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

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  • Mijatović, Aleksandar
  • Pistorius, Martijn

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.

Suggested Citation

  • Mijatović, Aleksandar & Pistorius, Martijn, 2015. "Buffer-overflows: Joint limit laws of undershoots and overshoots of reflected processes," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 2937-2954.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:8:p:2937-2954
    DOI: 10.1016/j.spa.2015.02.007
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    References listed on IDEAS

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    1. Bertoin, J. & van Harn, K. & Steutel, F. W., 1999. "Renewal theory and level passage by subordinators," Statistics & Probability Letters, Elsevier, vol. 45(1), pages 65-69, October.
    2. Bertoin, J. & Doney, R. A., 1994. "Cramer's estimate for Lévy processes," Statistics & Probability Letters, Elsevier, vol. 21(5), pages 363-365, December.
    3. Mijatović, Aleksandar & Pistorius, Martijn R., 2012. "On the drawdown of completely asymmetric Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 122(11), pages 3812-3836.
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    2. Griffin, Philip S., 2022. "Path decomposition of a reflected Lévy process on first passage over high levels," Stochastic Processes and their Applications, Elsevier, vol. 145(C), pages 29-47.

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