Lévy-driven polling systems and continuous-state branching processes

Onno Boxma,Kamil Kosiński,Michel Mandjes,Jevgenijs Ivanovs

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 \textit{polling model}. Each of the queues is fed by a non-decreasing L\'evy process, which can be different during each of the consecutive periods within the server's cycle. The $N$-dimensional L\'evy 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\'evy input) polling systems and multi-type Ji\v{r}ina 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.

Highlights

Consider a queueing model consisting of multiple queues attended by a single server, visiting the queues one at a time in a cyclic order
Our analysis heavily relies on establishing a link between fluid (Levy input) polling systems and multi-type Jirina processes
The generalization from compound Poisson processes (CPPs) to Levy input implies that we can no longer speak of notions such as customers and queue lengths; this explains why we focus on the workload process

Summary

Stochastic Systems

Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org. The N -dimensional Levy processes obtained in this fashion are described by their (joint) Laplace exponent, 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. Our analysis heavily relies on establishing a link between fluid (Levy input) polling systems and multi-type Jirina 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

Introduction

Similarly as above we obtain

On the intervals Vi we have