Random Closed Sets

2006 ◽  
pp. 109-130
Keyword(s):  
Closed Sets ◽  
Random Closed Sets

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

scholarly journals Poisson hail on a hot ground

2011 ◽  
Vol 48 (A) ◽  
pp. 343-366
Author(s):  
Francois Baccelli ◽  
Sergey Foss
Keyword(s):  
Euclidean Space ◽  
Service Time ◽  
Evolution Equations ◽  
Stationary Regime ◽  
Service Discipline ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Exponential Moments ◽  
First In First Out ◽  
Arrival Intensity

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.

Size–biased random closed sets

1999 ◽  
Vol 32 (9) ◽  
pp. 1631-1644 ◽  
Author(s):  
M.N.M. van Lieshout
Keyword(s):  
Closed Sets ◽  
Random Closed Sets

Random Closed Sets

2006 ◽  
pp. 55-64 ◽  
Author(s):  
Paul Brodhead ◽  
Douglas Cenzer ◽  
Seyyed Dashti
Keyword(s):  
Closed Sets ◽  
Random Closed Sets
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