Analytical approaches to CA for traffic flow: Approximations and exact solutions

Cellular automata have turned out to be important tools for the simulation of traffic flow. They are designed for an efficient impletmentation on the computer, but hard to treat analytically. Here we discuss several approaches for an analytical description of the Nagel-Schreckenberg (NaSch) model and its variants. These methods yield the exact solution for the special case $\vm=1$ of the NaSch model and are good approximations for higher values of the velocity ($\vm > 1$). We discuss the validity of these approximations and the conclusions for the underlying physics that can be drawn from the success or failure of the approximation.

[1]  D. Chowdhury,et al.  Particle-hopping models of vehicular traffic: Distributions of distance headways and distance between jams , 1997, cond-mat/9706094.

[2]  W. Knospe,et al.  Disorder Effects in CA-Models for Traffic Flow , 1997 .

[3]  Michael Schreckenberg,et al.  Garden of Eden states in traffic models , 1998 .

[4]  Stephen Wolfram,et al.  Theory and Applications of Cellular Automata , 1986 .

[5]  A. Schadschneider,et al.  Jamming transition in a cellular automaton model for traffic flow , 1998 .

[6]  H. Takayasu,et al.  1/f NOISE IN A TRAFFIC MODEL , 1993 .

[7]  A. Schadschneider,et al.  Traffic flow models with ‘slow‐to‐start’ rules , 1997, cond-mat/9709131.

[8]  Nagel,et al.  Discrete stochastic models for traffic flow. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  R. Barlovic,et al.  Metastable States in CA models for Traffic Flow , 1997 .

[10]  Michael Schreckenberg,et al.  A cellular automaton model for freeway traffic , 1992 .

[11]  Arnab Majumdar,et al.  Distributions of Time Headways in Particle-hopping Models of Vehicular Traffic , 1997 .

[12]  A. Crisanti,et al.  Products of random matrices in statistical physics , 1993 .

[13]  Michael Schreckenberg,et al.  Car-oriented mean-field theory for traffic flow models , 1996 .

[14]  A. Schadschneider,et al.  Cellular automation models and traffic flow , 1993, cond-mat/9306017.

[15]  R. Kikuchi.  The Path Probability Method , 1966 .

[16]  M. Fukui,et al.  Traffic Flow in 1D Cellular Automaton Model Including Cars Moving with High Speed , 1996 .

[17]  ben-Avraham,et al.  Mean-field (n,m)-cluster approximation for lattice models. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[18]  Jonathan D. Victor,et al.  Local structure theory for cellular automata , 1987 .