Cellular Automata For Traffic Flow: Analytical Results

TL;DR

This paper analytically investigates cellular automata models for traffic flow, deriving exact and approximate fundamental diagrams using mean-field and cluster approaches, and introduces a related modified automaton model.

Contribution

It presents two mean-field approaches for traffic cellular automata, including an exact solution for maximum velocity one and an improved cluster method for higher velocities.

Findings

Exact fundamental diagram for v_m=1 from car-oriented mean-field theory

Cluster approach yields results in excellent agreement with simulations for v_m>1

Modified automaton model related to a two-dimensional dimer model

Abstract

We use analytical methods to investigate cellular automata for traffic flow. Two different mean-field approaches are presented, which we call site-oriented and car-oriented, respectively. The car-oriented mean-field theory yields the exact fundamental diagram for the model with maximum velocity $\vm =1$ whereas in the site-oriented approach one has to take into account correlations between nearest-neighbour sites. Going beyond mean-field using the so-called $n$-cluster approach our results for $\vm >1$ are in excellent agreement with numerical simulations. We also present a modified cellular automaton which is closely related to a two-dimensional dimer model.