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In this paper we analyse Markov-modulated fluid processes over finite time intervals. We study the joint distribution of the level at time  and of the maximum level over [0, ], as well as the joint distribution of the level at time  and the minimum level over [0, ]. We approximate  by a random variable  with Erlang distribution and so use an approach different from the usual Laplace transform to compute the distributions. We present probabilistic interpretation of the equations and provide a numerical illustration.

Approximations for Time-Dependent Distributions in Markovian Fluid Models