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Loss Rate Asymptotics

We consider a Levy process St which is reflected at 0 and K > 0. The reflected process V K t is given as a solution to a Skorokhod problem, which implies a representation V K t = V0 + St + L 0 L K , where L 0 and L K are the local times at 0 at K, respectively. The regenerative structure of V K t yields a stationary distribution denoted K and the loss rate is defined as the mean of L K in the stationary situation. The loss rate was studied in Asmussen & Pihlsgard [1], where it was expressed in terms of the characteristic triplet of St and K, and asymptotics of the loss rate as K ! 1 was derived in the case of negative drift and light tails. Asymptotics for positive drift is straightforward by reversing the role of the barriers 0 and K and using a conservation law. We use the expression for the loss rate from [1] to derive asymptotics in the case of negative drift and heavy tails, as well as in the case of zero drift. In the zero drift case, functional limit theorems (with a Brownian or stable limit) play an important role and are based on continuity properties of the loss rate.