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Phase Transitions in Stochastic Network Dynamics

by Francois Baccelli (INRIA-ENS - Paris)

The talk will be centered on the use of ergodic theory to study the dynamics of two classes of stochastic networks involving both queuing and routing, with a special focus on their infinite dimensional versions. We will first consider the class of monotone, finite dimensional stochastic queuing/routing networks. We will review the ergodic theory tools allowing one to 1) analyze their 'stability' phase transition, which separates the region where there exists a finite stationary regime from that where there is no such regime; 2) give pathwise representations of the stationary regime(s). The main focus will be on recent advances extending this approach to a classe of infinite dimensional queuing networks without routing. We will then discuss joint queuing and routing dynamics on random graphs. We will focus on the class of SINR random graphs. In the infinite graph case, we will show how to use first passage percolation and ergodic theory in order to analyze the phase transition separating the ballistic regime, where the asymptotic velocity on a route is positive, from the sub-ballistic regime, where it is 0. We will conclude by listing a few open problems and research directions along these lines and by discussing related stochastic networks where such tools could be used.