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 Lévy 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 Cramér 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 Lévy 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.

11 pages, no figures

References in corpus (2)

• Asymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Equivalent Cases
• Joint asymptotic distribution of certain path functionals of the reflected process

Cited by in corpus (1)

• Sample path behavior of a Lévy insurance risk process approaching ruin, under the Cramér-Lundberg and convolution equivalent conditions