Queues with server vacations in urban traffic control

Abstract

A queuing system resulting from a semaphorized intersection regulated by semi-actuated control in a network urban traffic is considered. Modelization of the queue length and of the delay of vehicles is crucial in the study of the performance of intersections equipped with traffic signals. In these systems, the server (green signal) is desactivated (red signal) during a random period of time. Due to this particularity, models for classic queues such as M/M/1, M/G/1 and G/M/1 are not appropriate. In the urban traffic literature, the frequent desactivation of the server as well as the variation of the service period are not well formulated. In the present work a M/G/1 queue where the server occasionally takes vacations and the service discipline is a non-gated time-limited policy is analyzed. The present analysis follows [Leung and Eisenberg, 1991] who consider an application of these models in telecommunications. Their implementation, given its complexity, is made possible by using Laguerre functions when looking for an approximate solution of the differential equations involved. One concludes that the mean delays of vehicles given by this model are slightly smaller than those obtained by simulation procedures, but they are able to give us a good approximation for larger flows, which is of interest for traffic engineers, since, in that case, the approximations one can find in the traffic literature are known not to be adequate.