10

0.54

0.61

0.75

0.92

1.26

1.71

2.32

3.11

4.24

5.85

8.35

12.69

17.08

22.70

26.28

35.90

700-750

725

0.10

0.54

0.61

0.74

0.90

1.21

1.61

2.16

2.88

3.90

5.39

7.72

11.77

15.92

21.30

24.62

34.17

750-800

775

0.10

0.54

0.61

0.74

0.89

1.17

1.53

2.03

2.67

3.60

4.98

7.17

10.98

14.90

20.07

23.17

32.66

800-850

825

0.10

0.54

0.60

0.73

0.87

1.13

1.46

1.91

2.49

3.35

4.63

6.68

10.28

14.01

18.99

21.90

31.33

850-900

875

0.10

0.54

0.60

0.72

0.86

1.09

1.39

1.80

2.33

3.12

4.31

6.25

9.67

13.22

18.04

20.77

30.14 29.09

900-950

925

0.10

0.54

0.60

0.72

0.85

1.06

1.33

1.71

2.18

2.91

4.03

5.87

9.12

12.52

17.19

19.77

950-1000

975

0.10

0.54

0.60

0.71

0.84

1.04

1.28

1.63

2.05

2.73

3.78

5.53

8.62

11.89

16.42

18.86

28.13

1000-1050

1025

0.10

0.54

0.60

0.71

0.83

1.01

1.23

1.55

1.94

2.56

3.56

5.22

8.18

11.32

15.73

18.05

27.27

1050-1100

1075

0.10

0.54

0.59

0.70

0.82

0.99

1.19

1.48

1.83

2.41

3.35

4.93

7.77

10.80

15.10

17.31

26.49

1100-1150

1125

0.09

0.53

0.59

0.70

0.81

0.97

1.15

1.42

1.74

2.28

3.17

4.68

7.41

10.33

14.53

16.63

25.77

1150-1200

1175

0.09

0.53

0.59

0.69

0.81

0.95

1.11

1.36

1.65

2.15

3.00

4.45

7.07

9.90

14.01

16.02

25.11

1200-1250

1225

0.09

0.53

0.59

0.69

0.80

0.93

1.08

1.31

1.57

2.04

2.84

4.23

6.76

9.51

13.53

15.45

24.51

Note: Cells shaded in grey show the extrapolated traffic volume range.

264 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 3 of 3-lane freeway continued. Hourly traffic volume range

