Academic Journals Database
Disseminating quality controlled scientific knowledge

Lévy-driven polling systems and continuous-state branching processes

Author(s): Onno Boxma | Jevgenijs Ivanovs | Kamil Marcin Kosiński | Michel Mandjes

Journal: Stochastic Systems
ISSN 1946-5238

Volume: 1;
Issue: 2;
Start page: 411;
Date: 2011;
Original page

Keywords: Polling system | Lévy processes | branching processes

In this paper we consider a ring of N ≥ 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.
Affiliate Program      Why do you need a reservation system?