Finite-size effects in the Nagel-Schreckenberg traffic model

Power laws and phase transitions in heterogenous car following with reaction times

We study the effect of reaction times on the kinetics of relaxation to stationary states and on congestion transitions in heterogeneous traffic using simulations of Newell's model on a ring. Heterogeneity is modeled as quenched disorders in the parameters of Newell's model and in the reaction time of the drivers. We observed that at low densities, the relaxation to stationary state from a homogeneous initial state is governed by the same power laws as derived by E. Ben-Naim et al., Kinetics of clustering in traffic flow, Phys. Rev. E 50, 822 (1994)1063-651X10.1103/PhysRevE.50.822. The stationary state, at low densities, is a single giant platoon of vehicles with the slowest vehicle being the leader of the platoon. We observed formation of spontaneous jams inside the giant platoon which move upstream as stop-go waves and dissipate at its tail. The transition happens when the head of the giant platoon starts interacting with its tail, stable stop-go waves form, which circulate in the ring without dissipating. We observed that the system behaves differently when the transition point is approached from above than it does when approached from below. When the transition density is approached from below, the gap distribution behind the leader has a double peak and is fat-tailed but has a bounded support and thus the maximum gap in the system and the variance of the gap distribution tend to size-independent values. When the transition density is approached from above, the gap distribution becomes a power law and, consequently, the maximum gap in the system and the variance in the gaps diverge as a power law, thereby creating a discontinuity at the transition. Thus, we observe a phase transition of unusual kind in which both a discontinuity and a power law are observed at the transition density. These unusual features vanish in the absence of reaction time, i.e., when the vehicles react instantaneously to a perturbation ahead (e.g., automated driving). Overall, we conclude that the nonzero reaction times of drivers in heterogeneous traffic significantly change the behavior of the free flow to congestion transition while it doesn't alter the kinetics of relaxation to stationary state.

Mechanisms of jamming in the Nagel-Schreckenberg model for traffic flow

We study the Nagel-Schreckenberg cellular automata model for traffic flow by both simulations and analytical techniques. To better understand the nature of the jamming transition, we analyze the fraction of stopped cars P(v=0) as a function of the mean car density. We present a simple argument that yields an estimate for the free density where jamming occurs, and show satisfying agreement with simulation results. We demonstrate that the fraction of jammed cars P(v∈{0,1}) can be decomposed into the three factors (jamming rate, jam lifetime, and jam size) for which we derive, from random walk arguments, exponents that control their scaling close to the critical density.