Some Inequalities for the Queue GI/G/1

Abstract

One of the central problems in the theory of queues has been the study of the singleserver queue with various assumptions about the probabilistic nature of the input and service processes. In particular, the queue GI1//1 (in the notation of Kendall (1953)) has been studied by Lindley (1952), and a complete formal solution, under mild analytic conditions, for the equilibrium waiting-time distribution, has been obtained by Smith (1953). This solution is not, however, easy to evaluate except for very special distributions of interarrival and service times. The present paper is concerned with the mean and variance of the waiting time, which are quantities of some practical importance. Inequalities are derived for these moments, both in equilibrium and non-equilibrium conditions. Thus we consider a queueing system in which customers C0, C1, ... arrive and are attended to by a single server in the order of their arrival. The customer Cn waits in the queue for a time Wn, and is then served in a further time 8nWe denote by tn the time elapsing between the arrivals of successive customers Cn, Cn+?, and write n-i UnSntn Un E Ur. (1) r=O

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