Asymptotic Expansions of Moments of the Waiting Time in a Shared-Processor of an Interactive System

Abstract

An interactive computer system’ service is characterized by the random waiting (or response) time perceived by users. This paper presents a novel solution to the problem of efficiently computing the second moment of the waiting time of a class of large interactive systems. The physical system consists of a bank of terminals, each of which asynchronously alternates between “thinking” and waiting for service from a CPU which operates under the processor-sharing discipline. The problem of obtaining higher order waiting time moments is quite different from that of obtaining CPU queue statistics. Only the first moment of the waiting time is obtainable from the CPU queue statistics. The technique for arriving at the second moment of the waiting time consists of developing an asymptotic expansion in inverse powers of the number of terminals. Hence, as the system grows, quite fortuitously, fewer terms of the series require computation to achieve the desired accuracy. Beside its numerical advantages, the results give new insight since the leading terms of the series, which contain most of the information, are obtained explicitly. The novelty also rests on the fact that instead of solving matrix equations, the problem is turned into one of solving a second order differential equation. A simple two-dimensional recursion yields all the terms of the asymptotic expansion.