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Periodicity in the transient regime of exhaustive polling systems. (English) Zbl 1121.60098

Summary: We consider an exhaustive polling system with three nodes in its transient regime under a switching rule of generalized greedy type. We show that, for the system with Poisson arrivals and service times with finite second moment, the sequence of nodes visited by the server is eventually periodic almost surely. To do this, we construct a dynamical system, the triangle process, which we show has eventually periodic trajectories for almost all sets of parameters and in this case we show that the stochastic trajectories follow the deterministic ones a.s. We also show there are infinitely many sets of parameters where the triangle process has aperiodic trajectories and in such cases trajectories of the stochastic model are aperiodic with positive probability.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
37E05 Dynamical systems involving maps of the interval

References:

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