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

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.