A L\'{e}vy input model with additional state-dependent services

We consider a queuing model with the workload evolving between consecutive i.i.d. exponential timers $\{e_q^{(i)}{\}}_{i=1,2,\mathrm{.}\mathrm{.}\mathrm{.}}$ according to a spectrally positive L'{e}vy 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 $e_q^{(1)}+\mathrm{.}\mathrm{.}\mathrm{.}+e_q^{(i)}$ 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 $\{B_i{\}}_{i=1,2,\mathrm{.}\mathrm{.}\mathrm{.}}$ are i.i.d. nonnegative random variables. We analyse the steady-state workload distribution for this model.