Some time-dependent properties of symmetric M/G/1 queues

We consider an M/G/1 queue that is idle at time 0. The number of customers sampled at an independent exponential time is shown to have the same geometric distribution under the preemptive-resume last-in-first-out and the processor-sharing disciplines. Hence, the marginal distribution of the queue length at any time is identical for both disciplines. We then give a detailed analysis of the time until the first departure for any symmetric queueing discipline. We characterize its distribution and show that it is insensitive to the service discipline. Finally, we study the tail behavior of this distribution.