On doubly reflected completely asymmetric Lévy processes

Consider a completely asymmetric Lévy process X and let Z be X reflected at 0 and at a>0. In applied probability (e.g. The Single Server Queue, 2nd Edition, North-Holland, Amsterdam, 1982) the process Z turns up in the study of the virtual waiting time in an M/G/1-queue with finite buffer a or the water level in a finite dam of size a. We find an expression for the resolvent density of Z. We show Z is positive recurrent and determine the invariant measure. Using the regenerative property of Z, we determine the asymptotic law of for an appropriate class of functions f. Finally, the long time average of the local time of Z in x[set membership, variant][0,a] is studied.