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Unformatted text preview: Modeling Drivers' Acceleration and Lane
Changing Behavior
by Kazi Iftekhar Ahmed B. Sc. Eng. Civil
Bangladesh Univ. of Eng. and Technology BUET, Dhaka, Bangladesh 1991
M.S. in Transportation
Massachusetts Institute of Technology, Cambridge, MA 1996
Submitted to the Department of Civil and Environmental Engineering
in partial ful llment of the requirements for the degree of Doctor of Science in
Transportation Systems and Decision Sciences
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 1999
c Massachusetts Institute of Technology 1999. All rights reserved. Author Certi ed by Certi ed by Accepted by ............................................
Department of Civil and Environmental Engineering
January 8, 1999
............................................
Moshe E. BenAkiva
Professor of Civil and Environmental Engineering
Thesis Supervisor
............................................
Dr. Haris N. Koutsopoulos
Operations Research Analyst
Thesis Supervisor
............................................
Andrew J. Whittle
Chairman, Departmental Committee on Graduate Studies Modeling Drivers' Acceleration and Lane
Changing Behavior
by
Kazi Iftekhar Ahmed
Submitted to the Department of Civil and Environmental Engineering
on January 8, 1999, in partial ful llment of the
requirements for the degree of
Doctor of Science in
Transportation Systems and Decision Sciences Abstract This thesis contributes to the development of microscopic tra c performance models
which includes the acceleration and lane changing models. It enhances the existing
models and develops new ones. Another major contribution of this thesis is the
empirical work, i.e., estimating the models using statistically rigorous methods and
microscopic data collected from real tra c.
The acceleration model de nes two regimes of tra c ow: the car following regime
and the free ow regime. In the car following regime, a driver is assumed to follow his her leader, while in the free ow regime, a driver is assumed to try to attain his her desired speed. A probabilistic model, that is based on a time headway
threshold, is used to determine the regime the driver belongs to. Heterogeneity across
drivers is captured through the headway threshold and reaction time distributions.
The parameters of the car following and free ow acceleration models along with the
headway threshold and reaction time distributions are jointly estimated using the
maximum likelihood estimation method.
The lane changing decision process is modeled as a sequence of three steps: decision to consider a lane change, choice of a target lane, and gap acceptance. Since
acceptable gaps are hard to nd in a heavily congested tra c, a forced merging model
that captures forced lane changing behavior and courtesy yielding is developed. A
discrete choice model framework is used to model the impact of the surrounding tra c
environment and lane con guration on drivers' lane changing decision process.
The models are estimated using actual tra c data collected from Interstate 93 at
the Central Artery, located in downtown Boston, MA, USA. In addition to assessing
the model parameters from statistical and behavioral standpoints, the models are validated using a microscopic tra c simulator. Overall, the empirical results are
encouraging, and demonstrate the e ectiveness of the modeling framework.
Thesis Supervisor: Moshe E. BenAkiva
Title: Professor of Civil and Environmental Engineering
Massachusetts Institute of Technology
Thesis Supervisor: Dr. Haris N. Koutsopoulos
Title: Operations Research Analyst
Volpe National Transportation Systems Center
Cambridge, MA, USA. 4 To
Abbu, Ammu,
my son, Sabih,
and
my wife, Lubna 5 Thesis Committee
Moshe E. BenAkiva Chairman Professor
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology Haris N. Koutsopoulos Operations Research Analyst
Volpe National Transportation Systems Center Ismail Chabini Assistant Professor
Department of Civil and Environmental Engineering
Massachusetts Institute of Technology Mithilesh Jha Research Associate
Center for Transportation Studies
Massachusetts Institute of Technology 6 Acknowledgments
I acknowledge with deep sense of gratitude the guidance, invaluable advice, and constant inspiration provided by my supervisors Prof. Moshe Ben Akiva and Dr. Haris
Koutsopoulos during the course of my studies. I feel privileged to get the opportunity
to work with them for the last ve years. I have learned a lot from them during the
course of this research.
I am grateful to the other members of my dissertation committeeProf. Ismail
Chabini and Dr. Mithilesh Jha, for their advice, feed back, and inspiration during
the course of this research.
My special thanks goes to the following individuals without whose contribution
this thesis could not be completed: Dr. Qi Yang, Dr. Kalidas Ashok, Prof. Rabi
Mishalani, Prof. Michel Bierlaire, Alan Chachich, Dave Cuneo, Masroor Hasan, Dr.
Owen Chen, Russel Spieler, Tania Amin, Khwaja Ehsan, Shahnaz Islam, and Prof.
Sha qul Islam.
I am also thankful to the CA T project at the ITS Research Program for nancially supporting my ve years of studies at MIT.
I would like to thank my friends, fellow students, and administrative sta at
the CEE Department, CTS, and ITS O ce, that made my life at MIT an enjoyable
experience, especially, Adriana, Amalia, Andras, Atul, Bruno, Cheryl, Chris, Cynthia,
Denise, Didier, Dinesh, Dale, Deiki, Dong, Hari, Hong, Je , Jessei, John, Jon, Joan,
Juli, Krishna, Lisa, Mark, Masih, Nagi, Niranjan, Pat, Paula, Peter, Prodyut Da,
Shenoi, Scott, Sreeram, Sridevi, Sudhir, Susan, Tomer, Winston, and Yan.
Thanks are due to fellow Bangladeshis Adnan, Fahria and Zeeshan, Minu and
Monjur, Oni and Arif, Rima, Rita and Mukul, Rumi and Saquib, Sabah and Mahmood, and Shampa and Sabet, for their friendship and support.
Finally, I wish I knew a better way to express my indebtedness to my wife, Lubna,
my three year old son, Sabih, for their unconditional support, endless love, to my
parents for their encouragement and inspiration throughout my life that helped me
outgrow again and again.
7 Contents
1 Introduction
1.1
1.2
1.3
1.4
1.5 The Problem . . . .
Motivation . . . . . .
Thesis Objectives . .
Thesis Contributions
Thesis Outline . . . . .
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. 2 Literature Review 2.1 Acceleration Models . . . . . . . . . . . . . . .
2.1.1 Car Following Models . . . . . . . . . .
2.1.2 General Acceleration Models . . . . . . .
2.1.3 Estimation of the Brake Reaction Time .
2.2 Lane Changing Models . . . . . . . . . . . . . .
2.2.1 Gap Acceptance Models . . . . . . . . .
2.3 Summary . . . . . . . . . . . . . . . . . . . . . 3 The Acceleration Model 3.1 Introduction . . . . . . . . . . . . . . . . . .
3.2 The Acceleration Model . . . . . . . . . . .
3.2.1 The Car Following Model . . . . . .
3.2.2 The Free Flow Acceleration Model .
3.2.3 The Headway Threshold Distribution
3.2.4 The Reaction Time Distribution . . .
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. 18
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57 3.3 Likelihood Function Formulation . . . . . . . . . . . . . . . . . . . .
3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 The Lane Changing Model 4.1 Introduction . . . . . . . . . . . . . . . .
4.2 The Lane Changing Model . . . . . . . .
4.2.1 Conceptual Framework . . . . . .
4.2.2 Model Formulation . . . . . . . .
4.2.3 Likelihood Function Formulation
4.2.4 Discussions . . . . . . . . . . . .
4.3 The Forced Merging Model . . . . . . . .
4.3.1 Conceptual Framework . . . . . .
4.3.2 Model Formulation . . . . . . . .
4.3.3 Likelihood Function Formulation
4.3.4 Discussion . . . . . . . . . . . . .
4.4 Conclusions . . . . . . . . . . . . . . . . .
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. 5 Data Requirements for Estimating Driver Behavior Models .
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. 59
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84 85 5.1 Methodology for Estimating Instantaneous Speed and Acceleration
from Discrete Trajectory Data . . . . . . . . . . . . . . . . . . . . . . 86
5.1.1 The Local Regression Procedure . . . . . . . . . . . . . . . . . 87
5.2 Data Collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5.2.1 Description of the Data Collection Site . . . . . . . . . . . . . 91
5.2.2 Video Processing Software . . . . . . . . . . . . . . . . . . . . 93
5.2.3 Processing the Tra c Data . . . . . . . . . . . . . . . . . . . 94
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6 Estimation Results 107 6.1 Estimation Results of the Acceleration Model . . . . . . . . . . . . . 107
6.1.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Estimation Results of the Lane Changing Model . . . . . . . . . . . . 121
9 6.2.1 Estimation Results of the Discretionary Lane Changing Model
6.2.2 Estimation Results of the Mandatory Lane Changing Model .
6.2.3 Estimation Results of the Forced Merging Model . . . . . . .
6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Model Validation Using a Microscopic Tra c Simulator
7.1 MITSIM: a Microscopic Tra c Simulator . .
7.1.1 The Acceleration Model . . . . . . .
7.1.2 The Lane Changing Model . . . . . .
7.2 Validation Methodology . . . . . . . . . . .
7.2.1 Number of Replications . . . . . . .
7.2.2 Measures of Goodness of t . . . . .
7.3 Case Study . . . . . . . . . . . . . . . . . .
7.3.1 The Network . . . . . . . . . . . . .
7.3.2 Tra c Data . . . . . . . . . . . . . .
7.3.3 O D Estimation from Tra c Counts
7.3.4 MITSIM Modi cations . . . . . . . .
7.3.5 Experimental Design . . . . . . . . .
7.3.6 Validation Results . . . . . . . . . .
7.4 Conclusions . . . . . . . . . . . . . . . . . . 8 Conclusions and Future Research Directions
8.1 Summary of Research . . . . . . . . .
8.1.1 The Acceleration Model . . .
8.1.2 The Lane Changing Model . .
8.1.3 Validation by Microsimulation
8.2 Contributions . . . . . . . . . . . . .
8.3 Future Research Directions . . . . . .
8.3.1 Modeling . . . . . . . . . . .
8.3.2 Estimation and Validation . .
8.4 Conclusion . . . . . . . . . . . . . . .
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. 121
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180 A Speci cation of the Random Utility Model Appropriate for Panel
Data
181
B Calibration of the Simulation Model Parameters 183 Bibliography 185 11 List of Figures
21 The subject and the front vehicle. . . . . . . . . . . . . . . . . . . . .
22 De nition of reaction time corresponding to the four actions source:
Ozaki, 1993. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23 The subject, lead, lag, and front vehicles, and the lead and lag gaps. . 25 31 The subject and the front vehicle. . . . . . . . . . . . . . . . . . . . .
32 Impact of the relative speed on drivers' acceleration decision. . . . . . 47
50 41 The lane changing model structure. . . . . . . . . . . . . . . . . . . .
42 The subject, lead, lag, and front vehicles, and the lead and lag gaps. .
43 The lane changing decision tree for a driver driving in a two lane roadway and possible states of the driver. . . . . . . . . . . . . . . . . . .
44 De nition of the adjacent gap. . . . . . . . . . . . . . . . . . . . . . .
45 The forced merging model structure. . . . . . . . . . . . . . . . . . .
46 Initial state of the driver for the forced merging model for di erent cases.
F
47 De nition of n M t for the forced merging model. . . . . . . . . . . . 65
69 51 An example of estimation of instantaneous speed and acceleration from
discrete position measurements. . . . . . . . . . . . . . . . . . . . . .
52 The weight function and the tted curve for an observation at time
period 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53 Schematic diagram of the I 93 southbound data collection site gure
not drawn to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54 Flow, density, and average speed of the I 93 southbound trajectory data.
12 32
40 74
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95 55 Histograms of the absolute values of the position estimation error using
di erent window sizes. . . . . . . . . . . . . . . . . . . . . . . . . . .
56 Estimated speed and acceleration pro les using di erent window sizes.
57 Examples of curve tting by local regression. . . . . . . . . . . . . . .
58 Histograms of the acceleration, subject speed, relative speed, time and
space headway, and density in the data used for estimating the acceleration model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59 The subject and the front, lead, and lag vehicles. . . . . . . . . . . .
510 The subject, lead, lag, and front vehicles, and the lead and lag gaps. .
61 The likelihood function as a function of . . . . . . . . . . . . . . . .
62 Sensitivity of di erent factors on the car following acceleration and
deceleration decisions. . . . . . . . . . . . . . . . . . . . . . . . . . .
63 Comparison between the car following acceleration and deceleration
estimated in this thesis with those obtained by Subramanian 1996. .
64 The headway threshold distribution and the probability of car following
as a function of time headway. . . . . . . . . . . . . . . . . . . . . . .
65 Comparison between the estimated mean headway threshold and the
61 meters threshold suggested by Herman and Potts 1961. . . . . .
66 The probability density function and the cumulative distribution function of the reaction time. . . . . . . . . . . . . . . . . . . . . . . . . .
67 Schematic diagram of the I 93 southbound data collection site gure
not drawn to scale. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68 The decision tree for a driver considering a discretionary lane change
with the current and the left lanes as choice set. . . . . . . . . . . . .
69 The subject and the front, lead, and lag vehicles. . . . . . . . . . . .
610 The estimated probability of acceptance of gaps that were acceptable
and merging were completed. . . . . . . . . . . . . . . . . . . . . . .
611 The median lead and lag critical gaps for discretionary lane change as
a function of relative speed. . . . . . . . . . . . . . . . . . . . . . . .
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132 612 The decision tree for a driver merging from an on ramp to the adjacent
mainline lane. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
613 The subject, lead, lag, and front vehicles, and the lead and lag gaps. .
614 The probability of responding to MLC as a function of delay. . . . .
615 The estimated probability of acceptance of gaps that were acceptable
and merging were completed. . . . . . . . . . . . . . . . . . . . . . .
616 The mean lag critical gap for mandatory lane change as a function of
lag relative speed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
617 Comparison between the estimated critical gap lengths under DLC and
MLC situations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
618 Remaining distance versus explanatory variable remaining distance impact, the utility function, and the estimated probability of being in
state M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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140 143 The network used in the validation exercise. . . . . . . . . . . . . . . 157
Schematic diagram of the on ramp and Storrow Drive merging area. . 158
Flow of tra c entering the network. . . . . . . . . . . . . . . . . . . . 160
O D estimation from tra c counts for the case study. . . . . . . . . . 161
Comparison of average speeds obtained from di erent versions of MITSIM
for p = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Comparison of average speeds obtained from di erent versions of MITSIM for p = 85. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
Comparison of average speeds obtained from di erent versions of MITSIM
for p = 70. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Comparison of the real tra c counts with those obtained from di erent
versions of MITSIM for p = 100. . . . . . . . . . . . . . . . . . . . 170
Comparison of the real tra c counts with those obtained from di erent
versions of MITSIM for p = 85. . . . . . . . . . . . . . . . . . . . . 171
Comparison of the real tra c counts with those obtained from di erent
versions of MITSIM for p = 70. . . . . . . . . . . . . . . . . . . . . 172
14 B1 Model parameter calibration approach. . . . . . . . . . . . . . . . . . 183 15 List of Tables
2.1 Estimation results of the model developed by Gazis et al. 1959. . .
2.2 Estimation results of the GM Model by May and Keller 1967. . . .
2.3 Estimation results of the GM Model by Subramanian 1996. . . . . . 28
31
37 4.1 Possible decision state sequences of observing a lane change by forced
merging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.1 Description of the collected tra c video. . . . . . . . . . . . . . . . .
5.2 Statistics of the data used for estimating the acceleration model. . . .
5.3 Statistics of the discretionary lane changing model data corresponding
to the gaps that the drivers merged into. . . . . . . . . . . . . . . . .
5.4 Statistics of the mandatory lane changing model data corresponding
to the gaps that the drivers merged into. . . . . . . . . . . . . . . . .
5.5 Statistics of the data used for estimating the forced merging model. .
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8 93
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103
104
105 Estimated likelihood function for di erent values of h ; h ; and max .108
min max
Estimation results of the acceleration model. . . . . . . . . . . . . . . 110
Estimation results of the acceleration model for = 1. . . . . . . . . . 112
Comparison between the reaction time distribution parameters obtained from di erent sources. . . . . . . . . . . . . . . . . . . . . . . 120
Estimation results of the discretionary lane changing model. . . . . . 125
Estimation results of the discretionary lane changing model. . . . . . 127
Estimation results of the mandatory lane changing model. . . . . . . 134
Estimation results of the mandatory lane changing model. . . . . . . 136
16 6.9 Estimation results of the forced merging model. . . . . . . . . . . . . 141
6.10 Estimation results of the forced merging model. . . . . . . . . . . . . 142
7.1 The cumulative distribution of speed that is added to the posted speed
limit to obtain the desired speed. . . . . . . . . . . . . . . . . . . . . 149
7.2 Maximum acceleration m s2. . . . . . . . . . . . . . . . . . . . . . . 150
7.3 Summary statistics of the comparison of the eld observed counts with
those obtained from di erent versions of MITSIM using three di erent
O D sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 17 Chapter 1
Introduction
1.1 The Problem
Tra c congestion in and around the urban areas of the world is a major problem.
Congestion during peak hours extends for longer periods each day. Congestion adversely a ects mobility, safety, and air quality. These cause direct economic losses
due to delays and accidents, and indirect economic losses due to environmental impact. In most cases, the capacity of the existing roadway systems cannot be increased
by adding additional lanes due to space, resource, or environmental constraints. Potential ways to address the congestion problem are to improve the utilization of the
existing systems through better tra c management and operations strategies, and
improve the geometric design of roads and highways.
Tra c operations in the congested sections of roadways is very complex, since
di erent drivers employ di erent techniques to travel through such sections while
interacting with other drivers. To understand the occurrence of bottlenecks and to
devise solutions for it, a comprehensive analysis of vehicle to vehicle interactions is
essential. This requires the development of tra c theories to explain driver behavior
at the microscopic level, the main elements of which are the acceleration and lane
changing dimensions.
Drivers' acceleration behavior, when they are in the car following regime, has
been studied extensively since the 1950s. In this regime, drivers are assumed to
18 follow their leaders. However, estimation of these models using microscopic data,
for example, speed of a subject and its leader, gap length, acceleration applied by
the subject, collected from real tra c has not received much attention. On the other
hand, researchers started paying attention to the acceleration behavior in the free ow
regime beginning early 1980s. In the free ow regime, drivers are not close to their
leaders and therefore, have the freedom to attain their desired speed. The parameters
of the general acceleration model, that captures drivers' acceleration decision in both
the car following and free ow regimes, have not been estimated.
The principal focus of research in modeling drivers' lane changing behavior has
been on modeling the gap acceptance behavior at stop controlled T intersections.
The gap acceptance phase is a part of the lane changing process. Researchers started
paying attention to the lane changing model as microscopic tra c simulation emerged
as an important tool for studying tra c behavior and developing and evaluating different tra c control and management strategies. However, the existing lane changing
models are rule based and do not explicitly capture variability within driver and between drivers. Furthermore, the model parameters have not been estimated formally.
In this thesis, we present a comprehensive framework for modeling drivers' acceleration and lane changing behavior. This includes enhancing existing models, developing new ones, providing framework for model estimation, and nally, estimating
the models using statistically rigorous method and microscopic data collected from
real tra c. 1.2 Motivation
Research in Intelligent Transportation Systems ITS is being performed to develop
tra c management and operations strategies to deal with problems associated with
congestion. The number of strategies needed to be tested for a transportation system
may be large and eld testing would be prohibitively expensive. For this purpose,
`microscopic tra c simulation' is a suitable tool. An important element of a tra c
simulator is the set of driver behavior models that is used to move vehicles in the
19 network. This includes the acceleration1 and lane changing models. Reliability of
simulation results depends heavily on these underlying driver behavior models.
Near on and o ramps or weaving sections, drivers often change to the lanes
that are connected to their destinations. These areas are potential locations for
bottleneck formation when the fraction of drivers trying to change lanes is high.
Lane changing operations are critical in selecting geometric con guration of such
areas AASHTO 1990. Drivers' lane changing behavior has direct in uence on the
capacity and safety of such areas HCM 1985. Therefore, a detailed understanding
of drivers lane changing behavior is necessary.
Tra c engineers use the mean of the minimum acceptable gap length at intersections to estimate the capacities of and delays at intersections and pedestrian crossings. Therefore, the mean has to be estimated as accurately as possible; this requires
a thorough understanding of the gap acceptance process. Gap acceptance behavior
also a ects the design of the length of an acceleration lane which is an important
design element from capacity and safety perspectives.
Microscopic driver behavior models play a very important role in the analysis of
tra c ow characteristics in the presence of ITS technologies, such as lane use sign,
variable message signs, tra c control, and route guidance. Macroscopic speed ow
density relationship assumes homogeneous speed and density for a given freeway
segment and treats capacity as an exogenous parameter. In the presence of ITS
technologies, these assumptions may not be realistic Yang 1993. Capacity can be
in uenced not only by drivers' acceleration pattern, but also by the number of lane
changes taking place. A better understanding of driver acceleration and lane changing
behavior is, therefore, essential to model the impact of the ITS Technologies on the
tra c ow relationships.
Rear end collision accounted for 2.2 million automobile crashes in 1990, which
was 19 of the total number of crashes in the US in that year NSC 1992. NSC
1992 also reported that nearly half of these crashes were due to drivers following
their leaders too closely. In such cases, drivers are not able to decelerate fast enough
1 Acceleration refers to both acceleration and deceleration unless deceleration is mentioned. 20 when their leaders decelerate at unexpectedly high rates. Studying safety in the
car following situations is, therefore, very important to the design of an Automated
Highway Systems and Intelligent Cruise Control Chen 1996. To evaluate safety in
car following situations, a detailed understanding of drivers car following behavior
and braking reaction time is required.
In conclusion, there is a need for improving the current understanding of drivers'
acceleration and lane changing behavior at a microscopic level. 1.3 Thesis Objectives
The main objective of this thesis is to advance the state of the art in modeling drivers'
acceleration and lane changing behavior. The models need to be estimated using real
driver data and have to be assessed from statistical and behavioral standpoints.
The acceleration model should capture drivers acceleration behavior in both the
car following and free ow regimes. In the car following regime, drivers follow their
leaders and try to match their leaders' speed, whereas, in the free ow regime, they
try to attain their desired speed. The headway threshold, that is used to determine
the regime a driver belongs to, should be modeled as a random variable to capture
variability between drivers. In addition, the reaction time or the time lag of response
to stimulus should be modeled to be sensitive to the tra c conditions. Furthermore,
the sensitivity of di erent factors on the car following acceleration and deceleration
decisions may not be same, di erent set of parameters should be allowed while estimating the models.
Modeling a lane changing decision process is very complex due to its latent nature
and the number of factors a driver considers before reaching a decision. The only
observable part is a successful lane change operation. The exact time at which a driver
decides to change lanes cannot be observed except in a few specialized situations, for
example, turning left right at an intersection. In addition, the in uence of past lane
changes, such as time elapsed since the most recent lane change, on the current lane
changing behavior further complicates the modeling of such a process. Therefore, the
21 modeling e ort should nd a balance between simplicity in modeling and representing
reality. 1.4 Thesis Contributions
This thesis advances the state of the art in modeling drivers' acceleration and lane
changing behavior. It enhances the existing models and develops new ones. Another
major contribution of this thesis is the empirical work, i.e., estimating the models
using statistically rigorous methods and microscopic data collected from real tra c.
More speci cally,
Contribution to the modeling framework:
The car following model, which captures drivers' acceleration behavior
when they are following their leader, is extended by assuming that the
stimulus is a nonlinear function of the lead relative speed and capturing
the impact of tra c conditions ahead of the driver. These are signi cant
improvements over the existing models that restrict the impact of the lead
relative speed the stimulant on the acceleration response to be linear and
do not model the impact of tra c conditions ahead of the driver except
for the position and speed of the leader.
The existing models restricts the lead relative speed the stimulant and
other factors such as subject speed, gap in front of the subject that
a ect the acceleration decision to be observed at the same time. This
corresponds to an assumption that drivers base their decisions on the tra c
environment at the time they were stimulated into action. We relax this
assumption by allowing drivers to update their perception of the tra c
environment during the decision making process.
A headway threshold distribution is introduced that allows any driver behavior to be captured aggressive or conservative. The headway threshold
22 de nes whether a driver is following its leader or trying to attain its desired
speed.
An individual driver speci c reaction time is introduced which is allowed
to be sensitive to the tra c situations under consideration.
A probabilistic lane changing model is developed that captures drivers'
lane changing behavior under both the mandatory and discretionary lane
changing situations. This is a signi cant improvement over the existing
deterministic rule based lane changing models.
The proposed lane changing model allows for di erent gap acceptance
model parameters for mandatory and discretionary lane changing situations. It also captures the variability within driver and between drivers in
the lane changing decision process.
A forced merging model is proposed that captures merging in heavily congested tra c by gap creation either through force or through courtesy
yielding.
Contribution to model estimation:
A methodology to estimate instantaneous speed and acceleration that is
required for model estimation from discrete trajectory data that can be
obtained from real tra c is developed.
All the components of the acceleration model are estimated jointly using
real microscopic tra c data. The component models are the car following
acceleration and deceleration models, the free ow acceleration model, and
the headway threshold and reaction time distributions. Estimation results
demonstrate the robustness of the modeling framework.
Separate car following model parameters under acceleration and deceleration situations are allowed in the estimation. This captures the fact that,
the sensitivity of di erent factors on drivers' acceleration behavior may
not be same under these two situations.
23 Separate gap acceptance models for the mandatory and discretionary lane
changing situations are estimated.
The proposed lane changing model and the forced merging model are estimated using the maximum likelihood estimation method. 1.5 Thesis Outline
In Chapter 2, a literature review of the existing acceleration and lane changing models is presented. The acceleration and the lane changing models are presented in
chapters 3 and 4 respectively. In Chapter 5, data needs of this research is presented.
First, a methodology to estimate instantaneous speed and acceleration from discrete
trajectory data is presented. Then, the data source and the data extracted from this
source to estimate di erent driver behavior models are presented. Estimation results
of all the models described in chapters 3 and 4 are presented in Chapter 6. In Chapter
7, validation of the acceleration model and a part of the lane changing model, using
a microscopic tra c simulator, is presented. Conclusions and directions for future
research are presented in Chapter 8. 24 Chapter 2
Literature Review
In this chapter, a literature review of the acceleration and lane changing models is
presented. Findings from this review are summarized at the end of the chapter. 2.1 Acceleration Models
The models capturing drivers' acceleration behavior can be classi ed as:
Car following models,
General acceleration models.
The car following models capture acceleration behavior in the car following regime.
In this regime, the drivers are close to their leaders and follow their leaders see
Figure 21. The general acceleration models capture acceleration behavior in both
subject front vehicle
or leader space headway Figure 21: The subject and the front vehicle. 25 the car following and free ow regimes. In the free ow regime, drivers are not close
to their leaders and therefore, have the freedom to attain their desired speed.
Drivers' acceleration behavior, when they are in the car following regime, has been
studied extensively since the 1950s. Estimation of these models using microscopic
data, for example, speed of a subject and its leader, gap length, acceleration applied
by the subject, has not received much attention. Simple correlation analysis was used
to estimate the models in most cases.
Researchers started paying attention to the acceleration behavior in the free ow
regime in the early 1980s as microscopic simulation emerged as an important tool for
studying tra c behavior and developing and evaluating di erent tra c control and
management strategies. However, the parameters of a general acceleration model,
that captures drivers' acceleration behavior in both the car following and free ow
regimes, have not been estimated.
Previous research on each of these categories and the estimation of the brake
reaction time is presented next. 2.1.1 Car Following Models
The general form of the car following models developed in the late 1950s is as follows: responsent = sensitivityn t , n stimulusnt , n 2.1 where, t = time of observation,
n = reaction time for driver n,
responsent = acceleration applied at time t.
The reaction time, n, includes the perception time time from the presentation of
the stimulus until the foot starts to move and the foot movement time. The front
26 relative speed1 see Figure 21 is generally considered as the stimulus and sensitivity
is a proportionality factor that may be a function of factors such as subject speed,
space headway.
Chandler et al. 1958 developed the rst car following model that is a simple
linear model. Mathematically, the model can be expressed as ant = Vnfrontt , n 2.2 where, an t =
=
Vnfront t , n =
Vnt , n =
Vnfront t , n = acceleration applied by driver n at time t,
constant, Vnfront t , n , Vnt , n : stimulus,
subject speed at time t , n,
leader or front vehicle speed at time t , n . A driver responds to the stimulus at time t , n by applying acceleration at time
t. The same sensitivity terms are used for both the acceleration and deceleration
situations. They estimated the model using the correlation analysis method and
microscopic car following data. The data was collected from a sample of 8 drivers
driving test vehicles in a two lane two way road in real tra c for 20 to 30 minutes.
For each driver, the data included discrete measurements of the acceleration, speed,
space headway, and relative speed over the time of observation. For di erent values
of and , correlations between the observed and the estimated accelerations were
computed. The values of and that yielded the highest correlation were used as
the estimates of and for each driver. The estimated and averaged over all
samples were 1.5 seconds and 0.37 second,1 respectively.
A major limitation of the above model is the assumption of a constant sensitivity
In this thesis, relative speed with respect to another vehicle is de ned as the speed of that vehicle
less the speed of the subject.
