One-Dimensional Traffic Flow Problems: A Microscopic Approach

Abstract

A microscopic approach to one-dimensional (1D) cellular automaton trafficflow models is presented. Starting from an equation describing the timeevolution of the Boolean variable, which describes whether a site isoccupied by a car or empty, defined on each site and using adecoupling approximation, an equation relating the average speeds v(t+1)and v(t) is obtained. This relation can be treated as a dynamicalmapping between the average speeds v(t+1) and v(t).The average speed in the long time limit can be obtained by studying theattractors of the mapping and their stabilities. The approach is alsoapplied to models with random delays. Results that are in agreement withsimulation data and previously proposed mean field theories using macroscopicconsiderations are obtained. Our approach thus represents a microscopicmethod of deriving mean field theories with the advantage that controllableapproximations can be introduced step by step.