USA Banner

Official US Government Icon

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure Site Icon

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

U.S. Department of Transportation U.S. Department of Transportation Icon United States Department of Transportation United States Department of Transportation

Public Roads - Jan/Feb 1999

Traffic-Flow Theory

by Henry Lieu

Traffic-flow theories seek to describe in a precise mathematical way the interactions among vehicles, drivers, and the infrastructure. The infrastructure consists of the highway system and all its operational elements, including control devices, signage, and markings. These theories are an indispensable element of all traffic models and analysis tools that are being used in the design and operation of streets and highways.

The scientific study of traffic flow had its beginnings in the 1930s with the application of probability theory to the description of road traffic and with the pioneering studies conducted by Bruce D. Greenshields at the Yale Bureau of Highway Traffic on the study of models relating volume and speed and the investigation of performance of traffic at intersections. After World War II, with the tremendous increase in the use of automobiles and the expansion of the highway system, there was also a surge in the study of traffic characteristics and the development of traffic-flow theories.

In December 1959, the First International Symposium on the Theory of Traffic Flow was held at the General Motors Research Laboratories in Warren, Mich. This was the first of what has become a series of triennial symposia on the theory of traffic flow and transportation. A glance through the proceedings of these symposia will provide a good indication of the tremendous developments over the last 40 years in the understanding and the treatment of traffic-flow processes.

The field of traffic-flow theory and transportation has become too diffuse to be covered by any single type of meeting, and numerous other symposia and specialty conferences about a variety of traffic-related topics are held on a regular basis. Yet, even as traffic-flow theory is increasingly better understood and more easily characterized through advanced computation technology, the fundamentals are just as important today as in the early days. They form the foundation for all the theories, techniques, and procedures that are being applied in the design, operation, and development of advanced transportation systems.

This article outlines the revised Monograph on Traffic Flow Theory, which can be viewed on the Turner-Fairbank Highway Research Center Web site ( This report is an updated and expanded version of two previous works that were sponsored by the Transportation Research Board (TRB) and its predecessor, the Highway Research Board (HRB). The first monograph was published as HRB Special Report 79 in 1964. A completely rewritten monograph was published as TRB Special Report 165 in 1975. This volume is now out of print, and in 1987, the TRB Committee on Traffic Flow Theory and Characteristics recommended that a new monograph be prepared as a joint effort of committee members and other authors.

While many of the basic theories may not have changed much, the significant developments since 1975 merited the writing of a new version of the monograph. The Federal Highway Administration (FHWA) supported this effort through an interagency agreement with the Oak Ridge National Laboratory. TRB is currently reviewing the revised monograph provided by FHWA, and the monograph will be published as a formal TRB report sometime in 1999.


The Revised Monograph

The general philosophy of the previous two reports was retained, but the text was completely rewritten and reorganized into 10 chapters. The primary reasons for such a major revision were: to bring the material up-to-date, to include new developments in traffic-flow theory (e.g., network models), to ensure consistency among chapters and topics, and to emphasize the applications or practical aspects of the theory.

The 10 chapter titles are: (1) Introduction, (2) Traffic Stream Characteristics, (3) Human Factors, (4) Car-Following Models, (5) Continuum Flow Models, (6) Macroscopic Flow Models, (7) Traffic Impact Models, (8) Unsignalized Intersection Theory, (9) Traffic Flow at Signalized Intersections, and (10) Traffic Simulation.

Chapter 2 presents the various models that have been developed to characterize the relationships among the traffic stream variables: speed, flow, and concentration. Most of the relationships are concerned with uninterrupted traffic flow, primarily on freeways or expressways. The chapter stresses the link between theory and measurement capability because, to a large extent, development of the former depends on the latter.

Chapter 3, Human Factors, discusses salient performance aspects of the human element in the context of the person-machine system ( i.e., the motor vehicle). The chapter describes discrete components of performance, including: perception-reaction time; control movement time; how different segments of the population differ in performance; and responses to traffic control devices, to the movement of other vehicles, and to hazards in the roadway. Next, the kind of control performance that underlies steering, braking, and speed control - the primary control functions - is described. Applications of open-loop and closed-loop vehicle control to specific maneuvers such as lane-keeping, car-following, overtaking, gap acceptance, lane closures, and sight distances are also described. To round out the chapter, a few other performance aspects of the driver-vehicle system are covered, including speed limit changes, distractions on the highway, and responses to real-time driver information. The most obvious application of human factors is in the development of car-following models.

Chapter 4 on car-following models examines the manner in which individual vehicles (and their drivers) follow one another. In general, they are developed from a stimulus-response relationship, where the response of successive drivers in the traffic stream is to accelerate or decelerate in proportion to the magnitude of the stimulus at time t after a time lag T. Car-following models form a bridge between the microscopic behavior of individual vehicles and the macroscopic characteristics of a single-lane traffic stream with its corresponding flow and stability properties.

Chapter 5 deals with continuum flow models. Because traffic involves flows, concentrations, and speeds, there is a natural tendency to attempt to describe traffic in terms of fluid behavior. Car-following models recognize that traffic is made up of discrete particles and determine the interactions between these particles. Continuum models are concerned more with the overall statistical behavior of the traffic stream rather than with the interactions between the two particles.