Vehicles per hour

1250-1300 1300-1350

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1275

0.09

0.53

0.59

0.69

0.79

0.91

1.05

1.26

1.49

1.93

2.70

4.03

6.48

9.14

13.08

14.92

23.94

1325

0.09

0.53

0.58

0.68

0.79

0.90

1.02

1.21

1.43

1.83

2.56

3.85

6.21

8.80

12.67

14.44

23.42

1350-1400

1375

0.09

0.53

0.58

0.68

0.78

0.88

0.99

1.17

1.36

1.74

2.44

3.68

5.97

8.49

12.29

13.99

22.94

1400-1450

1425

0.09

0.53

0.58

0.68

0.78

0.87

0.97

1.13

1.30

1.66

2.32

3.53

5.74

8.20

11.93

13.57

22.49

1450-1500

1475

0.09

0.53

0.58

0.67

0.77

0.85

0.94

1.10

1.25

1.58

2.22

3.38

5.53

7.93

11.60

13.18

22.06

1500-1550

1525

0.09

0.53

0.58

0.67

0.77

0.84

0.92

1.06

1.20

1.51

2.12

3.24

5.33

7.68

11.29

12.81

21.67

1550-1600

1575

0.09

0.53

0.58

0.67

0.76

0.83

0.90

1.03

1.15

1.44

2.03

3.12

5.15

7.44

11.00

12.47

21.29

1600-1650

1625

0.09

0.53

0.57

0.67

0.76

0.82

0.88

1.00

1.10

1.38

1.94

3.00

4.98

7.22

10.73

12.15

20.94

1650-1700

1675

0.09

0.53

0.57

0.66

0.75

0.81

0.87

0.97

1.06

1.32

1.86

2.88

4.81

7.01

10.47

11.85

20.61

1700-1750

1725

0.09

0.52

0.57

0.66

0.75

0.80

0.85

0.95

1.02

1.26

1.78

2.78

4.66

6.81

10.23

11.56

20.29

1750-1800

1775

0.09

0.52

0.57

0.66

0.74

0.79

0.83

0.92

0.99

1.21

1.71

2.68

4.51

6.62

10.00

11.29

20.00

1800-1850

1825

0.09

0.52

0.57

0.65

0.74

0.78

0.82

0.90

0.95

1.16

1.64

2.58

4.38

6.45

9.78

11.04

19.71

1850-1900

1875

0.09

0.52

0.57

0.65

0.74

0.77

0.80

0.88

0.92

1.11

1.57

2.49

4.25

6.28

9.58

10.80

19.44

1900-1950

1925

0.09

0.52

0.56

0.65

0.73

0.77

0.79

0.86

0.88

1.07

1.51

2.41

4.13

6.12

9.38

10.57

19.19

1950-2000

1975

0.09

0.52

0.56

0.65

0.73

0.76

0.78

0.84

0.85

1.02

1.46

2.33

4.01

5.97

9.20

10.35

18.94

2000-2050

2025

0.09

0.52

0.56

0.65

0.73

0.75

0.76

0.82

0.83

0.98

1.40

2.25

3.90

5.83

9.02

10.14

18.71

2050-2100

2075

0.09

0.52

0.56

0.64

0.72

0.74

0.75

0.80

0.80

0.95

1.35

2.18

3.79

5.69

8.85

9.94

18.48

2100-2150

2125

0.09

0.52

0.56

0.64

0.72

0.74

0.74

0.78

0.77

0.91

1.30

2.11

3.69

5.56

8.69

9.75

18.27

2150-2200

2175

0.09

0.52

0.56

0.64

0.72

0.73

0.73

0.76

0.75

0.88

1.25

2.05

3.60

5.44

8.54

9.57

18.06

2200-2250

2225

0.09

0.51

0.56

0.64

0.71

0.72

0.72

0.75

0.73

0.84

1.21

1.99

3.51

5.32

8.39

9.40

17.87

2250-2300

2275

0.09

0.51

0.55

0.63

0.71

0.72

0.71

0.73

0.70

0.81

1.16

1.93

3.42

5.21

8.25

9.24

17.68

2300-2350

2325

0.09

0.51

0.55

0.63

0.71

0.71

0.70

0.72

0.68

0.78

1.12

1.87

3.34

5.10

8.11

9.08

17.49

2350-2400

2375

0.09

0.51

0.55

0.63

0.70

0.71

0.69

0.70

0.66

0.75

1.08

1.81

3.26

5.00

7.99

8.93

17.32

2400-2450

2425

0.09

0.51

0.55

0.63

0.70

0.70

0.68

0.69

0.64

0.72

1.05

1.76

3.18

4.90

7.86

8.78

17.15

2450-2500

2475

0.09

0.51

0.55

0.63

0.70

0.70

0.67

0.68

0.62

0.70

1.01

1.71

3.11

4.80

7.74

8.64

16.99

Note: Cells shaded in grey show the extrapolated traffic volume range.

265 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 1 of 4-lane freeway continued. Hourly traffic volume range

