A Class of Approximations for the Waiting Time Distribution in a GI/G/1 Queueing System

Single server queueing models arise quite naturally in the study of a wide variety of stochastic server systems. A particularly important class is single processor computer systems, such as stored program control switching systems or nodes in a data communication network. In many such applications, it is important to keep as much of the structure of the interarrival and service time processes as possible in order to obtain realistic results; that is, the simplifying assumptions of exponential distributions cannot be made. While the resulting GI/G/l queue may be extremely difficult to analyze, one is often content with reasonable approximations that incorporate the main features of the problem. In addition, one often desires results that are reasonably simple analytically, since the behavior of the G//G/1 295 queueing model may be the input to the analysis of a more complex system. Our main purpose here is to present a class of approximations for the waiting time distribution, W(jc), for such GI/G/l systems which allows the analyst to use as much (or little) of the structure of the relevant input and service processes as desired. The resulting approximations can be extremely simple in form (e.g., a single exponential) or, with additional effort, more complex. In particular, we give relatively simple expressions for constants C and a, such that WA(x) = 1 -- Ce~ax provides a good fit to the probability of delay, PD = [1 -- W(0)] and the mean delay, w. Similar approximations are developed which provide a good fit to the tails of the delay distribution.