Bunching of cars in asymmetric exclusion models for freeway traffic

One-dimensional cellular automaton (CA) models are presented to simulate bunching of cars in freeway traffic. The CA models are three extended versions of the asymmetric simple-exclusion model with parallel dynamics. In model I, the inherent velocities of individual cars are taken into account. It is shown that bunching of cars occurs since the car with low velocity prevents the car with high velocity from going ahead. The mean interval 〈Δx〉 of consecutive cars scales as 〈Δx〉≊${\mathit{t}}^{0.47\pm 0.03}$ where t is time. In model II, the asymmetric exclusion model is extended to take into account the dependence of the transition probability T upon the interval Δx of consecutive particles (cars): T=Δ${\mathit{x}}^{\mathrm{-}\mathrm{\alpha }}$ (α≥0). It is shown that the mean interval 〈Δx〉 of consecutive particles scales as 〈Δx〉≊${\mathit{t}}^{1/(1+\mathrm{\alpha })}$ by bunching of cars. In model III, the velocity v of a car depends on the interval Δx of consecutive cars in such a manner that the transition probability T=1 for Δx>${\mathit{x}}_{\mathit{c}}$ (${\mathit{x}}_{\mathit{c}}$≥1), and for Δx≤${\mathit{x}}_{\mathit{c}}$, T=(Δx/${\mathit{x}}_{\mathit{c}}$${)}^{\mathrm{\alpha }}$. It is shown that a transition from laminar traffic flow (uncongested traffic flow) to congested traffic flow occurs with increasing density p of cars.