Comments on the Transient Solution of Stochastic Fluid Models

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 transientregime fulfills a partial linear differential system and many techniques of resolution in the literature have beenproposed. But the uniqueness of the solution has been never studied. In this paper, the boundary conditions toensure uniqueness were precisely specified. The jumps with specific values in some critical points are necessary toguarantee the uniqueness of the solution. This work showed also that some methods using the Laplace transformdo not take into account of the discontinuity property leading in that way to different mathematical mistakesdepending on the used approach. Finally, this paper showed that the approach using the uniformization techniquegives the exact solution and its proof is completed.