Article ID Journal Published Year Pages File Type
10527311 Stochastic Processes and their Applications 2015 18 Pages PDF
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.
Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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