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 {eq(i)}i=1,2,… according to a spectrally positive Lévy process Y(t)Y(t) which is reflected at 00. When the exponential clock eq(i) ends, the additional state-dependent service requirement modifies the workload so that the latter is equal to Fi(Y(eq(i))) at epoch eq(1)+⋯+eq(i) for some random nonnegative i.i.d. functionals FiFi. In particular, we focus on the case when Fi(y)=(Bi−y)+Fi(y)=(Bi−y)+, where {Bi}i=1,2,…{Bi}i=1,2,… are i.i.d. nonnegative random variables. We analyse the steady-state workload distribution for this model.