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Generalizations of the stochastic decomposition results for single server queues with vacations. (English) Zbl 0701.60094

The paper demonstrates that the sample paths comparison, used by the author in J. Appl. Probab. 22, 419-428 (1985; Zbl 0566.60090) to prove the stochastic decomposition of the steady state waiting time for GI/G/1 queues with exhaustive service and multiple vacations, is applicable to show the validity of this decomposition results for more general arrival and service processes. Lucantoni, Meier-Hellstern and Neuts, in an appearing paper, give a detailed analytic treatment of such systems under a class of non-renewal (MAP) processes which raises several questions:

1. Can the stochastic decomposition be shown by simple sample path arguments and do we need the assumption that the vacations are independent and identically distributed? 2. Do stochastic decomposition results also hold for semi-Markov arrival processes? 3. If the second question has a positive answer, are then the results valid for more general classes of arrival, service or vacation processes?

The paper shows that the answers are positive for the exhaustive service, multiple vacation models as long as the vacation lengths are independent of the arrival and service processes. Especially, the stochastic decomposition of the steady state waiting time and virtual waiting time is valid under the assumption that the basic queueing system (without vacations) and the vacation sequence are stationary and the system is work conserving.

If the arrival process is semi-Markov and the service times are i.i.d., the waiting time and virtual waiting distributions have unique ergodic limits which have the usual decomposition property. A matrix integral form of the stochastic decomposition for the queue length at departure epochs and at an arbitrary time in steady state is proved for this case. The basic approach is to exploit the generality of the relationships between the sample paths of the virtual waiting time process with and without vacations.

Reviewer: L.Lakatos
MSC:
60K25Queueing theory
90B22Queues and service (optimization)