An exhaustive Lévy storage process with intermittent output

We consider a Lévy storage model in which the output is shut off every time the system reaches zero, for a length of time which is determined by an arbitrary stopping time with respect to a filtration which may be richer than the one generated by the Lévy process. The main contribution is in showing that under suitable conditions, the steady state distribution associated with such a model exhibits a new decomposition property. Namely it can be written as an independent sum of two random variables. One has the distribution associated with a standard reflected Lévy process and the other has the steady state distribution of some clearing process (in contrast to excess lifetime or excess number of customers arriving during a vacation). A special case of such a model is the workload process in an M/G/1 queue, in particular with the well known T, N and D-policies.