A Lévy input model with additional state-dependent services

We consider a queuing model with the workload evolving between consecutive i.i.d. exponential timers according to a spectrally positive Lévy process Y(t) which is reflected at 0. When the exponential clock ends, the additional state-dependent service requirement modifies the workload so that the latter is equal to at epoch for some random nonnegative i.i.d. functionals Fi. In particular, we focus on the case when Fi(y)=(Bi-y)+, where {Bi}i=1,2,... are i.i.d. nonnegative random variables. We analyse the steady-state workload distribution for this model.