Vehicles per hour

2-50 50-100

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

25

0.11

1.87

4.28

6.51

12.61

26.72

39.85

66.23

100.18

152.74

202.75

262.51

345.46

429.30

446.34

558.35

507.76

75

0.11

1.00

1.80

2.61

4.63

9.25

13.65

22.15

33.09

49.91

66.36

86.59

115.24

144.30

153.95

190.70

185.15

100-150

125

0.10

0.81

1.28

1.79

2.97

5.63

8.23

13.06

19.27

28.75

38.29

50.36

67.78

85.51

93.48

114.74

118.08

150-200

175

0.10

0.73

1.07

1.46

2.30

4.16

6.02

9.35

13.62

20.09

26.80

35.53

48.37

61.48

68.80

83.72

90.76

200-250

225

0.10

0.69

0.95

1.27

1.92

3.34

4.79

7.28

10.48

15.29

20.43

27.30

37.59

48.12

55.06

66.47

75.53

250-300

275

0.10

0.66

0.87

1.15

1.68

2.81

4.01

5.97

8.48

12.23

16.37

22.07

30.73

39.62

46.31

55.48

65.79

300-350

325

0.10

0.64

0.82

1.07

1.51

2.45

3.46

5.06

7.10

10.12

13.57

18.45

25.99

33.73

40.24

47.86

59.01

350-400

375

0.10

0.63

0.78

1.01

1.39

2.18

3.07

4.39

6.09

8.57

11.52

15.79

22.50

29.41

35.78

42.26

54.01

400-450

425

0.10

0.61

0.74

0.96

1.29

1.98

2.76

3.88

5.32

7.39

9.95

13.77

19.84

26.11

32.36

37.97

50.15

450-500

475

0.10

0.60

0.72

0.92

1.21

1.82

2.52

3.48

4.71

6.46

8.71

12.17

17.74

23.50

29.65

34.58

47.09

500-550

525

0.10

0.59

0.70

0.89

1.15

1.68

2.32

3.15

4.21

5.71

7.72

10.88

16.04

21.38

27.45

31.83

44.58

550-600

575

0.10

0.58

0.68

0.86

1.10

1.58

2.16

2.88

3.81

5.09

6.89

9.81

14.63

19.63

25.62

29.55

42.49

600-650

625

0.10

0.58

0.66

0.84

1.06

1.48

2.02

2.65

3.46

4.57

6.20

8.91

13.45

18.17

24.09

27.63

40.72

650-700

675

0.10

0.57

0.65

0.82

1.02

1.40

1.90

2.46

3.17

4.13

5.62

8.15

12.44

16.91

22.77

26.00

39.19

700-750

725

0.10

0.56

0.64

0.80

0.98

1.34

1.80

2.30

2.92

3.75

5.11

7.50

11.58

15.83

21.63

24.58

37.86

750-800

775

0.10

0.56

0.63

0.78

0.96

1.28

1.71

2.15

2.71

3.42

4.67

6.93

10.82

14.89

20.64

23.34

36.69

800-850

825

0.10

0.55

0.62

0.77

0.93

1.22

1.64

2.02

2.52

3.13

4.29

6.43

10.16

14.06

19.76

22.25

35.65

850-900

875

0.09

0.55

0.61

0.76

0.91

1.18

1.57

1.91

2.35

2.87

3.95

5.99

9.57

13.32

18.98

21.28

34.71

900-950

925

0.09

0.54

0.60

0.74

0.88

1.13

1.51

1.81

2.20

2.65

3.65

5.59

9.05

12.67

18.28

20.41

33.87

950-1000

975

0.09

0.54

0.59

0.73

0.87

1.10

1.45

1.72

2.06

2.44

3.38

5.24

8.58

12.08

17.64

19.63

33.10

1000-1050

1025

0.09

0.54

0.59

0.72

0.85

1.06

1.40

1.64

1.94

2.26

3.14

4.92

8.15

11.55

17.07

18.92

32.40

1050-1100

1075

0.09

0.53

0.58

0.71

0.83

1.03

1.35

1.56

1.83

2.10

2.92

4.64

7.77

11.06

16.55

18.28

31.75

1100-1150

1125

0.09

0.53

0.57

0.70

0.82

1.00

1.31

1.50

1.73

1.95

2.72

4.38

7.42

10.62

16.07

17.69

31.15

1150-1200

1175

0.09

0.52

0.57

0.70

0.80

0.98

1.27

1.43

1.64

1.81

2.54

4.14

7.10

10.22

15.63

17.15

30.60

1200-1250

1225

0.09

0.52

0.56

0.69

0.79

0.95

1.24

1.38

1.56

1.69

2.37

3.92

6.80

9.85

15.22

16.65

30.08

Note: Cells shaded in grey show the extrapolated traffic volume range.

266

Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 1 of 4-lane freeway continued. Hourly traffic volume range

