# CAR FOLLOWING MODELS: FORMULATION ISSUES AND PRACTICAL CONSIDERATIONS

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## CHAPTER TWO:LITERATURE REVIEW

Introduction

The importance of transportation in the present day need hardly be emphasized. Today’s highway system carries significantly more vehicle miles of travel than ever before. The demands on the transportation system continue to grow faster than improvements can be made. Consequently, the ability to understand and apply the traffic flow fundamentals is very essential towards improving the transportation system. Traffic flow theories seek to describe in a precise mathematical way, the interactions between the vehicles and their operators and the infrastructure. The scientific study of traffic flow had its beginnings in the 1930’s with the application of probability theory to the description of road traffic and the pioneering studies conducted by Bruce D. Greenshields at the Yale Bureau of Highway Traffic. After World War II, with tremendous increase in use of automobiles and the expansion of the highway systems, there was also a surge in the study of various traffic characteristics. The 1950’s saw theoretical developments based on a variety of approaches – car-following, traffic waves and queuing theories. By the 1970’s the field of traffic flow and transportation has become so diffuse that it can no longer be covered under the same heading. Further research continues till date, improving/altering the previously held notions and formulating new theories that describe traffic variables more realistically. The following sections describe the existing models in the areas of car-following and traffic stream modeling and issues in capacity and discharge headways at signalized intersections. The sections trace historically the development of concepts and ideas in the
above mentioned fields.

### Traffic Stream (Continuum Flow) Models

Traffic stream models describe the motion of a traffic stream by approximating for the flow of a continuous compressible fluid. These models relate three traffic stream variables, namely: the traffic stream flow rate, traffic stream density, and traffic stream
space-mean-speed. Since there are three traffic variables, three relationships must be established among them in order to compute the magnitude of these variables. The first relationship among flow (q), density (k) and speed (u) is inherent in the quantity definition:
q = ku [2-1] The second relationship, the continuity of vehicles, is expressed as follows:
Where S(x,t) = Generation (dissipation) rate of vehicles per time per unit length. Solution of the continuity equation as it applies to traffic flow was first proposed by Lighthill and Whitham (1955). Both of the above relationships are true for all fluids,including traffic and there is no controversy as to their validity. However, it is the controversy about the third equation that gives a one-to-one relationship between speed and density or between flow and density, which has led to the development of many continuum flow (traffic stream) models.

#### Simple Continuum Models

These models use a deterministic form of speed-density relationship as the third equation. They state that the average traffic speed is a function of traffic density. The models of this type were developed in the earlier times and assumed a single regime phenomenon over the complete range of flow conditions including free-flow and congested regimes. Later models attempted to improve on the earlier ones by considering two separate regimes. Many such relationships have been mentioned in the literature and the important ones among them are mentioned below. Single-Regime Models The first single-regime model was developed by Greenshields in 1935, based on observing speed-density measurements obtained from an aerial photographic study. His conclusion was that, speed (u) is a linear function of density (k) and can be expressed mathematically as .Greenberg (1958) developed the second single regime model by treating the traffic stream as a continuous fluid. He utilized the theory developed by Lighthill and Whitham (1955) for his model. Starting with the equation of motion of a one-dimensional fluid.and the two equations 2-1 and 2-2, the basic equation of traffic flow and the equation of continuity, he developed a model of the form.The important consequence of this model was the discovery of the bridge between macroscopic and microscopic models in the work by Gazis et al a year later.The third model was proposed by Underwood (1961) as a result of traffic studies on Merritt Parkway in Connecticut. He proposed this new model mainly to overcome the deficiency in the Greenberg model – values of speed going to infinity at very low density values. Hence, his model was of the form.

ABSTRACT
ACKNOWLEDGEMENTS
LIST OF TABLES
LIST OF FIGURES
NOMENCLATURE
CHAPTER ONE:  INTRODUCTION
1.1 Background
1.2 Problem Definition
1.3 Thesis Objectives
1.4 Thesis Contributions
1.5 Thesis Layout and Approach
CHAPTER TWO:  LITERATURE REVIEW
2.1 Introduction
2.2 Traffic Stream (Continuum Flow) Models
2.2.1 Simple Continuum Models
2.2.2 High-Order Continuum Models
2.3 Car-Following Theories
2.3.1 Gazis – Herman – Rothery (GHR) Model
2.3.2 Linear Models
2.3.3 Collision Avoidance models
2.3.4 Other Models
2.4 Capacity and Discharge Headway at Signalized Intersections
2.4.1 Capacity Drop
2.5 Conclusions
CHAPTER THREE:  CAR FOLLOWING MODELS: FORMULATION ISSUES AND PRACTICAL CONSIDERATIONS
3.1 Introduction
3.1.1 Link between Car-following and Traffic Stream Models
3.1.2 Standard Car-Following Notation
3.1.3 Illustrative Example
3.2 Overview of Selected Models
3.2.1 Pipes or GM – 1 Car Following Model
3.2.2 Greenshields Model
3.2.3 Greenberg Model
3.2.4 Van Aerde Model
3.3 Formulation Issues
3.3.1 Speed Formulation
3.3.2 Acceleration Formulation
3.4 Practical Issues Related to Car-Following Model Implementation
3.4.1 Discrete time step movements of vehicles
3.4.2 Acceleration Constraint
3.4.3 Collision Avoidance Constraint
3.4.4 Car Following Behavior Formulation
3.5 Conclusions
CHAPTER FOUR:  COMPARISON OF CAR-FOLLOWING MODELS
4.1 Introduction
4.2 Overview
4.2.1 Car-following Models
4.2.2 Scenarios Considered
4.2.3 General Assumptions
4.3 Car-Following Behavior Comparison
4.3.1 SCENARIO I: Uninterrupted Flow Conditions
4.3.2 SCENARIO II: Interrupted Flow Conditions
4.5 Capacity Drop Issue
4.5.1 Signalized Intersection
4.5.2 Lane Drop
4.5.3 Effect of Typical Acceleration Factor
4.6 Conclusions
CHAPTER FIVE:
SUMMARY, CONCLUSIONS AND RECOMMENDATIONS
5.1 Summary
5.2 Conclusions
5.3 Recommendations for Further Research
REFERENCES
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FORMULATIONS, ISSUES, AND COMPARISON OF CAR-FOLLOWING MODELS