In the fluid-flow analogy, the traffic stream is treated as a one-dimensional compressible fluid. This leads to two basic assumptions: (1) Traffic flow is conserved, and this leads to the conservation or continuity equation. (2) A one-to-one relationship exits between speed and density or between flow and density.

The simple continuum model consists of the conservation equation and the equation of state (speed-density or flow-density relationship). If these equations are solved together with the basic traffic-flow equation (flow equals density times speed), we can obtain the speed, flow, and density at any time and at any point in the roadway. By knowing these basic traffic-flow variables, we know the state of the traffic system and can derive measures of effectiveness, such as delays, stops, travel time, total travel, and other measures that allow the analysts to evaluate how well the traffic system is performing. In this chapter, both simple and high-order models are presented along with analytical and numerical methods for their implementation.

Chapter 6, Macroscopic Flow Models, discards the microscopic view of traffic in terms of individual vehicles or individual system components (such as links or intersections) and adopts instead a macroscopic view of traffic in a network. A variety of models are presented together with empirical evidence of their applicability. Variables that are being considered, for example, include the traffic intensity (the distance traveled per unit area), the road density (the length or area of roads per unit area of city), and the weighted space mean speed. The development of such models extends traffic-flow theory into the network level and can provide traffic engineers with the means to evaluate systemwide control strategies in urban areas. Furthermore, the quality of service provided to motorists can be monitored to assess the city's ability to manage growth. Network performance models could also be used to compare traffic conditions among different cities in order to determine the allocation of resources for transportation system improvements.


Chapter 7 addresses traffic impact models. Issues of traffic safety and environmental impacts, such as fuel consumption and air pollution, were not specifically addressed in the previous monographs. Traffic safety is always the number one issue when dealing with traffic operations and management, and since the Clean Air Act Amendments became law in 1990, fuel consumption and air quality have become critical issues when dealing with transportation and traffic management projects. The following models are specifically discussed: traffic and safety models, fuel-consumption models, and air-quality models.

Chapter 8 is on unsignalized intersection theory. Unsignalized intersections give neither positive indication nor control to the driver. The driver alone must decide when it is safe to enter the intersection. Typically, he looks for a safe opportunity or "gap" in the conflicting traffic. This model of driver behavior is called "gap acceptance." At unsignalized intersections, the driver must also respect the priority of other drivers. This chapter discusses in detail the gap-acceptance theory and the headway distributions used in gap-acceptance calculations. It also discusses the intersections among two or more streams and provides calculations of capacities and quality of traffic operations based on queuing modeling.

Chapter 9 discusses traffic flow at signalized intersections. The statistical theory of traffic flow is presented to provide estimates of delays and queues at isolated intersections, including the effect of upstream traffic signals. This leads to the discussion of traffic bunching, dispersion, and coordination at traffic signals. The fluid (shock-wave) approach to traffic signal analysis is not covered in this chapter; it is treated to some extent in chapter 5. Both pretimed and actuated signal-control theory are presented in some detail. Further, delay models that are founded on steady-state queue theory, as well as those using the so-called coordinate transform method, are covered. Adaptive signal control is discussed only in a qualitative manner because this topic pertains primarily to the development of optimal signal-control strategies, which is outside the scope of this chapter.

The last chapter, chapter 10, is on traffic simulation. Simulation modeling is an increasingly popular and effective tool for analyzing a wide variety of dynamic problems that are not amenable to study by other means. These problems are usually associated with complex processes that cannot readily be described in analytical terms. To provide an adequate test bed, the simulation model must reflect with fidelity the actual traffic-flow process. This chapter describes the traffic models that are embedded in simulation packages and the procedures that are being used to build traffic simulation models and conduct simulation experiments.


The revised monograph includes new developments in traffic-flow theory since 1975. Chapter 3 (Human Factors) and chapter 5 (Continuum Flow Models) are two completely new chapters in this report. All chapters deal with issues on a much broader basis than the previous reports. More importantly, issues of, and application to, intelligent transportation systems (ITS) are discussed in the chapters to the extent possible. For example, the Human Factors chapter includes three levels of driving tasks, including the knowledge-based behavior that becomes increasingly more important to traffic-flow theories as ITS mature. The Traffic Simulation chapter is expanded from the microscopic simulation model to the mesoscopic and macroscopic simulation models, and the simulation-based traffic assignment is expanded to address time-dependent traffic assignment issues.

To ensure the highest degree of reliability, accuracy, and quality in the content of this report, the collaboration of a large number of specialists was enlisted, and this report presents their cooperative efforts. A serious and commendable effort has been made by the contributing authors and reviewers of this report to present fundamental traffic-flow theories and information that will have enduring value. It is hoped that this report will be useful to the ITS community, graduate students, researchers, and practitioners, and others in the transportation profession.

This article was adapted from the revised Monograph on Traffic Flow Theory (November 1997) edited by Nathan H. Gartner, Carroll J. Messer, and Ajay K. Rathi.

Dr. Henry Lieu is a research highway engineer in the Intelligent Systems and Technology Division of FHWA's Office of Safety and Traffic Operations Research and Development. He has been working on FHWA's traffic simulation models since 1987. He received a bachelor's degree in civil engineering from National Taiwan University, a master's degree in transportation engineering from the University of Mississippi, and a doctorate from the University of Maryland.