Computable Throughput and Metastability in a Cellular Automaton Model for Traffic Flow

A cellular automaton model for traffic flow is analyzed. For this model, it is shown that under ergodic initial configurations, the distribution of cars will converge in time to a mixture of free flow and solid blocks. Furthermore, the nature of the free flow and solid block distributions is fully described, thus allowing for a specific computation of throughput in terms of the parameters. The model is also shown to exhibit a hysteresis phenomenon, which is similar to what has been observed on actual highways. 1. Introduction and Description of the Model 1.1. Introduction There have been various cellular automaton models introduced to model traffic flow [1–3]. Many of these models gain computational advantage over older so-called car-following, fluid dynamical, and kinetic (gas-type) models by discretizing both space and time (see [4] for an overview of various models). For these discrete models, simple rules are developed to govern car movement. While, on a small scale, the rules oversimplify traffic behavior, the goal is that large scale traffic phenomena, such as the formation and persistence of traffic jams, present themselves in this simplified approach. The model used in this paper is a discrete time probabilistic cellular automaton model developed by Gray and Griffeath in [2]. We will be concerned with macroscopic limiting phenomena on an infinitely long one-dimensional highway. In this paper, we show the existence of a limiting throughput (flux) of cars and describe these regions explicitly. For traffic densities above a critical value, we are able to show that the traffic organizes itself into regions of free flow and regions of traffic jam, both of which will be given precise mathematical definitions in this context. We also observe the existence of metastable states: conditions which allow certain ergodic traffic distributions to have higher throughput than others with the same density of cars. The existence of metastable states has been sought after [5] due to the fact that such states have been shown to be exhibited in real-world traffic flow [6]. These metastable states exhibit a hysteresis phenomenon in the sense that minor perturbations of the cars in these states may eventually lead to a drastic change in the traffic throughput. As mentioned in [2], a property which may be related to the hysteresis phenomenon encountered with the metastable states is the so-called slow-to-start feature, which may be the key element which gives realistic macroscopic behavior to the cellular automaton model. Other slow-to-start models can be found in [3,