22 March - 26 March 2010

Isaac Newton Institute for Mathematical Sciences, Cambridge, UK

**Workshop Organisers:** Takis Konstantopoulos (*Heriot-Watt*) and Kavita Ramanan (*Brown University*).

**Workshop Programme Committee:** Vivek Borkar (*Tata Institute of Fundamental Research*), Serguei Foss (*Heriot-Watt University*), Peter Glynn (*Stanford*),

Bruce Hajek (*University of Illinois*), Frank Kelly (*Cambridge*), P.R. Kumar (*University of Illinois*), Tom Kurtz (*University of Wisconsin-Madison*), Jean Mairesse (*LIAFA*), Philippe Robert (*INRIA*), John Tsitsiklis (*MIT*) and Ruth Williams (*University of California, San Diego*)

in association with the Newton Institute programme

Stochastic Processes in Communication Sciences (11 January to 2 July 2010) Participants | Application | Accommodation and CostTitle: The Evolution of a Spatial Stochastic Network

The asymptotic behavior of a stochastic network represented by a birth and death processes of particles is analyzed. Births: Particles are created at rate $\l_+$ and their location is independent of the current configuration. Deaths are due to negative particles arriving at rate $\l_-$. The death of a particle occurs when a negative particle arrives in its neighborhood and kills it. Several killing schemes are considered. The arriving locations of positive and negative particles are assumed to have the same distribution. By using a combination of monotonicity properties and invariance relations it is shown that the configurations of particles converge in distribution for several models. The problems of uniqueness of invariant measures and of the existence of accumulation points for the limiting configurations are also investigated. It is shown for several natural models that if $\l_+<l_-$ then the asymptotic configuration has a finite number of points with probability $1$. Examples with $\l_+<\l_-$ and an infinite number of particles in the limit are also presented.

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