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arXiv · 1006.0384

Lévy-driven polling systems and continuous-state branching processes

Onno Boxma·Jevgenijs Ivanovs·Kamil Marcin Kosiński·Michel Mandjes
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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 \textit{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 \textit{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.

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BibTeXRIS

Onno Boxma, Jevgenijs Ivanovs, Kamil Marcin Kosiński, Michel Mandjes. 2015-05-21. Lévy-driven polling systems and continuous-state branching processes. https://doi.org/10.1214/10-ssy008

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