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

Aleksandar Mijatović and Martijn Pistorius

Stochastic Processes and their Applications, 2015, vol. 125, issue 8, 2937-2954

Abstract: 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.

Keywords: Reflected Lévy process; Asymptotic undershoot and overshoot; Cramér condition; Queueing (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (2)

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DOI: 10.1016/j.spa.2015.02.007

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