A Markov chain model of a polling system with parameter regeneration

We study a model of a polling system i.e.\ a collection of $d$ queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with...

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Bibliographic Details
Published in:The Annals of applied probability 2007-10, Vol.17 (5/6), p.1447-1473
Main Authors: Macphee, Iain, Menshikov, Mikhail, Petritis, Dimitri, Popov, Serguei
Format: Article
Language:English
Subjects:
Mathematics
Probability
Online Access:Get full text
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Summary:We study a model of a polling system i.e.\ a collection of $d$ queues with a single server that switches from queue to queue. The service time distribution and arrival rates change randomly every time a queue is emptied. This model is mapped to a mathematically equivalent model of a random walk with random choice of transition probabilities, a model which is of independent interest. All our results are obtained using methods from the constructive theory of Markov chains. We determine conditions for the existence of polynomial moments of hitting times for the random walk. An unusual phenomenon of thickness of the region of null recurrence for both the random walk and the queueing model is also proved.
ISSN:1050-5164
2168-8737
DOI:10.1214/10505167000000212