1 27 for all situations. Gazis et al. 1959 address it by incorporating the space headway
see Figure 21 between the two vehicles in the sensitivity term. Their model is as
follows: ant = X t , Vnfront t , n
n
n 2.3 where, Xnt , n denotes the space headway at time t , n . The model was
estimated using microscopic data collected from the car following experiments in the
Holland Tunnel and the Lincoln Tunnel in New York and at the General Motors test
track. The parameters and were estimated for each driver of each data set using
correlation analysis. For each data set, the values of the parameters averaged over all
samples were reported as the estimates. Table 2.1 summarizes the estimation results.
Table 2.1: Estimation results of the model developed by Gazis et al. 1959.
Data
Number
collection site of drivers mph second
GM Test Track
8
27.4
1.5
Holland Tunnel
10
18.3
1.4
Lincoln Tunnel
16
20.3
1.2
The mean reaction time measured at the test track varied from 1.0 to 2.2 seconds.
Edie 1961 pointed out that, the model given by Equation 2.3 su ers from two
limitations. First, from a behavioral standpoint, the follow the leader theory is not
applicable at low densities. Second, the macroscopic speed density relationship derived from Equation 2.3 yields in nite speed as the density approaches zero. De ne, u = speed of a stream of tra c at density k,
c = integration constant,
kj = jam density.
Assuming that tra c is in a steady state and ignoring the reaction time, integrating
28 both sides of Equation 2.3 yields:
Z a dt
u
u
at k = kj ; u = 0
u Z = X V dt
= c + ln X 1
= c + ln k
c = ln! j
k
= ln kj
k 2.4 In this equation, corresponds the stream speed at maximum ow. This equation is
the macroscopic speed density relationship developed by Greenberg 1959. It does
not yield free ow speed at zero density.
Edie addressed the above mentioned limitations by changing the sensitivity term
and the model is as follows: t
ant = Vnt, n2 Vnfront t , n
Xn , n 2.5 Sensitivity is now proportional to the speed and inversely proportional to the square
of the headway. Equation 2.5 can be integrated as was done to obtain the model
given by Equation 2.4 to obtain a model that yields free ow speed as the density
approaches zero. This model performed better than the model proposed by Gazis
et al. 1959 at low densities. However, the stimulus term is still a function of the
front relative speed, which is not realistic at low densities, in particular, when the
headways are high.
Instead of using the sensitivity stimulus formulation to explain the car following
acceleration decision, Newell 1961 suggested the following relationship between the
speed and the headway: Vnt = Gn Xnt , n 29 2.6 where, Gn is a function whose form determines the speci cation of the car following
models that are presented above. Di erent forms of Gn were assumed for the acceleration and deceleration decisions. Although, the model had the advantage of
integrability to obtain di erent macroscopic speed ow density relationships, no attempt was reported to obtain a quantitative result to validate the model.
The car following model developed by Gazis et al. 1961, known as the General
Motors Nonlinear Model, is the most general one. The model is given by:
n
ant = XVtt Vnfront t , n
n ,n 2.7 where, , , and are model parameters. The sensitivity is proportional to the speed
raised to the power and inversely proportional to the headway raised to the power
. The parameter is a constant and the front relative speed is the stimulus. The
models developed earlier by Chandler et al. 1958and Gazis et al. 1959 can be
derived from this model as special cases. It should be mentioned that the macroscopic
ow speed relationship developed by Greenshields 1934 can be derived from the GM
Model by setting = 0 and = 2. No rigorous framework for estimating the model
was provided.
Bexelius 1968 suggested that instead of following only the immediate leader,
drivers in a car following situation also follow the vehicles ahead of the leader. Mathematically, the model is given by: ant = N
X
i=1 i Vnit , n , Vnt , n 2.8 where, i and Vnit , n are the sensitivity and speed associated with the ith front
vehicle and N is the number of drivers. However, the model was not estimated and
validated.
May and Keller 1967 estimated the GM Model Equation 2.7 using a macroscopic relationship between speed and density that was derived by Gazis et al. 1961.
In addition to using integer values of and , May and Keller 1967 also used non
30 integer values and found higher correlation coe cients for the non integer cases. The
estimated parameters are presented in Table 2.2. Since they used a macroscopic
relationship between speed and density, reaction time could not be identi ed.
Table 2.2: Estimation results of the GM Model by May and Keller 1967.
Parameter Estimates with
integer and
1.35 10,4
1.0
3.0
48.7 free speed uf , mph
jam density kj , vpm
1
optimum speed, mph
29.5
optimum density ko, vpm
60.8
maximum ow, vph
1795
macroscopic model
u = uf e,0:5k=ko 2 Estimates with
non integer and
1.33 10,4
0.8
2.8
50.1
220
29.6
61.1
1810 1:8 5
u = uf 1 , kkj Leutzbach 1968 proposed a psycho physical spacing model that addresses two
limitations of the car following models from a behavioral standpoint. First, drivers
do not follow their leaders at large spacings, and second, drivers cannot perceive
small di erences in front relative speeds and therefore, do not react to such di erences. Leutzbach introduced the term perceptual threshold" to de ne a relative
speed threshold which is a function of the space headway. The threshold is smaller
at low space headways and gradually increases with space headway. A driver reacts
to the stimulus, the front relative speed, only when the stimulus exceeds the perceptual threshold. At a certain large space headway, the threshold becomes in nity,
i.e., a driver no longer follows its leader beyond that space headway. An important
nding of his research is that the perceptual threshold for negative relative speed
is smaller than that for positive relative speed. This implies that the sensitivity of
spacing and front relative speed on drivers' acceleration and deceleration decisions
are di erent. Leutzbach, however, did not provide any mathematical formulation of
the proposed model, nor provided any direction as to how the perceptual threshold
can be estimated.
31 Recently, Ozaki 1993 estimated the GM Model Equation 2.7 parameters. He
used regression analysis to estimate a model for drivers' reaction time and correlation
analysis to estimate parameters ; and .
Ozaki listed four actions to identify reaction time. Figure 22 shows the de nition
of reaction time corresponding to these actions. The actions are:
T T
Action D Action A ∆V
time an
time T
Action B T Action C Figure 22: De nition of reaction time corresponding to the four actions source:
Ozaki, 1993. Action A start of deceleration: time elapsed since the relative speed became zero and the subject, who was accelerating at that instant of time, started decelerating; Action B maximum deceleration: time elapsed since the relative speed reached its
minimum value negative and the subject applied the maximum deceleration; Action C start of acceleration: time elapsed since the relative speed became zero and
the subject, who was decelerating at that instant of time, started accelerating;
32 Action D maximum acceleration: time elapsed since the relative speed reached its
maximum value positive and the subject applied the maximum acceleration. These de nitions of reaction time are not consistent with those suggested by earlier
car following model researchers and the Tra c Engineering Literature Gerlough
and Huber 1975. These researchers de ned the reaction time as the summation of
perception and foot movement times. Depending on the deceleration capability of a
vehicle, its driver may start reacting at di erent times. For example, a driver driving a
vehicle with powerful brakes may not decelerate, even after realizing that its leader is
slower, until the driver gets very close. This does not imply that the driver's reaction
time is larger as suggested by Ozaki. He, however, made an important observation:
tra c conditions, such as the headway and the acceleration of the leader, in uence
the reaction time.
To estimate the car following model parameters, he rst identi ed the reaction
time using the de nition of reaction time for di erent actions listed above. Then, the
correlation between the observed acceleration and estimated acceleration obtained
by using the explanatory variables lagged by the reaction time and setting the car
following model parameters to present numbers was calculated for di erent values of
the parameters. The combination that yielded the maximum correlation was reported
as the estimates.
Ozaki assumed a di erent set of parameters for the acceleration and the deceleration decisions; this captures the fact that di erent factors, such as subject speed, front
relative speed, and headway, may not have the same impact on driver's acceleration
and deceleration decisions. The parameters ; and were estimated to be 1.1, 0.2,
and 0.2 respectively for the acceleration model, and 1.1, 0.9, and 1.0 respectively for
the deceleration model.
Aycin and Benekohal 1998 developed a car following model which estimates the
acceleration rate at any instant of time. Acceleration for the next time instant is
then computed by adding the product of the acceleration rate estimate and the time
di erence to the current acceleration. This guarantees continuity in the acceleration
pro le for a given driver. Equations of laws of motion are used to compute the
33 acceleration rate required for a driver to attain its leader's speed while maintaining
a preferred time headway. The preferred time headway is de ned as a headway the
driver wants to maintain under steady state car following conditions. For each driver
in the car following data set that traveled at speeds within 5 ft sec of its leader's
speed, the discrete time headways measurements over time were averaged. Then,
the average was taken as the driver's preferred time headway. The preferred time
headway values ranged from 1.1 to 1.9 seconds with a mean of 1.47 seconds. The
e ect of reaction time is explicitly modeled. According to this model, drivers are
assumed to be in the car following regime if the clear gap see Figure 21 is less
than 250 feet. This rule ignores variability between drivers. The reaction time was
not estimated using a rigorous method. It was assumed to be 80 of the estimated
preferred time headway. 2.1.2 General Acceleration Models
The models presented above apply to the car following regime only. When the headways are large, drivers do not follow their leader, instead they try to attain their desired speeds. Developing an appropriate acceleration model for the free ow regime
is important for microscopic simulation models.
Gipps 1981 developed the rst general car following model that is applicable
to both the car following and free ow regimes. This model calculates a maximum
acceleration for a driver such that the speed would not exceed a desired speed, and
the clear gap would be at least a minimum safe distance. Mechanical limitations
of vehicles were captured by using the parameters maximum acceleration and most
severe deceleration. Equations of laws of motion were used in the above computations.
The parameters of the models were not estimated rigorously and the reaction time
was set arbitrarily for all drivers.
Benekohal and Treiterer 1988 developed a car following simulation model, called
CARSIM, to simulate tra c in both normal and stop and go conditions. The acceleration for a vehicle is calculated for ve di erent situations and the most binding
acceleration is used to update the vehicle's speed and position. These situations are
34 the subject i.e., the following vehicle is moving but has not reached its desired
speed;
the subject has reached its desired speed;
the subject was stopped and starts from a standstill position;
the subject's movement is governed by the car following algorithm in which a
space headway constraint is satis ed; and,
the subject is advancing according to the car following algorithm with a non
collision constraint.
Equations of laws of motion are used in the above computations. In addition, a comfortable and a maximum allowable deceleration are assumed to limit the output from
the acceleration models within a reasonable boundary. The reaction times of drivers
are randomly generated, and shorter reaction times are assigned at higher densities.
No rigorous framework for parameter estimation was presented and the reaction time
distribution parameters were adopted from Johansson and Rumer 1971 which is
presented in Section 2.1.3.
Yang and Koutsopoulos 1996 developed a general acceleration model that is used
in MITSIM, a microscopic tra c simulator. Based on headway, a driver is assigned
to one of the three following regimes:
the emergency regime, if the current headway is less than a lower threshold;
the car following regime, if the current headway is greater than the lower threshold but less than an upper threshold; and nally,
the free ow regime, if the current headway is greater than the upper threshold.
In the emergency regime, a driver applies the necessary deceleration to avoid colliding
with its leader and increase headway. The GM Model Equation 2.7 is used to determine the acceleration rate in the car following regime. Di erent set of parameters are
used for positive and negative relative speed cases. In the free ow regime, a driver
35 tries to attain its desired speed by applying a maximum acceleration if the current
speed is less than the desired speed or a normal deceleration otherwise. The model
parameters were not estimated using eld data.
Subramanian 1996 developed a general acceleration model that captures drivers'
acceleration behavior in both the car following and free ow regimes. A space headway threshold distribution was assumed that determines which regime a driver is in
at any instant of time. In the car following regime, drivers are assumed to follow
their leader, and in the free ow regime, they are assumed to try to attain their desired speed. He, however, estimated only the car following model parameters using
data that was collected in 1983 from a section of Interstate 10 Westbound near Los
Angeles Smith 1985.
His speci cation of the car following model is an extension of the GM Model
Equation 2.7 and is given by: Vn ,
ant = Xtt , n Vnfront t , n + cf t
n
n
n 2.9 where, cf t is the random term associated with driver n at time t. He modeled
n
the reaction time as a random variable to capture the variability within driver and
between drivers. Variables cf t and n are assumed to be distributed normal and
n
truncated lognormal respectively.
He estimated separate models for acceleration and deceleration observations. The
estimation results are presented in Table 2.3. The estimated mean reaction time
was larger than those reported by Johansson and Rumer 1971 and Lerner et al.
1995, and the mean reaction time estimate for the deceleration decision was higher
than that for the acceleration decisiona counter intuitive result. He also estimated
the GM Model using di erent headway thresholds and concluded that the headway
threshold has signi cant impact on the parameter estimates. 36 Table 2.3: Estimation results of the GM Model by Subramanian 1996.
Model for
Model for
acceleration
deceleration
parameter
estimate
estimate
9.21
15.24
1.67
1.09
0.88
1.66
std. dev cf
0.780
0.632
mean , sec.
1.97
2.29
std. dev
1.38
1.42
2 , speed in ft sec, space in feet.
Note: acceleration in ft sec 2.1.3 Estimation of the Brake Reaction Time
In this section, we present the studies that were conducted to obtain the brake reaction
time of drivers driving in real tra c.
Johansson and Rumer 1971 estimated the distribution of the brake reaction time
from a sample of 321 drivers traveling in a real tra c. The subjects were instructed
to apply the brake pedal as soon as they hear a sound. The time elapsed from the
moment the sounds were made to the moment the drivers' brake light turned on were
recorded as the brake reaction time. The brake reaction time varied from 0.4 to 2.7
seconds with a median, mean, and standard deviation of 0.89, 1.01, and 0.37 seconds
respectively and a 90 percentile value of 1.5 seconds. These numbers may be biased
downwards, since the sound, to which the drivers reacted, might have reduced the
perception time, and hence the reaction time.
Recently, Lerner et al. 1995 estimated the reaction time distribution from a
sample of 56 drivers driving in real tra c. To estimate the brake reaction time for
unexpected situations to mimic real driving conditions, subjects were not informed
that they were participating in a brake reaction time study. When a subject reached
the test site at 40 mph speed, a large yellow highway crash barrel was released approximately 200 ft in front of the vehicle. The barrel was chained so that it was held
within the median. The time elapsed since a barrel is released to the instant a driver
37 applies brake was recorded as the driver's reaction time. The brake reaction time
varied from 0.7 to 2.5 seconds with a median, mean, and standard deviation of 1.44,
1.51, and 0.39 seconds respectively. 2.2 Lane Changing Models
In this section, a literature review of the lane changing models is presented followed
by a literature review of the gap acceptance models.
The principal focus of research in modeling drivers' lane changing behavior has
been on modeling the gap acceptance behavior at stop controlled T intersections.
The gap acceptance phase is a part of the lane changing process.
Gipps 1986 presented a lane changing decision model to be used in a microscopic
tra c simulator. The model was designed to cover various urban driving situations
where tra c signals, obstructions, and the presence of heavy vehicles for example,
bus, truck, semi trailer a ect a driver's lane selection decision. Three major factors
were considered in the lane changing decision process: necessity, desirability, and
safety. Di erent driving conditions were examined including the ones where a driver
may face con icting goals. However, di erent goals were prioritized deterministically, and inconsistency and non homogeneity in driver behavior were not modeled.
The terms inconsistency implies that a driver may behave di erently under identical
conditions at di erent times, while the term non homogeneity implies that di erent
drivers behave di erently under identical conditions. The model parameters were not
estimated formally.
CORSIM FHWA 1998 is a microscopic tra c simulator that uses FREESIM to
simulate freeways and NETSIM to simulate urban streets. In CORSIM, a lane change
is classi ed as either mandatory MLC or discretionary DLC. A driver performs
an MLC when the driver must leave the current lane and performs a DLC when the
driver perceives the driving conditions in the target lane to be better, but, a lane
change is not required. The necessity or desirability of changing lanes is determined
by computing a risk factor that is acceptable to a driver which is a function of a
38 driver's position relative to the object that gives rise to the need for a lane change. A
default set of model parameters are provided with the exibility of using user provided
parameters. The gap acceptance behavior is not modeled in a systematic manner.
Minimum gap lengths for di erent situations are listed and all drivers are assumed
to have identical gap acceptance behavior.
Yang and Koutsopoulos 1996 developed a rule based lane changing model that
is applicable only for freeways. Their model is implemented in MITSIM. A lane
change is classi ed as either mandatory MLC or discretionary DLC. Unlike Gipps
1986, they used a probabilistic framework to model drivers' lane change behavior
when they face con icting goals. A driver considers a discretionary lane change only
when the speed of the leader is below a desired speed, and checks neighboring lanes
for opportunities to increase speed. Two parameters, impatience factor and speed
indi erence factor, were used to determine whether the current speed is low enough
and the speeds of the other lanes are high enough to consider a DLC. They also
developed a gap acceptance model that captures the fact that the critical gap length
de ned as the minimum acceptable gap length under an MLC situation is lower
than that under a DLC situation. They pointed out that, for a case of merging into
a tra c parallel to the current lane, a gap is acceptable only when both the lead and
lag gaps are acceptable. However, no formal parameter estimation was done and a
framework to do so was not developed.
Recently, Ahmed et al. 1996 developed a framework for a general lane changing
model that captures lane changing behavior under both the MLC and DLC situations.
Lane change is modeled as a sequence of four steps: decision to consider a lane change,
choice of a target lane, acceptance of gaps in the target lane, and performing the
lane change maneuver. A discrete choice framework is used to model these decision
elements that allows for modeling impact of di erent tra c and roadway environment
on driver behavior. From a model estimation view point, the utilities capturing the
rst and the fourth steps cannot be uniquely identi ed in the absence of any indicator
available to the analyst di erentiating these two steps. They estimated parameters
of the model only for a special case: merging from a freeway on ramp. They used
39 the data collected in 1983 from a site at Interstate 95 northbound near the Baltimore
Washington Parkway Smith 1985.
In this case, it was assumed that drivers have already decided to change to the
adjacent freeway and therefore, the decision process involved acceptance of a gap
and the actual lane change maneuver. Following Yang and Koutsopoulos 1996, a
gap is considered acceptable only when both the lead and lag gaps are acceptable.
Figure 23 shows the de nition of the lead and lag gaps. The lead and lag critical
X total clear gap + vehicle length
lag gap Y lead gap lag vehicle lead vehicle subject front
vehicle X Y Figure 23: The subject, lead, lag, and front vehicles, and the lead and lag gaps.
gap lengths were assumed to be lognormally distributed and whether a lane change
will take place immediately, given the gap is acceptable, was modeled using a binary
logit model.
The estimated lead critical gap for driver n at time t is Gcr;leadt = exp 2:72 , 0:055 n +
n lead t
n 2.10 where, Gcr;leadt = lead critical gap feet,
n
n = driver speci c random term that is constant for a given driver,
assumed distributed standard normal,
lead t = random term that varies across di erent components of a gap
n
for a given driver, across di erent gaps for a given individual,
40 as well as across drivers, lead t N 0; 1:612:
n The estimated lag critical gap for driver n at time t is Gcr;lag t =
n
exp ,9:32 + 0:1170 minVnlag t; 10 + 0:1174 maxVnlag t , 10; 0 +
1
1:57 nstGapt + 1:88 lnLremt + 1:90 n + lnag t
2.11
n
where, Gcr;lag t = lag critical gap feet,
n
Vnlag t = lag vehicle speed subject speed mph,
8
1 if delaynt = 0
1stGap t =
n
: 0 otherwise.
delaynt = time elapsed since MLC conditions apply seconds,
Lremt = remaining distance to the point at which lane change must be
n
completed feet,
lag t N 0; 1:312:
n
The estimated model of changing lanes, given that both the lead and lag gaps are
acceptable, is:
Pn change lanes at timet j gap acc: = 1
2.12
1 + exp1:90 , 0:52 delaynt The gap acceptance model, however, cannot be applied to a case of forced merging or
merging through courtesy yielding. In this case, gaps of acceptable lengths may not
exist due to high congestion level, and in order to merge gaps have to be created. 41 2.2.1 Gap Acceptance Models
Di erent gap acceptance models were developed in the 1960s and 1970s based on the
assumption on the distribution of the critical gap length. Herman and Weiss 1961
assumed the critical gap to be exponentially distributed, Drew et al. 1967 assumed
a lognormal distribution, and Miller 1972 assumed a normal distribution. They,
however, did not capture the e ect of previously rejected gaps on the critical gap.
In general, data collected for estimating gap acceptance models is panel in nature,
i.e., it contains one or more observations from each individual. Di erent observations
from a given sample are likely to be correlated which may introduce bias in the
parameter estimates. Daganzo 1981 used a probit model formulation appropriate
for panel data to estimate the gap acceptance model parameters for drivers merging
from the minor leg of a stop controlled T intersection to the major leg. The critical
gap for driver n at time t is assumed to have the following functional form: Gcr t = Gn + cr t
n
n 2.13 where, Gn = component of critical gap attributable to driver n,
cr t = random term that varies across di erent gaps for a given driver as
n
well as across di erent drivers.
Gn, and cr t are assumed to be mutually independent. Further, he assumed Gn
n
2
N G; G and cr t N 0; 2. The individual speci c random term, Gn, captures
n
the correlation between di erent observations from driver n. The model has the
exibility to incorporate the impact of other factors on a driver's gap acceptance
behavior by varying the mean of the distribution of Gn. However, he had estimability
problems and the estimated critical gap lengths were not guaranteed to be non
negative.
Mahmassani and She 1981 used the data that Daganzo 1981 used and ad42 dressed the estimability problem by ignoring panel data formulation, i.e., they treated
the data as cross sectional data. They assumed the critical gap to be normally distributed. The mean of the critical gap was allowed to be a function of explanatory
variables, a framework that allows for incorporating the impact of di erent factors
on a driver's gap acceptance behavior. The variable number of gaps rejected, capturing the impatience factor, was found to have a signi cant impact on drivers gap
acceptance behavior.
The Highway Capacity Manual HCM 1985, or HCM, uses the mean critical gap
length of drivers at an intersection to estimate the delay at and the capacity of that
intersection. The HCM de ned the critical gap for a two way stop controlled intersection as the median of all acceptable gap lengths. A major limitation of this
de nition is that an observation of a large gap accepted by a driver provides no information about the minimum acceptable gap length. In the revised HCM procedure,
the critical gap is de ned as the largest observed rejected gap length. This de nition
is again awed, since one very conservative driver can greatly increase the estimate.
In addition, Cassidy et al. 1995 listed other de ciencies of this approach. First,
only a subset of the data is used and all accepted gaps shorter than the largest one is
not included in the estimation. Second, inconsistency in driver behavior accepting
a gap smaller than a previously rejected gap is addressed either by discarding or by
modifying the data. However, the bene t of using the HCM de nition of a critical
gap is ease in estimation.
Kita 1993 used a logit model to estimate the gap acceptance model for the case
of merging from a freeway on ramp. The impact of di erent factors on drivers' gap
acceptance behavior was modeled by using a random utility model. Although he used
panel data, he did not use an appropriate panel data formulation. In addition to the
gap length, relative speed of the subject with respect to the mainline vehicles and
the remaining distance of the acceleration lane were found to have impact on drivers'
gap acceptance behavior.
Cassidy et al. 1995 used Kita's approach to model the gap acceptance behavior
at a stop controlled T intersection. They, too, ignored the panel data formulation
43 and found that a gap acceptance function with disaggregate factors have signi cantly
more predictive power than a function that includes only the mean gap length. 2.3 Summary
A summary of the ndings from the literature review is presented below.
Modeling acceleration behavior:
Primary attention of the research has been on modeling drivers' acceleration behavior in the car following regime.
The impact of stimulus the front relative speed on the car following
acceleration was assumed to be linear.
The reaction time was modeled but not estimated rigorously in most cases.
Variability within driver and between drivers were not captured in most
cases.
The headway threshold, that determines whether a driver is in the car
following regime or in the free ow regime, is modeled deterministically in
most cases.
A general acceleration model was proposed by Subramanian 1996. The
model captures acceleration behavior in both the car following and free
ow regimes. It also captures the inconsistency in driver behavior. A
probabilistic framework was used to model variability in headway threshold
and reaction time. However, only the car following part of the model was
estimated.
Modeling lane changing behavior:
Modeling drivers' gap acceptance behavior has been the primary focus of
the research in modeling drivers' lane changing behavior.
44 A majority of the research in modeling drivers' gap acceptance behavior
used panel data. However, model formulation appropriate for panel data
was not used while estimating the parameters.
A model capturing drivers' lane change decision process was developed by
Ahmed et al. 1996. However, the model is not applicable to mandatory
lane changing situations in a heavily congested tra c where gaps of acceptable lengths are hard to nd. The parameters of the discretionary lane
changing model have not been estimated.
The impact of an acceleration decision, which determines drivers speed,
on the lane changing decision is modeled by using speed as an explanatory
variable in the lane changing model.
The acceleration model proposed in this thesis builds on the earlier work by Subramanian 1996 and extends his model. The impact of the stimulus on the car following
acceleration is allowed to be a nonlinear function of the lead relative speed and the
sensitivity term is extended to capture the impact of tra c conditions ahead of the
subject and its leader. In addition, all the components of the acceleration model
are estimated jointly using microscopic data collected from real tra c. On the other
hand, the lane changing model proposed in this thesis extends the model proposed
by Ahmed et al. 1996 to capture merging behavior in heavily congested tra c and
the model is estimated using statistically rigorous methods and real driver data. 45 Chapter 3
The Acceleration Model
In this chapter, a rigorous framework for estimating the parameters of the acceleration
model is presented that builds on the previous work by Subramanian 1996 and
extends it. The proposed model consists of two components: the car following model
and the free ow acceleration model. The car following model is applied when a
driver follows its leader i.e., the vehicle in front. The free ow acceleration model is
applied when a driver tries to attain its desired speed and is not following its leader.
This chapter starts with a presentation of the conceptual framework and specication of the model. Next, the likelihood function that is necessary for estimating
the model is formulated. 3.1 Introduction
Based on a headway threshold, a driver is assumed to be in one of the two following
regimes: the car following regime and the free ow regime. If the current headway
is less than the threshold, the driver is assumed to be in the car following regime
and follow its leader see Figure 31. Speed selection and hence the acceleration
decision is governed by the speed of the leader. Otherwise, the driver is assumed
to be in the free ow regime in which case speed selection is governed by its desired
speed.
The existing car following models for example, Gazis et al. 1961, Subramanian
46 subject front vehicle
or leader space headway Figure 31: The subject and the front vehicle.
1996 restrict the stimulus of the acceleration to be a linear function of the front
relative speed and do not capture the impact of tra c conditions ahead of the driver
except the speed and position of its leader. The front relative speed is de ned as the
speed of the leader less the subject speed. In this chapter, we use the terms front
relative speed and relative speed interchangeably.
In the car following regime, the sensitivity of di erent factors, such as speed,
headway, and front relative speed, on drivers' acceleration decision under acceleration
and deceleration situations may be di erent. For example, consider two cases: one
with a positive relative speed with a certain magnitude and the other with a negative
relative speed with the same magnitude, and all other factors are identical. The
acceleration in case one is likely to be less than the deceleration in absolute terms
in case two due to safety concerns involved.
The model proposed in this thesis relaxes the restriction that the car following
stimulus is a linear function of the front relative speed and captures the impact
of tra c conditions ahead of the subject on the car following sensitivity by using
as explanatory variable the density ahead of the subject. Separate car following
model parameters under acceleration and deceleration situations are allowed in the
estimation.
The model proposed in this thesis, however, does not explicitly capture the impact
of lane changing decisions on the acceleration decision. For example, a driver may
accelerate or decelerate to t into a gap in an adjacent lane. Instead, random terms
are used in all component models that capture the e ect of unobserved factors. This
is left as a subject for further research and is discussed in Chapter 8.
47 3.2 The Acceleration Model
A driver is assumed to be in the car following regime if the headway is less than a
threshold, and in the free ow regime otherwise. Mathematically, the acceleration
model can be expressed as:
8 ant = acf t if hnt , n h
n
n
: aff t otherwise
n 3.1 where, t=
n=
ant =
acf t =
n
aff t =
n
hnt , n =
h =
n time of observation,
reaction time of driver n,
acceleration at time t,
car following acceleration at time t,
free ow acceleration at time t,
time headway1 at time t , n,
unobserved headway threshold for driver n. Reaction time refers to the delay in a driver's response to a stimulus, or the response
lag. It includes both the perception time from the presentation of the stimulus
until the foot starts to move and foot movement times. Since these two cannot be
identi ed uniquely from the observed data, the term reaction time is used to designate
the summation of the two.
We de ne the headway threshold, h , in terms of time headway as opposed to
n
space headway for two reasons. First, previous research for example, Winsum and
1 Time headway is de ned as:
hn t = Xntt ;
Vn Vn t 0 where, Vn t and Xn t denote the subject speed and the space headway see Figure 31 at
time t respectively. In this research, headway is used to designate time headway unless otherwise
mentioned. 48 Heino 1996, Aycin and Benekohal 1998 indicates that drivers maintain certain
time headways independent of speed in a steady state car following situations. And
second, equal space headways have identical acceleration regimes car following versus
free ow although speeds may be very di erent, while the time headway does not
su er from this limitation.
