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A Markov chain model of a polling system with parameter regeneration

MacPhee, Iain M.; Menshikov, Mikhail; Petritis, Dimitri; Popov, Serguei

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Authors

Iain M. MacPhee

Mikhail Menshikov

Dimitri Petritis

Serguei Popov



Abstract

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.

Citation

MacPhee, I. M., Menshikov, M., Petritis, D., & Popov, S. (2007). A Markov chain model of a polling system with parameter regeneration. Annals of Applied Probability, 17(5/6), 1447-1473. https://doi.org/10.1214/105051607000000212

Journal Article Type Article
Publication Date Oct 3, 2007
Deposit Date Feb 25, 2008
Publicly Available Date Mar 29, 2024
Journal Annals of Applied Probability
Print ISSN 1050-5164
Publisher Institute of Mathematical Statistics
Peer Reviewed Peer Reviewed
Volume 17
Issue 5/6
Pages 1447-1473
DOI https://doi.org/10.1214/105051607000000212
Keywords Polling system, Hitting time moments, Random environment, Parameter regeneration, Stability, Time-inhomogeneous Markov chains, Recurrence, Lyapunov functions.

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