## Section: New Results

### Algorithms: Distributed Hash Table

Participants : Mathieu Feuillet, Philippe Robert.

The Distributed Hash Table (DHTs) consists of a large set of nodes connected through the Internet. Each file contained in the DHT is stored in a small subset of these nodes. Each node breaks down periodically and it is necessary to have back-up mechanisms in order to avoid data loss. 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. We modeled this problem thanks to standard queues in order to understand the behavior of a single file and the global dynamic of the system. 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 have been able to study different time scales of the system with an averaging phenomenon. An article is in preparation on this subject. A more sophisticated distributed model with mean field techniques is under investigation.

On the side of this project, we notably studied the distribution of hitting times of the classical Ehrenfest and Engset models by using martingale techniques, furthermore their asymptotic behavior has been analyzed when the size of the system increases to infinity [5] .