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Gas flow in pipeline networks
HTTP turbulence
1.  Département d'Informatique, ENS, 45 rue d'Ulm, 75005 Paris, France 
2.  Thomson Research 46, quai Alphonse Le Gallo, 92648 Boulogne Cedex, France 
3.  Alcatel Bell, Francis Wellesplein 1, B2018 Antwerp, Belgium 
4.  Department of Mathematics and Statistics, University of Ottawa, 585 King Edward Avenue, Ottawa, Ontario K1N 6M8, Canada 
We study the meanfield model obtained by letting the number of flows go to infinity. This meanfield limit may have two stable regimes: one with out congestion in the link, in which the density of transmission rate can be explicitly described, the other one with periodic congestion epochs, where the intercongestion time can be characterized as the solution of a fixed point equation, that we compute numerically, leading to a density of transmission rate given by as the solution of a Fredholm equation. It is shown that for certain values of the parameters (more precisely when the link capacity per user is not significantly larger than the load per user), each of these two stable regimes can be reached depending on the initial condition. This phenomenon can be seen as an analogue of turbulence in fluid dynamics: for some initial conditions, the transfers progress in a fluid and interactionless way; for others, the connections interact and slow down because of the resulting fluctuations, which in turn perpetuates interaction forever, in spite of the fact that the load per user is less than the capacity per user. We prove that this phenomenon is present in the Tahoe case and both the numerical method that we develop and simulations suggest that it is also be present in the Reno case. It translates into a bistability phenomenon for the finite population model within this range of parameters.
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