# Probability and Statistics Seminar

### Analysis of a Stochastic Model for Flash Crowd Scenarios

In this talk we investigate the performance of a file sharing principle similar to the one implemented by eMule and BitTorrent. For this purpose, we consider a system composed of N peers becoming active at exponential random times, thus modeling a flash crowd'' scenario where an initial burst of clients occurs. The system is initiated with only one server offering the desired file and the other peers try to download it after becoming active. Once the file has been downloaded by a peer, this one immediately becomes a server. While the system starts in a congested state where all servers available are saturated by incoming demands, it shifts to a state where a growing number of servers are idle. We are interested in the time needed for this shift to happen, which is closely related to the transient performance of this file sharing principle. In spite of its apparent simplicity, this queueing model (with a random number of servers) reveals quite difficult to analyze. A formulation in terms of an urn and ball model is proposed and corresponding scaling results are derived. These asymptotic results are then compared against simulations.