Power Law Condition for Stability of Poisson Hail

Sergey Foss (), Takis Konstantopoulos () and Thomas Mountford ()
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Sergey Foss: Heriot-Watt University
Takis Konstantopoulos: Uppsala University
Thomas Mountford: Ecole Polytechnique Fédérale de Lausanne

Journal of Theoretical Probability, 2018, vol. 31, issue 2, 684-704

Abstract: Abstract The Poisson hail model is a space-time stochastic system introduced by Baccelli and Foss (J Appl Prob 48A:343–366, 2011) whose stability condition is nonobvious owing to the fact that it is spatially infinite. Hailstones arrive at random points of time and are placed in random positions of space. Upon arrival, if not prevented by previously accumulated stones, a stone starts melting at unit rate. When the stone sizes have exponential tails, then stability conditions exist. In this paper, we look at heavy tailed stone sizes and prove that the system can be stabilized when the rate of arrivals is sufficiently small. We also show that the stability condition is, in a weak sense, optimal. We use techniques and ideas from greedy lattice animals.

Keywords: Poisson hail; Stability; Workload; Greedy lattice animals; 82B44; 82D30; 60K37 (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/s10959-016-0723-3

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