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Geometrical Optics Approach to Markov-Modulated Fluid Models

Diego Dominici·Charles Knessl
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Abstract

We analyze asymptotically a differential-difference equation, that arises in a Markov-modulated fluid model. Here there are N identical sources that turn "on" and "off", and when "on" they generate fluid at unit rate into a buffer, which process the fluid at a rate c < N. In the steady state limit, the joint probability distribution of the buffer content and the number of active sources satisfies a system of N + 1 ODEs, that can also be viewed as a differential-difference equation analogous to a backward/forward parabolic PDE. We use singular perturbation methods to analyze the problem for N large with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.

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Diego Dominici, Charles Knessl. 2003-06-17. Geometrical Optics Approach to Markov-Modulated Fluid Models. https://arxiv.org/abs/math/0306262

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