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Birth death processes with rational birth and death rates have been studied by Maki (1976). In this paper a fluid queue driven by a birth death process with infinite state space and absorption is studied where the birth and death rates are rational functions of linear polynomials. We obtain an explicit transient solution for the fluid queue model using continued fraction approach to solve the underlying system of partial differential equations. For specific value of the parameter the considered model reduces to the model discussed by Parthasarathy, and the results coincide. We also analyze the behavior of the fluid queue in a long run.

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