MacPhee, I. M. and Menshikov, M. V. and Popov, S. and Volkov, S. (2006) 'Periodicity in the transient regime of exhaustive polling systems.', Annals of applied probability., 16 (4). pp. 1816-1850.
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.
|Keywords:||Polling systems, Greedy algorithm, Transience, Random walk, Dynamical system, Interval exchange transformation, a.s. convergence.|
|Full text:||(VoR) Version of Record|
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|Publisher Web site:||http://dx.doi.org/10.1214/105051606000000376|
|Date accepted:||No date available|
|Date deposited:||17 May 2010|
|Date of first online publication:||01 January 1970|
|Date first made open access:||No date available|
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