# Workshop Programme

## for period 22 - 26 March 2010

### Stochastic Networks

22 - 26 March 2010

Timetable

 Friday 26 March Chair: Onno Boxma 09:30-10:30 Evans, S (UC, Berkeley) Go forth and multiply? Sem 1 Organisms reproduce in environments that change in both time and space. Even if an individual currently resides in a region that is typically quite favorable, it may be optimal for it to not put all its eggs in the one basket'' and disperse some of its offspring to locations that are usually less favorable to cover the possibility that the effect of poor conditions in one place will be fortuitously offset by good conditions in another. I will describe joint work with Peter Ralph and Sebastian Schreiber from Evolution and Ecology at U.C. Davis and Arnab Sen from Statistics at U.C. Berkeley that uses stochastic differential equations to explore the usefulness of such dispersal strategies. 10:30-11:00 Coffee 11:00-12:00 Robert, P (INRIA Paris - Rocquencourt) The Evolution of a Spatial Stochastic Network Sem 1 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 $\l_+ 12:15-13:30 Lunch at Wolfson Court (open for lunch from 12:15--13:30) Chair: James Martin 14:00-15:00 Ferrari, P (Buenos Aires) Slow-to-start traffic models, coalescing Brownian motions and M/M/1 queues Sem 1 "Cars" are points in the real line that move to the left and have two possible velocities: 0 or 1. Cars maintain order, so that when a moving car is blocked by a stopped car, it has to stop too. Unblocked cars with zero velocity wait a random time with exponential distribution to adquire velocity one (this is the slow-to-start rule). The initial distribution of cars is a Poisson process of parameter$\l\$. There are three regimes: supercritical (\l>1) where cars condensate in a random set of initial positions; critical (\l = 1) when there is condensation but the asymptotic speed of the cars is 1, and subcritical (\l<1) where each car get eventually unblocked. The rescaling of this process in the critical and supercritical regimes is related with coalescing Brownian motions. In the subcritical regime the final relative positions of the cars is given by a Poisson process. In this case the car trajectories can be mapped to the loading process of a M/M/1 queue. Joint work with Leo Rolla, Fredy Caceres and Eugene Pechersky. 15:00-15:30 Tea 15:30-16:00 Kelly, F (Cambridge) Closing Perspectives Lecture Sem 1 18:30-19:30 Dinner at Wolfson Court