Hostname: page-component-cb9f654ff-mwwwr Total loading time: 0 Render date: 2025-08-19T08:30:33.389Z Has data issue: false hasContentIssue false
  • English
  • Français

Shape theorems for Poisson hail on a bivariate ground

Part of: Limit theorems Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  10 June 2016

François Baccelli ,
Héctor A. Chang-Lara  and
Sergey Foss
François Baccelli*
Affiliation:
University of Texas
Héctor A. Chang-Lara*
Affiliation:
Columbia University
Sergey Foss*
Affiliation:
Heriot-Watt University and Sobolev Institute of Mathematics, Novosibirsk
*
* Postal address: Department of Mathematics, University of Texas, Austin, TX 78712, USA. Email address: baccelli@math.utexas.edu
** Postal address: Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, USA. Email address: changlara@math.columbia.edu
*** Postal address: School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK. Email address: s.foss@hw.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider an extension of the Poisson hail model where the service speed is either 0 or ∞ at each point of the Euclidean space. We use and develop tools pertaining to sub-additive ergodic theory in order to establish shape theorems for the growth of the ice-heap under light tail assumptions on the hailstone characteristics. The asymptotic shape depends on the statistics of the hailstones, the intensity of the underlying Poisson point process, and on the geometrical properties of the zero speed set.

Keywords

Point process theoryPoisson rainstochastic geometryrandom closed settime and space growthshapequeueing theorymax-plus algebraheapsbranching processsub-additive ergodic theory

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60F15: Strong theorems 60G55: Point processes

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 2 , June 2016 , pp. 525 - 543
Copyright
Copyright © Applied Probability Trust 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

[1]Baccelli, F. and Foss, S. (2011).Poisson hail on a hot ground. In New Frontiers in Applied Probability: A Festschrift for Søren Asmussen (J. Appl. Prob. Spec. Vol.48A),Applied Probability Trust,Sheffield, pp.343–366.Google Scholar
[2]Baccelli, F.,Borovkov, A. and Mairesse, J. (2000).Asymptotic results on infinite tandem queueing networks.Prob. Theory Relat. Fields 118,365–405.CrossRefGoogle Scholar
[3]Baccelli, F.,Cohen, G.,Olsder, G. J. and Quadrat, J.-P. (1992).Synchronization and Linearity: An Algebra for Discrete Event Systems.John Wiley,Chichester.Google Scholar
[4]Bhamidi, S.,van der Hofstad, R. and Hooghiemstra, G. (2010).First passage percolation on random graphs with finite mean degrees.Ann. Appl. Prob. 20,1907–1965.Google Scholar
[5]Cox, J. T.,Gandolfi, A.,Griffin, P. S. and Kesten, H. (1993).Greedy lattice animals. I. Upper bounds.Ann. Appl. Prob. 3,1151–1169.Google Scholar
[6]Foss, S.,Konstantopoulos, T. and Mountford, T. (2014).Power law condition for stability of Poisson hail. Preprint. Available at http://arxiv.org/abs/1410.0911.Google Scholar
[7]Gandolfi, A. and Kesten, H. (1994).Greedy lattice animals. II. Linear growth.Ann. Appl. Prob. 4,76–107.Google Scholar
[8]Halsey, T. (2000).Diffusion-limited aggregation: a model for pattern formation.Physics Today 53,36–41.Google Scholar
[9]Liggett, T. M. (1985).An improved subadditive ergodic theorem.Ann. Prob. 13,1279–1285.CrossRefGoogle Scholar
[10]Seppäläinen, T. (1998).Hydrodynamic scaling, convex duality, and asymptotic shapes of growth models.Markov Process. Relat. Fields 4,1–26.Google Scholar