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arXiv · 1603.01945

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

Guy Latouche·Giang T. Nguyen
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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.

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Guy Latouche, Giang T. Nguyen. 2016-03-07. Feedback control: two-sided Markov-modulated Brownian motion with instantaneous change of phase at boundaries. https://arxiv.org/abs/1603.01945

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