Speci cation of the car following and free ow acceleration models and the distributions of the headway threshold and reaction time are presented next. 3.2.1 The Car Following Model
Since, it is hypothesized that the expected value of the acceleration distribution is
greater than zero when the relative speed is positive, the model corresponding to a
positive relative speed is called the car following acceleration model. Similarly, the
model corresponding to a negative relative speed is called the car following deceleration model. The model can be expressed as follows:
cf;g
acf;g t = s Xn t , n f Vnt , n +
n cf;g t
n 3.2 where, g 2 facc; decg
cf;g
cf
s Xn t , n = sensitivity, a function of Xn t , n,
cf;g
Xn t , n = vector of explanatory variables a ecting the car following
acceleration sensitivity observed at time t , n,
2 0; 1 ; a parameter for sensitivity lag,
f Vnt , n = stimulus, a function of relative speed, Vnt , n, Vnt , n = Vnfront t , n , Vnt , n ;
Vnt , n = subject speed at time t , n ,
Vnfront t , n = front vehicle speed at time t , n ,
49 cf;g t
n = random term associated with the car following acceleration
of driver n at time t. The acceleration or deceleration applied by driver n at time t is proportional
to the stimulus, a function of the front relative speed at time t , n. The reaction
time, n, varies from driver to driver, and therefore, is modeled as a random variable.
The sensitivity term is the proportionality factor, a function of explanatory variables
discussed below observed n seconds earlier. The parameter for the sensitivity lag,
, varies between 0 and 1.
The stimulus term is a function of the relative speed. Figure 32 a shows the
expected e ect of the relative speed on drivers' acceleration decision. At low relative
acc or
dec acc or
dec V
V1 V V2 (a) Effect of relative speed on
acceleration/deceleration (b) linear impact of relative
speed acc or
dec acc or
dec V V (c) nonlinear impact of relative
speed, parameter < 1 (d) nonlinear impact of relative
speed, parameter > 1 Figure 32: Impact of the relative speed on drivers' acceleration decision.
speeds, drivers' acceleration response may not be signi cant as they may not be able
to perceive a small magnitude of the relative speed. For relative speeds beyond a
certain threshold, jV1j, drivers get a better sense of the stimulus and therefore,
acceleration increases at an increasing rate. Beyond another threshold, jV2 j, the
acceleration applied by a driver is limited by the acceleration capacity of the vehicle
50 and hence, acceleration increases at a decreasing rate until it reaches the maximum
acceleration.
The e ect of the relative speed on the car following acceleration discussed above
can be captured by assuming a piecewise nonlinear function of the relative speed of
the following form:
g g g f Vnt , n = V 1nt , n + V 2nt , n + V 3nt , n
1 2 3 3.3 where,
V 1nt , n = minjVnt , nj; jV1j
j j = absolute value,
V 2nt , n = max0; minjVnt , n j , jV1j; jV2j , jV1j
V 3nt , n = max0; jVnt , nj , jV2j
The breakpoints, jV1 j and jV2j, should be chosen such that they are reasonable
from a behavioral standpoint. In order to replicate the impact of the relative speed
on the acceleration as shown by the curve in Figure 32 a, both g and g should
1
2
g should be less than one. In addition, g should be less
be greater than one while 3
1
than g .
2
Figure 32 b shows the linear approximation of the impact of the relative speed
on acceleration which has been used by existing models. This implies the following: f Vnt , n = jVnt , nj 3.4 The functional form given by Equation 3.3 can be simpli ed by allowing only one
parameter: f Vnt , n = jVnt , n jg 3.5 Figures 32 c and d show the e ect of the relative speed on the acceleration using
51 this speci cation for g 1 and g 1 respectively. We apriori expect g to be
less than one for both the acceleration and deceleration due to the existence of a
maximum value for acceleration and deceleration that a driver can apply in reality.
The parameter g can be tested statistically to determine whether it is signi cantly
di erent from one. Note that, g = 1 corresponds to the speci cation given by
Equation 3.4.
The GM Model Equation 2.7 assumed the sensitivity term to be a nonlinear
function of the subject speed at time t and the space headway at time t , n . It
allowed a single set of parameters for both the acceleration and deceleration decisions.
Mathematically, this is given by:
cf
s Xn t , n = Vnt
Xnt , n 3.6 where, ; ; and are constant parameters, and is set to 0 and 1 for speed and
space headway respectively. We extend the GM Model by allowing di erent sets of
parameters for the car following acceleration and deceleration sensitivities, by incorporating the density of tra c as explanatory variable into the sensitivity term, and by
allowing the time at which the explanatory variables are observed to be a parameter
to be estimated:
cf;g
s Xn t , n Vnt , n g k t , g
= X t , g n
n
n
n
g 3.7 where,
g; g; g; g = constant parameters,
knt , n = density of tra c ahead of the subject within
its view at time t , n.
The parameter captures the fact that drivers may update their perception of the
tra c environment during the acceleration decision making process. Restricting
to be equal to one implies that drivers do not update their perception of the tra c
52 environment and react accelerate decelerate based on the tra c conditions at the
time they observe the stimulus. In other words, = 1 implies that the lag for
sensitivity and stimulus are equal, while, 1 implies that lag for sensitivity is
smaller than that for stimulus.
There are apriori expectations regarding the signs of the various parameters. The
constant g in the sensitivity term should be positive and negative for the acceleration
and deceleration models respectively. In the car following regime under acceleration
situations, drivers are likely to apply a lower acceleration at high speeds compared
to low speeds and therefore, the corresponding parameter acc should be negative.
On the other hand, under deceleration situations, drivers are likely to apply a higher
deceleration at high speeds compared to low speeds which implies that dec should
be positive. The sign of the parameter acc can either be negative or positive. Under
acceleration situations, drivers may apply a higher acceleration when space headways
are larger, implying a negative headway parameter, acc i.e., the space headway
should be in the numerator of the sensitivity. However, as the space headway increases, drivers may tend to follow the speed of the lead vehicle less and if this is the
case, acc would be positive. Under deceleration situations, drivers are likely to apply smaller decelerations at larger headways, implying a positive headway parameter,
dec .
Tra c conditions ahead of the subject and its leader are likely to change more
rapidly at high densities than at low densities. Due to this, higher uncertainty is
involved in predicting the position and speed of the leader in the near future. In
addition, high tra c density represents lack of maneuverability compared to low
tra c density for both the subject and its leader. As a result, drivers are expected to
be more conservative at high densities than at low densities. Hence, at high densities
the subject is likely to accelerate at a lower rate, while decelerate at a higher rate.
These imply that, acc and dec are expected to be negative and positive respectively.
The random term captures the e ect of omitted variables. It is assumed to be
independent for di erent decisions of a given driver as well as for di erent drivers.
The correlation between di erent acceleration decisions of a given driver is assumed
53 to be captured through the reaction time and headway threshold distributions. This
implies: cov cf;g t;
n cf;g t
n
cf;g0 t0 n0 N 0;
8
= 2cf;g 2cf;g if g = g 0 ; t = t0 ; n = n0 : 0 otherwise 3.8 3.2.2 The Free Flow Acceleration Model
When the headway is greater than the threshold, the driver has the freedom to attain
its desired speed. Hence, the acceleration applied by a driver in this regime is assumed
to have the following functional form: aff t = ff Vn t , n , Vnt , n +
n f f t
n 3.9 where, ff
Vnt , n
Vnt , n , Vnt , n
f f t
n = constant sensitivity,
= desired speed of the driver,
= stimulus,
= random term associated with the free ow
acceleration of driver n at time t. The desired speed of a driver is de ned as the speed the driver wants to maintain
after considering the speed limit of the section it is traveling, vehicle's mechanical
capability, the e ect of surrounding tra c, the roadway and weather conditions, and
the geometry of the roadway section. The desired speed is assumed to have the
following functional form: 54 DS
Vn t , n = Xn t , n DS 3.10 where,
DS
Xn t , n = vector of explanatory variables a ecting the desired speed DS ,
DS = constant parameters. Replacing the speci cation of Vnt , n in Equation 3.9, the free ow acceleration
model becomes
h DS
aff t = ff Xn t , n
n i
ff
n t , n + n t DS , V 3.11 In this model, the acceleration at time t is assumed to be proportional to the stimulus
the di erence between the driver's desired speed and current speed at time t , n.
The sensitivity term is assumed to be a constant.
If the desired speed is higher than the current speed, drivers are expected to
accelerate and vice versa. The magnitude of the applied acceleration deceleration
depends on the di erence between the current and the desired speeds.
Important explanatory variables a ecting the desired speed of a driver include
geometry of the roadway curvature, grade, lane width, pavement surface quality
roughness, presence of pot holes, weather conditions, the speed limit of the roadway
section, density of tra c ahead of the subject, speed of the vehicles ahead of the
subject which is also a proxy for density and maneuverability, type of the vehicle,
and characteristics of the driver. For example, in a curved road or roadway with
grades particularly upgrade or in a roadway with rough pavement, vehicles tend
to slow down thus the desired speed of drivers reduce even when there is no lead
vehicle. Similarly, drivers often set their desired speed relative to the speed limit of
the roadway section. To estimate models with these site speci c factors, data from
di erent sites is necessary.
55 High density of tra c ahead of a driver within the driver's view, or a lower speed
of the lead vehicle reduces desired speed, as might be expected. And nally, heavy
vehicles for example, bus, truck, semi trailer etc. that have length greater than 9.14
meters or 30 ft AASHTO 1990 have lower acceleration and deceleration capability
and hence respond slowly to free ow conditions.
We further assume that, f f t is normally distributed with zero mean and a
n
variance 2f f , i.e., f f t N 0; 2f f , and f f t is independent of the random
n
n
cf;acct and cf;dec t for a given driver2 .
terms n
n
Note that, since the desired speed of a driver cannot be observed, extending the
free ow acceleration model to have di erent sensitivity under acceleration and deceleration situations cannot be done without increasing the complexity of the current
framework. 3.2.3 The Headway Threshold Distribution
The headway threshold, h , is assumed to be truncated normally distributed with
n
truncation on both sides. This distribution is given by:
8 f h =
n : 1 h , nh hh max
h ,h , hmin ,h if h h h
min
n
max 0 otherwise h h 3.12 where, h; h = constant mean and standard deviation of the untruncated
distribution, h ; h = minimum and maximum value of h parameters to be estimated,
min max
n
= probability density function of a standard normal random variable,
As mentioned above, the correlation between the car following and the free ow acceleration
decisions is assumed to be captured through the reaction time and headway threshold distributions.
2 56 = cumulative distribution function of a standard normal random
variable.
The advantage of using a truncated normal distribution with mean, variance, and
the truncation ends h and h as parameters is that the distribution is not
min
max
restricted to be skewed to a particular direction. For instance, a distribution skewed
to the left implies that, the probability of a driver being aggressive is higher than
that of being conservative, since, an aggressive driver is expected to have a shorter
headway threshold compared to a conservative driver. The above treatment of the
headway distribution is a generalization over Subramanian 1996 who used a shifted
truncated lognormal distribution that restricts the distribution to be skewed to the
left.
Using Equation 3.12, the probability that driver n, who is hnt behind its leader,
is in the car following regime is given by:
Pncar following at time t
= P hnt h
n
8
1 1, =
: 0
hn th,h , hminh,h h , h , max h
min h h , h if hnt h
min
if h
min hn t hmax 3.13 otherwise At very large headways, it is unlikely that a driver would be in a car following regime.
Hence, the corresponding probability is zero for headways greater than h . Simimax
larly, at very low headways it is unlikely that a driver would be in a free ow regime
and the corresponding probability of car following is one for headways less than h .
min 3.2.4 The Reaction Time Distribution
The reaction time is assumed to be truncated lognormally distributed i.e., skewed
to the left as suggested by Subramanian 1996. This implies that the probability of
57 a driver having a smaller reaction time is higher than that of having a larger reaction
time. This was also supported by Johansson and Rumer 1971 and Lerner et al.
1995. Truncation is assumed since reaction time is nite. The distribution is as
follows:
8 ln f n =
: 1 max , n pe
2 , 1 ln n ,
2 2 if 0 0 n max 3.14 otherwise where,
= reaction time of driver n,
= mean of the distribution of ln n,
n max = standard deviation of the distribution of ln n,
= upper bound of the distribution of n parameter to be estimated. The mean, median, and variance of the above distribution are as follows: ln max , , mean = exp + 0:5 2
ln max ,
,1 median = exp +
variance = e2 + 0:5 ln ln e ,1
2 3.15
!!! max , max , ln ,2 max , 3.16
3.17 The mean of the distribution of ln n, , is assumed to be a function of explanatory
variables: = Xn
58 3.18 where, Xn = vector of explanatory variables,
= model parameters.
Important factors a ecting the reaction time include age, mental condition, visibility, weather conditions, roadway geometry, vehicle characteristics, vehicle speed,
and tra c conditions. Older drivers are expected to have longer reaction times. Poor
visibility increases driving di culty and drivers are expected to be more alert. This
implies a reduction in reaction time. During rain or snow drivers are expected to be
more alert compared to good weather conditions. Roadway sections with high curvature and or high grade make driving more di cult and hence would make drivers
more alert. At high speeds drivers are expected to be more alert compared to low
speeds due to safety reason. Tra c conditions such as the density of tra c and the
gap in front of the subject may also a ect reaction time. Drivers may be more alert
in congested tra c compared to free ow tra c due to higher uncertainty involved
in predicting future tra c conditions. 3.3 Likelihood Function Formulation
Using Equation 3.2 and the hypothesis that a driver in the car following regime will
accelerate if the leader is faster and vice versa, the distribution of the car following
acceleration, conditional on n, is given by: f acf t j
n n = f acf;acct j
n n Vn t, n f acf;dect j
n n 1, Vn t, n 3.19
where, 8 Vnt , n = : 1 if Vnt , n 0
0 otherwise
59 Using Equations 3.2 and 3.8, the distribution of the car following acceleration, conditional on n , are as follows: f acf;g t j
n n = 1
cf;g cf;g
cf;g
an t , s Xn t , n f Vnt , n cf;g ! 3.20 where, g 2 facc; decg. The free ow acceleration distribution, conditional on n , is
given by: f aff t j n
n = 1
ff ff
an t , ff DS
Xn t , n
ff DS , Vn t , n ! 3.21 Combining Equations 3.19 and 3.21, and using Equation 3.1, the distribution of
acceleration for driver n at time t, conditional on h and n , is as follows:
n f ant j h ; n = f acf t j
n
n n hn t, n f aff t j
n n 1, hn t, n 3.22
where, 8 hnt , n = : 1 if hnt , n h
n
0 otherwise As mentioned above, the reaction time and the headway threshold capture the correlation between di erent acceleration decisions at di erent times for a given driver.
This implies that, conditional on n and h , the Tn di erent observations of driver n
n
are independent. Therefore, the conditional joint density of observing an acceleration
pattern associated with driver n, an1; an2; : : : ; anTn, can be expressed as the
product of the conditional densities of each element of the pattern. Mathematically,
this can be expressed as follows: f an1; an2; : : : ; anTn j h ; n n = Tn
Y
t=1 f ant j h ; n
n 3.23 The unconditional distribution that constitutes the likelihood function for driver
60 n is:
f an1; an2; : : : ; anTn =
Z max Z h
max
f an1; an2; : : : ; anTn j h ; f h f dh d
0 hmin 3.24 Finally, assuming that the acceleration observations from di erent drivers in the sample are independent, the log likelihood function is given by: L= N
X
n=1 ln f an1; an2; : : : ; anTn 3.25 Maximizing the likelihood function would provide the MLE estimate of the model
parameters. 3.4 Conclusions
In this chapter, a rigorous framework for specifying and estimating the general acceleration model is presented that allows for joint estimation of all component models.
The component models are the car following acceleration and deceleration models,
the free ow acceleration model, and the headway threshold and reaction time distributions.
The proposed model builds on the earlier work by Subramanian 1996 and extends
it. First, separate model parameters under acceleration and deceleration situations in
the car following regime are allowed in the likelihood function formulation. Second,
the sensitivity of the car following acceleration is extended to capture the e ect of
tra c conditions ahead of the driver, in addition to the relative position and speed
of its leader. Third, it allows the time at which the explanatory variables of the car
following acceleration sensitivity are observed to be a parameter to be estimated as
opposed to restricting it to be the time at which the stimulus is observed. Fourth,
the stimulus of the car following acceleration is extended by making it a nonlinear
function of the lead relative speed. And nally, a more general headway threshold
61 distribution is used that allows any driver behavior to be captured aggressive or
conservative. 62 Chapter 4
The Lane Changing Model
In this chapter, the lane changing model is presented. Lane changes are classi ed
as either mandatory or discretionary. When a lane change is required due to, for
example, a lane drop, the operation is called a mandatory lane change MLC. On
the other hand, when lanes are changed by a driver to improve perceived driving
conditions, the operation is called a discretionary lane change DLC.
The proposed mandatory lane changing model extends the work by Ahmed et al.
1996 by developing a new model for heavily congested tra c. Under heavily congested tra c, gaps of acceptable lengths are hard to nd. Hence, a forced merging
model is proposed which captures merging by gap creation either through courtesy
yielding of the lag vehicle in the target lane or through the subject forcing the lag
vehicle to slow down.
This chapter begins by presenting the conceptual framework of the proposed lane
changing model. The model along with the likelihood function formulation is presented next. Then, the complexities associated with modeling the impact of past lane
changing decisions on the current lane changing decision are discussed. This chapter concludes by presenting the conceptual framework, the model, and the likelihood
function formulation of the forced merging model. 63 4.1 Introduction
A lane change decision process is assumed to have the following three steps:
decision to consider a lane change either a DLC or an MLC,
choice of a target lane, and
acceptance of a gap in the target lane.
Modeling such a process is extremely complicated. First, the entire lane change
decision process is latent in nature. All that is observed is the execution of the
lane change decisionthe nal acceptance of a gap. Second, the time at which a lane
change decision is made cannot be observed in general1. Furthermore, once a decision
to change lanes is made, a driver may continue to search for gaps or may change its
mindall of which are unobserved. Finally, the lane changing decision is continuous
in nature.
To simplify the modeling, time is discretized. Furthermore, drivers are assumed to
make decisions about lane changes at every discrete point in time irrespective of the
decisions made during earlier time periods. In other words, we do not explicitly model
the impact of past lane changing decisions on the current lane changing decision. The
complexities associated with capturing such behavior are discussed in Section 4.2.4.
The impact of past decisions on the current decision, however, is captured in the
proposed forced merging model. Due to their di erent structures, the lane changing
and forced merging models are presented separately.
Merging from an on ramp to a freeway is a notable exception, since as soon as a driver arrives at
the merging point, the driver would recognize the necessity of performing a mandatory lane change.
1 64 4.2 The Lane Changing Model
4.2.1 Conceptual Framework
The lane changing model structure is shown in Figure 41. As mentioned above, except for the completion of the execution of the lane change, the whole decision process
is latent in nature. The latent and observable parts of the process are represented by
ovals and rectangles respectively.
Start MLC MLC
driving
conditions not
satisfactory driving
conditions
satisfactory other
lanes Left Lane Right Lane Left Lane current
lane Right Lane Gap
Accept Gap
Reject Gap
Accept Gap
Reject Gap
Accept Gap
Reject Gap
Accept Gap
Reject Left
Lane Current
Lane Right
Lane Current
Lane Left
Lane Current
Lane Right
Lane Current
Lane Current
Lane Current
Lane Figure 41: The lane changing model structure.
The MLC branch in the top level corresponds to the case when a driver decides
to respond to the MLC condition2. Explanatory variables that a ect such decision
include remaining distance to the point at which lane change must be completed,
the number of lanes to cross to reach a lane connected to the next link, delay time
elapsed since the MLC conditions apply, and whether the subject vehicle is a heavy
When a mandatory lane changing situation does not apply, the probability of responding to
MLC is set to zero.
2 65 vehicle bus, truck, semi trailer etc.. Drivers are likely to respond to the MLC
situations earlier if it involves crossing several lanes. A longer delay makes a driver
more anxious and increases the likelihood of responding to the MLC situations. And
nally, due to lower maneuverability and larger gap length requirement of heavy
vehicles as compared to their non heavy counterparts, they have a higher likelihood
of responding to the MLC conditions.
The MLC branch corresponds to the case where either a driver does not respond
to an MLC condition, or that MLC conditions do not apply. A driver then decides
whether to perform a discretionary lane change DLC . This comprises of two decisions: whether the driving conditions are satisfactory, and if not satisfactory, whether
any other lane is better than the current lane. The term driving conditions satisfactory implies that the driver is satis ed with the driving conditions of the current lane.
Important factors a ecting the decision whether the driving conditions are satisfactory include the speed of the driver compared to its desired speed, presence of heavy
vehicles in front and behind the subject, if an adjacent on ramp merges with the
current lane, whether the subject is tailgated etc. If the driving conditions are not
satisfactory, the driver compares the driving conditions of the current lane with those
of the adjacent lanes. Important factors a ecting this decision include the di erence
between the speed of tra c in di erent lanes and the driver's desired speed, the density of tra c in di erent lanes, the relative speed with respect to the lag vehicle in
the target lane, the presence of heavy vehicles in di erent lanes ahead of the subject
etc. In addition, when a driver considers DLC although a mandatory lane change
is required but the driver is not responding to the MLC conditions, changing lanes
opposite to the direction as required by the MLC conditions may be less desirable.
If a driver decides not to perform a discretionary lane change i.e., either the driving
conditions are satisfactory, or, although the driving conditions are not satisfactory,
the current is the lane with the best driving conditions the driver continues in the
current lane. Otherwise, the driver selects a lane from the available alternatives and
assesses the adjacent gap in the target lane.
The lowest level of ovals in the decision tree shown in Figure 41 corresponds
66 to the gap acceptance process. When trying to perform a DLC , factors that a ect
drivers' gap acceptance behavior include the gap length, speed of the subject, speed
of the vehicles ahead of and behind the subject in the target lane, and the type of
the subject vehicle heavy vehicle or not. For instance, a larger gap is required
for merging at a higher travel speed. A heavy vehicle would require a larger gap
length compared to a car due to lower maneuverability and the length of the heavy
vehicle. In addition to the above factors, the gap acceptance process under the MLC
conditions is in uenced by factors such as remaining distance to the point at which
lane change must be completed, delay which captures the impatience factor that
would make drivers more aggressive etc.
Note that, delay cannot be used as an explanatory variable except for very specialized situations, for example, merging from an on ramp. This is because the very
inception of an MLC condition is usually unobserved. The speci cation of the complete model is presented next. 4.2.2 Model Formulation
The decisions in the hierarchy shown in Figure 41 can be modeled using the random
utility approach BenAkiva and Lerman 1985. The model formulation must explicitly capture the fact that, the available data for lane changing model estimation is
panel data. Model formulation appropriate for panel data is presented in Appendix
A. The Lane Selection Model
As mentioned above, the lane selection process consists of the top four levels of the
decision hierarchy shown in Figure 41. The top level, whether to respond to a
mandatory lane change MLC condition or not MLC , can be modeled using a
discrete choice model, for example, a binary logit model. Using the formulation of
random term appropriate for panel data see Appendix A, the probability that driver
n at time t will respond to MLC , conditional on the individual speci c random term,
67 n, is given by:
Pt MLC j n = 1 + exp,X MLC t1
n M LC , M LC n 4.1 where,
MLC
Xn t = vector of explanatory variables a ecting decision to respond to the
MLC conditions discussed in Section 4.2.1,
M LC = vector of parameters,
n = individual speci c random term assumed to be distributed
standard normal,
M LC = parameter of .
n The individual speci c random term, n, is introduced to capture the correlation
between di erent observations from a given driver. If the correlation is not captured
it may introduce bias in the parameter estimates. The larger the product of the
parameter M LC and the value of individual speci c random variable, n, the higher
is the probability that the driver would respond to an MLC condition earlier.
If a driver decides not to respond to an MLC condition, or MLC conditions do not
apply, a discretionary lane change DLC may be considered. The binary decision,
whether the driving conditions are satisfactory or not, can be modeled using a binary
logit model,
1
Pt DCNS j n = 1 + exp,X DCNS t
n DCNS , DCNS n 4.2 where, superscript DCNS denotes driving conditions not satisfactory. Generally, we
expect M LC and DCNS to have opposite signs, or, the two corresponding utilities
should have a negative correlation see Equation A.4. This implies that, a driver
postponing a response to an MLC condition to be an aggressive driver and hence,
may have a higher propensity to perform a discretionary lane change.
68 If the driving conditions are not satisfactory, drivers are assumed to compare the
driving condition of the current lane with the better among the left and right adjacent
lanes. The utilities of perceiving the driving conditions unsatisfactory and selecting
the other lanes over the current lane are expected to be positively correlated. Since
there is an e ort hassle associated with changing lanes which is not explicitly captured, the utility of the adjacent two lanes are likely to be correlated. The nested logit
model BenAkiva and Lerman 1985 is a natural choice to capture such phenomenon.
First, the utilities of the two adjacent lanes are compared the `left lane' versus `right
lane' decision under the `other lanes' oval in Figure 41. Then, the utility of the
`other lanes' is compared to the utility of the `current lane' to decide if the current
lane is the desired lane.
The output from the lane selection model is the probability of selecting each of the
three lanes in question. If the left or right lane is chosen, a driver seeks an acceptable
gap in the target lane. The gap acceptance model is presented next. The Gap Acceptance Model
The gap acceptance model captures drivers assessment of gaps as acceptable or unacceptable. Drivers are assumed to consider only the adjacent gap. An adjacent gap
is de ned as the gap in between the lead and lag vehicles in the target lane see
Figure 42. For merging into an adjacent lane, a gap is acceptable only when both
X total clear gap + vehicle length
lag gap Y lead gap lag vehicle lead vehicle subject front
vehicle X Y Figure 42: The subject, lead, lag, and front vehicles, and the lead and lag gaps.
lead and lag gaps are acceptable.
69 Drivers are assumed to have minimum acceptable lead and lag gap lengths which
are termed as the lead and lag critical gaps respectively. These critical gaps vary not
only among di erent individuals, but also for a given individual under di erent tra c
conditions. The critical gap for driver n at time t is assumed to have the following
functional form3:
g
Gcr;g t = expXn t g + g n + g t
n
n 4.3 where, g 2 flead; lagg;
g = parameter of for g 2 flead; lag g,
n
g t = generic random term that varies across all three dimensions, i.e.,
n
g; t; and n:
The exponential form of the critical gap guarantees that the estimated critical gap will
always be nonnegative. The individual speci c random term, n, and its parameter
capture the correlation between the lead and lag critical gaps for a driver. This
correlation, especially under MLC conditions, is expected to be positive.
A conservative driver is expected to have a larger lead lag critical gaps compared
to its aggressive counterpart. A larger product of g and n , g 2 flead; lagg implies
a larger critical gap length requirement, and hence, represents a conservative driver.
The lead lag critical gaps are expected to be positively correlated with the utility
of responding to an MLC condition and negatively correlated with the utilities of
perceiving the driving conditions as unsatisfactory and selecting the other lanes over
the current lane.
Assuming g t N 0; 2g , i.e., the critical gap lengths are lognormally disn
tributed, the conditional probability of acceptance of a gap is given by:
3 Adopted from Ahmed et al. 1996. 70 Pt gapAcc j n
= Pt lead gap acceptablej n Pt lag gap acceptablej n
= PGlead t Gcr;leadt j n PGlag t Gcr;lag t j n
n
n
n
n
= PlnGleadt lnGcr;leadt j n PlnGlag t lnGcr;lag t j n
n
n
n lead
!n
lead
lnGn t , Xn t lead , lead n
=
lead lag
lag
lnGn t , Xn t
lag lag , lag n ! 4.4 where, Gleadt and Glag t denote the lead and lag gaps see Figure 42 respectively
n
n
and denotes the cumulative distribution function of a standard normal random
variable. In addition to 's and 's, lead and lag are parameters that can be identi ed. Normalization of lead or lag is not necessary since the variables `lnGlead t'
n
lag t' in Equation 4.4 do not have any coe cient.
and `lnGn 4.2.3 Likelihood Function Formulation
Drivers are assumed to consider the entire lane change decision process Figure 41
at every discrete point in time, for example, every second. Let the sequence of lane
changes performed by driver n be denoted as follows: Jn1; Jn2; : : : ; JnTn
where, J
L
R
C
Tn 2 fL; R; C g
= change to the left lane,
= change to the right lane,
= continue in the current lane,
= number of time periods driver n is observed.
71 4.5 As mentioned above, the individual speci c random term captures the correlation
between di erent decisions at di erent times for a given driver. Therefore, conditional
on the individual speci c random term, the probability of observing a pattern for a
given driver can be expressed as the product of probabilities of observing each element
of the pattern. Mathematically, this can be expressed as follows:
PJ1n; J2n; : : : ; JTnn j n =
= Tn
Y
t=1
Tn
Y
t=1 PJtn j n
L
R
LR
Pt L j n tn Pt R j n tn Pt C j n1, tn , tn 4.6 where,
8 J
tn = : 1 if driver n changes to J at time t J 2 fL; Rg
0 otherwise. 4.7 The unconditional probability of observing a pattern for a given driver is given by:
PJ1n ; J2n; : : : ; JTnn = Z1 ,1 PJ1n; J2n; : : : ; JTnn j f d where, f denotes the distribution of .
