M/M/$\infty$ queues in semi-Markovian random environment

Details M/M/$\infty$ queues in semi-Markovian random environment M/M/$\infty$ queues in semi-Markovian random environment

2007-01-29

arxiv.org/abs/math/0701842v2

Queueing Systems (2008) Volume 58, Number 3, 221-237

Mathematics

Probability

17 pages, 2 figures

Scientific paper

10.1007/s11134-008-9068-7

In this paper we investigate an M/M/$\infty$ queue whose parameters depend on an external random environment that we assume to be a semi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the raw moments of the measure of a bidimensional random set. Finally the case when the random environment has only two states is deeper analyzed. We obtain an explicit formula to compute the above mentioned factorial moments when at least one of the two states has sojourn time exponentially distributed.

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