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