Assuming that the observations from di erent drivers in the sample are independent, the likelihood function for all drivers is: L= N
X
n=1 ln PJ1n ; J2n; : : : ; JTnn 4.8 where, N denotes the number of drivers.
The probability of staying in the current lane or changing to the left or right lanes
can be formulated using the decision tree of Figure 41. A driver may change to the
left lane when he she:
responds to MLC conditions, the left lane is chosen, and the lead and lag gaps
in the left lane are acceptable; or,
72 does not respond to MLC conditions or MLC conditions do not apply, perceives the driving conditions as unsatisfactory, selects the other lanes over the
current lane, selects the left lane, and the lead and lag gaps in the left lane are
acceptable.
Therefore, the probability of an observation of change to the left lane, conditional on
n, is:
Pt L j n =
Pt gap acceptable j left lane chosen; MLC; n
Pt left lane chosen j MLC; n Pt MLC j n +
Pt gap acceptable j left lane chosen; other lanes; driving conditions not
satisfactory; MLC; n
Pt left lane chosen j other lanes; driving conditions not satisfactory;
MLC; n
Pt other lanes j driving conditions not satisfactory; MLC; n
Pt driving conditions not satisfactory j MLC; n Pt MLC j n
Similarly, the conditional probability of changing to the right lane or continuing in
the current lane can be formulated. 4.2.4 Discussions
Complexities Associated with Capturing the Impact of Past Lane Changing Decisions in the Lane Changing Model
In this section, the complexities associated with modeling the impact of past lane
changing decisions on the current lane changing decision are discussed with the help of
a simple example. Consider a vehicle that is observed in a two lane roadway for three
consecutive time periods during which time it did not change lanes and mandatory
lane changing conditions do not apply. To simplify the discussion further, we combine
73 the two levels, `driving conditions satisfactory or not' and `other lane' or `current lane',
of the decision tree shown in Figure 41 into one level, DLC versus DLC . Here, DLC
implies that the driving conditions of the current lane is not satisfactory and another
lane the left lane is better than the current lane. Therefore, for this driver, the lane
changing decision tree shown in Figure 41 reduces to the one shown in Figure 43
a. Since the driver did not change lanes, he she may be in state `DLC and gap
reject given DLC ' or in state `DLC ' during these three time periods see Figure 43
b.
As shown in Figure 43 b, there are 23 = 8 possible state sequences that can
explain the three observations from the driver. If the driver is observed for Tn time
periods, the number of state sequences becomes 2Tn , i.e., the number of possible state
sequences increases exponentially with the number of times the driver is observed.
Therefore, the number of state sequences to explain a particular pattern of lane
time
lane
period change Start possible states
(arrows show state to state transitions) 1 DLC no DLC & gapRej  DLC DLC 2 no DLC & gapRej  DLC DLC 3 no DLC & gapRej  DLC DLC DLC gap
Acc gap
Rej Target
Lane Same
Lane Same
Lane (b) possible states during the
three successive time periods (a) decision tree at any
instant of time Figure 43: The lane changing decision tree for a driver driving in a two lane roadway
and possible states of the driver.
changing by a driver is prohibitively large from an estimation point of view. Further
research with various modeling approaches and approximations is necessary to capture
the impact of past lane changing decisions on the current lane changing decision. 74 Limitations of the Proposed MLC Model
Vehicles in heavily congested tra c travel at low speeds, with low space headways.
In such situations, it is likely that a driver, trying to change lanes, will not nd a gap
that is larger than the driver's minimum acceptable gap length. In order to merge,
gaps have to be created either through the lag vehicle's courtesy yielding or through
the subject forcing the lag vehicle to slow down. The mandatory lane changing model
presented above, however, assumes that drivers would ultimately nd an acceptable
gap. A forced merging model that captures driver decisions leading to a gap creation
is proposed in the following section. 4.3 The Forced Merging Model
We assume that a driver has decided to change to the adjacent lane see Figure 44.
The merging process involves the driver's decision as to whether he she intends to
merge into the adjacent gap and perception as to whether his her right of way is
established, and nally moving into the target lane. An adjacent gap is de ned as the
gap behind the lead vehicle in the target lane. Establishment of right of way means
that an understanding between the subject and the lag vehicle in the target lane has } lag
vehicle adjacent gap
for the subject target
lane
lead
vehicle subject Figure 44: De nition of the adjacent gap.
been reached such that the lag vehicle would allow the subject to be in front of it.
The conceptual framework of the proposed model is presented next. 75 4.3.1 Conceptual Framework
The tree diagram in Figure 45 summarizes the proposed structure of the forced
merging model. As before, the ovals correspond to the latent part of the process
that involves decisions and the rectangles correspond to the events that are directly
observable.
MLC do not start
forced merging
(M) start
forced merging
(M) Same
Lane Target
Lane Same
Lane Figure 45: The forced merging model structure.
At every discrete point in time, a driver is assumed to a evaluate the tra c
environment in the target lane to decide whether the driver intends to merge in front
of the lag vehicle in the target lane and b try to communicate with the lag vehicle to
understand whether the driver's right of way is established. If a driver intends to merge
in front of the lag vehicle and right of way is established, the decision process ends
and the driver gradually moves into the target lane. We characterize this instant by
state M , where M denotes start forced merging. This process may last from less than
a second to a few seconds. This is shown by the arrow below the left `same lane' box.
If right of way is not established, the subject continues the evaluation communication
process i.e., remains in state M during the next time instant.
76 4.3.2 Model Formulation
Let, Snt denote the state of driver n at time t. Using a binary logit model and
the random utility speci cation appropriate for panel data see Appendix A, the
probability of switching to state M from state M , conditional on n, is given by:
PfSn t = M j Snt , 1 = M; ng = 1 + e,Xn 1
FM t F M , F M n 4.9 where, superscript FM implies forced merging. Important explanatory variables
include:
lead relative speed only when the lead vehicle is slower: when the lead vehicle
is slower, the subject is more likely to slow down to match its speed with the
speed of the lead vehicle rst so as to focus exclusively on the interaction with
the lag vehicle; this reduces the probability of being in state M ;
lag relative speed: when the lag vehicle is faster, the subject is more likely to
speed up before attempting to establish right of way and hence this reduces the
probability of being in state M ;
remaining distance to the point at which lane change must be completed by: as
the remaining distance decreases, drivers become more concerned about merging
and hence more aggressive. As a result, the probability of being in state M also
increases;
delay time elapsed since the mandatory lane change conditions apply: higher
delay makes a driver more frustrated and hence more aggressive, i.e., the probability of being in state M increases with additional delays4 .
total clear gap equal to the sum of the lead and lag gaps, see Figure 42: a large
clear gap makes merging relatively easier and hence increases the probability of
being in state M ;
As explained in Section 4.2.1, delay can be used as an explanatory variable only when the
starting point is well de ned, for example, merging from an on ramp to the freeway.
4 77 indicator for heavy vehicles for example, bus, truck, semi trailer: due to lower
maneuverability and larger gap length requirement of heavy vehicles as compared to their non heavy counterparts, they have a higher probability of being
in state M under similar conditions.
The likelihood function formulation is presented next. 4.3.3 Likelihood Function Formulation
At any discrete point in time, a driver may be in state M or M see Figure 45. Once
a vehicle is in state M , by de nition, the decision process ends and the remaining
process is placing the vehicle in front of the lag vehicle, and the state of the driver
cannot return to M . The time taken in placing the vehicle in front of the lag vehicle
is captured by the arrow below the left `same lane' rectangle in Figure 45. This
implies the following:
PfSnt0 = M j Snt = M; n g = 1 8 t0 t
PfSnt0 = M j Snt = M; n g = 0 8 t0 t 4.10
4.11 We also assume that the initial state of the driver is M . Di erent cases in which
forced merging can occur are shown in Figure 46. Time period 1 denotes the rst
time period considered in the forced merging model and time period 0 denotes the
preceding time period. In the rst two cases Figures 46 a and b, i.e., merging
from an on ramp and exiting, at time period 0 it is practically impossible to start
merging. Therefore, the initial state is M . On the other hand, in the last two cases
Figures 46 c and d, the driver could be in state M or M at time 0. If a driver
is already in state M , then the sequence observed for the driver does not involve any
decision and therefore, the probability of the observing the sequence is not a function
of the model parameters to be estimated. In such cases, assuming drivers' state to be
M would be reasonable only if the length of the section measured from the diverging
point toward the upstream direction is large enough. A reasonable way to de ne a
section large enough is 200 meters or more. Note that, in the last two cases delay
78 merging
point merging
point Target
Lane 0 1 1 0
0 (a) merging from onramp
initial state (at time 0): M upstream end of the
data collection site
0 Target
Lane (b) exiting, 1 lane change required
initial state (at time 0): M upstream end of the
data collection site Target
Lane Target
Lane 1
0 (c) exiting, 2 lane changes required
initial state (at time 0): M or M 1 (d) connecting to the next link
initial state (at time 0): M or M Figure 46: Initial state of the driver for the forced merging model for di erent cases.
cannot be used as an explanatory variable since the time instant at which the driver's
state became M for the rst time cannot be observed.
Since, n is assumed to capture the correlation between the utilities of di erent
states at di erent times, conditional on n, the probability of being in state M at
time t, given all earlier states were M , is also given by Equation 4.9. Mathematically,
PfSnt = M j Snt0 = M; t0 = 0; 1; : : : ; t , 1; ng = 1 FM
1 + e,Xn t F M , F M n 4.12 The impact of being in state M during the earlier time periods on the probability of
being in state M at time t is captured through the explanatory variable delay.
Let, Tn be the number of time periods driver n was observed in the original lane.
There are Tn possible state sequences that may lead to observing driver n in the
target lane at time Tn + 1. These sequences are listed in Table 4.1. Sequence 1 in
Table 4.1 implies that driver n reached state M at time period 1 and it took another
79 Table 4.1: Possible decision state sequences of observing a lane change by forced
merging.
Time Observed
Possible state sequences
Period
lane
1 2 3 ::: t :::
1
SL
MMM
M
2
SL
MMM
M
3
SL
MMM
M
...
...
... ... ...
...
t,1
SL
MMM
M
t
SL
MMM
M
...
...
... ... ...
...
Tn , 1
SL
MMM
M
Tn
SL
MMM
M
Tn + 1
TL
Note: SL = same lane, TL = target lane Tn
M
M
M
...
M
M
...
M
M Tn , 1 seconds to execute the lane changing process. Sequence 2 corresponds to the
case where driver n was in state M at time period 1, and in state M during the time
interval 2 to Tn. Similarly, Sequence t corresponds to the case that the driver was in
state M during the time interval 1 to t , 1, and in state M during the time interval
t to Tn. Note that, these sequences are mutually exclusive.
As mentioned above, the individual speci c random term, n, captures the correlation between di erent decision elements at di erent times for a given driver. Therefore, conditional on n, the probability of observing a particular state sequence for a
given driver can be expressed as the product of probabilities of observing each state
of the sequence. The conditional probability of observing the tth state sequence for
driver n is, 80 Pn fstate sequencet j ng = PfSnTn = M j Snt0 = M; 8t0 = t; : : : ; Tn , 1; Snt00 = M; 8t00 = 0; 1; : : : ; t , 1; ng
PfSnTn , 1 = M j Snt0 = M; 8t0 = t; : : : ; Tn , 2;
Snt00 = M; 8t00 = 0; 1; : : : ; t , 1; ng : : :
PfSnt = M j Snt0 = M; 8t0 = 0; 1; : : : ; t , 1; ng : : :
PfSn2 = M j Sn1 = M; Sn0 = M; ng
PfSn1 = M j Sn0 = M; ng 4.13 Using Equations 4.11 and 4.12,
Pn fstate sequencet j ng
= PfSnt = M j Snt0 = M; 8t0 = 0; 1; : : : ; t , 1; ng : : :
PfSn2 = M j Sn1 = M; Sn0 = M; ng
PfSn1 = M j Sn0 = M; ng
= PfSnt = M j Snt0 = M; 8t0 = 0; 1; : : : ; t , 1; ng
tY
,1 t0 =1 PfSnt0 = M j Snt00 = M; 8t00 = 1; : : : ; t0 , 1; ng 4.14 Since, an observed lane change by a driver can be explained by any one of the
mutually exclusive state sequences listed in Table 4.1, the conditional likelihood function is the sum of the probabilities of observing all the sequences. This is given by: 81 Ln
=
= FM; FM j
n
Tn
X
Pnfstate sequencet j ng
t=1
Pnfstate sequence1 j n g + Pnfstate Pnfstate sequenceTn j ng sequence2 j ng + : : : + = PfSn1 = M j Sn0 = M; ng +
PfSn2 = M j Sn1 = M; Sn0 = M; ngPfSn1 = M j Sn0 = M; ng + : : : + PfSnTn = M j Snt0 = M; 8t0 = 0; 1; : : : ; Tn , 1; ng
TY1
n,
t0 =1 PfSnt0 = M j Snt00 = M; 8t00 = 1; : : : ; t0 , 1; ng Let us now introduce variable
8 FM
n t =
: FM
tn 4.15 de ned as: 1 if the adjacent gap at time t is the same gap
driver n ultimately merged into
0 otherwise 4.16 F
Figure 47 illustrates the meaning of the variable n M t with the help of an example.
The subject vehicle C was observed for 4 time periods in the original lane the right
lane. Only during the 3rd and 4th time periods, vehicle C was adjacent to the gap
between vehicles A and B that it ultimately merged into. Therefore, the sequence of
F M for this driver is f0,0,1,1g. Since, the driver was not adjacent to the gap between
vehicles A and B during the rst two time periods, communication with vehicle B
cannot be established during these time intervals. Therefore, the driver cannot be
in state M during time periods 1 and 2. This implies that, the rst two sequences
listed in Table 4.1 do not apply to this driver. While forming the likelihood function
for this driver, the rst two sequences must be taken out of the likelihood function
Equation 4.15. A convenient way of incorporating this into the likelihood function
for a general case is as follows: Ln FM; FM j n = Tn
X
t=1 Pnfstate sequencet j ng
82 FM
n t 4.17 time = 1
delta = 0
A B C subject
time = 2
delta = 0
B A C time = 3
delta = 1
B A
C time = 4
delta = 1
A B
C time = 5
Lane change completed
B Figure 47: De nition of FM
n t C A for the forced merging model. The unconditional likelihood function for driver n is Ln FM; FM = Z 1 X
Tn ,1 t=1 Pn fstate sequencet j g FM
n t ! f d 4.18 where, f denotes, as before, the probability density function of the random variable
.
Assuming that the observations from di erent drivers in the sample are independent, the log likelihood function for all observations is given by: L FM; FM =
Z Tn
N
1X
X
n=1 ln ,1 t=1 Pnfstate sequencet j g 83 !
F M t f d
n 4.19 4.3.4 Discussion
As mentioned above, in order to merge in heavily congested tra c, drivers must
create gaps either through force or through courtesy yielding. A reasonable way to
de ne heavy tra c congestion is level of service F de ned by the Highway Capacity
Manual HCM 1985. The HCM characterizes this level of service as tra c conditions
in which a breakdown of ow occurs and queues form behind breakdown points. At
this level of congestion, the probability of nding acceptable gaps is very low and in
order to merge gaps have to be created. For level of services A through E, drivers are
assumed to merge by the gap acceptance process presented in Section 4.2.2.
The boundary level of services F versus A through E that is used to determine
whether to apply the usual gap acceptance process or the forced merging process is
rather arbitrary. Although the boundary can also be estimated formally e.g. like the
headway threshold in the acceleration model, the process the drivers actually follow
may be di erent. For example, drivers rst search for acceptable gaps and consider
forced merging only when they perceive the probability of nding acceptable gaps to
be very low. Further research is necessary to combine the mandatory lane changing
and forced merging models into a single framework which would apply to all level of
services. We leave this as a subject for future research. 4.4 Conclusions
In this chapter, a framework for modeling drivers' lane changing behavior was developed. A signi cant enhancement to the state of the art is the development of the
forced merging model that captures merging behavior under heavy tra c congestion.
This model is based on the assumption that in heavily congested tra c, gaps of acceptable lengths are rare, and therefore, for a vehicle to merge, gaps must be created
either through courtesy yielding of the lag vehicle in the target lane or through the
subject forcing the lag vehicle to slow down.
84 Chapter 5
Data Requirements for Estimating
Driver Behavior Models
In this chapter, the data required to estimate the acceleration and lane changing
models and the data that was obtained from real tra c are presented. In addition,
a methodology for estimating instantaneous speed and acceleration that are required
for model estimation from discrete trajectory data that can be obtained from the eld
is developed.
Data required to estimate the acceleration and lane changing models include the
position, speed, acceleration, and length of a subject vehicle and the vehicles ahead of
and behind the subject in the current lane as well as in adjacent lanes. Data on gap
lengths, headways, density of tra c, etc. can be extracted from the above mentioned
data by simple addition and subtraction operations. In addition, to capture the impact of site speci c factors, such as the speed limit of a section, geometry curvature,
grade, and lane con guration and whether the roadway section is a tunnel or not,
data from di erent sites is required.
Typically, such data is collected using photographic and video equipment see, for
example, Smith 1985. The raw data collected through such devices is processed to
obtain useful information such as vehicle location at discrete points in time. Instantaneous speed and acceleration data, that is required for estimation of the models,
have to be inferred from the trajectory data.
85 This chapter begins with a description of the method that is used to estimate
instantaneous speed and acceleration from discrete trajectory data. Then the data
collection strategy and the actual processing of the data is presented. 5.1 Methodology for Estimating Instantaneous Speed
and Acceleration from Discrete Trajectory Data
As mentioned above, the data usually available includes discrete measurements of
vehicle positions over time. A continuous function describing the vehicle trajectory
can be estimated from the discrete position observations using the local regression
procedure developed by Cleveland and Devlin 1988. Once the trajectory function,
X t, is estimated, the rst and second derivatives of the estimated trajectory function
at time t, provide estimates of the speed and acceleration at time t respectively.
Mathematically, V t = dX t
dt
d2X t
at = dt2 5.1 where, V t = speed at time t,
at = acceleration at time t.
In general, vehicles frequently stop in congested tra c, often for signi cant durations. Whether a vehicle is stopped or not, cannot be ascertained from the observed
trajectory, as there are measurement errors while collecting and processing the data.
A very high order polynomial would be necessary to t a curve to the trajectories of
such vehicles. This gives rise to computational problems as the objective function of 86 such problems becomes nearly singular1. Furthermore, even a high order polynomial
may not t the data well during the instances when a vehicle is stopped. The local
regression procedure addresses some of these problems by tting local curves using
the observations around the time period of interest which is described in more detail
in Section 5.1.1.
Local regression can be used to estimate a wide class of functions. Three major
uses of local regression were listed by Cleveland and Devlin 1988. First, a local
regression estimate can be used as a graphical exploratory tool to study the structure
of the data. This would help in choosing an appropriate functional form that ts
the data. Second, it can be used to validate an already estimated model that used
a parametric class of models. And nally, local regression estimates can be used instead of regular regression estimates when dealing with data that require very exible
functional form. The application in this research falls in the third category. 5.1.1 The Local Regression Procedure
The local regression procedure has three basic elements: weight assignment, function
speci cation, and neighborhood or window size. A unit weight is assigned to the
trajectory observation at the time period of interest t and a gradually decreasing
weight is assigned to the other points, depending on their distance from the tth observation. The window size around time t determines the number of points that are
used for tting a polynomial curve of a suitable degree.
Weight Assignment. A tricube weight function with the following functional form
is used: wto; t = 1 , uto; t33 5.2 1
For polynomials of order 10 or above, the hessian of the objective function becomes nearly
singular as the independent variables polynomial of time vary from tens and twenties to billions.
This also depends on the precision of the computer. Nearly singular hessian makes the estimation
process computationally intensive and time consuming as the convergence rate reduces signi cantly. 87 where, t = time period at which speed acceleration estimates are desired,
wto; t = weight for observation at time to , to 2 f1; 2; : : :; t; : : :g,
uto; t = distance function for an observation at to
= j t , to j ;
d
dt = distance from t to the farthest point + constant.
A small constant is added to d so that the distance function, u, is less than one for the
observation farthest from t. This guarantees a nonzero weight for that observation.
Note that, 0 uto; t 1 and 0 wto; t 1 8to.
Function speci cation. As mentioned above, the trajectory function, X t, is
assumed to be a polynomial of time. The parameters of a polynomial are uniquely
identi ed if the order of the polynomial is at most one less than the number of
observations trajectory points. A perfect t is obtained when the order of the
polynomial is one less than the number of points.
Window Size. The window size determines the number of points used in each local
regression. For example, a window size of 7 implies that the 7 closest in terms of
time of observation position measurements including the measurement at the time
period of interest t are considered for local tting of data. The bias variance of
the estimated position increases decreases with increasing window size Cleveland
et al. 1988. Depending on the type of application, the window size should be selected
such that either the bias or the variance or the mean square error of the estimates
is minimized. Note that, the mean square error is the sum of the bias squared
and the variance. Since the proposed curve t algorithm uses inequality constraints
discussed below, a close form solution for estimating the bias or variance does not
exist. Instead, a sensitivity analysis can be conducted to evaluate the impact of the
window size on the quality of the results e.g. magnitude of the position estimation
errors, the speed and acceleration pro les etc..
88 Mathematically, the curve t problem using the local regression can be stated as:
tminm
;s 2 X t; s , T t; s t;s 0 W t; s X t; s , T t; s t;s 5.3 where, t = time period at which speed acceleration estimates are desired,
s = window size,
X t; s = vector of discrete trajectory observations corresponding to time
period t and window size s,
T t; s = matrix of independent variables, constant; time; time2 ; : : : ; timem ,
t;s = vector of parameters corresponding the tth time period and window
size s,
m = order of the polynomial,
W t; s = weight function, a diagonal matrix.
The ith diagonal element of W t; s corresponds to the weight assigned to the ith
trajectory observation obtained by using Equation 5.2.
However, it is very common that in the data due to measurement errors, the measured position of a vehicle at two successive time periods may be decreasing. Hence,
a curve tted to these points may yield an unrealistic negative speed and or acceleration estimates. To guarantee that the speed estimates are non negative and
acceleration estimates are within the acceleration and deceleration capacities, Equation 5.3 has to be minimized subjected to the following set of constraints over the
range of time periods considered in a particular local regression:
speed 0
acceleration maximum deceleration
acceleration maximum acceleration
89 5.4 The curve t algorithm Equation 5.3 subjected to constraints 5.4 is repeated for
each driver for each instant of time at which position, instantaneous speed and or
acceleration are desired. Then for each instant of time, the tted value of the polynomial is used as an estimate of position and the rst and second derivatives of the
polynomial as the speed and acceleration respectively.
Figure 51 shows an example of estimating the speed and acceleration at time
220 200 original position
estimated trajectory function 180 Position of vehicle, m 160 140 120 100 80 measured position at time period 8 = 91.0480 m
estimated position at time period 8 = 90.8427 m 60 estimated speed at time period 8 = 0.82 m/s
estimated acceleration at time period 8 = −2.03 m/s2 40 20 0 5 10 15
time period, seconds 20 25 30 Figure 51: An example of estimation of instantaneous speed and acceleration from
discrete position measurements.
period 8 from discrete position measurements of a vehicle using the local regression
methodology. A window of size 9 was used in this exercise. Figure 52 shows the
weight function used in this exercise and tted curve. The trajectory function was
tted from the discrete position measurements around time period 8 time periods 4
to 12. As shown Figure 52, a very good t of the data was obtained except for time
periods 8 and 9. Vehicle position at time periods 8 and 9 was measured at 91.05 and
91.04 meters respectively. Since, vehicle position cannot decrease which implies an
unrealistic negative average speed, the position measurement either at time period
90 1 weight 0.8
0.6
0.4
0.2
0 4 5 6 7 8
9
time period, seconds 10 11 12 Position of the vehicle, meters 100 90 80 70 60 original position
estimated position
4 5 6 7 8
9
time period, seconds 10 11 12 Figure 52: The weight function and the tted curve for an observation at time period
8.
8 or at time period 9 must be erroneous. When the local regression procedure is
used, the speed non negativity constraint Equation 5.4 takes care of this problem.
Using the local regression procedure, the position at time period 8 was estimated to
be 90.84 meters. Observe that, tting a single curve to the observations in Figure 51
would require a polynomial of a very high degree. 5.2 Data Collection
Data was collected using standard video equipment. The video tapes were analyzed
using VIVA, an image processing software specially designed for tra c application
described in Section 5.2.2. 5.2.1 Description of the Data Collection Site
Video data of tra c ow was collected on Interstate 93 at the Central Artery, located
in downtown Boston the rectangle area in Figure 53. The video was processed using
the VIVA software package described in Section 5.2.2. The manual and automatic
features of VIVA were used to process congested and uncongested to semi congested
91 lane 1 lane 2 lane 3 lane 4 200 m I93 SB 402 m
(1/4 mile) South Station
Exit 402 m
(1/4 mile) China Town
Exit Mass. Pike
Exit Figure 53: Schematic diagram of the I 93 southbound data collection site gure not
drawn to scale.
tra c respectively to obtain discrete measurements of vehicle lengths and positions
over time. The processed data was then used to obtain vehicle trajectories.
The section has a three lane mainline lanes 1 to 3 and a weaving lane lane 4.
The mainline lanes continue into an underground tunnel. The weaving section leads
to Exit 22 The South Station Exit. There are two more exits further downstream
from this section. The rst exit is 1 4 mile away Exit 21, The Kneeland Street and
Chinatown Exit and the second exit is 1 2 mile away Exit 20, The Massachusetts
Turnpike and Albany Street Exit.
Trajectory and vehicle length data was extracted for vehicles only when they were
within the rectangular area shown in Figure 53. The length of the recorded section
varied from 150 to 200 meters as di erent zooms were used during the lming process
see Table 5.1. Data was collected for 2 hours starting at 10:26 a.m. tape 1 on
92 Table 5.1: Description of the collected tra c video.
Date
tape 1
tape 2
tape 3
tape 4
tape 5 8
12
12
12
12 9 95
10 97
10 97
10 97
10 97 Time
Length of the
hrs
Section meters
10:26 to 12:26
200
12:30 to 13:00
165
13:09 to 13:39
190
13:47 to 14:17
150
14:25 to 14:55
180 August 9, 1995, and for 30 minutes each starting at 12:30 p.m. tape 2, 1:09 p.m.
tape 3, 1:47 p.m. tape 4, and 2:25 p.m. tape 5 on December 10, 1997. On both
days of recording, the sky was overcast with periodical sunshine.
Vehicles that traveled in the mainline lanes and made no lane changes provide samples for estimating the acceleration model. Vehicles from lanes 2 or 3, that changed to
the left adjacent lane or did not change lanes within the data collection site, provide
samples for estimating the discretionary lane change model see discussion on page
121. Vehicles that traveled from the on ramp and merged with the mainline provide
samples for estimating the mandatory lane change and forced merging models. 5.2.2 Video Processing Software
VIVA2 Video Tra c Analysis System is an image processing software developed at
Universitat Kaiserslautern, Germany. It is capable of measuring positions of vehicles
from video images. It has both an automatic and a manual feature. The automatic
feature extracts positions of all vehicles within the video image in real time. However,
in heavily congested tra c, due to lack of spacing between vehicles, the software
runs into di culty in identifying front and rear bumpers of closely spaced vehicles
and hence the position estimates become unreliable. In such situations, the manual
feature can be used to identify vehicle positions by clicking on the screen and the
software generates the coordinates of vehicle positions. The manual process, however,
Information about this software package may be found in the World Wide Web at the URL
http: transport.arubi.uni kl.de ViVAtra c English.
2 93 is very time consuming. An initial testing indicated that the manual feature requires
approximately 30 person hours to process one minute of video data.
The accuracy associated with the position measurements from the video images
depends on the sharpness of the images and the scale of the images. In the automatic
feature, VIVA uses the contrast between the image of a vehicle and that of the underlying pavement to identify the vehicle. In the manual process, the user identi es the
vehicle. Therefore, a sharper image compared to a blurred image and a larger scale3
compared to a smaller scale would increase the accuracy with which the bumpers of
the vehicles can be identi ed on which the accuracy of position measurement depends.
The position measurement error for the video that we collected was estimated to be
1 meter. 5.2.3 Processing the Tra c Data
Description of the Trajectory Data
The trajectory data obtained from processing the video data with VIVA included
vehicle position recorded at discrete time points. The methodology described in
Section 5.1 was used to develop vehicle trajectories and subsequently speed and acceleration pro les for each vehicle.
The rst row of plots in Figure 54 shows minute by minute tra c ow at the
upstream end of the data collection site, and the second and the third row of plots
show second by second density and average speed of all vehicles of the mainline lanes
1 to 3 respectively. The rst column of plots corresponds to tape 1 data, the second
column of plots corresponds to tape 2 data and so on.
Nine minutes of trajectory data was extracted from tape 1 using the manual
feature of VIVA as tra c was extremely congested and at times stopped and the
software's automatic data extraction feature would not work in such tra c conditions.
Using the automatic feature, for the other four tapes that had less congestion, 30, 30,
21, and 18 minutes of trajectory data were extracted.
