Jackson network in a random environment: strong approximation

TL;DR

This paper studies a Jackson network with random environmental effects causing server breakdowns and repairs, and proves a strong approximation of queue lengths by a reflected Brownian motion.

Contribution

It introduces a model of Jackson networks influenced by a random environment and establishes a strong approximation theorem for queue lengths.

Findings

Queue lengths approximated by reflected Brownian motion

Model incorporates independent random environment effects

Provides theoretical foundation for analyzing such networks

Abstract

We consider a Jackson network with regenerative input flows in which every server is subject to a random environment influence generating breakdowns and repairs. They occur in accordance with two independent sequences of i.i.d. random variables. We establish a theorem on the strong approximation of the vector of queue lengths by a reflected Brownian motion in positive orthant.