  We investigate in this paper the performance of a simple file sharing
principle. For this purpose, we consider a system composed of N peers becoming
active at exponential random times; the system is initiated with only one
server offering the desired file and the other peers after becoming active try
to download it. Once the file has been downloaded by a peer, this one
immediately becomes a server. To investigate the transient behavior of this
file sharing system, we study the instant when the system shifts from a
congested state where all servers available are saturated by incoming demands
to a state where a growing number of servers are idle. In spite of its apparent
simplicity, this queueing model (with a random number of servers) turns out to
be 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.