3 A larger scale compared to a smaller scale implies that a vehicle would appear larger. 94 Note: 1350 hrs implies 1:50pm; tape 1 implies data collected from tape 1
tape 1 tape 2 tape 3 tape 4 tape 5 main line flow in 100, veh/hr/lane 20 20 20 20 20 15 15 15 15 15 10 10 10 10 10 5 5 5 5 5 0
1025 1030 0
1035 1230 1240 1250 time in hrs 0
1310 1320 1330 time in hrs tape 1 0 time in hrs tape 2 1350 1360 0 time in hrs tape 3 tape 4 tape 5 80 80 80 80 70 70 70 70 70 60 60 60 60 60 50 50 50 50 50 40 40 40 40 40 30 30 30 30 30 20 20 20 20 20 10 density, veh/lane−km 1440 time in hrs 80 10 10 10 10 0
1025 1030 0
1035 1230 1240 1250 time in hrs 0
1310 1320 1330 time in hrs tape 1 0 time in hrs tape 2 1350 1360 0 time in hrs tape 3 1430 1440 time in hrs tape 4 tape 5 35 35 35 35 35 30 30 30 30 30 25 25 25 25 25 20 20 20 20 20 15 15 15 15 15 10 10 10 10 10 5 average speed, m/s 1430 5 5 5 5 0
1310 1320 1330 0 0
1025 1030 time in hrs 0
1035 1230 1240 1250 time in hrs time in hrs 1350 1360 time in hrs 0 1430 1440 time in hrs Figure 54: Flow, density, and average speed of the I 93 southbound trajectory data.
Although in the rst 9 minutes of data, ow was in the order of 800 to 1300
vehicles hr lane compared to a capacity of approximately 2000 vehicles hr lane,
due to conditions downstream of the data collection site, tra c moved very slowly.
This may have been in part due to a high volume of tra c trying to perform a lane
change to take the two exits a quarter mile and a half mile downstream. The density,
as shown in the 1st plot of row 2 in Figure 54, was always above 41 vehicles km lane
which corresponds to level of service F HCM 1985. The average speed of tra c
across the mainline lanes varied from 3 to 10 m s meters per second which is also
95 indicative of the heavy congestion.
Tra c in the last four tapes varied from free ow to semi congested with level of
service between A and E. The last four columns of the rst row of plots in Figure 54
show that the ow varied from 1000 to almost 2000 vehicles hr lane. Density, shown
in the last four plots of the second row, varied from 0 to 40 vehicles km lane and
never exceeded 41 vehicles km lane. The average speed of the mainline tra c varied
from 15 to 33 m s. Estimation Results using the Local Regression Procedure
Window Size Selection
As mentioned above, depending on the type of application, the window size should
be selected such that either the bias or the variance or the mean square error of
the estimates is minimized. However, a close form solution for estimating the bias
or variance does not exist since the curve t algorithm uses inequality constraints
Equation 5.4. We, instead, conducted a sensitivity analysis to evaluate the impact
of window size on the quality of the results. For this analysis we have used odd
number of window sizes e.g. 7, 9, 11 etc. to make the number of observations
before and after the time period of interest equal. The minimum window selected
was 7. The reason for this choice is the following: with window size equal to 5, the
order of the polynomial cannot exceed 4. As a result, the order of the polynomial
representing the acceleration pro le would be 2 since the acceleration is obtained
by taking the second derivative of the trajectory function. This implies that, the
curvature of the acceleration pro le i.e., its second derivative is restricted to be a
constant. Windows of size 7 and above do not su er from such a limitation.
The curve t procedure was repeated for a subset of vehicles using window sizes
7, 9, 11, 13, and 15. Figure 55 shows the histograms of the absolute values of the
position estimation error corresponding to the measured position of a driver using different window sizes. Although, in this case the mean of the absolute errors increased
with the window size, such phenomenon was not observed when the sensitivity analysis was performed on the trajectories of other vehicles. Considering the 1 meter
96 Pecentage 0.4
window size 7
0.2 Pecentage 0 Pecentage 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 window size 9
0.2 mean error = 0.250 m
0 0.1 0.2 0.3 0.4 0.5 0.6
0.7 0.8 0.9 1 0.4
window size 11
0.2
0 Pecentage 0 0.4 0 mean error = 0.316 m
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4
window size 13
0.2
0 Pecentage mean error = 0.223 m mean error = 0.346 m
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4
window size 15
0.2
0 mean error = 0.355 m
0 0.1 0.2 0.3 0.4
0.5
0.6
0.7
Absolute position estimation error, m 0.8 0.9 1 Figure 55: Histograms of the absolute values of the position estimation error using
di erent window sizes.
accuracy associated with recording vehicle trajectories, the magnitude of errors for
di erent window sizes is within a reasonable range.
The estimated speed and acceleration pro les of the vehicle for di erent window
sizes are shown in Figure 56. Except for the rst and last time periods, the speed and
acceleration estimates do not di er signi cantly. Therefore, speed and acceleration
estimates at these boundary points should not be used in estimating di erent driver
behavior models. On the basis of these plots, any window size between 7 to 15 can be
considered acceptable. Of these sizes, 9 was chosen since it strikes the best balance
between accuracy and computational e ort. Errors for window size greater than or
equal to 9 were very similar especially for the case of stopped vehicles.
Examples
Figure 57 shows two examples of curve tting to the whole trajectory by applying
the local regression procedure described above. In the second example second plot
of Figure 57, the vehicle was stopped for a few seconds. As shown in the gure, a
97 14
12 window size 7
window size 9
window size 11
window size 13
window size 15 Speed, m/s 10
8
6
4
2
0 0 5 10 15
Time period, seconds 20 25 30 25 30 Acceleration, m/s2 5 0
window size 7
window size 9
window size 11
window size 13
window size 15 −5 0 5 10 15
Time period, seconds 20 Figure 56: Estimated speed and acceleration pro les using di erent window sizes.
very good t of the model was obtained in both cases. Data for Estimating the Acceleration Model
The data required for estimating the acceleration model includes acceleration, speed,
headway, and type of the subject vehicle, speed and type of its leader's vehicle, density
ahead4 of the subject, roadway curvature, grade, speed limit of the roadway section,
and pavement surface quality.
The trajectory information described above was used to estimate acceleration,
Although the data collection section is fairly straight, a visibility distance of 100 meters ahead
of the driver was used while computing the explanatory variable density ahead. If, however, the
distance from a vehicle' current position to the downstream end of the data collection site is less
than 100 meters, the density of tra c ahead computed while this distance was greater than 100
meters is used as the density for this case.
4 98 220 200 180 Position of vehicle, meters 160 140 120 100 80 original position
estimated position 60 40 20 0 5 10 15
time period, seconds 20 25 30 220 200 180 Position of vehicle, meters 160 140 120 100 80 60 40
original position
estimated position 20 0 5 10 15 20
25
time period, seconds 30 35 40 Figure 57: Examples of curve tting by local regression.
speed, and space headways. The tapes were also used to collect data on vehicle
length. Since the data was collected only from one site, site speci c explanatory
variables for example, geometric characteristics cannot be used. Observations for
a driver were recorded from the instant the driver reached the upstream end of the
data collection site. A sample of 1647 observations from 402 drivers was used for
estimation.
An acceleration observation was recorded at a time instant such that the data
on the tra c conditions max seconds the maximum reaction time earlier can be
obtained. We adopted the maximum of the range of max to be considered while
estimating the model equal to 4 seconds. This is because, 4 seconds is the most
conservative value suggested in the literature see, for example, Johansson and Rumer
99 percent of cases 1971, Lerner et al. 1995, and Homburger and Kell 1988. Therefore, the rst
acceleration observation was recorded at the 5th second. Furthermore, since reaction
time cannot vary for a given driver observed over a short period of time, explanatory
variables for the reaction time model for example, average front vehicle speed to be
used as a proxy for vehicle travel speed were obtained by averaging the observations
recorded during the rst ve seconds, i.e., before the rst acceleration observation
recorded for the driver.
Figure 58 shows the histograms of the acceleration, subject speed, front relative
20 10 percent of cases 0
−4 percent of cases 0
acceleration, m/s/s 1 2 3 4 0 5 10 15 20 25 speed, m/s 40
20 −5 −4 −3 20 −2
−1
0
1
2
front vehicle speed − subject speed, m/s 3 4 5 10 0 percent of cases −1 60 0 0 2 4 40 6
time headway, sec 8 10 12 20 0 percent of cases −2 10 0 percent of cases −3 20 5 15 25 35 45
55
space headway, meters 65 75 85 95 5 15 25 35 45
55
density, veh/km/lane 65 75 85 95 30
20
10
0 Figure 58: Histograms of the acceleration, subject speed, relative speed, time and
space headway, and density in the data used for estimating the acceleration model.
speed, time and space headways, and density of tra c ahead of the subject of all
100 vehicles in the data. The second plot of Figure 58 shows two regimes: regime one
represents heavily congested tra c with speeds varying between 0 and 12 m s and the
tra c density varying between 40 and 102 veh km lane, and regime two represents
semi congested to uncongested tra c with speeds varying between 12 and 32 m s
and density below 40 veh km lane. Table 5.2 shows more statistics of the data.
Table 5.2: Statistics of the data used for estimating the acceleration model.
accefront
time
space
density
leration speed
Vehicle
headway headway
ahead
m s s m s speed m s sec
m
veh km lane
maximum 7.28
27.1
32.3
15.1
152.1
102.0
minimum 5.73
1.01
0.4
0.1
0.1
0.0
mean
0.12
7.2
7.5
2.4
17.2
64.4
median
0.19
5.7
5.8
2.2
12.1
66.7
std. dev.
1.29
5.2
6.0
1.3
17.2
19.8
number of drivers = 402
number of observations = 1647
percent of acceleration observations = 56.0
mean and standard deviation of all acceleration observations: 1.02, 0.78
mean and standard deviation of all deceleration observations: 1.02, 0.81
percent of heavy vehicle = 19 The speed of the subject vehicle varied from 1 to 27 m s. The tra c density varied
between 7 and 103 vehicles km lane. The time headway varied from a fraction of a
second to 15 seconds while the space headway varied from less than 5 meters to 152
meters. 54.0 of the observations were acceleration observations while the rest were
deceleration observations. 19 of the vehicles were heavy vehicles length greater
than 9.14 meters or 30 feet. The data, therefore, represents a wide range of tra c
conditions. Data for Estimating the Discretionary Lane Changing Model
The data used for estimating the discretionary lane changing model consists of observations from 843 drivers. The total number of gaps observed was 4335, and the
number of discretionary lane changes was 75
101 For each gap and driver, the data provides information on the lead, lag, and front
gaps, vehicle length, speed, and acceleration of the subject, front, lead and lag vehicles
see Figure 59 for de nition of di erent vehicles and gaps, density of tra c in the
total clear gap + vehicle length
lag gap lead gap lag vehicle lead vehicle subject front
vehicle Figure 59: The subject and the front, lead, and lag vehicles.
current and target lanes, and whether the driver merged into this gap.
In some cases, there was no lead vehicle within the data collection site rectangle
area in Figure 53. To obtain the lead gap, the trajectory data from earlier time
periods was searched to nd the last vehicle that crossed the downstream boundary
of the site from the target lane. Assuming that the vehicle continued in the same
lane with the same speed, the lead vehicle's position was extrapolated to the time
period in question to calculate the lead gap. Similarly, when there was no lag vehicle
in the target lane, trajectory data from later time periods was searched to nd the
rst vehicle that entered the upstream end of the target lane. Again, assuming that
the lag vehicle traveled with the speed of its rst appearance during this time span,
the lag vehicle's position was extrapolated backwards to the time period in question
to calculate the lag gap.
Statistics corresponding to the gaps that the drivers merged into i.e., the gaps
that were acceptable to the drivers are shown in Table 5.3. The density of tra c in
the target lane varied from 0 to 85 vehicles km lane. Vehicle speeds varied from 2 to
38 m s with a mean around 18 m s. As before, this represents a wide range of tra c
conditions. 102 Table 5.3: Statistics of the discretionary lane changing model data corresponding to
the gaps that the drivers merged into.
current
target
lead lag front
lead lag front
lane
lane
veh. veh. veh.
gap gap gap density
density speed speed speed speed
m m m veh km veh km m s m s m s m s
max.
192.5 232.5 166.6
84.9
75.8
28.1 37.9 25.7 37.3
min.
1.7
2.5
0.1
0.0
0.0
3.5
5.4
1.9
2.0
mean
35.1 36.1 29.5
24.8
22.3
16.9 20.1 16.8 17.4
median 22.3 26.9 16.2
18.2
16.9
18.5 22.8 19.4 19.3
std. dev. 41.8 34.5 35.0
21.4
16.1
7.0
8.1
6.5
7.9
number of drivers = 843
number of observations = 4335
number of lane change observations = 75
percent of heavy vehicle vehicle longer than 9.14 m = 22 Data for Estimating the Mandatory Lane Changing Model
This data consists of observations from vehicles merging from the on ramp to the
mainline. As mentioned in Section 4.3.4, we assumed that drivers merge by gap
acceptance when the level of service of the roadway section is between A and E and
by gap creation i.e., forced merging when level of service is F. Therefore, to estimate
the mandatory lane changing model, observations were recorded only when the level of
service was between A and E, i.e., density was less than 41 vehicles km lane. A total
of 500 observations was recorded from 202 drivers. For each driver, the observation
includes a series of gaps. The last gap observed by each driver before he she changed
lanes, was considered acceptable, since at the next time period the driver was observed
in the target lane.
The variables of interest for each gap and driver include the lead, lag, and front
gaps, vehicle length, speed, and acceleration of the subject, front, lead and lag vehicles, delay or time elapsed since the subject crossed the merging point between the
on ramp and the freeway section X X in Figure 510, remaining distance to point at
which the lane change must be completed section Y Y in Figure 510, and density
of tra c in the target lane.
103 X total clear gap + vehicle length
lag gap Y lead gap lag vehicle lead vehicle subject front
vehicle X Y Figure 510: The subject, lead, lag, and front vehicles, and the lead and lag gaps.
Similar to the discretionary lane change data, in some cases there was no lead
and or lag vehicle within the data collection site. The technique described in Section 5.2.3 was applied here as well to infer the lead lag gaps for such cases.
Statistics corresponding to the gaps that the drivers merged into i.e., the gaps
that were acceptable to the drivers are shown in Table 5.4. The maximum and
minimum delays in merging were 5 and 0 seconds respectively with a mean and
Table 5.4: Statistics of the mandatory lane changing model data corresponding to
the gaps that the drivers merged into.
rem. adjac.
lead lag
dislane
gap gap delay tance density speed
m m sec. m veh km m s
max.
302.7 188.1 5.0 179.9 33.96
25.90
min
1.2
2.1
0.0 47.6
0.00
9.09
mean
44.8 36.2 1.8 129.5 15.83
18.48
median 28.3 27.4 2.0 135.4 15.92
18.75
std. dev. 44.6 31.4 1.2 26.4
8.36
3.14
number of drivers = 202
number of observations = 500
percent of heavy vehicle vehicle longer than 9.14 m = 4.5 lead
veh.
speed
m s
35.92
6.52
20.86
20.36
4.71 lag
veh.
speed
m s
29.03
7.49
18.10
18.23
3.11 front
veh.
speed
m s
30.95
9.07
20.78
20.94
4.43 standard deviation of 1.8 and 1.2 seconds respectively. The density of tra c in the
target lane varied from 0 to 34 vehicles km lanewhich is rather low and explains
why the drivers experienced lower delay in merging. Vehicle speeds varied from 7 to
104 36 m s with a mean around 20 m s. The remaining distance to the point at which
lane change must be completed varied from 48 to 180m with a mean, median, and
standard deviation of 130, 135, and 26 meters respectively. This implies that, for
a majority of the drivers in this data, remaining distance may not have signi cant
in uence on the merging process. Data for Estimating the Forced Merging Model
The data consists of observations from vehicles merging from the on ramp to the
adjacent mainline. As mentioned in Section 4.3.4, we assumed that drivers merge by
gap creation i.e., forced merging when the level of service of the roadway section is
F. Therefore, observations were recorded only when the level of service was F, i.e.,
the tra c density was more than 41 vehicles km lane. A total of 998 observations
was recorded from 79 drivers. Descriptive statistics of the data corresponding to the
observations for the forced merging model are presented in Table 5.5.
Table 5.5: Statistics of the data used for estimating the forced merging model.
rem. mainline target
lead lag
lead lag
dislane
lane
veh. veh.
gap gap delay tance density
density speed speed speed
m m sec. m veh km veh km m s m s m s
max.
31.8 56.9 28.0 154
72.5
75.8
15.1 12.8 11.6
min
13.2 12.2 0.0
23
41.0
20.2
0.0
0.0
0.0
mean
4.3 7.8 9.5 103
57.9
58.8
5.1
5.2
5.0
median
3.7 3.6 8.0 102
59.0
60.7
4.8
5.1
4.8
std. dev. 6.0 11.9 6.3
27
7.9
10.1
2.8
2.5
2.6
number of drivers = 79
number of observations = 566
percent of heavy vehicle vehicle longer than 9.14 m = 5.1 The variables in the data, for each gap and driver, are the lead and lag gaps,
vehicle length, speed, and acceleration of the subject, lead and lag vehicles, time
elapsed since the subject crossed the merging point between the on ramp and the
freeway section X X in Figure 510, remaining distance to point at which the lane
105 change must be completed section Y Y in Figure 510, and density of tra c in the
mainline lanes and the target lane.
The lead gap varied from 13.25 to 32 meters with a mean of 4.3 meters. The
lag gap varied from 12 to 57 meters with a mean of 8 meters. The mean delay
experienced by the drivers was 9.5 seconds compared to 1.8 seconds observed in the
mandatory lane changing data Table 5.4. The mainline tra c density varied from
40 to 73 vehicles km lane with a mean of 58 vehicles km lane representing a very
congested tra c. The average speed of vehicles was around 5 m s. 5.3 Conclusions
In this chapter, a methodology to estimate instantaneous speed and acceleration that
are required for model estimation from trajectory data that can be obtained from
the eld using video technology is described. The methodology is based on the local
regression procedure of Cleveland and Devlin 1988. The main advantage of this
procedure, over the conventional regression, is that it allows for estimating position,
speed, and acceleration pro les that, otherwise, would require tting polynomials of
a very high order. Although local regression estimates are less e cient, the exibility
of the method outweighs this disadvantage.
The characteristics of the freeway trajectory data collected in the Central Artery,
Boston are also described. The data represents a wide range of tra c conditions,
from very congested stop and go tra c to free ow. The tra c density varied from
no vehicles within the data collection site to 90 vehicles km lane. In addition to the
length of each vehicle that traveled in the section, the data contains position, speed,
and acceleration of every vehicle for every second. Finally, descriptive statistics of
the data used to estimate the acceleration model, the discretionary and mandatory
lane changing models, and the forced merging model are presented. In this case, the lead vehicle and the subject were overlapping and the lead vehicle was a heavy
vehicle.
5 106 Chapter 6
Estimation Results
In this chapter, estimation results of the acceleration and lane changing models, using
the data described in Chapter 5, are presented. Along with the estimation results,
assessment of the parameter estimates from statistical and behavioral standpoints are
also presented.
In addition, parameter estimates of the car following model, the headway threshold and reaction time distributions, and the gap acceptance model under mandatory
lane changing situations are compared to those estimated by other researchers. No
such comparison can be made for parameters of the free ow acceleration model, the
discretionary lane changing model, and the forced merging model since these have
not been estimated before. 6.1 Estimation Results of the Acceleration Model
Given the complexity of the likelihood function, the estimation of the parameters was
simpli ed by estimating the values of h , h , and max non parametrically. The
min max
overall estimation approach was based on the following algorithm:
1. Set fh ; h ;
min max max g to reasonable initial values. 2. Using the current values of fh ; h ; max g specify and estimate the model
min max
using the maximum likelihood method Equation 3.25.
107 3. Estimate the parameters of the model speci cation in step 2 for di erent set
of values of fh ; h ; max g. Through this grid search procedure obtain the
min max
; h ; max g, i.e., the one with the highest likelihood value.
best fhmin max
4. Iterate between steps 2 and 3 until the same set of fh ; h ;
min max max g is obtained. The parameters h ; h ; and max were initially set to 0, 8, and 3 seconds remin max
spectively. Using these values, step two was executed. In this step, we investigated different model speci cations and simultaneously varied the parameter Equation 3.7
between the 0 to 1 range. The likelihood function attained its maximum at = 0.
Step three was performed next by varying h , h , and max . Table 6.1 shows
min max
the values of the maximized likelihood function at di erent values of h , h , and
min max
Table 6.1: Estimated likelihood function for di erent values of h ; h ; and
min max
h h
max min = 0
max = 3 2255.24
max = 3:5 2256.00
max = 4 2263.95 =6 h
min = 0:5
2252.17
2257.49
2266.09
=8
hmax
h
= 0 h = 0:5
min
min
2258.63
max = 3 2254.61
2256.41
max = 3:5 2257.50
2274.23
max = 4 2263.47
max . max . h
min =1
2257.69
2256.25
2259.95 h
min =1
2257.69
2262.38
2265.05 The likelihood function attained the maximum value for max = 3 seconds,
h = 0:5 second, and h = 6 seconds. In the next iteration of step two, the same
min
max
model speci cation was obtained as was used while executing step three. The parameter was varied again between the range 0 to 1. Figure 61 shows the value of the
likelihood function as a function of . As before, the likelihood function attained its
maximum value at = 0. Since, experience with the model estimation indicated that
the likelihood function may not be globally concave, we reestimated the model using
di erent starting values of the parameters to obtain the best possible local maxima.
We obtained the same solution for di erent starting values of the parameters.
108 −2252 −2254 the likelihood function −2256 −2258 −2260 −2262 −2264 −2266 0 0.1 0.2 0.3 0.4 0.5
ξ 0.6 0.7 0.8 0.9 1 Figure 61: The likelihood function as a function of .
Table 6.2 summarizes the estimation results. All the parameters, with the exception of the car following acceleration sensitivity parameters, have the expected signs.
Note that, a positive sign of the parameters of speed and headway in the car following
acceleration or deceleration model Equations 3.2 and 3.7 implies that the variables
are in the numerator and denominator respectively. The signs of the explanatory
variables speed and density of the car following acceleration model are not what
we anticipated. Their t statistics are highly signi cant. This indicates that Boston
drivers may behave di erently than the way we anticipated see Section 3.2.1. The
high acceleration sensitivity at high speeds and high densities indicate that drivers are
more aggressive in this situations. This may be in part due to the drivers e ort not
to let anyone infront of them from an adjacent lane. The positive sign of the space
headway parameter for the car following acceleration model indicates that drivers
tend to follow the speed of the lead vehicle less as the space headway increases.
In the car following acceleration and deceleration models, except the constants,
all the parameters have signi cant tstatistics at the 1 level of signi cance. The
speed parameter for the car following deceleration model had counterintuitive sign
with a t statistic of 0.64, and therefore, was dropped from the speci cation. The
explanatory variable density has signi cant tstatistic for both the acceleration and
deceleration models. Therefore, the proposed enhancement of the sensitivity term
was supported by the data for both the acceleration and deceleration models. 109 Table 6.2: Estimation results of the acceleration model.
Variable Parameter t stat.
Car following acceleration
constant
0.0225
1.08
speed m s
0.722
4.67
space headway m
0.242
6.31
density veh km lane
0.682
4.20
relative speed m s
0.600
7.20
ln cf;acc
0.193
2.64
Car following deceleration
constant
0.0418
1.20
space headway m
0.151
5.32
density veh km lane
0.804
4.21
relative speed m s
0.682
10.71
ln cf;dec
0.221
5.44
Free ow acceleration
sensitivity constant
0.309
7.37
constant
3.28
6.83
front veh. speed m s
0.618
10.04
heavy veh. dummy
0.670
1.54
indicator for density 19 veh km lane
7.60
5.51
ln f f
0.126
1.99
6
Headway threshold distribution, 0:5 h
mean sec
3.17
13.90
h
0.870
3.82
Reaction time distribution, 0
3
constant
0.272
7.62
ln
1.55
9.59
number of drivers = 402
number of observations = 1647
L0 = 2727.75
Lc = 2561.26
L ^ = 2252.17
2 = 0.167
Note: Density 19 veh km lane implies level of services A
through C HCM 1985. 110 The test statistic for the null hypothesis that the stimulus is a linear function of
the lead relative speed for the car following acceleration model i.e., acc = 1 is given
by:
^
acc , 1 = 0:600 , 1 = ,4:79
0:0834
^
varacc q 6.1 Therefore, the null hypothesis can be rejected at the 1 level of signi cance. Similar
null hypothesis for the car following deceleration model i.e., dec = 1 can also be
rejected at the 1 level of signi cance the tstatistic is equal to ,4:99. These imply
that the proposed extension of the the stimulus term to be a nonlinear function of
the lead relative speed is supported by the data.
The free ow acceleration model parameters with one exception have signi cant t
statistics at the 1 level of signi cance. The parameter for the heavy vehicle dummy
does not have a signi cant t statistic. Both the headway threshold distribution parameters are statistically signi cant at the 1 level of signi cance. The mean and
standard deviation of the reaction time distribution are statistically signi cant at the
1 level of signi cance. The adjusted t of the model1 was 0.167.
The acceleration model estimation results for the case of = 1 is presented in
Table 6.3. = 1 implies that the sensitivity is a function of the tra c conditions
at the moment the driver perceives the stimulus and decides that he she should
respond to it. Hence, the explanatory variables a ecting the acceleration sensitivity
are observed at the time instant at which the stimulus is observed. As shown in
Figure 61, this model has a signi cantly lower t than the one in which has the
best t. By relaxing to be a parameter to be estimated, the likelihood function
improved by 11.2 units for the = 0 case over this model the = 1 case. It
is interesting to note that, although the car following acceleration and deceleration
model parameters are of di erent orders of magnitude, the free ow acceleration
model and the headway threshold and reaction time distribution parameters are of
12 ^
parameters .
= 1 , L ,no. of0
L 111 Table 6.3: Estimation results of the acceleration model for = 1.
Variable Parameter t stat.
Car following acceleration
constant
0.468
1.56
speed m s
0.129
1.11
space headway m
0.194
5.44
density veh km lane
0.188
1.57
relative speed m s
0.670
10.48
ln cf;acc
0.253
5.48
Car following deceleration
constant
0.0470
1.25
space headway m
0.179
5.68
density veh km lane
0.791
4.31
relative speed m s
0.749
11.12
ln cf;dec
0.235
6.32
Free ow acceleration
sensitivity constant
0.316
8.31
constant
3.12
7.50
front veh. speed m s
0.611
10.80
heavy veh. dummy
0.638
1.57
indicator for density 19 veh km lane
7.58
5.96
ln f f
0.170
3.59
6
Headway threshold distribution, 0:5 h
mean sec
3.28
13.11
1.08
5.00
h
Reaction time distribution, 0
3
constant
0.307
8.89
ln
1.34
13.68
number of drivers = 402
number of observations = 1647
L0 = 2727.75
Lc = 2561.26
L ^ = 2263.39
2 = 0.163
Note: Density 19 veh km lane implies level of services A
through C HCM 1985. 112 the same order of magnitude. We adopt the model presented in Table 6.2 which has
a signi cantly higher t. 6.1.1 Discussion
The Car Following Models
The estimated car following acceleration model is
0:722
acf;acct = 0:0225 Vntt0:242 knt0:682 jVnt , n j0:600 +
n
X
n cf;acc t
n 6.2 where, t=
n=
Vnt =
Xnt =
knt =
Vnt , n =
cf;acct
n current time period,
reaction time for driver n,
subject speed at time t m s,
space headway at time t m,
density of tra c ahead of the subject veh km lane,
front vehicle speed subject speed m s, N 0; 0:8252: The estimated car following deceleration model is
1
acf;dect = ,0:0418 X t0:151 knt0:804 jVnt , n j0:682 +
n
n cf;dec t
n 6.3 where, cf;dect N 0; 0:8022.
n
Figure 62 shows the sensitivity of di erent factors on the car following acceleration and deceleration. Acceleration increases with speed, density, and relative speed,
and decreases with space headway. On the other hand, deceleration in absolute
term increases with density and relative speed in absolute term, and decreases
with headway.
113 Note: unless varied, speed = 10 m/s, headway = 20 m, density = 40 veh/km/lane, relative speed = 3 m/s acceleration, m/s 2 3
2.5
2
1.5
1
0.5 5 10 15 20 speed, m/s
−1
deceleration, m/s acceleration, m/s 2 2 2.5 2 1.5 1 0 10 20
headway, m 30 −1.2
−1.4
−1.6
−1.8 40 20
headway, m 30 40 30
40
50
density, veh/km/lane 60 2 −0.8 1.6
1.4
1.2
1
0.8
20 30
40
50
density, veh/km/lane −1
−1.2
−1.4
−1.6
20 60 −0.5
deceleration, m/s 2 2 2.5
acceleration, m/s 10 −0.6
deceleration, m/s acceleration, m/s 2 2
1.8 0 2
1.5
1
0.5 1 2 3
4
relative speed, m/s −1 −1.5 −2
−5 5 −4 −3
−2
relative speed, m/s −1 Figure 62: Sensitivity of di erent factors on the car following acceleration and deceleration decisions.
