Section: New Results

Miscellaneous

Participants : Florian Simatos, Philippe Robert, Danielle Tibi.

Metastability and communication networks

Danielle Tibi. In Antunes et al.   [25] , in the context of mobility, a model of a network that converges, as the number N of nodes gets large, to a mean-field limit with several stable equilibrium points has been exhibited. A similar phenomenon had been observed for a network under a rerouting procedure in Gibbens et al.   [30] . The analysis in  [25] relies on the derivation of an explicit Lyapunov function for the limiting dynamical system. Such a function is not available for the rerouting procedure.

The question of metastability - meaning exponential growth of exit times from neighborhoods of stable points - is not addressed in these papers, though this phenomenon is highly suspected to occur for both models (particularly in view of simulation results). But the non-reversibility is here an obstacle to using standard approaches to metastability.

These models are now re-examined in the light of Freidlin and Wentzell's large deviations approach of randomly perturbed dynamical systems  [28] . It can be shown that both models essentially fit the scheme developed in this reference for a certain class of Markov jump processes. For these processes, it provides large deviations results for exit times, points and paths from neighborhoods of stable points. The central quantity involved in the action functionals is the quasi-potential , that can be interpreted as the energy function associated to the system in the limit N .

These results can be adapted to our context. For the model in   [25] , it can also be proved that the above mentioned Lyapunov function is equal to the quasi-potential of a simplified process, which, contrary to the model in  [29] , has the particularity of being asymptotically reversible.

Continuous-state branching process in random environment

Florian Simatos, joint work with Vincent Bansaye (University Pierre et Marie Curie). Continuous state branching processes (CSBP) are continuous time, continuous state space Markov processes which satisfy the branching property. Until now, mainly time-homogeneous CSBP have been studied since they naturally appear as limits of Galton-Watson branching processes.

On the other hand, a growing interest has been recently devoted to Galton-Watson branching processes in a random environment: these processes are relevant from a modeling perspective, and it turns out that they have many interesting features from a mathematical standpoint as well. The goal of this collaboration is twofold: define CSBP in a varying environment as the solution of a certain martingale problem. Once these processes are defined, we want to let the environment be random and study the basic properties of CSBP in a random environment. Performance properties of interest are the extinction probability and the existence of a genealogy, for instance.

Load balancing via random local search.

Florian Simatos, joint work with A. Ganesh, S. Lilenthal, D. Manjunath and A. Proutière, Microsoft Cambridge. In traditional approaches to load balancing in parallel server systems, e.g., Join-the-Shortest-Queue and variants thereof, clients make smart server selections upon arrival. In this work we analyze a different approach, where load balancing is performed by random load resampling and migration strategies. Clients initially attach to an arbitrary server, but may switch server independently at random instants of time in an attempt to improve their service rate. Load resampling is particularly relevant in scenarios where clients cannot predict the load of a server before being actually attached to it. An important example is in wireless spectrum sharing where clients try to share a set of frequency bands in a distributed manner.

In  [21] we analyze closed systems, where we derive tight estimates of the time it takes to balance the load across servers. We also study open systems where clients arrive according to a random process and leave the system upon service completion. In this scenario, we analyze how client migrations within the system interact with the system dynamics induced by client arrivals and departures.

The Evolution of a Spatial Stochastic Network

Philippe Robert. Motivated by bandwidth allocation algorithms in wireless networks, the asymptotic behavior of a stochastic network represented by a birth and death processes of particles has been analyzed in [24] . Particles are created at rate ₊ and their location is independent of the current configuration. Deaths are due to negative particles arriving at rate _- . 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 have been investigated. It has been shown for several natural models that if ₊ <_- then the asymptotic configuration has a finite number of points with probability 1. It has also been shown that systems with ₊ <_- and an infinite number of particles in the limit exist.

A Scaling analysis of a Cat and Mouse Markov chain

Philippe Robert, joint work with Nelly Litvak (University of Twente, Holland). An original on-line page-ranking algorithm has been investigated in Litvak and Robert [23] : starting from an arbitrary Markov chain (C_n) on a discrete state space , a Markov chain (C_n, M_n) on the product space , the cat and mouse Markov chain, is constructed. The first coordinate of this Markov chain behaves like the original Markov chain and the second component changes only when both coordinates are equal. The asymptotic properties of this Markov chain are investigated. A representation of its invariant measure is in particular obtained. When the state space is infinite it is shown that this Markov chain is in fact null recurrent if the initial Markov chain (C_n) is positive recurrent and reversible. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple random walks in and , reflected simple random walk in and also in a continuous time setting. For several of these processes, a time scaling with rapid growth gives an interesting asymptotic behavior related to limit results for occupation times and rare events of Markov processes.