Geometrical Optics Approach to Markov-Modulated Fluid Models

We analyze asymptotically a differential-difference equation, that arises in a Markovmodulated 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 → ∞, with appropriate scalings of the two state variables. In particular, the ray method and asymptotic matching are used.