## 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.

#### The Time Scales of a Transient Network

A large distributed system where users' files are duplicated on unreliable data servers is investigated. Due to a server breakdown, a copy of a file can be lost, it can be retrieved if another copy of the same file is stored on other servers. In the case where no other copy of a given file is present in the network, it is definitely lost. In order to have multiple copies of a given file, it is assumed that each server can devote a fraction of its processing capacity to duplicate files on other servers to enhance the durability of the system.

A trade-off is necessary between the bandwidth and the memory used for this back-up mechanism and the data loss rate. Back-up mechanisms already exist and have been studied thanks to simulation. To our knowledge, no theoretical study exists on this topic. With a very simple centralized model, we have been able to emphasise a trade-off between capacity and life-time with respect to the duplication rate. From a mathematical point of view, we are currently studying different time scales of the system with an averaging phenomenon.

We have used scaling methods with different time scales to derive some asymptotic results on the decay of a simplified network: it is assumed that any copy of a given file is lost at some fixed rate and the total processing capacity of the system is devoted to duplicate the file with least number of copies. We start from the optimal initial state: each file has the maximum number of copies. Due to random losses, the state of the network is transient and all files will be eventually lost. There is a stability assumption for the system having a critical time scale of decay. When the stability condition is not satisfied, i.e. when it is initially overloaded, we have shown that the state of the network converges to an interesting local equilibrium. We are currently studying a more general case which the duplication depends on the structure of the system. See [7] .