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2011 Lévy-driven polling systems and continuous-state branching processes
Onno Boxma, Jevgenijs Ivanovs, Kamil Kosiński, Michel Mandjes
Stoch. Syst. 1(2): 411-436 (2011). DOI: 10.1214/10-SSY008

Abstract

In this paper we consider a ring of $N\ge 1$ queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a polling model.

Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server’s cycle. The $N$-dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch.

Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multi-type Jiřina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the branching property for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.

Citation

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Onno Boxma. Jevgenijs Ivanovs. Kamil Kosiński. Michel Mandjes. "Lévy-driven polling systems and continuous-state branching processes." Stoch. Syst. 1 (2) 411 - 436, 2011. https://doi.org/10.1214/10-SSY008

Information

Published: 2011
First available in Project Euclid: 24 February 2014

zbMATH: 1291.60185
MathSciNet: MR2949545
Digital Object Identifier: 10.1214/10-SSY008

Subjects:
Primary: 60K25
Secondary: 90B22

Rights: Copyright © 2011 INFORMS Applied Probability Society

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Vol.1 • No. 2 • 2011
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