Fluid approach to two-sided reflected Markov-modulated Brownian motion
Date
2015
Authors
Latouche, G.
Nguyen, G.
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Queueing Systems: Theory and Applications, 2015; 80(1-2):105-125
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Guy Latouche, Giang Nguyen
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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).
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© Springer Science+Business Media New York 2015