Time-Limited and k-Limited Polling Systems: A Matrix Analytic Solution

TL;DR

This paper introduces a matrix analytic method to analyze complex polling systems with time-limited and k-limited disciplines, providing a new approach where traditional methods fail due to the lack of the branching property.

Contribution

It develops an iterative scheme based on absorbing Markov chains to derive joint queue-length distributions in non-branching polling systems, extending to tandem queue networks.

Findings

Provides a closed-form relation between queue lengths at different server visit points.

Demonstrates the method's applicability to tandem queueing networks.

Offers a new analytical tool for previously intractable polling disciplines.

Abstract

In this paper, we will develop a tool to analyze polling systems with the autonomous-server, the time-limited, and the k-limited service discipline. It is known that these disciplines do not satisfy the well-known branching property in polling system, therefore, hardly any exact result exists in the literature for them. Our strategy is to apply an iterative scheme that is based on relating in closed-form the joint queue-length at the beginning and the end of a server visit to a queue. These kernel relations are derived using the theory of absorbing Markov chains. Finally, we will show that our tool works also in the case of a tandem queueing network with a single server that can serve one queue at a time.