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

TL;DR

This paper studies the limiting joint distribution of undershoot and overshoot for reflected Lévy processes at high levels, providing explicit formulas and applications to queueing systems at buffer overflow.

Contribution

It establishes the existence of weak limits for undershoot and overshoot in reflected Lévy processes under Cramér and positive drift conditions, with explicit joint distribution formulas.

Findings

Derived explicit joint CDFs for undershoot and overshoot.

Established weak limit existence under different drift conditions.

Applied results to analyze buffer overflow in M/G/1 queues.

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.