Products of random matrices and queueing system performance evaluation

TL;DR

This paper introduces a novel algebraic approach to analyze the performance of acyclic fork-join queueing networks using products of random matrices in max-plus algebra, providing bounds on cycle times.

Contribution

It develops an extended max-plus algebra technique to study the properties and limiting behavior of matrix products in network performance evaluation.

Findings

Derived bounds on the max-plus algebra maximal Lyapunov exponent.

Provided a new algebraic framework for queueing network analysis.

Enhanced understanding of cycle times in acyclic fork-join networks.

Abstract

We consider (max,+)-algebra products of random matrices, which arise from performance evaluation of acyclic fork-join queueing networks. A new algebraic technique to examine properties of the product and investigate its limiting behaviour is proposed based on an extension of the standard matrix (max,+)-algebra by endowing it with the ordinary matrix addition as an external operation. As an application, we derive bounds on the (max,+)-algebra maximal Lyapunov exponent which can be considered as the cycle time of the networks.