Asymptotic analysis of the sojourn time of a batch in a M [X]/M/1 processor sharing queue

• Published in: Stochastic models Vol. 40; no. 4; pp. 659 - 684
• Main Authors: Guillemin, Fabrice, Simonian, Alain, Nasri, Ridha, Rodriguez, Veronica Quintuna
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis 01-10-2024; Taylor & Francis Ltd
• Subjects: Asymptotic properties; Distribution functions; Laplace transform; Laplace transforms; Microprocessors; processor sharing discipline; queue with batch arrivals; Queues; sojourn time; tail asymptotics
• ISSN: 1532-6349; 1532-4214
• DOI: 10.1080/15326349.2024.2317210

In this article, we exploit recent results for the Laplace transform of the sojourn time Ω of an entire batch in the
M
[
X
]
/
M
/
1
processor sharing queue in order to derive the asymptotic behavior of the distribution function of Ω, namely the behavior of ℙ(Ω > x) when x tends to infinity. Specifically, we show that, up to a multiplying factor, ℙ(Ω > x) has the same order of magnitude as ℙ(ω > x) for large x, where ω is the sojourn time of an arbitrary job in the system. From a practical point of view, this means that if a system has to be dimensioned to guarantee the processing time for jobs, then the system can also guarantee processing times for entire batches by introducing a marginal amount of processing capacity.