# On Sojourn Times in the Finite Capacity $M/M/1$ Queue with Processor Sharing

## TL;DR

This paper derives an exact Laplace transform expression for the sojourn time distribution in a finite capacity processor shared M/M/1 queue and analyzes its tail behavior as capacity grows large.

## Contribution

It provides a novel exact formula for sojourn times in finite capacity queues and characterizes tail behavior in the limit of infinite capacity.

## Key findings

- Exact Laplace transform for sojourn time distribution
- Asymptotic tail behavior as capacity approaches infinity
- Insights into queue performance limits

## Abstract

We consider a processor shared $M/M/1$ queue that can accommodate at most a finite number $K$ of customers. We give an exact expression for the sojourn time distribution in the finite capacity model, in terms of a Laplace transform. We then give the tail behavior, for the limit $K\to\infty$, by locating the dominant singularity of the Laplace transform.