Asymptotic analysis of the sojourn time of a batch in an M^[X]/M/1 Processor Sharing Queue
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In this paper, we exploit results obtained in an earlier study for the Laplace transform of the sojourn time Ω 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(Ω>x) when x tends to infinity. We precisely show that up to a multiplying factor, the behavior of P(Ω>x) for large x is of the same order of magnitude as P(ω>x), where ω 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.
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