Vehicles per hour

1250-1300 1300-1350

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1275

0.09

0.52

0.56

0.68

0.78

0.93

1.20

1.33

1.48

1.57

2.22

3.72

6.53

9.50

14.84

16.19

29.60

1325

0.09

0.51

0.55

0.67

0.77

0.91

1.17

1.28

1.41

1.47

2.08

3.53

6.28

9.19

14.49

15.76

29.14

1350-1400

1375

0.09

0.51

0.55

0.67

0.76

0.89

1.14

1.23

1.34

1.37

1.95

3.36

6.05

8.89

14.16

15.36

28.72

1400-1450

1425

0.09

0.51

0.54

0.66

0.75

0.87

1.12

1.19

1.28

1.28

1.83

3.20

5.83

8.62

13.86

14.98

28.31

1450-1500

1475

0.09

0.50

0.54

0.65

0.74

0.85

1.09

1.15

1.22

1.20

1.72

3.05

5.63

8.36

13.57

14.63

27.93

1500-1550

1525

0.09

0.50

0.53

0.65

0.73

0.83

1.07

1.12

1.17

1.12

1.62

2.92

5.44

8.12

13.30

14.31

27.57

1550-1600

1575

0.09

0.50

0.53

0.64

0.72

0.82

1.04

1.08

1.12

1.05

1.52

2.79

5.27

7.90

13.04

14.00

27.22

1600-1650

1625

0.09

0.49

0.52

0.64

0.71

0.80

1.02

1.05

1.08

0.98

1.43

2.67

5.10

7.69

12.80

13.71

26.90

1650-1700

1675

0.09

0.49

0.52

0.63

0.70

0.79

1.00

1.02

1.03

0.92

1.34

2.56

4.95

7.49

12.58

13.43

26.58

1700-1750

1725

0.09

0.49

0.52

0.63

0.69

0.78

0.99

0.99

0.99

0.86

1.27

2.45

4.80

7.30

12.36

13.17

26.28

1750-1800

1775

0.09

0.49

0.51

0.62

0.69

0.77

0.97

0.96

0.95

0.80

1.19

2.35

4.66

7.12

12.15

12.92

26.00

1800-1850

1825

0.09

0.48

0.51

0.62

0.68

0.75

0.95

0.94

0.92

0.75

1.12

2.26

4.53

6.95

11.96

12.69

25.72

1850-1900

1875

0.09

0.48

0.50

0.61

0.67

0.74

0.93

0.91

0.88

0.70

1.06

2.17

4.41

6.79

11.77

12.47

25.45

1900-1950

1925

0.09

0.48

0.50

0.61

0.67

0.73

0.92

0.89

0.85

0.65

0.99

2.08

4.29

6.64

11.59

12.25

25.20

1950-2000

1975

0.08

0.48

0.50

0.60

0.66

0.72

0.90

0.87

0.82

0.61

0.94

2.00

4.18

6.50

11.42

12.05

24.95

2000-2050

2025

0.08

0.47

0.49

0.60

0.65

0.71

0.89

0.85

0.79

0.57

0.88

1.93

4.08

6.36

11.26

11.86

24.72

2050-2100

2075

0.08

0.47

0.49

0.60

0.65

0.70

0.88

0.83

0.77

0.53

0.83

1.86

3.98

6.23

11.11

11.67

24.49

2100-2150

2125

0.08

0.47

0.49

0.59

0.64

0.69

0.86

0.81

0.74

0.49

0.78

1.79

3.88

6.11

10.96

11.49

24.27

2150-2200

2175

0.08

0.47

0.49

0.59

0.64

0.68

0.85

0.79

0.71

0.46

0.73

1.73

3.79

5.99

10.81

11.32

24.05

2200-2250

2225

0.08

0.46

0.48

0.58

0.63

0.68

0.84

0.78

0.69

0.42

0.69

1.66

3.70

5.87

10.67

11.16

23.84

2250-2300

2275

0.08

0.46

0.48

0.58

0.63

0.67

0.83

0.76

0.67

0.39

0.64

1.61

3.62

5.76

10.54

11.00

23.64

2300-2350

2325

0.08

0.46

0.48

0.58

0.62

0.66

0.82

0.74

0.65

0.36

0.60

1.55

3.54

5.66

10.41

10.85

23.44

2350-2400

2375

0.08

0.46

0.47

0.57

0.62

0.65

0.81

0.73

0.63

0.33

0.57

1.50

3.46

5.56

10.29

10.71

23.25

2400-2450

2425

0.08

0.45

0.47

0.57

0.61

0.64

0.80

0.72

0.61

0.31

0.53

1.45

3.39

5.46

10.17

10.57

23.07

2450-2500

2475

0.08

0.45

0.47

0.57

0.61

0.64

0.79

0.70

0.59

0.28

0.49

1.40

3.32

5.37

10.05

10.43

22.89

Note: Cells shaded in grey show the extrapolated traffic volume range.

267 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 2 of 4-lane freeway continued. Hourly traffic volume range

