Feedback control: two-sided Markov-modulated Brownian motion with instantaneous change of phase at boundaries

TL;DR

This paper introduces a flexible Markov-modulated Brownian motion model with instantaneous phase changes at boundaries, analyzing its first exit times and stationary distribution using regenerative methods, extending classical boundary behavior analysis.

Contribution

It develops a novel model with phase changes at boundaries and applies regenerative techniques to analyze its properties, expanding beyond traditional boundary conditions.

Findings

Derived first exit time characteristics for the process.

Obtained the stationary distribution of the flexible process.

Demonstrated the effectiveness of regenerative approaches for complex boundary behaviors.

Abstract

We consider a Markov-modulated Brownian motion $\{Y(t), \rho(t)\}$ with two boundaries at $0$ and $b > 0$, and allow for the controlling Markov chain $\{\rho(t)\}$ to instantaneously undergo a change of phase upon hitting either of the two boundaries at semi-regenerative epochs defined to be the first time the process reaches a boundary since it last hits the other boundary. We call this process a flexible Markov-modulated Brownian motion. Using the recently-established links between stochastic fluid models and Markov-modulated Brownian motions, we determine important characteristics of first exit times of a Markov-modulated Brownian motion from an interval with a regulated boundary. These results allow us to follow a Markov-regenerative approach and obtain the stationary distribution of the flexible process. This highlights the effectiveness of the regenerative approach in analyzing Markov-modulated Brownian motions subject to more general boundary behaviors than the classic regulated boundaries.