On a Processor Sharing Queue That Models Balking

TL;DR

This paper analyzes a processor sharing queue with balking behavior, deriving explicit and asymptotic distributions for a tagged customer's sojourn time under various system load conditions.

Contribution

It provides explicit expressions and asymptotic analysis for the sojourn time distribution in an $M/M/1$-PS queue with a specific balking function, extending understanding of such systems.

Findings

Explicit sojourn time distribution derived for the balking queue.

Asymptotic behavior differs significantly from standard $M/M/1$-PS queue.

Perturbation method proposed for general balking functions.

Abstract

We consider the processor sharing $M/M/1$-PS queue which also models balking. A customer that arrives and sees $n$ others in the system "balks" (i.e., decides not to enter) with probability $1-b_n$. If $b_n$ is inversely proportional to $n+1$, we obtain explicit expressions for a tagged customer's sojourn time distribution. We consider both the conditional distribution, conditioned on the number of other customers present when the tagged customer arrives, as well as the unconditional distribution. We then evaluate the results in various asymptotic limits. These include large time (tail behavior) and/or large $n$, lightly loaded systems where the arrival rate $\lambda\to 0$, and heavily loaded systems where $\lambda\to\infty$. We find that the asymptotic structure for the problem with balking is much different from the standard $M/M/1$-PS queue. We also discuss a perturbation method for deriving the asymptotics, which should apply to more general balking functions.