Macroscopic Traffic Models with Behavior Variation Driven by Noise

Abstract

Hyperbolic PDE can be used to describe the macroscopic dynamics of traffic flow. Models of traffic flow with conservation laws begin with conservation of mass. Equilibrium models (also called first order models) are scalar conservation laws with an explicit closure relation, called the Fundamental Diagram (FD), describing velocity as a function of density. The FD is largely approximated from observation, and in general velocity is a non-increasing function of density. Driver behavior, however, differs among drivers and over time; this variability is not captured by deterministic models. Real data suggests that while one may identify `mean' driver behavior due to non-equilibrium effects and general variability in driving style there is some distribution around the mean. To model driver variability, we introduce a driver-related parameter that describes deviation from the mean and create a family of fundamental diagrams that provide velocity as a function of both density and this parameter. The parameter is governed by an advection diffusion equation with white noise forcing and a relaxation to mean behavior. The resulting models adhere to accepted principles for traffic modeling and are capable of reproducing a richer set of traffic flow phenomena. Most notably, they illustrate that small perturbations may grow into large coherent wave structures, including the formation of jams and emergence of stop-and-go flow patterns, in equilibrium models. Dynamic generalizations have been proposed by numerous authors and describe a velocity that does not instantaneously adjust to traffic density, but instead is governed by an additional equation. In addition to modeling driver variation as an auxiliary variable, these models also permit variation of velocity through a direct modification to the velocity equation. In the present work equilibrium and non-equilibrium traffic models with a stochastic behavior variable are presented, as is a direct stochastic velocity perturbation.