Metastable states in cellular automata for traffic flow

Barlovic, R.; Santen, L.; Schadschneider, A.; Schreckenberg, M. · 1998 · OpenAlex-citations

DOI: 10.1007/s100510050504

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Summary

This paper addresses the absence of metastable states and hysteresis effects in the Nagel-Schreckenberg (NaSch) cellular automaton model, despite their presence in empirical traffic data. The authors aim to identify the minimal modifications required for the NaSch model to reproduce these phenomena, specifically focusing on the prerequisites for hysteresis in the flow-density relation. The study investigates whether such states arise from realistic braking rules, continuum descriptions, or specific stochastic parameters. The authors analyze three variants of the NaSch model incorporating "slow-to-start" (s2s) rules, which reduce the outflow from jams compared to maximum possible flow. First, they examine the T2 model, which uses a spatial s2s rule where standing cars accelerate with lower probability than moving cars. Second, they review the Benjamin-Johnson-Hui (BJH) model, which employs a temporal s2s rule requiring memory of previous braking events. Third, they propose a new generalization of the NaSch model featuring velocity-dependent randomization (VDR), where the braking probability $p$ is higher for stopped cars ($p_0$) than for moving cars ($p$). Simulations were conducted using parallel dynamics on periodic lattices, with system sizes ranging from 32 to 10,000 cells, and maximum velocities $v_{max}$ of 1 and 5. The authors utilized two methods to detect metastability: adiabatic density changes and initialization from distinct states (homogeneous vs. megajam). The results demonstrate that all three models exhibit metastable states and hysteresis loops in the fundamental diagram for $v_{max} > 1$. The T2 and VDR models show phase separation, where high-density states consist of a large jam and a free-flowing region, while low-density states remain homogeneous. The VDR model, in particular, allows independent control of free-flow and congested flow properties. The authors find that the reduction of outflow from jams is the critical mechanism enabling phase separation; if outflow is maximal, small jams dissolve immediately. For $v_{max}=1$, the T2 model exhibits a geometric phase transition leading to complete blockage, whereas the VDR model shows no phase separation unless deterministic limits are approached. The VDR model’s phenomenological approach accurately predicts the flow branches, confirming that reduced outflow destabilizes clusters in the outflow region. The significance of this work lies in demonstrating that metastable states in traffic flow do not require complex, realistic braking rules or continuous space-time descriptions. Instead, they emerge from simple stochastic modifications, specifically the reduction of jam outflow via slow-to-start behaviors. This finding validates the flexibility of cellular automata in capturing fine-structure traffic phenomena, such as hysteresis, and provides a theoretical basis for traffic control strategies aimed at stabilizing homogeneous flow to maximize throughput.

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• traffic density