Traffic models

Summary: Mathematical models and statistical analysis of traffic flow suggest solutions.

Traffic flow is studied using mathematical and statistical techniques and computer simulations in order to better understand the movement of vehicles on roads and highways. Americans drive their vehicles almost 3 trillion miles per year on approximately 4 million miles of public roads. Mathematical models have shown that the behavior of even a single driver can have a broad impact on overall traffic flow in this dynamic system. As every driver knows, traffic patterns can often be unpredictable and frustrating, leading to driver stress, accidents, pollution, wasted fuel, and wasted time. Mathematical analysis of traffic congestion can provide transportation engineers with insights leading to improvements in efficiency and safety in the transportation of goods and people. A mathematical understanding of traffic flow patterns can also provide guidance for the design of roadways and provide more accurate calculations of trip itineraries and real-time driving times. These can be disseminated to the public and used in intelligent transportation systems.

The use of mathematics to describe traffic flow patterns slowly originated in the 1930s in order to study road capacity and also to begin to address traffic-related questions, such as how does traffic move through intersections. The mathematical investigations of vehicular traffic increased rapidly in the 1950s, mainly because of the expansion of the highway system after World War II. In the twenty-first century, theoretical models of traffic are utilized by high-performance computers, which can simulate the motions of vehicles on virtual road networks of entire cities and regions.

Traffic engineers distinguish between uninterrupted traffic flow situations (for example, traffic streams on highways and other limited-access roads) and interrupted flow circumstances (for example, where two or more traffic streams meet at a road intersection). The methods suited to analyze a particular traffic scenario depend on whether the flow is interrupted or uninterrupted. When formulating a mathematical description or model of traffic, one must attempt to account for the interplay between the vehicles and the drivers, the layout of the road system, traffic lights, road signs, and other factors.

Queuing theory, which is essentially the mathematical theory of waiting lines, is a probabilistic framework used for analyzing various traffic flow problems, such as optimizing vehicle passage through an intersection or traffic circle, calculating vehicle waiting times at tollbooths, and other similar waiting problems. On the other hand, car-following models and hydrodynamic modeling are deterministic approaches for analyzing traffic flow on long stretches of road.

Car-Following Traffic Models

Car-following models, also known as microscopic models, are considered from the point of view of tracking the movements of a line of n=1,…, N individual cars driving in the same direction down a road in order to try to predict their exact positions x_n(t), velocities v_n(t), and accelerations a_n(t). The starting point for car-following problems is to model how the driver of a car reacts when the vehicle directly in front of it changes speed (it is assumed for simplicity that there no passing is allowed). As a first crude estimation, one could assume a driver adjusts instantaneously according to the relative speed of the driver’s car and the vehicle in front:

where C is a constant of proportionality, called the sensitivity parameter, which can be measured experimentally. A more realistic assumption would be that a driver adjusts with a lag response time of about one or two seconds, to a maneuver by the vehicle in front of it:

where T is the time lapse because of the driver’s delayed reaction. Equations with delays such as these are then solved to keep track of each vehicle as the traffic moves. Numerous additional assumptions and effects have been incorporated into more sophisticated theories of car-following, such as considering the impact of spacing between cars, the effect of aggressive or cautious driving, and the effect of drivers looking ahead in the road and reacting to the motions of multiple vehicles in front of it.

Hydrodynamic Traffic Models

Hydrodynamic modeling, also called “continuum modeling,” considers the flow of a traffic stream to be analogous to the flow of a compressible fluid in a pipe. Continuum traffic models do not keep track of the positions of individual vehicles, like car-following models, but track averaged, macroscopic quantities. For a long stretch of crowded road, such as an interstate highway, three important quantities of interest are flow rate (Q in vehicles per hour), vehicle speed (V in miles per hour), and vehicle density (ρ in number of vehicles per mile). These variables, of course, can vary along the stretch of road in both space and time, and their relationship is described algebraically as Q=ρV. Furthermore, based on observations of traffic patterns over the years, it has been posited that for a given stretch of road, there exists a direct relationship between the flow rate and density. What has essentially been observed is that, on a road having some maximum flow rate, there is a critical vehicle density below which speed is not severely impacted but above which speed reduces. As the density continues to increase, then eventually flow rate reduces, and traffic becomes completely congested. For a concrete example, Greenshield’s model postulates a simple linear relation between vehicle speed and density,

where the parameterV_free is the free flow speed of a vehicle that is unencumbered, and ρ_jam is the density corresponding to bumper-to-bumper traffic. Then, the flow-density relation would be given by

This parabolic function begins to capture some of the flow-density behavior that is observed on some real roads, although it is certainly an oversimplification. If the traffic density is zero (ρ=0), then the flow rate must also be zero (Q=0). Additionally, in bumper-to-bumper traffic (ρ=ρ_jam), the flow rate is zero, or very nearly zero in reality.

In the Lighthill–Whitham–Richards (LWR) theory of traffic, a long stretch of road is considered that has no entries or exits. On such a stretch of road, the number of vehicles must be conserved, and this fact combined with a flow-density relation gives rise to an equation, called a “conservation law,” that predicts how vehicle density varies along the stretch of road. When a traffic jam occurs, it manifests as a sudden disturbance, or shock-wave, in the vehicle density along the road. LWR theory and other much more sophisticated continuum models of traffic can predict conditions under which traffic jams will form, propagate, and dissipate. Common reasons for traffic jams are accidents, construction, lane merges, and other changes in road capacity. However (as all drivers have experienced) sometimes “phantom jams” occur on highways for no apparent reason. These phantom jams can also be explained by continuum traffic models.

Bibliography

Daganzo, Carlos F. Fundamentals of Transportation and Traffic Operations. Oxford, England: Pergamon-Elsevier, 1997.

Gazis, Denos C. Traffic Theory. Norwell, MA: Kluwer Academic Publishers, 2002.

May, Adolf D. Traffic Flow Fundamentals. Upper Saddle River, NJ: Prentice Hall, 1990.