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2015 The morphing of fluid queues into Markov-modulated Brownian motion
Guy Latouche, Giang T. Nguyen
Stoch. Syst. 5(1): 62-86 (2015). DOI: 10.1214/13-SSY133


Ramaswami showed recently that standard Brownian motion arises as the limit of a family of Markov-modulated linear fluid processes. We pursue this analysis with a fluid approximation for Markov-modulated Brownian motion. We follow a Markov-renewal approach and we prove that the stationary distribution of a Markov-modulated Brownian motion reflected at zero is the limit from the well-analyzed stationary distribution of approximating linear fluid processes. Thus, we provide a new approach for obtaining the stationary distribution of a reflected MMBM without time-reversal or solving partial differential equations. Our results open the way to the analysis of more complex Markov-modulated processes.

Key matrices in the limiting stationary distribution are shown to be solutions of a matrix-quadratic equation, and we describe how this equation can be efficiently solved.


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Guy Latouche. Giang T. Nguyen. "The morphing of fluid queues into Markov-modulated Brownian motion." Stoch. Syst. 5 (1) 62 - 86, 2015.


Received: 1 November 2013; Published: 2015
First available in Project Euclid: 23 December 2015

zbMATH: 1336.60151
MathSciNet: MR3442389
Digital Object Identifier: 10.1214/13-SSY133

Primary: 60B10, 60J25, 60J65

Rights: Copyright © 2015 INFORMS Applied Probability Society


Vol.5 • No. 1 • 2015
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