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- Title
Fluid approach to two-sided reflected Markov-modulated Brownian motion.
- Authors
Latouche, Guy; Nguyen, Giang
- 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 been shown recently 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. In so doing, we obtain a new representation for the stationary distribution. It is factorised into a vector determined by the phase behaviour when the fluid is either at the level 0 or the level $$b$$ , and a matrix expression characteristic of the process when the fluid is in the open interval $$(0,b)$$ .
- Subjects
MARKOV processes; STOCHASTIC processes; BIRTH &; death processes (Stochastic processes); BROWNIAN motion; MOTION
- Publication
Queueing Systems, 2015, Vol 80, Issue 1/2, p105
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-014-9432-8