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Geometrical Optics Approach to Markov‐Modulated Fluid Models
Author(s) -
Dominici Diego,
Knessl Charles
Publication year - 2005
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.0022-2526.2005.01540.x
Subject(s) - mathematics , ode , singular perturbation , mathematical analysis , markov process , fluid queue , partial differential equation , markov chain , limit (mathematics) , ordinary differential equation , differential equation , stationary distribution , statistics , queueing theory
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 processes 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.