At low speeds the mean acceleration is lower compared to those for higher speeds.
Tra c conditions ahead of the subject and its leader are likely to change more rapidly
at high densities than at low densities. Due to this, higher uncertainty is involved
in predicting the position and speed of the leader in the near future. As a result,
drivers are expected to be more conservative at high densities than at low densities.
Although the mean deceleration increases with density, the mean acceleration does
not decrease with density as we had expected.
The slopes of the acceleration and deceleration curves with respect to the relative
speed are decreasing. This captures the fact that, the acceleration deceleration
applied by a driver is limited by the acceleration deceleration capacity of the vehicle
and acceleration deceleration gradually reaches the capacity as the relative speed
114 increases.
Figure 63 shows a comparison between the estimated car following acceleration
and deceleration at di erent front gaps as a function of subject speed using the models
proposed in this thesis with those obtained by Subramanian 19962. The acceleration
Note: speed = 10 m/s, density = 40 veh/km/lane
5 0 4.5 −0.5
5 m/s 1 m/s 4 5 m/s −1.5
3 m/s deceleration, m/s2 2 3 m/s −1 3.5 acceleration, m/s 1 m/s 3 2.5
5 m/s 2 3 m/s 1.5 1 −2
3 m/s
−2.5 −3 5 m/s −3.5 −4 1 m/s
1 m/s 0.5 0
10 Acc(Estimated)
Acc(Subramanian)
15 20
front gap, m 25 −4.5 30 −5
10 Dec(Estimated)
Dec(Subramanian)
15 20
front gap, m 25 30 Figure 63: Comparison between the car following acceleration and deceleration estimated in this thesis with those obtained by Subramanian 1996.
and deceleration estimated in this thesis are generally smaller in magnitude compared
to those estimated by Subramanian. At low speeds, his acceleration and deceleration
estimates are too high. The acceleration estimated by the model proposed in this
thesis is smaller than expected, while the estimated deceleration is reasonable. This
may be due to lack of variability in the data with acceleration observations or may
be due to the in uence of the geometric characteristics of the Boston data collection
site.
The di erence between the two models may be due to several reasons. First,
Subramanian used data that was collected in 1983 from a section of Interstate 10
Westbound near Los Angeles, whereas, this research used data that was collected in
1995 and 1997 from a section of Interstate 93 Southbound in Boston. The di erent
data collection years and sites may have contributed to the di erences in the estimates.
2 The parameters estimated by Subramanian 1996 are presented in Table 2.3. 115 Second, the LA data collection site is a fairly straight section without any ramps,
whereas, the Boston data collection site has a weaving section adjacent to the freeway.
The geometry of the freeway and the number of lane changes taking place in the
Boston data may have an e ect on the estimates obtained in this research. Finally,
Subramanian assumed that all the drivers in the data were car following even at
large space headways. He further investigated the implication of this assumption and
concluded that the assumption on the headway threshold has signi cant impact on
the car following model estimates. The estimation results presented in this thesis do
not su er from such a limitation. The Free Flow Acceleration Model
The estimated free ow acceleration model is
h
aff t = 0:309 3:28 + 0:618 Vnfrontt , n , 0:670 neavy
n
+7:60 knt , n , Vnt , n + f f t
n 6.4 where, Vnfront t , n = front vehicle speed at time t , n m s,
8
1 if the subject vehicle is a heavy vehicle
heavy =
vehicle length 9.14 m or 30 ft
n
: 0 otherwise
8 1 if knt , n 19 veh km lane
: 0 otherwise
f f t N 0; 1:132:
n knt , n = The estimated free ow acceleration increases with front vehicle leader speed.
A higher acceleration for level of services A through C captures the e ect of higher
maneuverability at low densities compared to high densities. The impact of lower
maneuverability for the heavy vehicles compared its non heavy counterparts is cap116 tured by an indicator whether the subject vehicle is heavy. The standard deviation
of the free ow acceleration is high compared to its car following acceleration and
deceleration counterparts. The Headway Threshold Distribution
The headway threshold seconds, that de nes whether a driver is in the car following
regime or in the free ow regime, is distributed as follows see Figure 64:
0.5 h* pdf 0.4
0.3
0.2
0.1
0 0 1 2 3
*
h , sec. 4 5 6 0 1 2 3
headway, sec. 4 5 6 1 P(car−following) 0.8
0.6
0.4
0.2
0 Figure 64: The headway threshold distribution and the probability of car following
as a function of time headway.
8 f h = : 1 h,3:17 if 0:5
0:868
0:870 h 6
otherwise 0 6.5 For a given headway, hnt, the probability that driver n is in the car following regime
is given by:
8 1
if hnt 0:5 h t,3:17 1
Pcar following at time t = 1:00 , 0:998 n 0:870
if 0:5 hn t 6 6.6
:0
otherwise
117 The 5, 50, and 95 percentile values of the headway threshold are 1.75, 3.17, and 4.60
seconds respectively. These values are reasonable.
Figure 65 shows the mean of the headway threshold as a function of subject
100 90 space headway threshold, meter 80 70 60 50 40 30 20 10 estimated mean threshold
Herman and Potts (1961)
5 10 15 20 25 30 speed, m/s Figure 65: Comparison between the estimated mean headway threshold and the 61
meters threshold suggested by Herman and Potts 1961.
speed. In the Tra c Engineering Literature Herman and Potts 19613 a threshold
of 61 meters 200 ft is usually used to distinguish the free ow regime. As shown in
Figure 65, the 61 meters threshold is too high at low speeds while the threshold estimated in this thesis is high at high speeds. The two estimates are in close agreement
in the 15 to 23 m s speed range. The Reaction Time Distribution
The estimated distribution of reaction time is
8 f =
: 1p
0:212 2
0 ,0
, 1 ln 0.212:272
2
e 2 if 0 3 otherwise 6.7 Herman and Potts 1961 estimated this 61 meters thresholds based on an observation that, the
correlation between the observed accelerations and the accelerations estimated by the car following
model was low when the space headways were greater than 61 meters.
3 118 Figure 66 shows the probability density function and the cumulative distribution
function of the reaction time. The median, mean, and standard deviation of the
probability density function 1.5 1 0.5 cumulative distribution function 0 0 0.5 1 1.5
reaction time, sec. 2 2.5 3 0 0.5 1 1.5
reaction time, sec. 2 2.5 3 100
80
60
40
20
0 Figure 66: The probability density function and the cumulative distribution function
of the reaction time.
reaction time distribution are 1.31, 1.34, and 0.31 seconds respectively.
As discussed in Section 3.2.4, we apriori expect the surrounding tra c conditions
to a ect the reaction time of a driver. Explanatory variables capturing tra c conditions include the density of tra c ahead of the driver, the average front vehicle
speed that was used as a proxy for average travel speed of the subject, whether
the subject vehicle is a heavy vehicle, and an indicator for free ow tra c conditions
density 19 veh km lane. However, the t statistics of these explanatory variables
indicated that their impact on the reaction time were not signi cant and in some
cases the parameters had counterintuitive signs. Therefore, the model with only the
constant as an explanatory variable for the mean of the reaction time distribution
was adopted.
Finally, a comparison between the estimates of the reaction time distribution
parameters obtained in this thesis and those obtained by Johansson and Rumer 1971
and Lerner et al. 1995 is presented in Table 6.4. The median and mean estimated in
119 Table 6.4: Comparison between the reaction time distribution parameters obtained
from di erent sources.
source sample
size stimulus median mean std. dev.
sec sec sec speed
this thesis
402
di erence
1.31 1.34
0.31
Johansson and Rumer 1971 321
sound
0.89 1.01
0.37
unexpected
Lerner et al. 1995
56
object
1.44 1.51
0.39
Note: Speed di erence implies di erence between the target speed and the
current speed. this research are higher than those obtained by Johansson and Rumer 19714, while,
they are lower than those obtained by Lerner et al. 1995. The standard deviation
estimated in this research is smaller than those obtained by others.
The di erences between the reaction time estimates from di erent studies may be
due to the di erences in the time period of study, data collection site, or procedures
used in di erent studies. The acceleration and deceleration capacity of vehicles have
increased over the past 27 years which may have increased the reaction time of drivers
as better vehicle performance may have made driving more relaxing. Driving habits
at di erent locations may have also contributed to di erent reaction time estimates.
Finally, di erent stimulus were used in di erent studies. In the Johansson and Rumer
1971 study drivers responded to sound, in the Lerner et al. 1995 study drivers
responded to visualizing a rolling drum, while in this thesis, drivers responded to the
di erence between their target speeds desired speeds or the leaders' speeds depending
on the headways and the current speeds. Overall, the parameters of the reaction
time distribution estimated in this research are well within the typical range of other
studies.
In summary, the empirical work suggests that, the sensitivity term of the car
following acceleration is a function of the subject speed, the space headway, and the
As mentioned in Chapter 2, drivers responded to sound indicating them to press the brake pedal.
This may have reduced the perception time, and hence the reaction time.
4 120 density, while the sensitivity term of the car following deceleration is a function of the
space headway and the density of tra c. Furthermore, the sensitivity term comprises
of explanatory variables observed at the time of applying acceleration deceleration
while the stimulus term is lagged by the reaction time of the drivers. The impact of
the reaction time on the sensitivity was not supported by the data. The stimulus is a
nonlinear function of the front relative speed. The free ow acceleration is a function
of the subject's speed, the leader's speed, an indicator whether the subject vehicle is
a heavy vehicle, and an indicator whether the density of tra c is low. 6.2 Estimation Results of the Lane Changing Model
The discretionary and mandatory lane change models were estimated separately due
to lack of data over a long stretch of roadway approximately 1500 to 3000 meters
long. The data collection site, shown in Figure 67, used in this study has a length
of approximately 200m. If a driver in this site changes to the right lane and takes
the exit, it is unlikely that the driver is also performing a discretionary lane change.
However, if the remaining distance to the exit is 2000 meters as opposed to 200 meters, the probability of performing a discretionary lane change may not be negligible.
Therefore, a model that captures discretionary lane changing decision when the driver
is in a mandatory lane change situation cannot be estimated using this data.
Estimation results of the discretionary and the mandatory lane changing models are presented rst. Then, estimation results of the forced merging model are
presented. 6.2.1 Estimation Results of the Discretionary Lane Changing
Model
The discretionary lane changing model was estimated using observations from drivers
in the following two cases see Figure 67 for de nition of lanes 1 to 4:
drivers that changed from lanes 2 or 3 to the left, and
121 lane 1 lane 2 lane 3 lane 4 200 m I93 SB 402 m
(1/4 mile) South Station
Exit 402 m
(1/4 mile) China Town
Exit Mass. Pike
Exit Figure 67: Schematic diagram of the I 93 southbound data collection site gure not
drawn to scale.
drivers that traveled in lanes 2 or 3 without changing lanes.
If drivers from lanes 1 to 3 change to the right and take the exit at the downstream end
of the data collection site, the lane changes would be mandatory. Even if they do not
take this exit, since there are two exits a quarter mile and a half mile downstream, it
is likely that drivers would be changing lanes to take these exits. Since drivers are not
observed downstream of the data collection site, the upstream lane changes towards
lane 4 cannot be categorized with certainty as discretionary lane changes. Therefore,
the choice set for the discretionary lane change subjects includes the left adjacent and
the current lanes.
122 Thus, there are two observable states: change to the left lane and continue in the
current lane. The discretionary lane changing decision tree then reduces to the
decision tree shown in Figure 68. For this decision tree, the likelihood function given
by Equation 4.8 reduces to:
Start driving
conditions not
satisfactory left
lane driving
conditions
satisfactory current
lane Gap
Accept Gap
Reject Left
Lane Current
Lane Current
Lane Current
Lane Figure 68: The decision tree for a driver considering a discretionary lane change with
the current and the left lanes as choice set. L= N
X
n=1 ln Z Tn
1 Y
L
Pt L j tn Pt C
,1 t=1 L
j 1, tn ! f d 6.8 where,
8 L
tn = : 1 if driver n changes to the left lane at time t
0 otherwise. 6.9 Pt L j n =
Pt gap acceptable j left lane; driving conditions not satisfactory; n
Pt left lane j driving conditions not satisfactory; n
Pt driving conditions not satisfactory j n
Pt C j n = 1 , Pt L j n
123 6.10
6.11 The expression for the gap acceptance probability is given by Equation 4.4. The
conditional probability that the left lane is chosen is given by:
Pt left lane j driving conditions not satisfactory; n =
1
LL
1 + exp,Xn t LL , LLn 6.12 where, superscript `LL' denotes left lane. Finally, the conditional probability that
the driver is not satis ed with the driving condition of the current lane is given by:
Pt driving conditions not satisfactory j n =
1
DCNS t DCNS ,
1 + exp,Xn DCNS n 6.13 where, superscript `DCNS' denotes driving conditions not satisfactory. Estimation Results
Table 6.5 shows the estimation results of the discretionary lane changing model. At
convergence, the hessian of the the likelihood function did not invert since it was
nearly singular. The estimates of the standard deviation of the generic random terms
of the lead and lag critical gaps lead;dlc, lag;dlc were close to zero. Nearly singular
hessian and zero estimates of the standard deviations indicate identi cation problems
of the model. Next, we estimated a restricted version of the likelihood function in
which the serial correlation between di erent observations from a given driver is not
modeled. In this case, the likelihood function converged with a negative de nite
hessian at convergence as desired and the estimates of the standard deviation of the
generic random terms were reasonable. Further research is required to address the
identi cation problem mentioned above and this is left as a topic for future research.
The restriction of no serial correlation implies that, DCNS in Equation 6.13,
LL in Equation 6.12, and g ; g 2 flead; lag g in Equation 4.3 are restricted to be
zero, and the model formulation becomes a cross sectional one. The test statistic
124 Table 6.5: Estimation results of the discretionary lane changing model.
Variable Parameter Desired Speed Model
average speed, m s
0.727
Utility of Driving Conditions not Satisfactory
constant
0.0343
subject speed desired speed, m s
0.0757
heavy vehicle dummy
3.56
tailgate dummy
0.486
DCNS
1.11
Utility of the Left Lane
constant
1.87
lead veh. speed desired speed, m s
0.0328
front veh. speed desired speed, m s
0.158
lag veh. speed subject speed, m s
0.0960
LL
0.246
Lead Critical Gap
constant
0.665
min0, lead veh. speed subject speed, m s
0.412
lead
0.727
lead;dlc
ln
7.16
Lag Critical Gap
constant
1.69
min0, lag veh. speed subject speed, m s
0.172
max0, lag veh. speed subject speed, m s
0.177
lag
0.653
lag;dlc
ln
15.5
number of drivers = 843
number of observations = 4335
number of discretionary lane change observations = 75
L0 = 482.92
Lc = 360.05
L ^ = 326.51
2 = 0.282
Note: di erent vehicles and gaps are de ned in Figure 69. 125 ,2Lrestricted , Lunrestricted is distributed 2 with degrees of freedom equal to the number of restrictions.
The test statistic for the null hypothesis of no serial correlation, i.e.,
DCNS = LL = lead = lag = 0; 6.14 is given by: ,2Lrestricted , Lunrestricted = ,2 ,330:57 , ,326:51
= 8:12 6.15 The critical value of the 2 distribution with 4 degrees of freedom at the 5 level of
signi cance is 9.49. Therefore, the null hypothesis of no serial correlation cannot be
rejected and we adopt the model with no serial correlation.
Table 6.6 shows the parameter estimates obtained by maximizing the restricted
likelihood function. The factors a ecting a driver's decision whether the driving
conditions are satisfactory are the di erence between the subject speed and their
desired speed, an indicator whether the subject vehicle is a heavy vehicle, and an
indicator whether the subject is tailgated. See Figure 69 for de nition of di erent
vehicles and gaps.
total clear gap + vehicle length
lag gap lead gap lag vehicle lead vehicle subject front
vehicle Figure 69: The subject and the front, lead, and lag vehicles.
The desired speed model is assumed to have the following functional form:
DS
Vn t = Xn t 126 DS 6.16 Table 6.6: Estimation results of the discretionary lane changing model.
Variable Parameter t stat. Desired Speed Model
average speed, m s
0.768
Utility of Driving Conditions not Satisfactory
constant
0.225
subject speed desired speed, m s
0.0658
heavy vehicle dummy
3.15
tailgate dummy
0.423
Utility of the Left Lane
constant
2.08
lead veh. speed desired speed, m s
0.0337
front veh. speed desired speed, m s
0.152
lag veh. speed subject speed, m s
0.0971
Lead Critical Gap
constant
0.508
min0, lead veh. speed subject speed, m s 0.420
ln lead;dlc
0.717
Lag Critical Gap
constant
2.02
min0, lag veh. speed subject speed, m s
0.153
max0, lag veh. speed subject speed, m s
0.188
ln lag;dlc
0.642
number of drivers = 843
number of observations = 4335
number of discretionary lane change observations = 75
L0 = 482.92
Lc = 360.05
L ^ = 330.57
2 = 0.282
Note: di erent vehicles and gaps are de ned in Figure 69. 127 5.37
0.17
0.62
3.18
1.71
2.53
0.76
3.12
1.84
1.53
3.73
1.45
5.00
1.29
1.69
1.67 where,
DS
Xn t = explanatory variables a ecting the desired speed DS ,
DS = model parameters. Note that, a constant for the desired speed cannot be estimated since it is absorbed
into the constants of the utilities of the decisions `driving conditions not satisfactory'
and `left lane'. The explanatory variables used for the desired speed model include the
average speed of the vehicles ahead of the subject, the speed of the front vehicle, the
density of tra c ahead of the subject, and an indicator whether the subject vehicle
is a heavy vehicle. A higher average speed of the vehicles ahead or the front vehicle
speed and a lower density of tra c are expected to increase the desired speed of the
driver. Due to lack of maneuverability and safety concern, driver of a heavy vehicle
is expected to have lower desired speed than its non heavy counterpart. However,
except for the average speed, the t statistics of the other explanatory variables were
insigni cant and some of parameters had counterintuitive signs. In the nal model
only the average speed of the vehicles ahead of the subject was used which had a
signi cant t statistic. To capture the e ect of tailgating which cannot be observed
from the data, a proxy variable tailgate dummy is de ned as follows:
8 tailgate t =
n
: 1 if gap behind the subject's rear bumper 10 m and
tra c level of service is A, B, or C
0 otherwise 6.17 t
where, nailgate t denotes the tailgate dummy.
A speed above the desired speed implies satisfaction with the current lane, since
in this situation a driver has the exibility to adjust its speed. On the other hand,
a speed below the desired speed would motivate a driver to perform a discretionary
lane change. The corresponding parameter has the desired negative sign. Although,
its t statistic is not signi cant, it is included in the model due to its importance
from a behavioral standpoint. Due to lack of maneuverability, heavier vehicles are 128 hesitant toward changing lanes and the corresponding parameter has a signi cant
t statistic. Finally, when tailgated, drivers tend to seek discretionary lane change
and the corresponding parameter has the desired positive sign and its t statistic is
signi cant at the 10 level of signi cance.
Factors a ecting the decision whether the left lane is more desirable than the
current lane include the di erence between the lead vehicle's speed and the subject's
desired speed, the di erence between the front vehicle's speed and the subject's desired speed, and the di erence between the subject speed and the speed of the lag
vehicle. A higher lead or front vehicle's speed implies higher exibility for the subject
in the corresponding lanes. The lag relative speed captures the e ect of safety concern
to perform a lane changing decision and its parameter has a signi cant t statistic at
the 10 level of signi cance.
The only factor a ecting the discretionary lead critical gap is the lead relative
speed only when the lead vehicle is slower. Its parameter is statistically signi cant
at the 1 level of signi cance. For the lag critical gap, the lag relative speed is
the only important factor. To capture the di erent impact of the lag relative speed
depending on whether the lag vehicle is faster or not, a piecewise linear approximation
of the lag relative speed with a breakpoint at 0 m s is used. The variable max0,
lag vehicle speed  subject speed has a signi cant t statistic at the 10 level of
signi cance while the variable min0, lag vehicle speed  subject speed does not have
a signi cant t statistic. In spite this, the latter variable is included in the model due
to its importance from a behavioral standpoint. Higher sensitivity of the lag critical
gap when the lag vehicle is faster is captured by the higher parameter estimates of the
variable max0, lag vehicle speed subject speed compared to the variable min0,
lag vehicle speed subject speed.
The adjusted t of the model was 0.282. To test the null hypothesis that all the
parameters except the constants and standard deviations are zero, the likelihood ratio
test was used. The test statistic is given by: 129 ^
,2LC , L ^ = ,2,330:57 , ,360:05
= 58:96 6.18 The critical value of the 2 distribution with 9 degrees of freedom at the 5 level of
signi cance is 16.92. Hence, the null hypothesis can be rejected.
The estimated probability that driver n is not satis ed with the current lane
driving conditions not satisfactory at time t is given by:
Pt driving conditions not satisfactory =
1
,0:225+0:0658 Vn t,Vn t+3:15
1+e h
t
neavy ,0:423 nailgate t 6.19 The conditional probability of choosing the left lane over the current lane is given by:
Pt left lane j driving conditions not satisfactory =
1
2:08,0:0337 Vnlead t,Vn t+0:152 Vnfrontt,Vn t+0:0971 Vnlag t
1+e 6.20 where, Vnlag t denotes the lag vehicle speed minus the subject speed m s.
The estimated lead and lag critical gaps in meters for the discretionary lane
change case are Glead;dlct = exp 0:508 , 0:420 min0; Vnlead t + lnead;dlct
cr;n
Glag;dlct = exp 2:02 + 0:153 min0; Vnlag t +
cr;n
0:188 max0; Vnlag t + lnag;dlct
where,
Vnleadt = lead vehicle speed subject speed m s,
lead;dlc t N 0; 0:4882;
n
130 6.21
6.22 lag;dlc t
n N 0; 0:5262: Another way of assessing the estimated parameters is to compute the probability of
acceptance of gaps that drivers merged into and hence were acceptable to them. There
were 75 such cases. These estimates should be higher than 0.5 and close to 1.0. The
estimated probability had a mean of 0.83 and a standard deviation of 0.25. Figure 610
shows the histogram and cumulative distribution of the estimated probabilities. For
60 Percentage 50
40
30
20
10
0 0 0.1 0.2 0.3 0.4
0.5
0.6
Estimated Probability 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4
0.5
0.6
Estimated Probability 0.7 0.8 0.9 1 Cumulative Percentage 100
80
60
40
20
0 Figure 610: The estimated probability of acceptance of gaps that were acceptable
and merging were completed.
the majority of the gaps actually accepted, the estimated probability of acceptance
was close to one. On 88 of the cases, the estimated probabilities were greater than
0.5.
Finally, Figure 611 shows the median lead and lag critical gaps for DLC situations as a function of the lead and lag relative speeds. When both the lead and lag
relative speeds are zero, the median lead and lag critical gaps are 1.7 and 7.5 meters
respectively. This is intuitive since the lag gap acceptance process is more critical
than the lead gap acceptance process. The median lead critical gap decreases from
13.5 to 1.7 meters as the lead relative speed increases from 5 m s to 0 m s. As the
131 Median Lead Critical Gap Length, m 10 Median Lag Critical Gap Length, m 40 8
6
4
2
0
−5 −4 −3 −2
−1
0
lead vehicle speed − subject speed, m/s 1 2 30 20 10 0 −4 −2 0
2
4
lag vehicle speed − subject speed, m/s 6 8 Figure 611: The median lead and lag critical gaps for discretionary lane change as a
function of relative speed.
lag relative speed increases from 5 to 8 m s, the median lag critical gap increases
from 3.5 to 49.2 meters. These numbers are realistic from a behavioral standpoint.
In summary, drivers' decision to perform a discretionary lane change is modeled as
a two step decision process. First, drivers examine their satisfaction with the driving
conditions of the current lane. Important factors a ecting such decision include the
di erence between the current speed and the driver's desired speed, an indicator
whether the subject vehicle is a heavy vehicle, and an indicator whether the subject
is tailgated. If the driver is not satis ed with the the driving conditions of current
lane, he she compares the driving conditions of the current lane with those of the
other lanes. Such a decision is in uenced by the the speeds of the vehicles ahead in
di erent lanes compared to the subject's desired speed and the lag relative speed.
The lead critical gap is a function of the lead relative speed only when the leader is
slower while the lag critical gap is a function of the lag relative speed. The importance
on the decision to perform a discretionary lane change of other explanatory variables,
such as the relative density of tra c in di erent lanes, whether the lead or the lag
vehicle is heavy, and whether the lane is adjacent to an on ramp, was not supported
by the data.
132 6.2.2 Estimation Results of the Mandatory Lane Changing
Model
The mandatory lane changing model parameters are estimated using observations
from the drivers who merged from the on ramp lane 4 in Figure 67 to the adjacent
mainline lane lane 3. The data consists of observations from drivers when the level
of service of the roadway section was between A and E. For such drivers, the decision
tree shown in Figure 41 reduces to the decision tree shown in Figure 612.
Start MLC MLC Left Lane
Gap
Accept Gap
Reject Left
Lane Current
Lane Current
Lane Figure 612: The decision tree for a driver merging from an on ramp to the adjacent
mainline lane.
In this case, the left and the current lanes are the two choices. Thus, there are
two observable states: change to the left lane and continue in the current lane. For
this decision tree, the likelihood function given by Equation 4.8 reduces to: L= N
X
n=1 ln Z Tn
1Y
L
Pt L j tn Pt C
,1 t=1 L
j 1, tn ! f d 6.23 L
where, tn is de ned in Equation 6.9 and the conditional probability of an observation
of driver n changing to the left lane is given by: 133 Pt L j n = Ptgap acceptable j MLC; nPtMLC j n 6.24 The probabilities on the right hand side of Equation 6.24 are given by Equations 4.4
and 4.1 respectively. The probability of staying in the current lane is given by:
PtC j n = 1 , PtL j n 6.25 Estimation Results
The maximum likelihood estimation results of the mandatory lane changing model are
given in Table 6.7. All the parameters that capture the correlation between di erent
Table 6.7: Estimation results of the mandatory lane changing model.
Variable Parameter t stat.
Mandatory Lane Change Utility
constant
0.740
1.75
rst gap dummy
0.884
2.36
delay sec
0.749
1.36
M LC
0.685
0.65
Lead Critical Gap
constant
0.414
0.79
lead
0.676
1.21
ln lead;mlc
1.07
0.26
Lag Critical Gap
constant
0.663
0.97
min 0, lag veh. speed subject speed m s 0.0457
0.24
max 0, lag veh. speed subject speed m s
0.363
2.59
lag
0.330
0.69
lag;mlc
ln
0.0101
0.03
number of drivers = 202
number of observations = 500
L0 = 336.16
Lc = 334.83
L ^ = 288.19
2 = 0.107
134 observations from a given driver i.e., M LC in Equation 4.1, and g ; g 2 flead; lagg
in Equation 4.3 are statistically insigni cant. To test the null hypothesis of no serial
correlation, i.e.,
M LC = lead = lag = 0; 6.26 a restricted version of the likelihood function with no serial correlation was estimated.
These restrictions make the model formulation a cross sectional one. The test statistic
is ,2Lrestricted , Lunrestricted = ,2 ,288:45 , ,288:19
= 0:53 6.27 The critical value of the 2 distribution with 3 degrees of freedom at the 5 level of
signi cance is 7.81. Therefore, the null hypothesis of no serial correlation cannot be
rejected.
The parameters obtained by maximizing the restricted likelihood function are
shown in Table 6.8. Note that, except for the parameters that correspond to standard deviation of the random terms, the parameter estimates of the unrestricted and
restricted models are of the same order of magnitude. Factors a ecting a driver's
decision to respond to mandatory lane change situation MLC are delay time since
the driver crossed the merging point, section X X in Figure 613 and the indicator
for the rst gap when delay is equal to zero5 . The parameters have the expected
signs and signi cant t statistics at the 5 level of signi cance.
The lead critical gap was found to be insensitive to the tra c conditions, whereas,
the lag critical gap length is sensitive only to the lag relative speed. Similar to the
discretionary lag critical gap model, a piecewise linear lag relative speed variable
was used with a breakpoint at 0 m s was used. As expected, the parameter for the
As explained in Section 4.2.1, delay or the rst gap dummy cannot be de ned for general
mandatory lane changing cases unless the time at which the driver is in M LC state is well de ned.
5 135 Table 6.8: Estimation results of the mandatory lane changing model.
Variable Parameter t stat.
Mandatory Lane Change Utility
constant
0.654
2.27
rst gap dummy
0.874
2.50
delay sec
0.577
3.85
Lead Critical Gap
constant
0.384
0.63
ln lead;mlc
0.152
0.29
Lag Critical Gap
constant
0.587
0.79
min 0, lag veh. speed subject speed m s 0.0483
0.23
max 0, lag veh. speed subject speed m s
0.356
2.39
ln lag;mlc
0.0706
0.14
number of drivers = 202
number of observations = 500
L0 = 336.16
Lc = 334.83
L ^ = 288.45
2 = 0.115
explanatory variable max0, lag vehicle speed subject speed was higher than that
for min0, lag vehicle speed subject speed. The remaining distance did not a ect
a driver's decision processnot an intuitive result. This may be due to the fact
that, in the data the mean and median remaining distances were 130 and 135 meters
respectively and level of service HCM 1985 varied between A and C. As a result,
the remaining distance may not have a signi cant impact on the merging behavior
of the sample drivers in the data. The lead and lag critical gap parameters have low
t statistics except for the lag relative speed when the lag vehicle is faster.
