Growth and dissipation of a traffic jam

In this paper a mathematical method for analysing a highway traffic jam is formulated. While the problem of describing a traffic jam is related to several aspects of the theory of traffic flow (intersections, car-following and hydrodynamical theories in particular) it does not seem to have been considered explicitly in the literature to date, and this paper is an attempt to fill the gap. The method is also interesting in its own right as a generalization of the M/G/l queueing situation, the main extension being that the vehicles are assumed to have finite, variable lengths, so that the traffic jam queue changes in both position and length with the passage of time. We also assume that the vehicle causing the jam has a “departure period” which has a different distribution to that of the succeeding vehicles.

Having formulated the situation mathematically, we study various random variables which are of interest. Whereas writers in queueing theory have studied the distribution of individual customer waiting times, a quantity of greater importance in traffic theory is the total delay to all vehicles, since this is directly related to the financial cost of the traffic jam. This variable may be expressed as the integral of a stochastic process, and by imbedding a Markov chain in this process the expected total wait is obtained. Other variables of interest include the jam duration, the number of vehicles delayed and the spatial displacement, and distributions of these quantities are found. Some of the methods of queueing theory carry over with slight generalizations, while in some cases a new type of analysis is required. Some of the results were obtained under less general conditions by Jewel1 (1964).