On the Stability of a Polling System with an Adaptive Service Mechanism

TL;DR

This paper analyzes the stability of a cyclic polling system with an adaptive server rule, providing conditions for stability and discussing open problems related to the exact stability region, including models with different service disciplines.

Contribution

It introduces stability and instability results for a polling system with adaptive rules and explores the complex dependence of stability on service policies and primitive distributions.

Findings

Proved stability conditions for the polling system with limited service.

Identified open problems in characterizing the exact stability region.

Illustrated the impact of service discipline types on system stability.

Abstract

We consider a single-server cyclic polling system with three queues where the server follows an adaptive rule: if it finds one of queues empty in a given cycle, it decides not to visit that queue in the next cycle. In the case of limited service policies, we prove stability and instability results under some conditions which are sufficient but not necessary, in general. Then we discuss open problems with identifying the exact stability region for models with limited service disciplines: we conjecture that a necessary and sufficient condition for the stability may depend on the whole distributions of the primitive sequences, and illustrate that by examples. We conclude the paper with a section on the stability analysis of a polling system with either gated or exhaustive service disciplines.