Mean first passage times in fluid queues

A stochastic fluid queueing system describes the input–output flow of a fluid in a storage device, called a buffer. The rates at which the fluid enters and leaves the buffer depend on a random environment process. The external governing process is an irreducible CTMC and the fluid from the buffer is emptied at a constant rate μ. Let X(t) denote the buffer content at time t and I(t) denote the state of the random environment at time t. In this paper we present a method for computing the mean first passage times in the {X(t), t≥0} process, as well as in the bivariate {(X(t),I(t)), t≥0} process. We derive a system of first-order non-homogeneous linear differential equations for the mean first passage times which can easily be solved using well-known techniques. The method developed here can be readily implemented for computational purposes. We present two examples illustrating how to find explicitly the analytical solution to a small two-state problem and how to obtain numerical solutions to a multistate problem.