Stochastic Systems

The morphing of fluid queues into Markov-modulated Brownian motion

Guy Latouche and Giang T. Nguyen

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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.

Article information

Stoch. Syst., Volume 5, Number 1 (2015), 62-86.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J25: Continuous-time Markov processes on general state spaces 60J65: Brownian motion [See also 58J65] 60B10: Convergence of probability measures

Markov-modulated linear fluid models Markov-modulated Brownian motion weak convergence stationary distribution computational methods


Latouche, Guy; Nguyen, Giang T. The morphing of fluid queues into Markov-modulated Brownian motion. Stoch. Syst. 5 (2015), no. 1, 62--86. doi:10.1214/13-SSY133.

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