Journal of Applied Probability

Poisson hail on a hot ground

Francois Baccelli and Sergey Foss


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.

Article information

J. Appl. Probab., Volume 48A (2011), 343-366.

First available in Project Euclid: 18 October 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes 60K25: Queueing theory [See also 68M20, 90B22] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Poisson point process Poisson rain random closed set Euclidean space service stability backward scheme monotonicity branching process percolation hard-core exclusion process queueing theory stochastic geometry


Baccelli, Francois; Foss, Sergey. Poisson hail on a hot ground. J. Appl. Probab. 48A (2011), 343--366. doi:10.1239/jap/1318940476.

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