Comments on the transient solution of stochastic fluid models

(Hadi and Ali al-Alwan Nabli)

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Abstract

The joint probability distribution of the buffer content and the state of the Markov process controlling the input/output rates in a stochastic fluid model has been widely investigated. This probability distribution in transient regime fulfills a partial linear differential system and many techniques of resolution in the literature have been proposed. But the uniqueness of the solution has been never studied. In this paper, the boundary conditions to ensure uniqueness were precisely specified. The jumps with specific values in some critical points are necessary to guarantee the uniqueness of the solution. This work showed also that some methods using the Laplace transform do not take into account of the discontinuity property leading in that way to different mathematical mistakes depending on the used approach. Finally, this paper showed that the approach using the uniformization technique gives the exact solution and its proof is completed. Key Words: Laplace transform, Markov process, Partial differential equations, Stochastic fluid models.
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