L\'evy-driven polling systems and continuous-state branching processes

TL;DR

This paper analyzes a cyclic polling system with Le9vy process inputs, deriving steady-state workload distributions and establishing a connection to continuous-state branching processes, broadening understanding of complex queueing models.

Contribution

It introduces a novel link between fluid Le9vy-driven polling systems and multi-type continuous-state branching processes, extending analysis to general service disciplines.

Findings

Derived steady-state distributions at embedded and arbitrary epochs.

Established a broad branching property applicable to key service disciplines.

Linked fluid polling systems with continuous-state branching processes.

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.