Cellular automata models of single-lane traffic

The jamming transition in the stochastic cellular automation model (Nagel-Schreckenberg model [J. Phys. (France) I 2, 2221 (1992)]) of highway traffic is analyzed in detail by studying the relaxation time, a mapping to surface growth problems, and the investigation of correlation functions. Three different classes of behavior can be distinguished depending on the speed limit ${v}_{\mathrm{max}}.$ For ${v}_{\mathrm{max}}=1$ the model is closely related to the Kardar-Parisi-Zhang class of surface growth. For $1l{v}_{\mathrm{max}}l\ensuremath{\infty}$ the relaxation time has a well-defined peak at a density of cars \ensuremath{\rho} somewhat lower than the position of the maximum in the fundamental diagram: This density can be identified with the jamming point. At the jamming point the properties of the correlations also change significantly. In the ${v}_{\mathrm{max}}=\ensuremath{\infty}$ limit the model undergoes a first-order transition at $\ensuremath{\rho}\ensuremath{\rightarrow}0.$ It seems that in the relevant cases $1l{v}_{\mathrm{max}}l\ensuremath{\infty}$ the jamming transition is under the influence of a second-order phase transition in the deterministic model and a first-order transition for ${v}_{\mathrm{max}}=\ensuremath{\infty}.$