On doubly reflected completely asymmetric Lévy processes

Abstract

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 t −1 ∫ 0 t f(Z s) ds for an appropriate class of functions f. Finally, the long time average of the local time of Z in x∈[0, a] is studied.