One-dimensional traffic flow problems: A microscopic approach

A microscopic approach to one-dimensional (1D) cellular automaton traffic flow models is presented. Starting from an equation describing the time evolution of the Boolean variable, which describes whether a site is occupied by a car or empty, defined on each site and using a decoupling approximation, an equation relating the average speeds v(t + 1) and v(t) is obtained. This relation can be treated as a dynamical mapping between the average speeds v(t + 1) and v(t). The average speed in the long time limit can be obtained by studying the attractors of the mapping and their stabilities. The approach is also applied to models with random delays. Results that are in agreement with simulation data and previously proposed mean field theories using macroscopic considerations are obtained. Our approach thus represents a microscopic method of deriving mean field theories with the advantage that controllable approximations can be introduced step by step.