Vehicles per hour

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

2-50

25

0.11

3.61

4.26

10.13

16.53

33.35

51.90

76.53

102.76

136.64

177.46

239.61

344.64

410.91

475.38

593.00

537.05

50-100

75

0.11

1.54

1.81

3.77

5.92

11.46

17.57

25.62

34.26

45.40

58.90

79.38

113.87

136.51

159.28

196.87

187.56

100-150

125

0.11

1.12

1.32

2.49

3.79

7.08

10.70

15.44

20.56

27.16

35.19

47.34

67.73

81.63

96.05

117.65

117.59

150-200

175

0.11

0.94

1.10

1.94

2.88

5.20

7.76

11.08

14.69

19.34

25.03

33.61

47.96

58.11

68.95

83.70

87.54

200-250

225

0.11

0.84

0.98

1.64

2.37

4.15

6.12

8.65

11.43

14.99

19.39

25.98

36.98

45.05

53.88

64.85

70.81

250-300

275

0.11

0.77

0.90

1.44

2.05

3.49

5.08

7.11

9.36

12.23

15.80

21.13

30.00

36.74

44.30

52.85

60.13

300-350

325

0.10

0.72

0.85

1.30

1.82

3.02

4.35

6.04

7.92

10.32

13.32

17.78

25.16

30.99

37.66

44.55

52.71

350-400

375

0.10

0.69

0.81

1.20

1.65

2.68

3.82

5.25

6.86

8.92

11.50

15.32

21.63

26.77

32.78

38.46

47.24

400-450

425

0.10

0.66

0.77

1.12

1.52

2.42

3.42

4.65

6.06

7.84

10.10

13.44

18.92

23.55

29.06

33.81

43.04

450-500

475

0.10

0.64

0.75

1.06

1.42

2.22

3.10

4.18

5.42

7.00

9.00

11.95

16.79

21.00

26.11

30.14

39.71

500-550

525

0.10

0.62

0.72

1.00

1.33

2.05

2.83

3.80

4.91

6.31

8.12

10.75

15.06

18.95

23.73

27.17

36.99

550-600

575

0.10

0.60

0.70

0.96

1.26

1.91

2.62

3.48

4.48

5.75

7.38

9.76

13.64

17.25

21.76

24.72

34.73

600-650

625

0.10

0.59

0.69

0.92

1.21

1.79

2.44

3.21

4.12

5.27

6.77

8.93

12.45

15.82

20.10

22.66

32.83

650-700

675

0.10

0.58

0.67

0.89

1.15

1.69

2.28

2.98

3.82

4.87

6.24

8.23

11.43

14.60

18.69

20.91

31.19

700-750

725

0.10

0.56

0.66

0.86

1.11

1.60

2.15

2.79

3.56

4.52

5.79

7.62

10.56

13.56

17.47

19.40

29.77

750-800

775

0.10

0.55

0.65

0.84

1.07

1.53

2.03

2.62

3.33

4.22

5.40

7.09

9.79

12.65

16.40

18.08

28.52

800-850

825

0.10

0.55

0.64

0.82

1.04

1.46

1.93

2.47

3.13

3.95

5.05

6.62

9.13

11.84

15.47

16.93

27.42

850-900

875

0.10

0.54

0.63

0.80

1.00

1.40

1.84

2.33

2.95

3.72

4.75

6.21

8.54

11.14

14.64

15.91

26.43

900-950

925

0.10

0.53

0.62

0.78

0.98

1.35

1.75

2.21

2.79

3.51

4.47

5.84

8.01

10.50

13.90

15.00

25.54

950-1000

975

0.10

0.52

0.61

0.76

0.95

1.30

1.68

2.11

2.65

3.32

4.23

5.51

7.54

9.94

13.24

14.18

24.74

1000-1050

1025

0.09

0.51

0.60

0.75

0.93

1.25

1.61

2.01

2.52

3.15

4.01

5.22

7.12

9.43

12.64

13.44

24.01

1050-1100

1075

0.09

0.51

0.59

0.73

0.90

1.21

1.55

1.92

2.40

2.99

3.81

4.95

6.73

8.96

12.09

12.78

23.34

1100-1150

1125

0.09

0.50

0.59

0.72

0.88

1.18

1.50

1.84

2.30

2.85

3.63

4.71

6.38

8.54

11.59

12.17

22.72

1150-1200

1175

0.09

0.50

0.58

0.71

0.87

1.14

1.45

1.77

2.20

2.73

3.46

4.48

6.07

8.16

11.14

11.62

22.15

1200-1250

1225

0.09

0.49

0.57

0.69

0.85

1.11

1.40

1.70

2.11

2.61

3.31

4.28

5.77

7.80

10.72

11.11

21.62

Note: Cells shaded in grey show the extrapolated traffic volume range.

268 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 2 of 4-lane freeway continued.

