Poisson Hail

Consider a queue where the server is the Euclidean space, the customers are random closed sets (RACS) of the Euclidean space. These RACS 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 RACS can be served simultaneously and service is in the First In First Out order (only the hailstones in contact with the ground melt at speed 1, whereas the other ones are queued; a tagged RACS waits until all RACS arrived before it and intersecting it have fully melted before starting its own melting). We prove that this queue is stable for a sufficiently small arrival intensity, provided the typical diameter of the RACS and the typical service time have finite exponential moments.

In Shape Theorems For Poisson Hail on a Bivariate Ground, H. Chang, S. Foss and F. Baccelli have extended this Poisson Hail model to the situation where the service speed is either zero or infinity at each point of the Euclidean space. Tools pertaining to sub-additive ergodic theory are used 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.