Bunching of cars in asymmetric exclusion models for freeway traffic

Nagatani, Takashi · 1995 · OpenAlex-citations

DOI: 10.1103/physreve.51.922

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Summary

This paper addresses the phenomenon of car bunching in freeway traffic by extending one-dimensional asymmetric simple-exclusion models using cellular automata (CA). While previous models, such as the Nagel-Schreckenberg model, successfully simulated transitions to start-stop waves, they failed to capture the continuous clustering of cars observed in real highway traffic, which exhibits a $1/f$ power spectrum. The author proposes three extended CA models with parallel dynamics to simulate this bunching behavior, incorporating inherent velocity differences, interval-dependent transition probabilities, and safety distance constraints. In Model I, the simulation accounts for the inherent velocity of individual cars by assigning each car a fixed transition probability $p_i$, uniformly distributed between 0.5 and 1.0. Cars with lower velocities block faster cars, causing clusters to grow over time without attractive forces. Simulations on a lattice of $10^5$ sites reveal that for low densities ($p < 0.1$), the mean interval $\langle \Delta x \rangle$ between consecutive cars scales as $t^{0.47 \pm 0.03}$, consistent with the analytical result of 0.5 derived from the Burgers equation. The cumulative interval distribution follows an exponential scaling form. Model II introduces a dependence of the transition probability $T$ on the interval $\Delta x$ between consecutive cars, defined as $T = \Delta x^{-\alpha}$ for $\alpha \ge 0$. This model demonstrates that cars with shorter intervals move faster, leading to increased bunching. The mean interval scales as $\langle \Delta x \rangle \approx t^\beta$, where the scaling exponent is determined by the relation $\beta = 1/(1+\alpha)$. The cumulative interval distribution in this model approximates a Gaussian scaling function. Model III incorporates a critical safety distance $x_c$. Cars move at maximal velocity if the interval $\Delta x > x_c$, but their transition probability decreases as $(\Delta x/x_c)^\alpha$ when $\Delta x \le x_c$. This model exhibits a dynamical phase transition from laminar (uncongested) flow to congested flow characterized by interacting start-stop waves. The transition point occurs at a critical density $p_c \approx 1/(x_c + 1)$. The velocity of the start-stop waves is inversely proportional to the critical distance ($v_w \approx 1/x_c$). The traffic current profiles vary significantly with $x_c$ and $\alpha$, with congested flow currents decreasing as the power $\alpha$ increases. The study concludes that car bunching can be effectively modeled through velocity heterogeneity (Model I) or interval-dependent interactions (Models II and III). Models I and II produce bunching similar to Burgers turbulence, where cars continue moving, whereas Model III reproduces the start-stop wave dynamics seen in other CA models. These findings provide microscopic explanations for macroscopic traffic phenomena, linking exclusion processes to hydrodynamic descriptions of traffic flow.

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• traffic density