Asymptotic Expansions for the Conditional Sojourn Time Distribution in the $M/M/1$-PS Queue

TL;DR

This paper derives asymptotic formulas for the conditional sojourn time distribution in an M/M/1 processor sharing queue, covering large times, large service requests, and heavy traffic conditions, extending previous results.

Contribution

It provides new asymptotic expansions for the conditional sojourn time distribution in the M/M/1-PS queue under various limiting regimes, extending prior work by Morrison and Flatto.

Findings

Asymptotic formulas for large time and service request regimes.

Results applicable to heavy traffic conditions.

Extension of previous asymptotic results by Morrison and Flatto.

Abstract

We consider the $M/M/1$ queue with processor sharing. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. The asymptotic formulas relate to, and extend, some results of Morrison \cite{MO} and Flatto \cite{FL}.