On a piece-wise deterministic Markov process model

We study a piece-wise deterministic Markov process having jumps of i.i.d. sizes with a constant intensity and decaying at a constant rate (a special case of a storage process with a general release rule). Necessary and sufficient conditions for the process to be ergodic are found, its stationary distribution is found in explicit form. Further, the Laplace transform of the first crossing time of a fixed barrier by the process is shown to satisfy a Fredholm equation of second kind. Solution to this equation is given by exponentially fast converging Neumann series; convergence rate of the series is estimated. Our results can be applied to an important reliability problem.