The adjusted t of the model was 0.115. A likelihood ratio test was conducted
to test the null hypothesis that all the parameters except the constants and standard
^
deviations are zero. The test statistic, ,2LC , L ^, is equal to 92.76 and the
critical value with 4 degrees of freedom at the 5 level of signi cance is 9.49. Hence,
the null hypothesis can be rejected.
136 X Y total clear gap + vehicle length
lag gap lead gap lag vehicle lead vehicle front
vehicle subject X Y Figure 613: The subject, lead, lag, and front vehicles, and the lead and lag gaps.
The estimated probability that a driver would respond to an MLC situation is
Pt MLC = 1
1
1 + exp0:654 , 0:577 delaynt + 0:874 nstGapt 6.28 where, delaynt = time elapsed since an MLC situation arises sec,
8
1 if delay = 0
1stGap t =
n
: 0 otherwise.
Figure 614 shows the probability of responding to MLC as a function of delay. The
1 0.9 0.8 Probability of MLC 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 2 3 4 5
6
delay, seconds 7 8 9 10 Figure 614: The probability of responding to MLC as a function of delay.
estimated probability approaches one as delay increases beyond 10 seconds and is
137 higher than expected. This may be due to lack of data with larger delay experienced
by the drivers in the data set the median and the maximum delay experienced by
the drivers in the data were 2 and 5 seconds respectively.
The estimated lead and lag critical gaps in meter for mandatory lane change
situations are Glead;mlct = exp0:384 + lnead;mlct
cr;n
Glag;mlct = exp0:587 + 0:0483 min0; Vnlag t +
cr;n
0:356 max0; Vnlag t + lnag;mlct 6.29
6.30 where, Vnlag t denotes the lag relative speed m s, lnead;mlct N 0; 0:8592, and
lag;mlc t N 0; 1:072.
n
The probability of acceptance of gaps that drivers merged into and hence were
acceptable to them was calculated using the estimated parameters. On 72 of the
cases, the estimated probabilities were greater than 0.9. Figure 615 shows the histogram and cumulative distribution of the estimated probabilities. The estimated
70
60 Percentage 50
40
30
20
10
0 0 0.1 0.2 0.3 0.4
0.5
0.6
Estimated Probability 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4
0.5
0.6
Estimated Probability 0.7 0.8 0.9 1 Cumulative Percentage 100
80
60
40
20
0 Figure 615: The estimated probability of acceptance of gaps that were acceptable
and merging were completed.
138 probability had a mean of 0.93 and a standard deviation of 0.12. These results are
satisfactory.
The median value of the lag critical gap for the mandatory lane change situation
was calculated for di erent values of lag relative speed, and the variation is shown in
Figure 616. When the subject is faster than the lag vehicle in the target lane, the
40 35 Median Lag Critical Gap Length, m 30 25 20 15 10 5 0 −4 −2 0
2
4
lag vehicle speed − subject speed, m/s 6 8 Figure 616: The mean lag critical gap for mandatory lane change as a function of
lag relative speed.
median lag critical gap is less than 2 meters. The median lag critical gap increases
at an exponential rate to 32 meters as the lag relative speed increases to 8 m s. The
median lead critical gap was 1.5 meters.
Ahmed et al. 1996 estimated the median lead and lag critical gaps to be 4.7 and
15.6 meters respectively assuming a 152 meters remaining distance and the gap is the
rst gap observed by the driver6 . Although, this research and Ahmed et al. 1996
used the same methodology to estimate the gap acceptance model, the di erences in
the estimates may be due to the di erences in the data collection years or sites. From
1983 to 1995 vehicle characteristics have improved KBB 1998 and driving habits of
the drivers of these two areas may be di erent which may have contributed to the
Ahmed et al. 1996 used a data collected in 1983 from a site at Interstate 95 Northbound near
the Baltimore Washington Parkway Smith 1985. The site is a two lane freeway with an adjacent
weaving section on the right. Density of tra c varied from 3 veh km lane to 56 veh km lane with
a mean of 27 veh km lane. The median length of the lead and lag gaps corresponding to the gaps
that the drivers found acceptable and completed the merge were 21 and 25 meters respectively.
6 139 Median Lead Critical Gap Length, m 10 Median Lag Critical Gap Length, m di erences in the critical gap length estimates.
The estimated median critical gap lengths under MLC situations are also compared to their DLC counterparts as shown in Figure 617. As expected, the median
critical lead lag gaps under MLC situations are smaller than their DLC counterparts. 40 DLC situation
MLC situation 8
6
4
2
0
−5 −4 −3 −2
−1
0
lead vehicle speed − subject speed, m/s 1 2 DLC situation
MLC situation 30 20 10 0 −4 −2 0
2
4
lag vehicle speed − subject speed, m/s 6 8 Figure 617: Comparison between the estimated critical gap lengths under DLC and
MLC situations. 6.2.3 Estimation Results of the Forced Merging Model
The forced merging model was estimated for the case of merging from an on ramp.
Since, the forced merging model is assumed to be applicable only in heavily congested
tra c, the data consisted of observations from drivers when the level of service of the
roadway section was F.
The estimation results are shown in Table 6.9. The parameter F M in Equation 4.9 that captures the correlation between di erent observations from the same
driver, was estimated to be 0.0012, which is very small. Its t statistic was 0.001.
This implies that, the dynamic aspect of driver behavior may be adequately captured by state dependence and that the random term of the utility speci cation is
independent over time, even for the same driver. The estimation results presented
in Table 6.10 correspond to the parameter estimates obtained by maximizing the
140 Table 6.9: Estimation results of the forced merging model.
Variable
Parameter t stat.
constant
3.16
10.59
min0, lead veh. speed subject speed m s
0.313
2.66
remaining distance impact 10
2.05
5.33
total clear gap 10 meters
0.285
2.85
FM
0.0012
0.001
number of drivers = 79
number of observations = 566
L0 = 306.5
Lc = 112.3
L^ = 88.5
2 = 0.695
likelihood function with no serial correlation.
The probability that the nth driver will switch from state M to state M at time
t is given by:
PSnt = M j Snt , 1 = M =
1 + exp 3:16 , 0:303 1 f Vnld t , 2:05 f Lrem t , 0:285 Gnt 6.31
n where, Snt
f Vnldt
Vnldt
f Lremt
n
Lrem t
n = state of driver n at time t,
= min0; Vnldt;
= lead vehicle speed subject speed m s,
= remaining distance impact;
= remaining distance to the point at which lane change must be completed by,
Gnt = lead gap plus lag gap m.
141 Table 6.10: Estimation results of the forced merging model.
Variable
Parameter t stat.
constant
3.16
10.59
min0, lead veh. speed subject speed m s
0.313
2.66
remaining distance impact 10
2.05
5.33
total clear gap divided by 10 meters
0.285
2.85
number of drivers = 79
number of observations = 566
L0 = 306.5
Lc = 112.3
L^ = 88.5
2 = 0.698
State M is de ned as the situation in which a driver intends to merge into the adjacent
gap in the adjacent lane and perceives that his her right of way has been established
and thus starts merging.
The explanatory variable min0, lead vehicle speed subject speed captures the
fact that, if the subject is interested in merging into the adjacent gap, it would slow
down to match the leader's speed to better focus on the interaction with the lag
vehicle. The variable should always be nonpositive and its estimate re ects such
behavior. The explanatory variable total clear gap has the desired positive sign.
The variable remaining distance impact, a function of the remaining distance,
is used to capture the fact that the remaining distance does not impact a driver's
merging behavior when it is greater than a certain threshold, while at small values,
drivers become more concerned and hence more aggressive. The variable remaining
distance impact for driver n at time t is assumed to have the following functional
form:
1
remaining distance impact = 1 , 1 + eFM Lremt
n 6.32 where, FM is parameter. For di erent values of FM the likelihood function Equation 4.19 was maximized and the value of FM that corresponds to the highest
142 maximum likelihood value was adopted. FM was estimated to be 0.027.
Figure 618 shows how the explanatory variable remaining distance impact, the
Rem. Distance Impact 5
4
3
2
1
0 0 20 40 60 80
100
120
Remaining Distance, m 140 160 180 200 0 20 40 60 80
100
120
Remaining Distance, m 140 160 180 200 0 20 40 60 80
100
120
Remaining Distance, m 140 160 180 200 10 utility 5 0 −5 1 P(state = M) 0.8
0.6
0.4
0.2
0 Figure 618: Remaining distance versus explanatory variable remaining distance impact, the utility function, and the estimated probability of being in state M .
utility, and the estimated probability of being in state M change as a function of the
remaining distance assuming a zero lead relative speed and a 5m clear gap. Drivers'
increasing desperation to complete the merge as remaining distance decreases is shown
in the middle plot of Figure 618the utility increases at an increasing rate as a driver
approaches the end of the acceleration lane and the probability of being in state M
approaches unity. The variable remaining distance impact, as shown in the rst plot
of Figure 618, increases from 0 to 5 as the remaining distance decreases from greater
than 150 to 0 meters.
All parameters have signi cant tstatistics. The adjusted t of the model was
0.698. In addition, a likelihood ratio test was used to test the null hypothesis that
the parameters of all the explanatory variables except the constant are zero. The
statistic, 2Lc , L^, is equal to 47.54. The chi square critical value for 3 degrees
143 of freedom at the 5 level of signi cance is 7.81. Hence, the null hypothesis can be
rejected.
Note that, the variable lag relative speed was not included in the model since the
sign of the corresponding parameter was counter intuitive. The use of the explanatory
variable delay time elapsed since crossing the merging point between the on ramp
and the freeway was also not supported by the data may be because the e ect is
captured by the explanatory variable remaining distance impact. In addition, the
vehicle type heavy vehicle or not did not have any impact on the forced merging
behavior. This may be due to lack of observations since only 4 vehicles out of the 79
samples were heavy vehicles. 6.3 Conclusions
The estimation work presented in this chapter shows that, the sensitivity term of
the car following acceleration is a function of the subject speed, the space headway,
and the density of tra c, and the sensitivity term of the car following deceleration
is a function of the space headway and the density of tra c. The sensitivity lag
was estimated to be zero. In other words, the sensitivity is a function of tra c
conditions observed at the time instant at which acceleration is applied. In both
cases, the stimulus is a nonlinear function of the front relative speed. Enhancements
of the car following sensitivity and stimulus proposed in this thesis were supported
by the data. The mean of estimated free ow acceleration increases with front vehicle
speeds. Heavier vehicles tend to apply slower acceleration due to physical limitations.
In free ow tra c conditions i.e., density 19 veh km lane or level of service A
through C, drivers are expected to apply a higher acceleration and the parameter
of the indicator for free ow tra c conditions has the desired sign and a signi cant
t statistic.
The headway threshold distribution has a mean and standard deviation of 3.17 and
0.87 seconds respectively. The median, mean, and standard deviation of the reaction
time distribution are estimated to be 1.31, 1.34 and 0.31 seconds respectively.
144 The parameters of the discretionary and mandatory lane changing models were
estimated separately due to lack of appropriate data. The estimated median lead and
lag critical gap lengths under mandatory lane changing situations were lower their
discretionary lane changing situation counterparts.
Important factors that a ect forced merging behavior include the lead relative
speed only when the lead vehicle is slower, remaining distance to the point at which
the lane change must be completed by, and total clear gap reduced by the subject
vehicle length.
Finally, it must be stated that the estimation results presented in this section
were obtained using data from a particular freeway segment. Due to the curvature
of the roadway upstream of the data collection site, presence of the weaving section,
and two exits downstream, the behavior of drivers, while driving in this area, may
be in uenced by these conditions. This can be addressed by estimating the models
using data from di erent sites with di erent geometrical con gurations. This will
also address the issue of applicability of the estimation results to general networks. 145 Chapter 7
Model Validation Using a
Microscopic Tra c Simulator
In this chapter, the acceleration and lane changing models are evaluated through
their use in a microscopic tra c simulator, MITSIM1 . Some basic information about
MITSIM is given in Section 7.1. Two new versions of MITSIM were created: MITSIM
with only the acceleration model replaced MITSIM ONE, and MITSIM with both
the acceleration and lane changing models replaced MITSIM TWO. Tra c in a
small network in Boston was simulated using the original version of MITSIM and
MITSIM ONE and TWO. Actual tra c counts at di erent locations of the network
were collected during the morning peak hours. The actual counts, aggregated over
ve minute intervals, were compared to their simulated counterparts to assess the
performance of the estimated models.
This chapter begins with a brief description of MITSIM with emphasis on the
acceleration and lane changing models implemented in the original version. The
validation methodology and the case study are presented next.
A detailed description of MITSIM can be found in the World Wide Web at the URL
http: its.mit.edu products mitsim mitsim.html, or in Yang and Koutsopoulos 1996 or Yang
1997.
1 146 7.1 MITSIM: a Microscopic Tra c Simulator
MITSIM is a microscopic tra c simulation model that represents the road network,
surveillance system, tra c signs and signals, and the control logic in detail. Each
lane is represented with its geometric characteristics for example, curvature, grade,
connectivity, its functional classi cation for example, freeway, ramp, local street,
tunnel or at grade, speed limit, and lane use regulations. Loop detectors, lane
use signals, and variable message signs are simulated in MITSIM. The control logic
supported by MITSIM includes ramp metering, mainline metering, urban control,
etc.
In addition to the network, a time dependent origin destination trip table and
the tra c control and route guidance logic are input to the simulator. Vehicles travel
through the network between their origins and destinations. The simulator collects
the sensor readings, individual vehicle speci c trajectory and trip information, and
average link and path travel times to provide measures of e ectiveness required for
system evaluation. The sensor readings include tra c counts, occupancies, and speeds
at a given frequency e.g. every 5 or 10 minutes.
The travel behavior of the driver is captured by a route choice model. The route
choice model captures drivers' route selection process which is in uenced by tra c
information through variable message signs, highway advisory radio, on board navigation systems, etc. In the route selection process, drivers take lane use regulations
into consideration.
Two main driving behavior models are used to simulate vehicle movements in a
network:
the acceleration model, and
the lane changing model.
The acceleration model calculates the acceleration that drivers apply in response
to various situations and factors. The most restrictive acceleration is implemented.
The factors that trigger acceleration include:
147 car following,
desired speed,
signs and signals,
connection to appropriate downstream link,
speed limit,
incidents, and
courtesy yielding.
The lane changing model captures lane selection and gap acceptance behavior. A
driver rst checks for the necessity desirability of changing lanes. Subsequently, the
driver selects a lane from the available choices and assesses the adjacent gap in the
target lane. Lane change takes place when the driver perceives the gap in the target
lane as acceptable.
Description of the acceleration and the lane changing models implemented in the
original version of MITSIM that are replaced in the case study with the corresponding
models estimated in this thesis are presented next. 7.1.1 The Acceleration Model
Based on time headway, a driver is categorized to be in one of the three regimes: emergency regime: if the current headway is less than a lower threshold;
car following regime: if the current headway is greater than the lower threshold
but less than an upper threshold; and nally, free ow regime: if the current headway is greater than the upper threshold.
The default thresholds are 0.5 and 1.36 seconds respectively. They were estimated
by using engineering judgment in combination with a sensitivity analysis of these
parameters on the simulator performance. 148 In the emergency regime, drivers apply the minimum of the deceleration necessary
to avoid collisions with the leader and a normal" deceleration. The normal deceleration depends on the speed of the vehicle and was adopted from ITE 1982. The
value is 2.38 m s2 for speeds up to 6.1 m s, 2.0 m s2 for speeds within the range 6.1
to 12.2 m s, and 1.5 m s2 for speeds greater than 12.2 m s.
The acceleration in the car following regime is calculated using the GM Model
Equation 2.7. Di erent sets of parameters are allowed for positive and negative
relative speed cases. These parameters ; ; and were adopted from Subramanian
1996 and are presented in Table 2.3.
In the free ow regime, a driver does not accelerate decelerate if the current speed
is equal to its desired" speed Table 7.1. If the current speed is less than the desired
speed , he she applies maximum" acceleration Table 7.2, otherwise, he she applies
a normal deceleration.
Table 7.1: The cumulative distribution of speed that is added to the posted speed
limit to obtain the desired speed.
Percentile speed above the
speed limit m s
5
4.67
15
1.73
25
0.20
35
0.97
45
1.98
55
3.05
65
4.06
75
5.33
85
6.71
95
8.94 7.1.2 The Lane Changing Model
In MITSIM, lane changes are classi ed as either discretionary DLC or mandatory
MLC . The implementation is as follows:
149 Table 7.2: Maximum acceleration m s2.
vehicle
Speed m s
class
6.1 6.1 12.2 12.2 18.3 18.3 24.4
24.4
high performance car 3.05
2.41
1.71
1.22
1.22
low performance car
2.65
1.58
1.35
0.88
0.61
bus
2.13
1.52
1.22
0.46
0.30
heavy single unit truck 0.85
0.76
0.46
0.30
0.15
trailer trucks
0.49
0.44
0.27
0.14
0.12
Source: adjusted based on FHWA 1980, FHWA 1994, and Pline 1992.
1. check if a lane change is desired required and de ne the type of lane change,
2. select a target lane, and
3. check if the gap in the target lane is acceptable.
To capture di erent driver behavior under DLC and MLC situations, di erent gap
acceptance model parameters are allowed under DLC and MLC situations. The
models are presented next. The Discretionary Lane Changing Model
As mentioned in Section 2.2, MITSIM Yang and Koutsopoulos 1996 uses a rule
based discretionary lane changing model. A driver considers a discretionary lane
change DLC only if the driver cannot accelerate more than 85 of the maximum
acceleration Table 7.2 or if the current speed is less than an impatience factor times
the driver's desired speed Table 7.1. The impatience factor varies from 0.8 to 1.0.
Once a driver decides to to perform a DLC , he she selects a desired lane by
comparing the speeds of the left and right adjacent lanes with that of the current
lane. A parameter speed indi erence factor 10 is used to check whether the current
speed is low enough and the speeds in the adjacent lanes are high enough to consider
a lane change and start performing a gap assessment. 150 The Mandatory Lane Changing Model
Drivers consider a mandatory lane change in order to:
connect to the downstream link of their path,
bypass a lane blockage downstream,
avoid entering a restricted use lane, or,
respond to lane use or variable message signs.
At each time step of the simulation, a probabilistic model is used to decide when
a driver is in MLC state. The probability is a function of the remaining distance to
the point at which lane change must be completed by Lremt, the number of lanes
n
to cross to be in the target lane mnt, and the tra c density. The probability is
given by:
8 PnMLC t =
where, M LC t rem t,
exp Ln MLC 97:5
t
1
2 : 2 if Lrem t L0
n
otherwise. 7.1 is de ned as follows:
M LC t = 402:3 1 + 0:5 mn t + 1:0 knt=kj 7.2 where, knt and kj denote the tra c density of the segment and the jam density
130 veh km lane respectively. Once a vehicle is tagged MLC , it keeps the tag until
it performs the lane change operation or moves into a downstream link. An MLC
tagged driver then searches for an acceptable gap in the target lane. The Gap Acceptance Model
In the gap assessment phase, drivers compare the lead and lag gaps in the target lane
to the critical lead and lag gaps respectively. The speci cations for the lead and lag
critical gaps under mandatory and discretionary lane changing situations for driver
151 n at time t are:
Gcr;lead t = 0:5 max0:914; 0:914 + 0:05 Vnt , 0:10 Vnlead t 7.3
n;DLC
Gcr;lag t = 0:5 max1:524; 1:524 + 0:10 Vnt + 0:30 Vnlag t 7.4
n;DLC
where, Gcr;lead t
n;DLC
Gcr;lag t
n;DLC
Vnt
Vnleadt
Vnlag t = lead critical gap for DLC m,
= lag critical gap for DLC in m,
= subject vehicle's speed m s,
= lead veh. speed less subject speed m s,
= lag veh. speed less subject speed m s. At 10 m s speed, the DLC lead critical gap increases from 0.46 to 0.96 meters
as the lead relative speed decreases from 10 m s to 5 m s. Similarly, the DLC lag
critical gap increases from 0.76 to 2.76 meters as the lag relative speed increases from
5 m s to 10 m s. These values are rather small from a behavioral standpoint.
We expect drivers to be more aggressive under MLC situations compared to DLC
situations. To capture this, the lead lag critical gaps under MLC situations are
assumed to decrease with decreasing remaining distance to the point at which lane
change must be completed by. The speci cations for the MLC lead and lag critical
gaps are given by: Gcr;lead t = max0:914;
n;MLC
n
0:914 + 0:05 Vnt , 0:10 Vnleadt 1 , e,2:5E,5 Lrem t 7.5
Gcr;lag t = max1:524;
n;MLC
n
1:524 + 0:10 Vnt + 0:30 Vnlag t 1 , e,2:5E,5 Lremt 7.6 152 where, Gcr;lead t = lead critical gap for MLC m,
n;MLC
Gcr;lag t = lag critical gap for MLC in m.
n;MLC 7.2 Validation Methodology
7.2.1 Number of Replications
MITSIM is a stochastic simulation model. As a result, the output from one simulation
run may be di erent from another. Each output represents a sample and a number
of simulations are required to get statistics with a prespeci ed accuracy. Hence, an
important aspect of the validation methodology is the determination of the number
of replications required to obtain reliable estimates of the measures of interest.
s
Let, yr be an output from the r th run of the simulator corresponding to a eld
s
observation y. Therefore, yr is a realization of the random variable ys corresponding
to the actual observation y. An unbiased estimator of ys is the mean of the R
s
observations of yr from R di erent simulation runs. Mathematically, ys = R
^1 R
X
r=1 yrs 7.7 where, ys = an estimator of ys,
^
R = number of replications.
Assume that, the R di erent realizations of the random variable ys are distributed
iid normal with an unknown variance2 . Then, the number of replications required to
Although, the independence assumption may be violated due to the stochastic nature of the
simulator, we still make these assumptions in order to get a closed form solution to estimate the
required number of replications.
2 153 obtain a certain accuracy e at a certain level of signi cance is given by: Rreqd !
s t =2 2
=s
ye
^ 7.8 where, s = an estimate of the standard deviation of ys,
e = allowable error,
= desired level of signi cance,
t =2 = critical value of the t distribution at a level of signi cance .
Generally, output from the simulation includes speeds, ows and other quantities
that have spatial as well as temporal dimensions. For each of these types of output
for each time space point, the required number of replications needs to be calculated.
Then, the desired number of replications would be the most conservative value  in
other words, the maximum number of replications required by all the output elements. 7.2.2 Measures of Goodness of t
In this section, di erent measures of goodness of t to compare the simulated data
with their eld counterparts are presented see, for example, Pindyck and Rubinfeld,
1981.
Let, yis be a simulation estimate corresponding to a eld observed quantity yi,
where, subscript i denotes a time space point. For each time space point i, the
percent error, di is given by: ,
di = yi y yi 100
s i 7.9 A positive percent error represents an overprediction whereas a negative percent error
represents an underprediction. To evaluate systemwide performance, a useful measure
is the bias or mean percent error over all the time space points. The mean percent
154 error, b, is:
I
1X d
b= I
i
i=1 7.10 where, I is the total number of time space points. To identify whether an overprediction or an underprediction dominates the bias, a useful measure is the mean positive
and negative percent error. The mean positive percent error, bp, is given by:
J
1X
bp = J dj 7.11 j =1 where, j = index representing the time space point at which the percent error was
positive,
J = total number of positive percent error observations.
Similarly, the mean negative percent error is de ned for all the observations showing
underpredictions.
To penalize larger errors at a higher rate, the root mean square error is used. The
root mean square error RMS is given by: RMS v
u
I
u1 X
= t I yis , yi2
i=1 7.12 The RMS percent error is another measure that takes the scale of yi into account and
is given by: RMS percent error v
u
Is
u 1 X yi , yi
t
=I
yi
i=1 100 !2 7.13 Another useful measure of t is the Theil's inequality coe cient Pindyck and 155 Rubinfeld 1981, de ned as
q U 1 PI y s , yi 2
i
= q 1 PI I si=1 q 1 PI
2+
2
I i=1 yi
I i=1 yi 7.14 The value of U will always fall between 0 and 1. A value of U equal to 0 implies
a perfect t. Related to the Theil's inequality coe cient are three proportions: the
bias U M , the variance U S , and the covariance U C proportions. The proportions
are given by UM
US
UC ys , y2
= 1 PI s
2
I i=1 yi , yi
s , 2
= 1 PI s
2
I i=1 yi , yi
1
= 1 2PI , s s 2
I i=1 yi , yi 7.15
7.16
7.17 where, ys; y; s; and are the means and standard deviations of the simulated and
the original series respectively, and denotes the correlation between the two series.
Basically, these proportions allow us to determine the contribution of the bias and
the variance in the simulation error. Note that, UM + US + UC = 1 7.18 The bias proportion U M re ects the systematic error. The variance proportion
U S indicates how well the uctuation in the original data is replicated by the simulation. Therefore, lower values close to zero of U M and U S are desired. Finally, the
covariance proportion U C measures the unsystematic error. This is the remaining
error after the deviations from the average values have been accounted for and it is
of less worrisome as we desire smaller U M and U S .
In addition to these statistical measures, a plot of the real and simulated quantities
over time and space may be useful in identifying any systematic under or overpredictions at any particular time and or space.
156 7.3 Case Study
7.3.1 The Network
A small network in the Boston area, for which actual ow observations exist, was
used in the case study see Figure 71. The network is a 2.7 km 1.68 miles long
segment of the Storrow Drive in Boston. The network is a two lane freeway with two
section
A 390m section
B 400m section
C 545m section
D 770m 600m Note: Figure not drawn to scale Figure 71: The network used in the validation exercise.
on ramps and two o ramps. Video of tra c during the morning peak 7:30am to
9:15am on February 10, 1998 was recorded at four locations marked sections A to D
in Figure 71.
The part of the network between sections A and D, that are 1.7 km apart, was
simulated. Three 30 meters long dummy links were added to the upstream of section
A to load vehicles in appropriate lanes based on the O Ds estimated from the counts.
The simulated freeway was extended arbitrarily by 100 meters beyond section D.
Tra c sensors were placed at all four sections in the simulated network to collect
aggregate counts and average speeds. The speed limit for the freeway was 17.9 m s
40 mph.
The above network was selected for the following reasons:
1. Tra c was light at the beginning and at the end of the data collection time
and was congested in between. Therefore, initial conditions for congestion oc157 currence could be simulated accurately,
2. The bottleneck formation originated due to conditions within the network trafc merging from the on ramp near section C and was not a ected by conditions
downstream, and
3. The input ows for the network was not a ected by spillbacks from the bottleneck, and therefore, the input ows represent the demand exactly.
A limitation of the network though, is that due to the geometric con guration of
the on ramp and the freeway merge area, the mandatory lane changing and forced
merging models could not be validated. Although few mandatory lane changes take
place between sections A and B, the number of mandatory lane changes is too small
to validate the model. Figure 72 shows the schematic diagram of the merging area
adjacent
vehicle X Y mainline Onramp front
vehicle subject X Y Figure 72: Schematic diagram of the on ramp and Storrow Drive merging area.
between the on ramp and Storrow Drive downstream from section C in Figure 71
where two lanes merge into one lane and no lane change takes place.
In geometric con gurations like this, the mainline vehicles have priority over the
on ramp vehicles. Two vehicles can overlap laterally at this location since the lane
width is more than that of a single lane see location of the subject and the adjacent
vehicle in Figure 72 but less than that of two lanes. In this case, vehicles from the
two upstream lanes merge without any lane changing taking place. In addition, the
subject vehicle is not necessarily following its leader as is assumed in developing the
158 car following model, since the headway is negative. Therefore, the acceleration and
lane changing models developed in this thesis or those implemented in the original
version of MITSIM do not apply here.
In MITSIM, lanes are discrete and such geometry cannot be represented. An ad
hoc merging model Yang and Koutsopoulos 1996 is implemented to capture the
merging phenomenon in such areas. The on ramp vehicle checks whether there is
any vehicle from the adjacent mainline and executes the merge only if the gap is
acceptable. 7.3.2 Tra c Data
Minute by minute tra c counts were collected from video tapes for 1 hour 40 minutes
beginning at 7:33am. At section A, counts were collected for the left two lanes
combined and the rightmost lane that directly feeds into the rightmost lane of the
o ramp 400 meters away. At section B, the mainline the two leftmost lanes counts
and the o ramp the two rightmost lanes counts were collected. At section C, before
the merge, the mainline counts and the on ramp counts were collected. Finally, at
section D the mainline counts were collected.
