2013/07/26 by Aleksandar Mijatović, Mijatović, Aleksandar, Martijn Pistorius +1
Business, Management and Accounting · Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60F05 #60G17 #60G51 #Advanced Queuing Theory Analysis #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #math.PR #msc:60F05 #msc:60G17 #msc:60G51
paper · pdf · doi:10.48550/arxiv.1307.6947
11 pages, no figures
arxiv created 2013/07/26 · openalex publication_date 2013/07/26 · arxiv updated 2013/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.