Application of Storage Theory to Queues with Poisson Arrivals

Abstract

This paper is concerned with the waiting time process, $W(t)$, for the queueing system in which (1) there is only one counter, (2) the customers arrive at random and are served in the order of arrival, and (3) the service time distribution has a general form. It is observed that the Pollaczek-Khintchine formula for the transform of the limiting distribution of $W(t)$ is similar to the one occurring in the theory of continuous time storage processes, and it is inverted by the method used in that theory. Further, $W(t)$ is shown to be a special case of the storage process, and known methods and results of the storage theory are used to obtain the transition distribution function of $W(t)$.