Figure 73 shows the minute by minute ow through the left two lanes and the
rightmost lane at section A and through the on ramp near section C. The mean
ow values at these locations were 1650, 700, and 900 veh hr lane respectively with
a standard deviation of 350, 300, and 250 veh hr lane respectively. Although, the
on ramp volume was not high, a bottleneck formed near the on ramp and freeway
merging area due to merging. A spillback from the bottleneck reached section B
brie y, but never reached section A nor the upstream end of the on ramp at section
C. Therefore, the input ows for the network were not a ected by spillbacks from the
bottleneck and they represent the demand accurately. 159 (a) Two left lanes, Section A flow, veh/hr/lane 2500 2000 1500 1000 500 0 10 20 30 40 50
time, minute 60 70 80 90 100 70 80 90 100 70 80 90 100 (b) Rightmost lane, Section A flow, veh/hr/lane 1500 1000 500 0 0 10 20 30 40 50
time, minute 60 (c) On ramp, Section C flow, veh/hr/lane 1500 1000 500 0 0 10 20 30 40 50
time, minute 60 Figure 73: Flow of tra c entering the network. 7.3.3 O D Estimation from Tra c Counts
A required input to the simulator is a time dependent O D matrix. The minute by
minute O D matrix was estimated from the minute by minute tra c counts at section
A, the exit counts at section B, and the on ramp counts at section C by using an ad
hoc method. Note that, the method is developed considering the geometry and counts
of this particular network, and is not applicable to a general network. It was assumed
that, a certain percentage p of the tra c from the rightmost lane at section A the
exit only lane takes the exit ramp near section B. Since p is unknown, di erent sets
of O D matrices were created by varying p from 70 to 100 to investigate the e ect
of the assumption of p on the validation results.
The O D matrix for a particular minute say, the tth minute is estimated from the
160 counts of the corresponding minute by using the following equations see Figure 74
c:
Legends {
{ D4 { C1 C2 { C3 C: counts O: origin D: destination
p: % of rightmost lane traffic exiting D2 (a) traffic counts from video tape, veh/min O3 { O1 (b) origin and desitination definitions O 3D 4 = C 3  O 3D 2 3 4 O 3D 2 = C 2  O 1D 2
O 1D 4 = C 1  O 1D 2
1 O 1D 2 = min(C 1 * p , C 2) 2 (c) OD estimation Figure 74: O D estimation from tra c counts for the case study. O1D2t
O1D4t
O3D2t
O3D4t = minC1t p; C2t + tt
= C1t , O1D2 t
= C2t + tt , O1D2t 7.19 = C3t , O3D2 t 7.22 161 7.20
7.21 where see Figures 74 a and b, t
tt
C1t
C2t
C3t
O1D2t = time period between minutes t and t + 1,
= average travel time between sections A and B,
= counts at the rightmost lane at section A,
= counts at the exit ramp at section B,
= counts at the left two lanes at section A,
= number of vehicles from the rightmost lane at section A exiting at section B,
O1D4t = number of vehicles from the rightmost lane at section A that continue
on the freeway, O3D2t = number of vehicles from the left two lanes at section A exiting at
section B,
O3D4t = number of vehicles from the left two lanes at section A that continue
on the freeway.
The count C2 was advanced by tt to take into account that a vehicle counted
at section A at time to would reach section B at time to + tt. It was assumed
that tt is the travel time between sections A and B for everyone. We conducted
a sensitivity analysis on tt to investigate its impact on the O D estimations. The
average travel time to reach section B from section A ranged from 19 to 28 seconds
observations from simulation. Within this range of tt, C2t + tt di ered by less
than 2 vehicles minute for 92 of the cases. Therefore, the assumption on tt does
not a ect the O D estimation signi cantly. The variable tt was set to 22.4 seconds
which assumes that vehicles traveled at the speed limit.
From the rightmost lane at section A, the minimum of a certain percent p of C1
and the exit ramp count at section B C2 is assigned to take the exit ramp at section
B O1D2. The remaining C1 is assigned to continue on the freeway O1D4. This will
guarantee that at least 1 , p percent of the rightmost lane drivers would perform a
162 mandatory lane change to continue on the freeway.
The number of vehicles from the left two lanes exiting section B O3D2 is estimated by deducting O1D2 from the o ramp counts at section B C2. The remaining
tra c counted at the left two lanes at section A i.e., C3 , O3D2 continues in the
freeway O3D4. Finally, all vehicles entering the network through the on ramp at
section C have only one destination, i.e., they travel through section D. 7.3.4 MITSIM Modi cations
As mentioned above, two additional versions of MITSIM were created by incorporating the models developed in this thesis. In MITSIM ONE, only the acceleration
model was replaced with the one estimated in this thesis. In MITSIM TWO, both
the acceleration and lane changing models were replaced with those estimated in this
thesis. A version of MITSIM with only the lane changing model replaced with the
one estimated in this thesis was not created due to the following reason.
Early testing using the original version of MITSIM indicated that, it was not
capable of handling the demand used in this case study due to over prediction of
congestion and the resulting spillback. Vehicles were queued outside of the network
and were loaded only when spaces to load them became available. As a result, vehicles
could not be loaded on time according to their O D and departure times. By replacing
the default lane changing model with the one developed in this thesis, the problem
still persisted. Therefore, the results from such simulation runs would not be reliable,
and MITSIM with only the lane changing model replaced was not tested.
In addition to replacing the acceleration and lane changing models with those
estimated in this thesis, the following changes to MITSIM parameters were introduced
see Appendix B for a general approach to calibrate the simulation model parameters:
The lower headway threshold of the acceleration model was set to 0.4 seconds
and the upper threshold was adopted from Equation 6.5. The lower threshold
was set after some trial and error to avoid vehicle to vehicle collisions. Compared
to the 1.36 seconds upper threshold used in the original version of MITSIM, the
163 upper threshold estimated in this thesis was much higher. For example, the 5
percentile and the median upper headway thresholds were 1.75 and 3.17 seconds
respectively.
Since the data collection site used to estimate the models in this thesis and the
network used for the validation case study have di erent geometric con gurations, the constant of the car following acceleration model was adjusted to make
the model predictions more realistic for the validation network. The constant of
the car following acceleration model acc in Equation 6.2 was increased from
0.023 to 0.040. 7.3.5 Experimental Design
The three versions of MITSIM were used under di erent scenarios with respect to
the O D ows values of p were set equal to 100, 85, and 70. In order to
determine the number of replications required, we need estimates of the mean and
standard deviation of the measures of interest discussed in Section 7.2.1. To get
estimates of mean and standard deviation, all three versions of the simulator were run
10 times using the three di erent sets of O D matrices. Then the required number of
replications for all the cases were estimated using Equation 7.8. The most conservative
estimate was 4 and in subsequent computations output from all 10 runs were used.
Tra c counts and speeds for each scenario were aggregated over 5 minute periods.
The counts were compared to the corresponding real tra c counts. Speeds predicted
by di erent versions of the simulators were also compared. The statistics reported in
Section 7.2.2 were used to measure the goodness of t of the various simulation runs. 7.3.6 Validation Results
Table 7.3 summarizes the comparison of the original MITSIM with the two revised
MITSIM versions using three di erent set of O D matrices assuming that 100, 85,
and 70 of the drivers from the rightmost lane at section A took the exit at section B.
Observations corresponding to the rst ve minutes were not used in computations
164 since vehicles were loaded into an empty network and it took approximately 2 minutes
to ll the network.
As evident in Table 7.3, by varying the values of p, the percent of the rightmost
drivers at section A exiting at section B, the statistics did not show signi cant variation. This may be due to low ow through the rightmost lane at section A and the
exit at section B. Therefore, based on the low sensitivity of the results to the value
of p, the conclusion drawn should be valid for the actual O D ows as well.
For all cases, MITSIM ONE and TWO performed consistently better than the
original MITSIM version. Tra c in the original MITSIM got jammed 15 minutes after
the beginning of simulation and continued to be jammed throughout the simulation
period. The congestion originated near the on ramp merge. Tra c spilled back all
the way up to section A and beyond, and a ected vehicle loading into the network.
Whereas, in reality, the merging area near section C was congested from 8:04am
to 9:05am and the e ect of spillback reached section B brie y but never reached
section A. MITSIM ONE and TWO performed much better in this respect. This is
also re ected by consistently low average speeds at sections B and C for the original
MITSIM as shown in Figures 75, 76, and 77.
Due to lack of speed observations from the eld, simulated speeds could not be
compared to the eld observations. At section B, the average speeds were around 12
to 14 m s for MITSIM ONE and TWO compared to 5 m s for the original MITSIM.
The speed limit of the freeway is 17.9 m s 40 mph. Average speeds from the original
MITSIM were signi cantly lower than expected.
The RMS Percent Error in counts for the original MITSIM was 9.08 which
reduced to 8.09 and 7.53 for MITSIM ONE and TWO respectively. The root mean
square error was 28, 24, and 22 vehicles per 5 minutes for the original MITSIM and
MITSIM ONE and TWO respectively. The Theil's inequality coe cient was 0.050,
0.042, and 0.039 for the original MITSIM and MITSIM ONE and TWO respectively.
Note that, a smaller coe cient implies a better t.
The mean percent error contributed signi cantly toward the RMS Percent Error
for the original MITSIM and is equal to 5.81 compared to 1.95 and 1.56 for
165 Table 7.3: Summary statistics of the comparison of the eld observed counts with
those obtained from di erent versions of MITSIM using three di erent O D sets.
Percent vehicles from the rightmost lane taking exit = 100
Revised
Revised
Statistical
Original MITSIM, MITSIM,
Measure
MITSIM Acc only Acc & LC
RMS percent error
9.08
8.09
7.53
RMS error veh. per 5 min 27.82
24.23
22.22
mean percent error
5.81
1.95
1.56
Theil's inequality coe cient 0.050
0.042
0.039
U M bias proportion
0.419
0.073
0.059
U S variance proportion
0.063
0.005
0.011
avg. positive error
4.53
6.78
5.99
no. of positive errors
12
18
20
max. positive error
9.13
20.85
20.49
avg. negative error
8.56
5.98
5.64
no. of negative errors
45
39
37
max. negative error
19.97
18.99
16.17
Percent vehicles from the rightmost lane taking exit = 85
Revised
Revised
Statistical
Original MITSIM, MITSIM,
Measure
MITSIM Acc only Acc & LC
RMS percent error
9.08
7.83
7.44
RMS error veh. per 5 min 27.76
23.44
22.02
mean percent error
5.77
2.13
1.96
Theil's inequality coe cient 0.049
0.041
0.038
U M bias proportion
0.415
0.090
0.088
U S variance proportion
0.059
0.008
0.012
avg. positive error
3.90
5.86
5.72
no. of positive errors
14
19
19
max. positive error
10.24
19.46
18.74
avg. negative error
8.92
6.12
5.81
no. of negative errors
43
38
38
max. negative error
20.10
17.28
16.10
Percent vehicles from the rightmost lane taking exit = 70
Revised
Revised
Statistical
Original MITSIM, MITSIM,
Measure
MITSIM Acc only Acc & LC
RMS percent error
9.17
8.13
7.71
RMS error veh. per 5 min 28.06
24.21
23.00
mean percent error
5.83
2.25
2.20
Theil's inequality coe cient 0.050
0.042
0.040
U M bias proportion
0.416
0.091
0.099
U S variance proportion
0.060
0.003
0.011
avg. positive error
4.41
6.85
6.65
no. of positive errors
12
17
16
max. positive error
8.97
20.72
19.69
avg. negative error
8.56
6.11
5.79
no. of negative errors
45
40
40
max. negative error
20.51
17.65
16.41 166 ORIGINAL MITSIM
MITSIM ONE
MITSIM TWO
16 speed, m/s 14
12
10
section B 8
6
4 2 4 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 14 speed, m/s 12
10
section C
8
6
4 2 4 15 speed, m/s 14.5
14 section D 13.5
13
12.5 2 4 Figure 75: Comparison of average speeds obtained from di erent versions of MITSIM
for p = 100.
MITSIM ONE and TWO respectively. The bias proportion U M for the original
MITSIM is 0.419 which is very high. The bias proportions for MITSIM ONE and
TWO were 0.073 and 0.059 respectively. The variance proportions were 0.063, 0.005,
and 0.011 for the original MITSIM and MITSIM ONE and TWO respectively.
As a result of the systematic underrepresentation of ow, the number of positive
errors for the original MITSIM was small and the corresponding average and maximum positive errors were low compared to those for the two other MITSIMs. The
mean positive percent errors were 4.53, 6.78, and 5.99 for the original MITSIM
and MITSIM ONE and TWO respectively. The negative mean percent error for the
original MITSIM was 8.56 compared to 5.98 and 5.64 for MITSIM ONE and
167 ORIGINAL MITSIM
MITSIM ONE
MITSIM TWO
16 speed, m/s 14
12
10
section B 8
6
4 2 4 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 14 speed, m/s 12
10 section C 8
6
4 2 4 15 speed, m/s 14.5
14 section D 13.5
13
12.5 2 4 Figure 76: Comparison of average speeds obtained from di erent versions of MITSIM
for p = 85.
TWO respectively.
Compared to MITSIM ONE and TWO, the performance of the original MITSIM
was poor at all three sections see Figures 78, 79, and 710. Performances of
MITSIM ONE and TWO at section D were not as good as they were at sections B
and C. This may be due to the fact that, section D is near the simulated network
boundary where all tra c exits the network. As a result, vehicles leave the network
at speeds higher than the real speed. Therefore, the uctuation in the ow for section
D could not be reproduced well.
Performance of the original MITSIM improved signi cantly after the acceleration
model was replaced with the one estimated in this thesis MITSIM ONE. Due to
168 ORIGINAL MITSIM
MITSIM ONE
MITSIM TWO
16 speed, m/s 14
12
10
section B 8
6
4 2 4 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 14 speed, m/s 12
10 section C 8
6
4 2 4 15 speed, m/s 14.5
14 section D 13.5
13
12.5 2 4 Figure 77: Comparison of average speeds obtained from di erent versions of MITSIM
for p = 70.
the high congestion level, drivers traveled with low headways most of the time. As
shown in Figure 63, the deceleration calculated by the deceleration model used in the
original MITSIM adopted from Subramanian 1996 is too high at low headways.
This may have contributed to vehicles moving slowly in the original MITSIM, and
thereby reducing the volume of tra c the network could handle, especially, near the
merging area.
Finally, MITSIM TWO performed better than MITSIM ONE with respect to all
the statistics except the variance proportion U S and the bias proportion U M for
the p = 70 case. However, the RMS and the mean percent errors for MITSIM TWO
were smaller than its MITSIM ONE counterparts. This may be due to the fact that
169 ORIGINAL MITSIM
MITSIM ONE
MITSIM TWO
real data flow, veh/hr/lane 2400
2200
2000
section B 1800
1600
1400
1200 2 4 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 flow, veh/hr/lane 2400
2200
2000
1800 section C 1600
1400
1200 2 4 flow, veh/hr/lane 2400
2200
2000
1800
section D 1600
1400
1200 2 4 Figure 78: Comparison of the real tra c counts with those obtained from di erent
versions of MITSIM for p = 100.
the variance and the bias proportions do not take into account the scale of the errors
the di erences between the simulated and the original tra c counts with respect to
the original counts. Therefore, MITSIM TWO demonstrates the e ectiveness of the
discretionary lane changing model. 7.4 Conclusions
The acceleration and discretionary lane changing models were tested using a microscopic tra c simulator, MITSIM. Tra c on a 1.83 km long segment of a freeway
with one on and one o ramps was simulated using di erent versions of MITSIM:
the original MITSIM, MITSIM with only the acceleration model replaced with the
170 ORIGINAL MITSIM
MITSIM ONE
MITSIM TWO
real data flow, veh/hr/lane 2400
2200
2000
section B 1800
1600
1400
1200 2 4 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 flow, veh/hr/lane 2400
2200
2000
1800 section C 1600
1400
1200 2 4 flow, veh/hr/lane 2400
2200
2000
1800
section D 1600
1400
1200 2 4 Figure 79: Comparison of the real tra c counts with those obtained from di erent
versions of MITSIM for p = 85.
model estimated in this thesis, and MITSIM with both the acceleration and the
lane changing model replaced with the corresponding models estimated in this thesis. Simulated counts aggregated over ve minute intervals at di erent locations were
compared to the corresponding eld observations.
Performance of the original MITSIM signi cantly improved after the acceleration
model was replaced with the one estimated in this thesis MITSIM ONE. It improved
further when the lane changing model of the original MITSIM was replaced with the
one estimated in this thesis in addition to replacing the acceleration model MITSIM
TWO. For a full evaluation, a wider set of experiments covering di erent weather,
geometry, and congestion conditions is needed.
171 ORIGINAL MITSIM
MITSIM ONE
MITSIM TWO
real data flow, veh/hr/lane 2400
2200
2000
section B 1800
1600
1400
1200 2 4 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 6 8 10
12
time interval 14 16 18 20 flow, veh/hr/lane 2400
2200
2000
1800 section C 1600
1400
1200 2 4 flow, veh/hr/lane 2400
2200
2000
1800
section D
1600
1400
1200 2 4 Figure 710: Comparison of the real tra c counts with those obtained from di erent
versions of MITSIM for p = 70. 172 Chapter 8
Conclusions and Future Research
Directions
This chapter summarizes the conceptual framework and estimation results of the
proposed acceleration and lane changing models. Major contributions of this thesis
are also discussed. Finally, suggestions for future research are presented. 8.1 Summary of Research
8.1.1 The Acceleration Model
A comprehensive framework for estimating a general acceleration model is developed
that is applicable to both congested and uncongested tra c. The model de nes two
regimes based on a time headway threshold: the car following regime and the free
ow regime. At headways less than the threshold, a driver is assumed to be in the
car following regime trying to match its leader's speed, and at headways larger than
the threshold, the driver is assumed to be in the free ow regime trying to attain its
desired speed.
A headway threshold distribution is assumed to capture the variability among
drivers. A reaction time distribution is also assumed which captures the e ect of
response lag to stimulus. The mean of the reaction time distribution depends on the
173 tra c environment.
Both the car following and free ow acceleration models employ the response
equal to the stimulus times the sensitivity structure. The car following model uses
the GM Nonlinear Model as a basis and extends it. The original model was modi ed
to include the e ect of density in the sensitivity term and allow the stimulus to be
a nonlinear function of the front relative speed i.e., leader speed less the subject
speed. In addition, the estimation allows for capturing the fact that, drivers may
update their perception of the tra c environment after they recognize the stimuli for
the car following acceleration the lead relative speed. In the free ow acceleration
model, the sensitivity term is a constant and the di erence between the desired speed
and the current speed provides the stimulus.
The parameters of all the component models were estimated jointly using the
maximum likelihood estimation method and microscopic data collected from the video
of real freeway tra c. The network is a part of Interstate 93, the Central Artery in
Boston. The section has a three lane mainline and a weaving lane.
The estimation results show that the impact of speed, space headway, and density
of tra c is di erent under acceleration and deceleration situations. The sensitivity
term of the car following acceleration is a function of the subject speed, the space
headway, and the density of tra c, while that of the car following deceleration is a
function of the space headway and the density of tra c. The stimulus is a nonlinear
function of the lead relative speed. The free ow acceleration is a function of the
subject speed, its leader speed, an indicator whether the subject vehicle is a heavy
vehicle i.e., vehicle length greater than 9.14 meters or 30 ft, and an indicator whether
tra c density is low level of services A through C.
The mean and standard deviation of the headway threshold distribution were
estimated to be 3.17 and 0.87 seconds respectively. The median, mean, and standard
deviation of the reaction time distribution were estimated to be 1.31, 1.34, and 0.31
seconds respectively. 174 8.1.2 The Lane Changing Model
The lane change model is based on a decision that proceeds in the following three
steps:
decision to consider a lane change,
choice of a target lane, and
acceptance of a gap in the target lane.
Modeling such a process is extremely complicated due to its latent nature. To
simplify, drivers are assumed to make decisions about lane changes at every discrete
point in time irrespective of the decisions made during earlier time periods.
The proposed gap acceptance model recognizes that for merging into an adjacent
lane, both the lead and lag gaps must be acceptable. Drivers are expected to be more
aggressive under mandatory lane changing situations compared to discretionary lane
changing situations. The proposed model captures this behavior by allowing di erent
parameters for the gap acceptance model under the two situations. The models were
estimated using the same data as in the estimation of the acceleration model.
Drivers' decision to perform a discretionary lane change is modeled as a two steps
decision process. First, drivers examine their satisfaction with the driving conditions
of the current lane. Important factors a ecting such decision include the di erence
between the current speed and the driver's desired speed, an indicator whether the
subject vehicle is a heavy vehicle, and an indicator whether the subject is tailgated.
If the driver is not satis ed with the driving conditions of the current lane, he she
compares the driving conditions of the current lane with those of the other lanes.
Such decision is in uenced by the the speeds of the vehicles ahead in di erent lanes
compared to the subject's desired speed and the lag relative speed.
Factors a ecting a driver's decision to respond to the mandatory lane change
situation MLC are delay time elapsed since MLC conditions apply and an indicator for the rst gap when delay is equal to zero. The estimated median lead and
175 lag critical gap lengths under MLC situations are lower than their DLC situations
counterparts, as expected.
A forced merging model is developed to capture drivers' lane changing behavior
in heavily congested tra c where gaps larger than their minimum acceptable length
are hard to nd. In such situations, it is assumed that a driver changes lanes either
through courtesy yielding the lag vehicle in the target lane or through the subject
forcing the lag vehicle to slow down. Important factors that a ect drivers' forced
merging behavior include lead relative speed only when the lead vehicle is slower,
remaining distance to the point at which the lane change must be completed by, and
total clear gap in excess of the subject vehicle's length. 8.1.3 Validation by Microsimulation
The acceleration and lane changing models were tested using a microscopic tra c
simulator, MITSIM. A 1.83 km long segment of a freeway with one on and one
o ramps was simulated using di erent versions of MITSIM: the original MITSIM,
MITSIM with only the acceleration model replaced with the one estimated in this
thesis, and MITSIM with both the acceleration and lane changing models replaced
with the corresponding models estimated in this thesis. Simulated counts at di erent
time intervals, aggregated over ve minutes, at di erent locations were compared to
the corresponding eld observed counts.
Performance of the original MITSIM signi cantly improved after the acceleration
model was replaced with the one estimated in this thesis. It improved further when
the lane changing model of the original MITSIM was replaced with the one estimated
in this thesis in addition to replacing the acceleration model. 8.2 Contributions
This thesis contributes to the state of the art in modeling drivers' acceleration and
lane changing behavior in two major areas: enhancing existing models and proposing
new models, and estimating the models using statistically rigorous methods and real
176 microscopic tra c data. Contributions in each of these two areas are listed below.
Contribution to the modeling framework:
The car following model is extended by assuming that the stimulus is a
nonlinear function of the lead relative speed and that the sensitivity term
is also a function of the tra c conditions ahead of the driver.
The existing models restricts the stimulus the lead relative speed and
other factors such as subject speed, gap in front of the subject that
a ect the acceleration decision to be observed at the same time. This corresponds to an assumption that drivers base their decisions on the tra c
environment at the time they were stimulated into action. The proposed
model relaxes this assumption by allowing drivers to update their perception of the tra c environment during the decision making process.
A headway threshold distribution is introduced that allows any driver behavior to be captured aggressive or conservative.
An individual driver speci c reaction time is introduced which is allowed
to be sensitive to the tra c situation under consideration.
A probabilistic lane changing model is developed that captures drivers'
lane changing behavior under both the mandatory and discretionary lane
changing situations. This is a signi cant improvement over the existing
deterministic rule based lane changing models.
The proposed lane changing model allows for di erent gap acceptance
model parameters for mandatory and discretionary lane changing situations. It also captures the variability within driver and amongst drivers in
the lane changing decision process.
A forced merging model is proposed that captures merging in a heavily
congested tra c by gap creation either through force or through courtesy
yielding.
177 Contribution to model estimation:
A methodology to estimate instantaneous speed and acceleration that is
required for model estimation from discrete trajectory data that can be
obtained from real tra c is developed.
All the components of the acceleration model are estimated jointly using
real microscopic tra c data. The component models are the car following
acceleration and deceleration models, the free ow acceleration model, and
the headway threshold and reaction time distributions. Estimation results
demonstrate the robustness of the modeling framework.
Separate car following model parameters under acceleration and deceleration situations are allowed in the estimation. This captures the fact that,
the sensitivity of di erent factors on drivers' acceleration behavior may
not be same under these two situations.
Separate gap acceptance models for the mandatory and discretionary lane
changing situations are estimated. This captures the fact that, driver are
expected to be more aggressive under mandatory lane changing situations
compared to discretionary lane changing situations.
The proposed models were estimated using the maximum likelihood estimation method and real microscopic tra c data. 8.3 Future Research Directions
8.3.1 Modeling
The proposed acceleration model should be extended to capture the impact of
lane changing decisions on the acceleration decision. For example, drivers may
need to accelerate or decelerate to t into a gap in the target lane. In such
cases, the headway and speed of the lead and lag vehicles in the target lane will
in uence drivers' acceleration decisions.
178 The proposed lane changing model does not capture the impact of past lane
changing decisions on the current lane changing decision and various modeling
approximations should be considered.
The forced merging model and the mandatory lane changing model should be
combined into a single framework. In reality, drivers consider forced merging
only when they perceive the probability of nding an acceptable gap to be very
low.
Models capturing driver behavior in a merging area where two lanes gradually
become one see Figure 72 for an example have to be developed. 8.3.2 Estimation and Validation
To enhance the ability of the models proposed in this thesis to predict drivers' acceleration and lane changing behavior, the models should be estimated with richer
data that has more variability than the one used in this thesis. For example, the
car following model proposed in this thesis predicts acceleration that is smaller than
expected which should be reestimated using richer data. In addition, from an estimation point of view, a major research activity is the estimation of the models using
richer data that provide the required variability to assess the impact of various factors,
such as geometric characteristics etc. More speci cally,
The impact of geometric characteristics of a roadway, for example, lane width,
curvature, grade, pavement surface quality, on driver behavior was not captured
due to lack of data. The models should be estimated using data from di erent
sites with di erent geometric characteristics.
The discretionary lane changing behavior, when mandatory lane changing situations apply, cannot be estimated due to lack of appropriate data. This requires
data collected over a long stretch of a roadway 1500 3000 meters long.
The discretionary lane changing model was estimated using a data set in which
the choice set was the current lane and one adjacent lane. Ideally, data set with
179 two adjacent lanes in the choice set would be preferable.
The identi cation problem that arose while estimating the discretionary lane
changing model in which serial correlation was captured should be further investigated.
Further validation, using more extensive networks, is also required. 8.4 Conclusion
A comprehensive framework for modeling drivers' acceleration and lane changing behavior was developed in this thesis. Both the acceleration and lane changing models
were estimated using real microscopic tra c data, and validated from a behavioral
standpoint as well as using microsimulation. Overall, the empirical results are encouraging and demonstrate the e ectiveness of the modeling framework. 180 Appendix A
Speci cation of the Random Utility
Model Appropriate for Panel Data
Panel data contains one or more observations for each individual driver. Di erent
observations from a given driver are likely to be correlated which may introduce bias
in the parameter estimates. To capture this correlation, the random disturbance
of the utility function used to model the decisions at various levels is assumed to
have two components: an individual speci c random term that does not vary for a
given individual, and a generic random term Heckman 1981. Hence, the utility
formulation associated with a decision d within the hierarchy is given by:
d
d
Un t = Xn t d + dn + d t
n where, n
t
d
Un t
d
Xn t
d = individual driver,
= time instance,
= unobserved utility of responding to decision d at time t,
= vector of explanatory variables,
= vector of parameters,
181 A.1 n = individual speci c random term assumed to be distributed standard
normal,
d = parameter of for decision d,
n
d t = generic random term that varies across all three dimensions, i.e.,
n
d; t; and n:
These assumptions on the random terms imply:
8 cov d t; d00 t0
n
n = 2d if t = t0 , n = n0 and d = d0 0 otherwise
0
d
covn ; d 0 t = 0; 8t; n; d; n0 ; d0
n
8
d2 + 2d if d = d0; n = n0 and t = t0
d2
if d = d0; n = n0 and 8t 6= t0
d
d
covUn t; Un00 t0 =
d d0
if d 6= d0; n = n0 and 8t
:0
otherwise
: A.2
A.3
A.4 where, 2d denotes the variance of d t. Conditional on n, di erent discrete choice
n
models can be obtained by making di erent assumptions on the distribution of d t,
n
such as a logit or a probit model. 182 Appendix B
Calibration of the Simulation
Model Parameters
Figure B1 illustrates a systematic approach to calibrate simulation model paramData Collection Estimation of
Individual Models Model
Refinement
Disaggregate
Data Evaluation Validation of
Simulation Model Aggregate
Data Evaluation Calibrated and
Validated
Simulation
Model Figure B1: Model parameter calibration approach.
eters. Data collection involves collecting both disaggregate microscopic and ag183 gregate macroscopic data. Chapter 5 provided a detailed description of the data
required to estimate driver acceleration and lane changing behavior and the actual
disaggregate data collected from real tra c. The disaggregate data are used to estimate individual models as is done in this thesis presented in Chapter 6. Then the
parameter estimates are evaluated both from statistical and behavioral standpoints.
This may suggest re nement of the model structure which is followed by reestimation
of the models.
In the next phase, aggregate data is used to validate the overall performance of the
simulation model. Examples of aggregate data include speeds, counts, occupancies
at di erent locations of a roadway aggregated over a period of time. At that point
further re nement and calibration may take place. For example, the constant of
the models estimated using disaggregate data, collected at other locations, may be
recalibrated using aggregate data from the location where the application takes place. 184 Bibliography
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