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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Acknowledgments
The authors thank the anonymous referees for their constructive criticism of an earlier version of the paper. They acknowledge the financial support of the Ministère de la Communauté française de Belgique through the ARC grant AUWB-08/13–ULB 5, and of the Australian Research Council through the Discovery Grant DP110101663
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Latouche, G., Nguyen, G. Fluid approach to two-sided reflected Markov-modulated Brownian motion. Queueing Syst 80, 105–125 (2015). https://doi.org/10.1007/s11134-014-9432-8
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DOI: https://doi.org/10.1007/s11134-014-9432-8
Keywords
- Markov-modulated linear fluid models
- Reflected two-sided Markov-modulated Brownian motion
- Weak convergence
- Stationary distribution