Refinements of Mean Field Approximation

The design of efficient algorithms is closely linked to the evaluation of their performances. My work focuses on the use of stochastic models for the performance evaluation of large distributed systems. I am interested in developing tools that can characterize the emergent behavior of such systems and improve their performance. This leads me to solve stochastic control and optimization problems, notably through operations research methods. These problems suffer from combinatorial explosion: the complexity of a problem grows exponentially with the number of objects that compose it. It is therefore necessary to design models but also algorithmic processes whose complexity does not increase too rapidly with the size of the system. This document summarizes a few of my contributions on the use and the refinement of mean field approximation to study the performance of distributed systems and algorithms. Résumé La conception d’algorithmes efficaces est étroitement liés à l’évaluation de leurs performances. Mon travail se concentre sur l’utilisation de modèles stochastiques pour l’évaluation des performances des grands systèmes distribués. Je développe des outils permettant de caractériser le comportement émergent de ces systèmes et d’améliorer leur performance. Cela me conduit à résoudre des problème d’optimisation et de contrôle stochastique. Ces problèmes générallement souffrent d’un problème d’explosion combinatoire: la complexité d’un problème croît de façon exponentielle avec le nombre des objets qui le composent. Il faut donc concevoir des modèles mais aussi des processus algorithmiques dont la complexité n’augmente pas ou peu avec la taille du système. Ce document résume quelques unes de mes contributions sur l’utilisation et le raffinement des méthodes de type champ moyen pour étudier la performances d’algorithmes de contrôle pour des systèmes distribués. A C K N O W L E D G M E N T S This document is a summary of a part of my research. The path leading to this document has not always been a straight line and I would like to thank here all the people that made this possible. First, I would like to thank Sem Borst, Kavita Ramanan and Philippe Robert to have accepted to serve as referees (rapporteurs) of my habilitation. They dedicated a large amount of their time to provide insightful comments about my work. I would also like to thank Urtzi Ayesta, Michel Benaïm, Nicolas Champagnat and Denis Trystram to have accepted to be part of the jury as examimateurs. It is an honor to have been accepted as peer by such renowned scientists. I would also like to thank three persons that guided me through my early scientific career: Francois Baccelli who I met during my master and who was the first person to introduce me to performance modeling and stochastic processes, and without whom I might have chosen a very different path; Bruno Gaujal, who accepted me as his PhD student and of whom I always admire the enthusiasm and curiosity about almost anything; and Jean-Yves Le Boudec, with whom I spent four rich years at EPFL and who tried to teach me scientific rigor and how to apply theoretical concepts to very practical problems. Coming to work everyday is a real pleasure thanks to the many colleagues in Grenoble that for most are also close friends (Annie, Arnaud, Bary, Bruno, Denis, Greg, Olivier, Patrick, Pierre, Jean-Marc,.) and thanks to all PhD students or post-docs with whom I have or had the pleasure to work with (Guillaume, Josu, Benoît, Taki, Mouhcine, Vitalii, Henry, Kimang, Chen, Thomas,.). I am also grateful to all my co-authors for fruitful scientific discussions. Last but not least, all this work would not have been possible without the support of my friends and family. A particular thought goes to my wife Mathilde and our children Elliot, Martin and Lise. Thanks you for being wonderful human beings and for all the joy that you bring, for giving me slack when I want but bringing me back to reality when needed. Thank you for being there.