Section: Research Program
Local Interactions and Transient Analysis in Adaptive Dynamic Systems
Participants : Nicolas Gast, Bruno Gaujal, Florence Perronnin, Jean-Marc Vincent, Panayotis Mertikopoulos.
Many systems can be effectively described by stochastic population models. These systems are composed of a set of entities interacting together and the resulting stochastic process can be seen as a continuous-time Markov chain with a finite state space. Many numerical techniques exist to study the behavior of Markov chains, to solve stochastic optimal control problems  or to perform model-checking . These techniques, however, are limited in their applicability, as they suffer from the curse of dimensionality: the state-space grows exponentially with .
This results in the need for approximation techniques. Mean field analysis offers a viable, and often very accurate, solution for large . The basic idea of the mean field approximation is to count the number of entities that are in a given state. Hence, the fluctuations due to stochasticity become negligible as the number of entities grows. For large , the system becomes essentially deterministic. This approximation has been originally developed in statistical mechanics for vary large systems composed of more than particles (called entities here). More recently, it has been claimed that, under some conditions, this approximation can be successfully used for stochastic systems composed of a few tens of entities. The claim is supported by various convergence results , , , and has been successfully applied in various domains: wireless networks , computer-based systems , , , epidemic or rumour propagation ,  and bike-sharing systems . It is also used to develop distributed control strategies ,  or to construct approximate solutions of stochastic model checking problems , , .
Within the POLARIS project, we will continue developing both the theory behind these approximation techniques and their applications. Typically, these techniques require a homogeneous population of objects where the dynamics of the entities depend only on their state (the state space of each object must not scale with the number of objects) but neither on their identity nor on their spatial location. Continuing our work in , we would like to be able to handle heterogeneous or uncertain dynamics. Typical applications are caching mechanisms  or bike-sharing systems . A second point of interest is the use of mean field or large deviation asymptotics to compute the time between two regimes  or to reach an equilibrium state. Last, mean-field methods are mostly descriptive and are used to analyse the performance of a given system. We wish to extend their use to solve optimal control problems. In particular, we would like to implement numerical algorithms that use the framework that we developed in  to build distributed control algorithms  and optimal pricing mechanisms .