Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
10527311 | Stochastic Processes and their Applications | 2015 | 18 Pages |
Abstract
Let Ï(x) be the epoch of first entry into the interval (x,â), x>0, of the reflected process Y of a Lévy process X, and define the overshoot Z(x)=Y(Ï(x))âx and undershoot z(x)=xâY(Ï(x)â) 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)
Authors
Aleksandar MijatoviÄ, Martijn Pistorius,