On the Sojourn Time Distribution in a Finite Population Markovian Processor Sharing Queue

TL;DR

This paper analyzes the sojourn time distribution in a finite population Markovian processor-sharing queue, providing asymptotic results and conditions under which it approximates the infinite population model.

Contribution

It introduces new asymptotic analysis methods for finite population PS queues and identifies when they can be approximated by infinite population models.

Findings

Derived asymptotic distributions for large populations

Identified parameter regimes for approximation validity

Applied spectral and singular perturbation methods

Abstract

We consider a finite population processor-sharing (PS) queue, with Markovian arrivals and an exponential server. Such a queue can model an interactive computer system consisting of a bank of terminals in series with a central processing unit (CPU). For systems with a large population $N$ and a commensurately rapid service rate, or infrequent arrivals, we obtain various asymptotic results. We analyze the conditional sojourn time distribution of a tagged customer, conditioned on the number $n$ of others in the system at the tagged customer's arrival instant, and also the unconditional distribution. The asymptotics are obtained by a combination of singular perturbation methods and spectral methods. We consider several space/time scales and parameter ranges, which lead to different asymptotic behaviors. We also identify precisely when the finite population model can be approximated by the standard infinite population $M/M/1$-PS queue.