- Stoch. Syst.
- Volume 5, Number 1 (2015), 62-86.
The morphing of fluid queues into Markov-modulated Brownian motion
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.
Stoch. Syst., Volume 5, Number 1 (2015), 62-86.
Received: November 2013
First available in Project Euclid: 23 December 2015
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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. https://projecteuclid.org/euclid.ssy/1450879281