%0 Journal Article
%T L¨¦vy-driven polling systems and continuous-state branching processes
%A Onno Boxma
%A Jevgenijs Ivanovs
%A Kamil Marcin Kosi¨½ski
%A Michel Mandjes
%J Stochastic Systems
%D 2011
%I Institute for Operations Research and the Management Sciences (INFORMS), Applied Probability Society
%X In this paper we consider a ring of N ¡Ý 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 L¨¦vy process, which can be different during each of the consecutive periods within the server's cycle. The N-dimensional L¨¦vy 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 (L¨¦vy input) polling systems and multi-type Ji ina 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.
%K Polling system
%K L¨¦vy processes
%K branching processes
%U http://www.i-journals.org/ssy/viewarticle.php?id=8&layout=abstract