Velocity Statistics of the Nagel-Schreckenberg Model

TL;DR

This paper analyzes the velocity distribution and correlations in the Nagel-Schreckenberg traffic model, identifying phase transition features and developing an approximation method that closely matches simulation results.

Contribution

It introduces a 3-body approximation linking velocity and headway distributions, providing new insights into traffic flow phase transitions.

Findings

Velocity-velocity correlations resemble second order phase transition features.

Probability of standing vehicles acts as an order parameter for flow transition.

The approximation method accurately predicts velocity PDFs from headway data.

Abstract

The statistics of velocities in the cellular automaton model of Nagel and Schreckenberg for traffic are studied. From numerical simulations, we obtain the probability distribution function (PDF) for vehicle velocities and the velocity-velocity (vv) correlation function. We identify the probability to find a standing vehicle as a potential order parameter that signals nicely the transition between free congested flow for sufficiently large number of velocity states. Our results for the vv correlation function resemble features of a second order phase transition. We develop a 3-body approximation that allows us to relate the PDFs for velocities and headways. Using this relation, an approximation to the velocity PDF is obtained from the headway PDF observed in simulations. We find a remarkable agreement between this approximation and the velocity PDF obtained from simulations.