Resonances in compound processes

The first-exit time of a compound process with strictly positive jumps reaching a horizontal barrier is considered. The first-exit time distribution for the specific case of Poisson arrivals and gamma distributed jump sizes is derived. If the jump size distribution converges weakly to a Dirac delta function as the variance tends to zero, the process tends to a compound process with constant jump size. In the case when the barrier is an exact multiple of the constant jump size a small peculiarity arises; the firstexit time distribution with general jumps does not tend to the first-exit time distribution with constant jumps. The first-exit time distribution for M/G/1 queues with gamma distributed service times is shown to have the same peculiarity.