Slowing time: Markov-modulated Brownian motion with a sticky boundary

TL;DR

This paper studies the stationary distribution of Markov-modulated Brownian motions with a sticky boundary at zero, extending existing models to account for slowed evolution at the boundary using a Markov-regenerative approach.

Contribution

It introduces a novel method to analyze the stationary distribution of MMBMs with sticky boundaries by extending Brownian motion constructions and employing Markov-regenerative techniques.

Findings

Derived the stationary distribution for sticky MMBMs.

Extended Brownian motion models to include sticky boundaries.

Revisited classical results for regulated MMBMs.

Abstract

We analyze the stationary distribution of regulated Markov modulated Brownian motions (MMBM) modified so that their evolution is slowed down when the process reaches level zero --- level zero is said to be {\em sticky}. To determine the stationary distribution, we extend to MMBMs a construction of Brownian motion with sticky boundary, and we follow a Markov-regenerative approach similar to the one developed in past years in the context of quasi-birth-and-death processes and fluid queues. We also rely on recent work showing that Markov-modulated Brownian motions may be analyzed as limits of a parametrized family of fluid queues. We use our results to revisit the stationary distribution of the well-known regulated MMBM.