Analytical results for the steady state of traffic flow models with stochastic delay

Exact mean field equations are derived analytically to give the fundamental diagrams, i.e., the average speed-car density relations, for the Fukui-Ishibashi one-dimensional traffic flow cellular automaton model of high speed vehicles ${(v}_{\max }=M>\mathrm{}1)$ with stochastic delay. Starting with the basic equation describing the time evolution of the number of empty sites in front of each car, the concepts of intercar spacings longer and shorter than M are introduced. The probabilities of having long and short spacings on the road are calculated. For high car densities $(\rho >~1/M),$ it is shown that intercar spacings longer than M will be shortened as the traffic flow evolves in time, and any initial configurations approach a steady state in which all the intercar spacings are of the short type. Similarly for low car densities $(\rho <~1/M),$ it can be shown that traffic flow approaches an asymptotic steady state in which all the intercar spacings are longer than $M-2.$ The average traffic speed is then obtained analytically as a function of car density in the asymptotic steady state. The fundamental diagram so obtained is in excellent agreement with simulation data.