Since the subject of traffic dynamics has captured the interest of physicists, many surprising effects have been revealed and explained. Some of the questions now understood are the following: Why are vehicles sometimes stopped by ``phantom traffic jams'' even though drivers all like to drive fast? What are the mechanisms behind stop-and-go traffic? Why are there several different kinds of congestion, and how are they related? Why do most traffic jams occur considerably before the road capacity is reached? Can a temporary reduction in the volume of traffic cause a lasting traffic jam? Under which conditions can speed limits speed up traffic? Why do pedestrians moving in opposite directions normally organize into lanes, while similar systems ``freeze by heating''? All of these questions have been answered by applying and extending methods from statistical physics and nonlinear dynamics to self-driven many-particle systems. This article considers the empirical data and then reviews the main approaches to modeling pedestrian and vehicle traffic. These include microscopic (particle-based), mesoscopic (gas-kinetic), and macroscopic (fluid-dynamic) models. Attention is also paid to the formulation of a micro-macro link, to aspects of universality, and to other unifying concepts, such as a general modeling framework for self-driven many-particle systems, including spin systems. While the primary focus is upon vehicle and pedestrian traffic, applications to biological or socio-economic systems such as bacterial colonies, flocks of birds, panics, and stock market dynamics are touched upon as well.

Traffic and related self-driven many-particle systems

This paper presents a simple representation of traffic on a highway with a single entrance and exit. The representation can be used to predict traffic's evolution over time and space, including transient phenomena such as the building, propagation, and dissipation of queues. The easy-to-solve difference equations used to predict traffic's evolution are shown to be the discrete analog of the differential equations arising from a special case of the hydrodynamic model of traffic flow. The proposed method automatically generates appropriate changes in density at locations where the hydrodynamic theory would call for a shockwave; i.e., a jump in density such as those typically seen at the end of every queue. The complex side calculations required by classical methods to keep track of shockwaves are thus eliminated. The paper also shows how the equations can mimic the real-life development of stop-and-go traffic within moving queues.

The cell transmission model: a dynamic representation of highway traffic consistent with the hydrodynamic theory

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

Urban Transportation Networks: Equilibrium Analysis With Mathematical Programming Methods

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Lectu re Notes in Economics and Mathematical Systems

It is assumed that the velocity of a car at time t is some nonlinear function of the spacial headway at time t-Δ, so the equations of motion for a sequence of cars consists of a set of differential-difference equations. There is a special family of velocity-headway relations that agrees well with experimental data for steady flow, and that also gives differential equations which for Δ = 0 can be solved explicitly. Some exact solutions of these equations show that a small amplitude disturbance propagates through a series of cars in the manner described by linear theories except that the dependence of the wave velocity on the car velocity causes an accleration wave to spread as it propagates and a deceleration wave to form a stable shock. These conclusions are then shown to hold for quite general types of velocity-headway relations, and to yield a theory that in certain limiting cases gives all the results of the linear car-following theories and in other cases all the features of the nonlinear continuum theories, plus a detailed picture of the shock structure.

Nonlinear Effects in the Dynamics of Car Following

A very simple “car-following” rule is proposed wherein, if an nth vehicle is following an (n−1)th vehicle on a homogeneous highway, the time-space trajectory of the nth vehicle is essentially the same as the (n−1)th vehicle except for a translation in space and in time It seems that such a rule is at least as accurate as any of the more elaborate rules of car-following that have been proposed over the last 50 years or so Actually, the proposed model could be interpreted as a special case of existing models but with fewer parameters and a different logic At least this should form a reasonable starting point for investigating other phenomena

A simplified car-following theory: a lower order model

In the theory of “kinematic waves,” as described originally by Lighthill and Whitham in 1955, the evaluation of the shock path is typically rather tedious. Instead of using this theory to evaluate flows or densities, one can use it to evaluate the cumulative flow A ( x , t ) past any point x by time t . It is shown here how a formal solution for A ( x, t ) can be evaluated directly from boundary or initial conditions without evaluation at intermediate times and positions. If there are shocks, however, this solution will be multiple-valued. The correct solution, which is the lower envelope of all such formal solutions, will automatically have discontinuities in slope describing the passage of a shock. To evaluate A ( x, t ) at any particular location x , it is not necessary to follow the actual path of the shock. The solution can be evaluated directly in terms of the boundary data by either graphical or numerical techniques.

A simplified theory of kinematic waves in highway traffic, part I: General theory

The results contained herein pertain to the problem of determining the equilibrium distribution of customers in closed queuing systems composed of M interconnected stages of service. The number of customers, N, in a closed queuing system is fixed since customers pass repeatedly through the M stages with neither entrances nor exits permitted. At the ith stage there are ri parallel exponential servers all of which have the same mean service rate µi. When service is completed at stage i, a customer proceeds directly to stage j with probability pij. Such closed systems are shown to be stochastically equivalent to open systems in which the number of customers cannot exceed N. The equilibrium equations for the joint probability distribution of customers are solved by a separation of variables technique. In the limit of N â ∞ it is found that the distribution of customers in the system is regulated by the stage or stages with the slowest effective service rate. Asymptotic expressions are given for the marginal distributions of customers in such systems. Then, an asymptotic analysis is carried out for systems with a large number of stages i.e., M â« 1 all of which have comparable effective service rates. Approximate expressions are obtained for the marginal probability distributions. The details of the analysis are illustrated by an example.

Closed Queuing Systems with Exponential Servers

For a freeway having various entrance and exit ramps, the methods described in Part I are used to relate the cumulative flow curve at any junction to the net cumulative entrance flow at this junction, and the cumulative flow curves for the freeway at the next upstream junction and/ or the next downstream junction. If the type of flow-density relations typical of freeway traffic are idealized by a triangular shaped curve with only two wave speeds, one for free-flowing traffic (positive) and the other for congested traffic (negative), then the relationship is easy to evaluate. The cumulative flow curve at the junction is simply the lower envelope of a translation of the cumulative curve from upstream and a different translation of the cumulative curve from downstream. This relationship is the basic building block for a freeway flow prediction model described in Part III.

Gordon F. Newell

Papers

Nonlinear Effects in the Dynamics of Car Following

A simplified car-following theory: a lower order model

A simplified theory of kinematic waves in highway traffic, part I: General theory

Closed Queuing Systems with Exponential Servers

A simplified theory of kinematic waves in highway traffic, part II: Queueing at freeway bottlenecks