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On two-queue Markovian polling systems with exhaustive service
Authors:Jan-pieter L Dorsman  Onno J Boxma  Robert D van der Mei
Institution:1. EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
2. Stochastics, Centrum Wiskunde & Informatica (CWI), P.O. Box 94079, 1090 GB, Amsterdam, The Netherlands
3. Department of Mathematics, VU University Amsterdam, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands
Abstract:We consider a class of two-queue polling systems with exhaustive service, where the order in which the server visits the queues is governed by a discrete-time Markov chain. For this model, we derive an expression for the probability generating function of the joint queue length distribution at polling epochs. Based on these results, we obtain explicit expressions for the Laplace–Stieltjes transforms of the waiting-time distributions and the probability generating function of the joint queue length distribution at an arbitrary point in time. We also study the heavy-traffic behaviour of properly scaled versions of these distributions, which results in compact and closed-form expressions for the distribution functions themselves. The heavy-traffic behaviour turns out to be similar to that of cyclic polling models, provides insights into the main effects of the model parameters when the system is heavily loaded, and can be used to derive closed-form approximations for the waiting-time distribution or the queue length distribution.
Keywords:
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