A L\'evy input fluid queue with input and workload regulation

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'evy process $Y_i(t)$ that is reflected at zero, and where the environment $i$ equals 0 or 1. When the exponential clock $e_q^{(i)}$ ends, the workload, as well as the L'evy input process, are modified; this modification may depend on the current value of the workload, the maximum and the minimum workload observed during the previous cycle, and the environment $i$ of the L'evy input process itself during the previous cycle. We analyse the steady-state workload distribution for this model. The main theme of the analysis is the systematic application of non-trivial functionals, derived within the framework of fluctuation theory of L'evy processes, to workload and queuing models.