Section: New Results
Scaling Methods
Participants : Philippe Robert, Wen Sun.
Fluid Limits in Wireless Networks
This is a collaboration with Amandine Veber (CMAP, École Polytechnique). The goal is to investigate the stability properties of wireless networks when the bandwidth allocated to a node is proportional to a function of its backlog: if a node of this network has $x$ requests to transmit, then it receives a fraction of the capacity proportional to $log(1+x)$, the logarithm of its current load. This year we completed the analysis of a star network topology with multiple nodes. Several scalings were used to describe the fluid limit behaviour.
Large Unreliable Stochastic Networks
The reliability of a large distributed system is studied. The framework is a system where files have several copies on different servers. When one of these servers breaks down, all copies stored on it are lost. These copies can be retrieved afterwards if there is another copy of the same files stored on other servers. In the case where no other copy of a given file is present in the system, it definitely lost. We study two math model on this problem.
In the first model, it is assumed that the duplication process is local, any server has a capacity to make copies to another server, but the capacity can only be used for the copies present on this server. We have studied the asymptotic behavior of this system, i.e. the number of servers is large, via mean field methods. We have shown that asymptotically, the load of each server can be described by a nonlinear Markov process. This limiting process can also give an exponential decay of the number of files. This is a joint work with Reza Aghajani, Brown University.
In the second model, two policies for the reassignment of files are studied. It is assumed that each server has a neighborhood, that consists of a set of servers in the system. When a server breaks down, it restarts immediately but empty. Copies on it are reassigned to other servers in the neighborhood, following “Random Choice” (RC) policy or “Power of choices” (PoC) policy.

(PoC) Each copy chooses several servers in the neighborhood at random, and joins the least loaded one.
The asymptotic behaviors of these two policies are investigated through mean field models. We have show that when the number of servers getting large, the load of each server can be approached by a linear (resp. nonlinear) Markov process for RC (resp. PoC) policy. The equilibrium distributions of these asymptotic processes are also given. This is a joint work with Inria/UPMC Team Regal.