On the infimum attained by a reflected Lévy process

Details On the infimum attained by a reflected Lévy process On the infimum attained by a reflected Lévy process

2010-12-04

arxiv.org/abs/1012.0936v3

Queueing Systems 70 (2012) 23-35

Mathematics

Probability

Scientific paper

10.1007/s11134-011-9257-7

This paper considers a L\'evy-driven queue (i.e., a L\'evy process reflected at 0), and focuses on the distribution of $M(t)$, that is, the minimal value attained in an interval of length $t$ (where it is assumed that the queue is in stationarity at the beginning of the interval). The first contribution is an explicit characterization of this distribution, in terms of Laplace transforms, for spectrally one-sided L\'evy processes (i.e., either only positive jumps or only negative jumps). The second contribution concerns the asymptotics of $\prob{M(T_u)> u}$ (for different classes of functions $T_u$ and $u$ large); here we have to distinguish between heavy-tailed and light-tailed scenarios.

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