Fluid approach to two-sided Markov-modulated Brownian motion

TL;DR

This paper extends renewal methods to two-sided Markov-modulated Brownian motion, proving convergence of stationary distributions and providing a new representation that separates boundary and interior behaviors.

Contribution

It introduces a novel approach to analyze two-sided reflected MMBM by proving weak convergence of fluid model approximations to the true process.

Findings

Stationary distributions of fluid models converge to the MMBM stationary distribution.

The new representation separates boundary effects from interior behavior.

Weak convergence holds for systems with two reflecting boundaries.

Abstract

We extend to Markov-modulated Brownian motion (MMBM) the renewal approach which has been successfully applied to the analysis of Markov-modulated fluid models. It has recently been shown that MMBM may be expressed as the limit of a parameterized family of Markov-modulated fluid models. We prove that the weak convergence also holds for systems with two reflecting boundaries, one at zero and one at $b >0$, and that the stationary distributions of the approximating fluid models converge to the stationary distribution of the two-sided reflected MMBM. Thus, we obtain a new representation for the stationary distribution, effectively separating the limiting behaviour of the process at the boundaries from its behaviour in the interior of $(0,b)$.