Asymptotic analysis of the sojourn time of a batch in an $M^{[X]}/M/1$ Processor Sharing Queue

TL;DR

This paper analyzes the asymptotic tail behavior of batch sojourn times in an M[X]/M/1 processor sharing queue, showing that large batch sojourn times decay similarly to individual job times, informing system dimensioning.

Contribution

It derives the asymptotic distribution of batch sojourn times in an M[X]/M/1 PS queue, extending previous Laplace transform results to practical tail behavior analysis.

Findings

Large batch sojourn time tail behavior matches that of individual jobs.

System capacity can be adjusted to guarantee batch processing times.

Asymptotic analysis informs system dimensioning for batch processing.

Abstract

In this paper, we exploit results obtained in an earlier study for the Laplace transform of the sojourn time $\Omega$ of an entire batch in the $M^{[X]}/M/1$ Processor Sharing (PS) queue in order to derive the asymptotic behavior of the complementary probability distribution function of this random variable, namely the behavior of $P(\Omega>x)$ when $x$ tends to infinity. We precisely show that up to a multiplying factor, the behavior of $P(\Omega>x)$ for large $x$ is of the same order of magnitude as $P(\omega>x)$, where $\omega$ is the sojourn time of an arbitrary job is the system. From a practical point of view, this means that if a system has to be dimensioned to guarantee processing time for jobs then the system can also guarantee processing times for entire batches by introducing a marginal amount of processing capacity.