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

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.

34 pages