Wave theories of traffic flow

Abstract Two types of mathematical models are presented to explain the empirical observation that a form of “wave phenomenon” occurs when a long line of vehicles on a crowded highway is stopped or started by a signal. The interruption of the steady flow of vehicles initiates “waves of stopping” or “waves of starting.” The first model considered is a car-following model based on a system of linear differential-difference equations that contains two physically significant parameters. The second model adopts a macroscopic viewpoint in which a stream of traffic on a single-lane road may be represented by a compressible “traffic fluid” of density or concentration k and a rate of flow q. The analysis is based on the equation of continuity of the “traffic fluid” and a novel empirical “flow-concentration” relation. The behavior of small disturbances in the vicinity of steady flow is examined by the method of perturbations. The wave phenomena predicted by the first and second models are compared.