Approximations for Time-Dependent Distributions in Markovian Fluid Models

In this paper we analyse Markov-modulated fluid processes over finite time intervals. We study the joint distribution of the level at time \(\theta < \infty \) and of the maximum level over 0, θ], as well as the joint distribution of the level at time θ and the minimum level over 0, θ]. We approximate θ by a random variable T 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.