Probabilistic Description of Traffic Flow

Abstract

The study of traffic flow is investigated by different means. Well established theories are (i) kinematic models based on partial differential equations to describe traveling density waves, and (ii) deterministic models using nonlinear car‐following equations to determine trajectories of moving cars, as well as (iii) large-scale simulation hopping models like cellular automata. An important intermediate approach is (iv) the stochastic or probabilistic attempt to understand phenomena like “Stau aus dem Nichts” (phantom jam) on long crowded roads. Initiated by the old argument that road traffic is a stochastic process, we develop a new probabilistic theory based on Markov processes to improve our understanding of traffic flow and its three phases (free flow, synchronized motion, wide moving jams) discovered by Kerner. As an introductory example, first we consider a dissolution of a car queue described by the stochastic master equation as a one-step decay process. Furtheron more realistic models are developed to investigate the nucleation, growth and condensation as well as dissolution of car clusters on a circular one-lane freeway. In analogy to usual aggregation phenomena such as the formation of liquid droplets in supersaturated vapour the clustering behavior in traffic flow is described by the master equation. At overcritical densities the transition from the initial free particle situation (free flow of vehicles) to the final congested state, where one or several big aggregates of cars have been formed, is shown. In dependence on the concentration of cars on the road the stationary solution of the master equation is derived analytically. The obtained fundamental diagram as flow-density-relation indicates clearly the different regimes of traffic flow (free jet of cars, coexisting phase of jams and isolated cars, highly viscous heavy traffic). In the (thermodynamic) limit of infinite number of vehicles on an infinite long road the analytical solution for the fundamental diagram is in agreement with experimental traffic flow data. As a particular example we take into account measurements from German highways presented by Kerner and Rehborn.