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

Abstract

Let $\tau(x)$ be the epoch of first entry into the interval $(x,\infty)$, $x>0$, of the reflected process $Y$ of a L\'evy 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 Cram\'{e}r and positive drift assumptions, the existence of the weak limit of $(z(x), Z(x))$ as $x$ tends to infinity and provide explicit formulae for their joint CDFs in terms of the L\'{e}vy measure of $X$ and the renewal measure of the dual of $X$. We apply our results to analyse the behaviour of the classical M/G/1 queueing system at the buffer-overflow, both in a stable and unstable case.