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

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.