Asymptotic Analysis of the Waiting-Time Distribution for a Large Closed Processor-Sharing System

Abstract

The service of an interactive computer system is characterized by the random waiting (or response) time perceived by users. The system consists of a bank of terminals, each of which asynchronously alternates between “thinking” and waiting for service from a central processing unit which operates under the processor-sharing discipline. It is assumed that there are $(N + 1)$ terminals, where N is large, and that $\rho = Np = O(1)$, where p is the ratio of the mean required service time to the mean think time. Asymptotic approximations to the equilibrium distribution of the waiting time are derived for: normal usage, $\rho < 1$; heavy usage, $\rho - 1 = O(N^{ - 1/2} )$; and very heavy usage, $\rho > 1$. Singular perturbation techniques are used, and the matching of the heavy-usage approximation with both the normal and very heavy-usage approximations is investigated. Numerical results are presented which indicate that the approximations are useful for $N\geqq 100$.