Approximation Methods for Queues with Application to the Fixed-Cycle Traffic Light

Abstract

Approximations based upon the representation of the queue as a continuous fluid with either deterministic or stochastic properties are applied to the analysis of models of a fixed-cycle traffic light. These approximations are based upon the use of a law of large numbers or a central limit theorem and are not very sensitive to the detailed stochastic structure of the arrival or departure processes. In the applications considered here these approximations give delays correct to within a few percent. Introduction. The main object of the following paper is to describe and illustrate some approximation methods that can be used to obtain rough estimates of queue lengths, delays, etc., for various queueing problemis, particularly highway traffic intersection problems, which may be too difficult to solve exactly, or if solved exactly give formulas that are more detailed than one needs for the purpose of making quick estimates. The types of approximation we have in mind are those that will apply when the average queue lengths are much larger than 1. They will be asymptotic approximations that strictly speaking are valid only in the limiit of infinite queues, but ones which still give rough estimates for finite but large queues (perhaps of the order of 10). There is an extensive literature on queueing theory including at least a half dozen books and several hundred papers, but the vogue in queueing theory has been to obtain exact solutions of highly idealized models of various processes (exact, however, only in the sense that one usually must invert a few generating functions or Laplace transforms to obtain the quantities one really wants). Consequently the practical value of queueing theory has been severely limited by the lack of approximation methods which one can use to analyse more difficult problems, to estimate errors introduced by the model, or even to compute numbers from very cumbersome exact formulas. A few attempts, however, have been mnade in recent years to break away from the now standard analytic techniques. Particularly noteworthy are the works of Kingman [1], who has shown that properties of nearly saturated queues are rather insensitive to the detailed form of the arrival or service distributions, and of Benes [2], who has shown that many of the results of queueing theory do not depend upon some of the customary statistical independence assumptions. Even here, however, mathematical elegance takes precedence over intuition or applications. In the literature on traffic theory there are already at least 20 references relating to the single problem of delays at a fixed-cycle traffic light, three quarters of which are primarily directed toward a fruitless pursuit of simple exact solutions * Received by the editors October 19, 1964, and in revised form November 19, 1964. t Division of Applied Mathematics, Brown University, Providence, Rhode Island. Part of this work was done while the author was a Fulbright professor at the University of Adelaide, South Australia. The work was also supported in part by the National Science Foundation at Brown University.