On Sojourn Times in the $M/M/1$-PS Model, Conditioned on the Number of Other Users

Abstract

We consider the $M/M/1$-PS queue with processor sharing. We study the conditional sojourn time distribution of an arriving customer, conditioned on the number of other customers present. A new formula is obtained for the conditional sojourn time distribution, using a discrete Green's function. This is shown to be equivalent to some classic results of Pollaczeck and Vaulot from 1946. Then various asymptotic limits are studied, including large time and/or large number of customers present, and heavy traffic, where the arrival rate is only slightly less than the service rate.

arXiv comment45 pages, 2 figures

References in corpus (3)

• On Sojourn Times in the Finite Capacity $M/M/1$ Queue with Processor Sharing
• Asymptotic Expansions for the Conditional Sojourn Time Distribution in the $M/M/1$-PS Queue
• On a Processor Sharing Queue That Models Balking

Cited by in corpus (3)

• On a Processor Sharing Queue That Models Balking
• Sojourn Times for the One Dimensional Grover Walk
• On the Sojourn Time Distribution in a Finite Population Markovian Processor Sharing Queue