Hourly traffic volume range

Vehicles per hour

1250-1300 1300-1350

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

1275

0.09

0.49

0.57

0.68

0.83

1.08

1.36

1.64

2.03

2.50

3.17

4.09

5.51

7.48

10.33

10.64

21.13

1325

0.09

0.48

0.56

0.67

0.82

1.05

1.31

1.58

1.95

2.40

3.04

3.92

5.26

7.18

9.98

10.20

20.68

1350-1400

1375

0.09

0.48

0.56

0.66

0.80

1.03

1.28

1.53

1.88

2.31

2.92

3.75

5.03

6.90

9.64

9.80

20.25

1400-1450

1425

0.09

0.47

0.55

0.65

0.79

1.00

1.24

1.48

1.82

2.22

2.81

3.61

4.82

6.64

9.34

9.43

19.84

1450-1500

1475

0.09

0.47

0.54

0.65

0.78

0.98

1.21

1.43

1.75

2.14

2.70

3.47

4.62

6.40

9.05

9.08

19.46

1500-1550

1525

0.09

0.46

0.54

0.64

0.77

0.96

1.18

1.39

1.70

2.07

2.61

3.34

4.43

6.17

8.78

8.76

19.11

1550-1600

1575

0.09

0.46

0.53

0.63

0.75

0.94

1.15

1.35

1.64

2.00

2.52

3.22

4.26

5.96

8.52

8.46

18.77

1600-1650

1625

0.09

0.45

0.53

0.62

0.74

0.92

1.12

1.31

1.59

1.93

2.43

3.10

4.10

5.77

8.29

8.17

18.45

1650-1700

1675

0.09

0.45

0.53

0.61

0.73

0.91

1.10

1.28

1.55

1.87

2.35

2.99

3.95

5.58

8.06

7.90

18.14

1700-1750

1725

0.09

0.45

0.52

0.61

0.72

0.89

1.07

1.24

1.50

1.81

2.28

2.89

3.81

5.41

7.85

7.65

17.85

1750-1800

1775

0.09

0.44

0.52

0.60

0.71

0.87

1.05

1.21

1.46

1.76

2.21

2.80

3.67

5.24

7.65

7.42

17.58

1800-1850

1825

0.09

0.44

0.51

0.59

0.70

0.86

1.03

1.18

1.42

1.70

2.14

2.71

3.55

5.09

7.46

7.19

17.31

1850-1900

1875

0.08

0.44

0.51

0.59

0.70

0.84

1.01

1.15

1.38

1.65

2.08

2.63

3.43

4.94

7.28

6.98

17.06

1900-1950

1925

0.08

0.43

0.50

0.58

0.69

0.83

0.99

1.12

1.35

1.61

2.02

2.55

3.31

4.80

7.11

6.78

16.82

1950-2000

1975

0.08

0.43

0.50

0.58

0.68

0.82

0.97

1.10

1.31

1.56

1.96

2.47

3.21

4.67

6.95

6.59

16.58

2000-2050

2025

0.08

0.43

0.50

0.57

0.67

0.80

0.95

1.07

1.28

1.52

1.91

2.40

3.11

4.54

6.80

6.41

16.36

2050-2100

2075

0.08

0.42

0.49

0.56

0.66

0.79

0.93

1.05

1.25

1.48

1.85

2.33

3.01

4.42

6.65

6.24

16.15

2100-2150

2125

0.08

0.42

0.49

0.56

0.66

0.78

0.92

1.03

1.22

1.45

1.81

2.27

2.92

4.31

6.51

6.08

15.94

2150-2200

2175

0.08

0.42

0.49

0.55

0.65

0.77

0.90

1.01

1.19

1.41

1.76

2.20

2.83

4.20

6.38

5.92

15.74

2200-2250

2225

0.08

0.41

0.48

0.55

0.64

0.76

0.89

0.99

1.17

1.37

1.71

2.15

2.75

4.10

6.25

5.77

15.55

2250-2300

2275

0.08

0.41

0.48

0.54

0.64

0.75

0.87

0.97

1.14

1.34

1.67

2.09

2.67

4.00

6.13

5.63

15.37

2300-2350

2325

0.08

0.41

0.47

0.54

0.63

0.74

0.86

0.95

1.12

1.31

1.63

2.04

2.59

3.91

6.01

5.50

15.19

2350-2400

2375

0.08

0.40

0.47

0.53

0.62

0.73

0.84

0.93

1.09

1.28

1.59

1.98

2.52

3.82

5.90

5.37

15.01

2400-2450

2425

0.08

0.40

0.47

0.53

0.62

0.72

0.83

0.91

1.07

1.25

1.56

1.94

2.45

3.73

5.79

5.24

14.85

2450-2500

2475

0.08

0.40

0.46

0.53

0.61

0.71

0.82

0.90

1.05

1.22

1.52

1.89

2.39

3.65

5.69

5.13

14.68

Note: Cells shaded in grey show the extrapolated traffic volume range.

99

100

269 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 4 of 4-lane freeway continued. Hourly traffic volume range

