The waiting time distribution for the random order service $M/M/1$ queue

Abstract

The $M/M/1$ queue is considered in the case in which customers are served in random order. A formula is obtained for the distribution of the waiting time w in the stationary state. The formula is used to show that $P9w > t) \sim \alpha t^{-5/6} \exp (-\beta t - \gamma t^{1/3})$ as $t \to \infty$, with the constants $\alpha, \beta$, and $\gamma$ expressed as functions of the traffic intensity $\rho$. The distribution of w for the random order discipline is compared to that of the first in, first out discipline.