Particle-hopping Models of Vehicular Traffic: Distributions of Distance Headways and Distance Between Jams

TL;DR

This paper analyzes the distributions of vehicle gaps and jam distances in a traffic model, revealing coexistence of free flow and jams at higher speeds and providing exact and numerical results for different maximum speeds.

Contribution

It introduces a novel transfer matrix method and extends analytical approaches to derive jam distance distributions in the Nagel-Schreckenberg traffic model.

Findings

Distribution of headways shows two peaks at higher speeds indicating coexistence of free flow and jams.

Exact analytical expression for jam distances in the $V_{max} = 1$ case.

Numerical simulations for jam distances when $V_{max} > 1$.

Abstract

We calculate the distribution of the distance headways (i.e., the instantaneous gap between successive vehicles) as well as the distribution of instantaneous distance between successive jams in the Nagel-Schreckenberg (NS) model of vehicular traffic. When the maximum allowed speed, $V_{max}$, of the vehicles is larger than unity, over an intermediate range of densities of vehicles, our Monte Carlo (MC) data for the distance headway distribution exhibit two peaks, which indicate the coexistence of "free-flowing" traffic and traffic jams. Our analytical arguments clearly rule out the possibility of occurrence of more than one peak in the distribution of distance headways in the NS model when $V_{max} = 1$ as well as in the asymmetric simple exclusion process. Modifying and extending an earlier analytical approach for the NS model with $V_{max} = 1$, and introducing a novel transfer matrix technique, we also calculate the exact analytical expression for the distribution of distance between the jams in this model; the corresponding distributions for $V_{max} > 1$ have been computed numerically through MC simulation.