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Poisson hail on a hot ground

Part of: Markov processes Special processes Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  14 July 2016

Francois Baccelli  and
Sergey Foss
Francois Baccelli
Affiliation:
INRIA-ENS, ENS-DI TREC, 45 rue d'Ulm, 75230 Paris, France. Email address: francois.baccelli@ens.fr
Sergey Foss
Affiliation:
Heriot-Watt University and Institute of Mathematics, Novosibirsk, Department of Actuarial Mathematics and Statistics, Heriot-Watt University, Edinburgh, EH14 4AS, UK
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Abstract

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We consider a queue where the server is the Euclidean space, and the customers are random closed sets (RACSs) of the Euclidean space. These RACSs arrive according to a Poisson rain and each of them has a random service time (in the case of hail falling on the Euclidean plane, this is the height of the hailstone, whereas the RACS is its footprint). The Euclidean space serves customers at speed 1. The service discipline is a hard exclusion rule: no two intersecting RACSs can be served simultaneously and service is in the first-in–first-out order, i.e. only the hailstones in contact with the ground melt at speed 1, whereas the others are queued. A tagged RACS waits until all RACSs that arrived before it and intersecting it have fully melted before starting its own melting. We give the evolution equations for this queue. We prove that it is stable for a sufficiently small arrival intensity, provided that the typical diameter of the RACS and the typical service time have finite exponential moments. We also discuss the percolation properties of the stationary regime of the RACS in the queue.

Keywords

Poisson point processPoisson rainrandom closed setEuclidean spaceservicestabilitybackward schememonotonicitybranching processpercolationhard-core exclusion processqueueing theorystochastic geometry

MSC classification

Primary: 60G55: Point processes 60K25: Queueing theory 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Type
Part 8. Point Processes
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 343 - 366
Copyright
Copyright © Applied Probability Trust 2011 

References

[1] Baccelli, F. and Brémaud, P., (2003). Elements of Queueing Theory, 2nd edn. Springer, Berlin.CrossRefGoogle Scholar
[2] Baccelli, F. and Foss, S., (1995). On the saturation rule for the stability of queues. J. Appl. Prob. 32, 494–507.CrossRefGoogle Scholar
[3] Baccelli, F. and Foss, S., (2004). Moments and tails in monotone-separable stochastic networks. Ann. Appl. Prob. 14, 612–650.CrossRefGoogle Scholar
[4] Biggins, J. D., (1976). The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Prob. 8, 446–459.CrossRefGoogle Scholar
[5] Baszczyszyn, B., Rau, C. and Schmidt, V., (1999). Bounds for clump size characteristics in the Boolean model. Adv. Appl. Prob. 31, 910–928.CrossRefGoogle Scholar
[6] Hall, P., (1985). On continuum percolation. Ann. Prob. 13, 1250–1266.CrossRefGoogle Scholar
[7] Loynes, R. M., (1962). The stability of a queue with nonindependent interarrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.CrossRefGoogle Scholar
[8] Penrose, M. D., (2008). Growth and roughness of the interface for ballistic deposition. J. Statist. Phys. 131, 247–268.CrossRefGoogle Scholar
[9] Stoyan, D., (1983). Comparison Methods for Queues and Other Stochastic Models. John Wiley, Chichester.Google Scholar
[10] Stoyan, D., Kendall, W. S. and Mecke, J., (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, Chichester.Google Scholar