Analytical Results For The Steady State Of Traffic Flow Models With Stochastic Delay

TL;DR

This paper derives exact analytical mean field equations for the steady state of a stochastic delay traffic flow cellular automaton model, providing fundamental diagrams that match simulation results.

Contribution

It introduces an analytical approach to determine the steady state and fundamental diagrams of a stochastic delay traffic model, extending understanding of traffic flow dynamics.

Findings

Analytical fundamental diagrams match simulation data.

Steady states characterized by inter-car spacing distributions.

Traffic speed depends on car density with clear asymptotic behaviors.

Abstract

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>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 inter-car 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 $(\rho \geq 1/M)$, it is shown that inter-car 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 inter-car spacings are of the short type. Similarly for low car densities $(\rho \leq 1/M)$, it can be shown that traffic flow approaches an asymptotic steady state in which all the inter-car 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.