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# HTTP turbulence

• In this paper, we consider a set of HTTP flows using TCP over a common drop-tail link to download files. After each download, a flow waits for a random think time before requesting the download of another file, whose size is also random. When a flow is active its throughput is increasing with time according to the additive increase rule, but if it su®ers losses created when the total transmission rate of the flows exceeds the link rate, its transmission rate is decreased. The throughput obtained by a °ow, and the consecutive time to download one file are then given as the consequence of the interaction of all the flows through their total transmission rate and the link's behavior.
We study the mean-field model obtained by letting the number of flows go to infinity. This mean-field 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 inter-congestion 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 interaction-less 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 bi-stability phenomenon for the finite population model within this range of parameters.
Mathematics Subject Classification: Primary: 60K25; Secondary: 60K20.

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