Transient analysis of reflected Lévy processes

In this paper, we establish a formula for the joint Laplace-Stieltjes transform of a reflected Lévy process and its regulator at an independent exponentially distributed time, starting at an independent exponentially distributed state. The Lévy process is general, that is, it is not assumed that it is either spectrally positive or negative. The resulting formulas are in terms of the one-dimensional distributions associated with the reflected process, and the regulator starting from zero and stopped at the exponential time. For the discrete-time case (that is, a random walk), analogous results are obtained where the exponentially distributed time is replaced by a geometrically distributed one. As an application, we explore what can be expected when the stationary distribution of the reflected process, when it exists, has a distribution which is a mixture of an exponential distribution and the constant zero. This is known to exist for the spectrally negative case and the case of a compound Poisson process with exponentially distributed jump size and a negative drift. The latter is the process associated with the workload process of an M/M/1 queue.