We analyze the transient behavior of stochastic fluid flow models in which the input and output rates are controlled by a finite homogeneous Markov process. Such models are used in asynchronous transfer mode (ATM) to evaluate the performance of fast packet switching and in manufacturing systems for the performance of producers and consumers coupled by a buffer. The transient analysis of such models have already been considered in earlier works and solutions have been obtained by the use of Laplace transform. We derive in this paper a new transient solution only based on recurrence relations. We show that this solution is particularly interesting for its numerical properties. The limiting behavior of the solution is also considered. We empirically show that the algorithm for computing the transient solution can be stopped when some stationary behavior is detected.