A Queuing Model for Car Passing

A model is developed for the flow of automobiles in one direction along a two-lane, country road. The model takes into account the fact that cars differ in speed, that slower cars accumulate queues behind them, and that the rate of escape from such a queue, by passing the lead car, depends on the speed of the lead car, on the nature of the road, and on the density of traffic going in the opposite direction. Equations for the stochastic steady state of the system reduce to an in-integral equation for the mean queue length, as a function of lead-car speed and of a queue-delay parameter. Solutions are obtained, with tables and graphs, for two different assumptions regarding the dependence of passing delay on lead-car speed. In both cases the model exhibits a sudden change from sparse-traffic conditions, where queuing is rare and delay of the faster cars is minimal, to the dense-traffic conditions where nearly all cars find themselves trapped in a slow queue, as the passing-delay parameter is increased beyond its transitional value. Such a sudden “phase-change” in traffic character is typical of actual traffic under the specified conditions. Measured dependence of queue length on lead-car speed checks nicely with the model, for sparse-traffic conditions.