Abstract
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.
Highlights
Our purpose is to construct and analyse a family of fluid queues converging to Markov-modulated Brownian motion (MMBM) with the intention of adapting, to the analysis of MMBM, tools and methods which have been developed in the context of fluid queues.Fluid queues are two-dimensional processes {X(t), φ(t) : t ≥ 0}, where {φ(t) : t ≥ 0} is a continous-time Markov chain on a finite state space M, tX(t) = X(0) + cφ(t) dt, and ci for i ∈ M are arbitrary real numbers
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
These are known as Markov-modulated linear fluid processes, with X referred to as the fluid level and φ as the phase: during intervals of time where the phase φ remains in Received November 2013. ∗The authors had interesting discussions with V
Summary
Publication details, including instructions for authors and subscription information: http://pubsonline.informs.org. Full terms and conditions of use: https://pubsonline.informs.org/Publications/Librarians-Portal/PubsOnLine-Terms-andConditions. Descriptions of, or references to, products or publications, or inclusion of an advertisement in this article, neither constitutes nor implies a guarantee, endorsement, or support of claims made of that product, publication, or service. For more information on INFORMS, its publications, membership, or meetings visit http://www.informs.org. 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 Markovmodulated 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. 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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