Vehicles per hour

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

2-50

25

0.10

0.50

3.08

4.48

11.10

24.83

49.81

73.75

102.56

136.46

181.65

257.03

336.60

417.83

468.54

613.48

569.79

50-100

75

0.10

0.50

1.33

1.86

3.98

8.42

16.37

24.17

33.67

45.01

60.08

84.89

112.47

140.48

159.85

206.67

203.17

100-150

125

0.10

0.49

0.98

1.33

2.56

5.14

9.69

14.26

19.90

26.72

35.77

50.47

67.64

85.00

98.08

125.30

129.76

150-200

175

0.10

0.49

0.82

1.10

1.94

3.73

6.82

10.01

14.00

18.88

25.35

35.72

48.43

61.22

71.60

90.41

98.24

200-250

225

0.10

0.49

0.74

0.98

1.60

2.95

5.23

7.66

10.72

14.53

19.56

27.53

37.75

48.00

56.87

71.03

80.68

250-300

275

0.10

0.48

0.68

0.89

1.39

2.45

4.22

6.16

8.64

11.77

15.88

22.31

30.96

39.59

47.49

58.68

69.47

300-350

325

0.10

0.48

0.65

0.84

1.23

2.11

3.52

5.12

7.20

9.85

13.34

18.71

26.25

33.76

40.99

50.13

61.68

350-400

375

0.10

0.48

0.62

0.79

1.12

1.85

3.01

4.36

6.14

8.45

11.47

16.07

22.80

29.49

36.21

43.86

55.94

400-450

425

0.09

0.48

0.59

0.76

1.04

1.66

2.62

3.78

5.33

7.38

10.04

14.05

20.16

26.22

32.55

39.05

51.53

450-500

475

0.09

0.47

0.57

0.73

0.97

1.51

2.31

3.32

4.70

6.53

8.92

12.45

18.08

23.63

29.66

35.26

48.02

500-550

525

0.09

0.47

0.56

0.71

0.91

1.38

2.06

2.95

4.18

5.85

8.01

11.17

16.39

21.54

27.31

32.18

45.17

550-600

575

0.09

0.47

0.55

0.69

0.87

1.28

1.85

2.65

3.76

5.28

7.25

10.10

15.00

19.80

25.36

29.64

42.80

600-650

625

0.09

0.47

0.53

0.68

0.83

1.19

1.68

2.39

3.40

4.81

6.62

9.21

13.83

18.35

23.73

27.50

40.79

650-700

675

0.09

0.46

0.53

0.66

0.79

1.12

1.53

2.17

3.10

4.40

6.09

8.45

12.83

17.11

22.33

25.68

39.06

700-750

725

0.09

0.46

0.52

0.65

0.76

1.05

1.41

1.99

2.84

4.06

5.62

7.80

11.97

16.04

21.12

24.11

37.57

750-800

775

0.09

0.46

0.51

0.64

0.74

1.00

1.30

1.82

2.61

3.76

5.22

7.23

11.22

15.10

20.06

22.73

36.25

800-850

825

0.09

0.46

0.50

0.63

0.72

0.95

1.20

1.68

2.42

3.49

4.87

6.73

10.56

14.28

19.13

21.52

35.08

850-900

875

0.09

0.46

0.49

0.62

0.70

0.91

1.11

1.56

2.24

3.26

4.56

6.29

9.98

13.55

18.31

20.45

34.04

900-950

925

0.09

0.45

0.49

0.61

0.68

0.87

1.04

1.44

2.08

3.05

4.28

5.89

9.45

12.90

17.57

19.50

33.10

950-1000

975

0.09

0.45

0.48

0.60

0.66

0.83

0.97

1.34

1.94

2.86

4.03

5.54

8.99

12.32

16.90

18.64

32.25

1000-1050

1025

0.09

0.45

0.48

0.59

0.65

0.80

0.91

1.25

1.82

2.69

3.80

5.22

8.56

11.79

16.30

17.86

31.47

1050-1100

1075

0.09

0.45

0.47

0.59

0.63

0.77

0.85

1.17

1.70

2.54

3.60

4.94

8.18

11.31

15.75

17.15

30.76

1100-1150

1125

0.09

0.44

0.47

0.58

0.62

0.74

0.80

1.10

1.60

2.40

3.41

4.67

7.83

10.87

15.24

16.51

30.10

1150-1200

1175

0.09

0.44

0.46

0.57

0.61

0.72

0.76

1.03

1.51

2.27

3.24

4.44

7.51

10.47

14.78

15.91

29.50

1200-1250

1225

0.09

0.44

0.46

0.57

0.60

0.70

0.72

0.97

1.42

2.16

3.09

4.22

7.22

10.10

14.36

15.37

28.94

Note: Cells shaded in grey show the extrapolated traffic volume range.

