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

Let τ(x) be the epoch of first entry into the interval (x,∞), x>0, of the reflected process  Y of a Lévy process X, and define the overshoot Z(x)=Y(τ(x))−x and undershoot z(x)=x−Y(τ(x)−) of Y at the first-passage time over the level x. In this paper we establish, separately under the Cramér 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 Lévy 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.