Branching processes, the max-plus algebra and network calculus

Branching processes can describe the dynamics of various queueing systems, peer-to-peer systems, delay tolerant networks, etc. In this paper we study the basic stochastic recursion of multitype branching processes, but in two non-standard contexts. First, we consider this recursion in the max-plus algebra where branching corresponds to finding the maximal offspring of the current generation. Secondly, we consider network-calculus-type deterministic bounds as introduced by Cruz, which we extend to handle branching-type processes. The paper provides both qualitative and quantitative results and introduces various applications of (max-plus) branching processes in queueing theory

Evolutionary Games

International audienceEvolutionary games constitute the most recent major mathematical tool for understanding, modelling and predicting evolution in biology and other fields. They complement other well establlished tools such as branching processes and the Lotka-Volterra [6] equations (e.g. for the predator - prey dynamics or for epidemics evolution). Evolutionary Games also brings novel features to game theory. First, it focuses on the dynam- ics of competition rather than restricting attention to the equilibrium. In particular, it tries to explain how an equilibrium emerges. Second, it brings new de nitions of stability, that are more adapted to the context of large populations. Finally, in contrast to standard game theory, players are not assumed to be \rational" or \knowledgeable" as to anticipate the other players' choices. The objective of this article, is to present founda- tions as well as recent advances in evolutionary games, highlight the novel concepts that they introduce with respect to game theory as formulated by John Nash, and describe through several examples their huge potential as tools for modeling interactions in complex systems