Response-Time Distribution for a Processor-Sharing System

Abstract

We consider the response time for jobs in a processor-sharing system with a Poisson arrival process and exponentially distributed required service time, i.e. an $M/M /1 - PS$ queue. The response time $\tilde W$ is the sum of the delay and the required service time. We derive an integral representation for the equilibrium response-time distribution $\Pr \{ \tilde W > t \}$, and evaluate this integral numerically for several values of the traffic intensity $\rho < 1$. We also investigate the behavior of $\Pr \{ \tilde W > t \}$ in the heavy-traffic case when $\rho $ is close to 1. For $\rho = 1 - \alpha ,0 < \alpha \ll 1$, it is shown that a singular perturbation occurs at $t - 0$, and we derive a composite approximation to $\Pr \{ \tilde W > t \}$. For $0.75\leqq \rho \leqq 0.99$, the relative error of this approximation is less than $0.62\alpha ^2 $ in the range for which $1\geqq \Pr \{ \tilde W > t \}\geqq 0.01$.