Buffer-overflows: Joint limit laws of undershoots and overshoots of reflected processes

Let tau(x) be the epoch of first entry into the interval (x, infinity), x > 0, of the reflected process Y of a Levy process X, and define the overshoot Z(x) = Y(tau(x)) - x and undershoot z(x) = x - Y(tau(x)-) of Y at the first-passage time over the level x. In this paper we establish, separately under the Cramer and positive drift assumptions, the existence of the weak limit of (z(x), Z(x)) as x tends to infinity and provide explicit formulas for their joint CDFs in terms of the Levy measure of X and the renewal measure of the dual of X. Furthermore we identify explicit stochastic representations for the limit laws. We apply our results to analyse the behaviour of the classical M/G/1 queueing system at buffer-overflow, both in a stable and unstable case. (C) 2015 Elsevier B.V. All rights reserved.