270 Appendix C: Extrapolated cumulative IAT (in sec) distribution table for lane 4 of 4-lane freeway continued. Hourly traffic volume range

Vehicles per hour

1250-1300

Cumulative Percentage (%) 0

1

2

5

10

20

30

40

50

60

70

80

90

95

98

99

100

1275

0.09

0.44

0.45

0.56

0.58

0.68

0.68

0.91

1.34

2.05

2.95

4.01

6.95

9.76

13.96

14.87

28.41

1300-1350

1325

0.09

0.44

0.45

0.56

0.58

0.66

0.64

0.86

1.27

1.95

2.81

3.83

6.70

9.44

13.60

14.40

27.92

1350-1400

1375

0.09

0.43

0.45

0.55

0.57

0.64

0.61

0.81

1.20

1.86

2.69

3.66

6.46

9.15

13.26

13.97

27.46

1400-1450

1425

0.09

0.43

0.44

0.55

0.56

0.62

0.58

0.77

1.14

1.78

2.58

3.50

6.25

8.88

12.94

13.57

27.03

1450-1500

1475

0.09

0.43

0.44

0.54

0.55

0.61

0.55

0.72

1.08

1.70

2.47

3.35

6.05

8.62

12.64

13.19

26.62

1500-1550

1525

0.09

0.43

0.44

0.54

0.54

0.59

0.52

0.68

1.02

1.63

2.37

3.21

5.86

8.38

12.36

12.84

26.23

1550-1600

1575

0.08

0.43

0.43

0.53

0.53

0.58

0.50

0.65

0.97

1.56

2.28

3.08

5.68

8.16

12.10

12.51

25.87

1600-1650

1625

0.08

0.42

0.43

0.53

0.53

0.57

0.48

0.61

0.93

1.49

2.20

2.96

5.52

7.95

11.85

12.20

25.52

1650-1700

1675

0.08

0.42

0.43

0.53

0.52

0.55

0.45

0.58

0.88

1.43

2.12

2.84

5.36

7.75

11.61

11.90

25.19

1700-1750

1725

0.08

0.42

0.42

0.52

0.51

0.54

0.43

0.55

0.84

1.38

2.04

2.74

5.21

7.57

11.39

11.63

24.87

1750-1800

1775

0.08

0.42

0.42

0.52

0.51

0.53

0.41

0.53

0.80

1.32

1.97

2.64

5.07

7.39

11.18

11.37

24.57

1800-1850

1825

0.08

0.42

0.42

0.52

0.50

0.52

0.40

0.50

0.76

1.27

1.90

2.54

4.94

7.22

10.98

11.12

24.28

1850-1900

1875

0.08

0.41

0.42

0.51

0.50

0.51

0.38

0.47

0.73

1.23

1.84

2.45

4.82

7.06

10.79

10.88

24.01

1900-1950

1925

0.08

0.41

0.41

0.51

0.49

0.50

0.36

0.45

0.70

1.18

1.78

2.37

4.70

6.91

10.61

10.66

23.74

1950-2000

1975

0.08

0.41

0.41

0.51

0.49

0.49

0.35

0.43

0.67

1.14

1.72

2.29

4.59

6.77

10.44

10.45

23.49

2000-2050

2025

0.08

0.41

0.41

0.50

0.48

0.49

0.33

0.41

0.64

1.10

1.66

2.21

4.48

6.63

10.27

10.24

23.24

2050-2100

2075

0.08

0.41

0.41

0.50

0.48

0.48

0.32

0.39

0.61

1.06

1.61

2.14

4.38

6.50

10.11

10.05

23.00

2100-2150

2125

0.08

0.40

0.40

0.50

0.47

0.47

0.31

0.37

0.58

1.02

1.56

2.07

4.28

6.38

9.96

9.86

22.78

2150-2200

2175

0.08

0.40

0.40

0.49

0.47

0.46

0.29

0.35

0.56

0.99

1.52

2.00

4.19

6.26

9.82

9.69

22.55

2200-2250

2225

0.08

0.40

0.40

0.49

0.46

0.45

0.28

0.33

0.53

0.96

1.47

1.94

4.10

6.14

9.68

9.52

22.34

2250-2300

2275

0.08

0.40

0.40

0.49

0.46

0.45

0.27

0.32

0.51

0.93

1.43

1.88

4.02

6.03

9.55

9.36

22.14

2300-2350

2325

0.08

0.40

0.39

0.48

0.46

0.44

0.26

0.30

0.49

0.90

1.39

1.82

3.94

5.93

9.42

9.20

21.94

2350-2400

2375

0.08

0.40

0.39

0.48

0.45

0.43

0.25

0.29

0.47

0.87

1.35

1.77

3.86

5.83

9.29

9.05

21.74

2400-2450

2425

0.08

0.39

0.39

0.48

0.45

0.43

0.24

0.27

0.45

0.84

1.31

1.72

3.79

5.73

9.17

8.91

21.55

2450-2500

2475

0.08

0.39

0.39

0.48

0.44

0.42

0.23

0.26

0.43

0.81

1.28

1.67

3.71

5.64

9.06

8.77

21.37

Note: Cells shaded in grey show the extrapolated traffic volume range.

Loading...