Motivated by models of queues with server vacations, we consider a Levy process modified to have random jumps at arbitrary stopping times. The extra jumps can counteract a drift in the Levy process so that the overall Levy process with secondary jump input can have a proper limiting distribution. For example, the workload process in an $M/G/1$ queue with a server vacation each time the server finds an empty system is such a Levy process with secondary jump input. We show that a certain functional of a Levy process with secondary jump input is a martingale and we apply this martingale to characterize the steady-state distribution. We establish stochastic decomposition results for the case in which the Levy process has no negative jumps, which extend and unify previous decomposition results for the workload process in the $M/G/1$ queue with server vacations and Brownian motion with secondary jump input. We also apply martingales to provide a new proof of the known simple form of the steady-state distribution of the associated reflected Levy process when the Levy process has no negative jumps (the generalized Pollaczek-Khinchine formula).
"Queues with Server Vacations and Levy Processes with Secondary Jump Input." Ann. Appl. Probab. 1 (1) 104 - 117, February, 1991. https://doi.org/10.1214/aoap/1177005983