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Periodicity in the transient regime of exhaustive polling systems

MacPhee, I.M.; Menshikov, M.V.; Popov, S.; Volkov, S.

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Authors

I.M. MacPhee

S. Popov

S. Volkov



Abstract

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.

Citation

MacPhee, I., Menshikov, M., Popov, S., & Volkov, S. (2006). Periodicity in the transient regime of exhaustive polling systems. Annals of Applied Probability, 16(4), 1816-1850. https://doi.org/10.1214/105051606000000376

Journal Article Type Article
Publication Date Nov 1, 2006
Deposit Date Feb 20, 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 16
Issue 4
Pages 1816-1850
DOI https://doi.org/10.1214/105051606000000376
Keywords Polling systems, Greedy algorithm, Transience, Random walk, Dynamical system, Interval exchange transformation, a.s. convergence.
Publisher URL http://www.maths.dur.ac.uk/~dma0imm/mmpv_AAP247.pdf

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