短期交互TCP流的平均场分析

F. Baccelli, A. Chaintreau, D. D. Vleeschauwer, D. McDonald
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引用次数: 24

摘要

在本文中,我们考虑一组使用TCP的HTTP流,通过一个公共的拖尾链接来下载文件。每次下载之后,流在请求下载另一个文件(其大小也是随机的)之前等待一段随机的思考时间。当一个流处于活动状态时,其吞吐量根据加性增长规律随时间增加,但如果流的总传输速率超过链路速率而造成损失,则其传输速率降低。一个流获得的吞吐量和下载一个文件的连续时间是所有流通过它们的总传输速率和链路行为相互作用的结果。我们研究了流的数量趋于无穷时得到的平均场模型。该平均场极限可以有两个稳定的状态:一个是链路中没有拥塞,其中传输速率密度可以显式描述,另一个是具有周期性拥塞时期,其中间拥塞时间可以表征为我们数值计算的不动点方程的解,从而得到传输速率密度,如Fredholm方程的解所示。结果表明,对于某些参数值(更准确地说,当每个用户的链路容量不明显大于每个用户的负载时),这两个稳定状态中的每一个都可以根据初始条件达到。这种现象可以看作是流体动力学中湍流的类似物:在某些初始条件下,传递以流体和无相互作用的方式进行;对于另一些人来说,连接交互并由于由此产生的波动而变慢,这反过来又使交互永远持续下去,尽管每个用户的负载小于每个用户的容量。我们证明了这种现象存在于太浩的情况下,我们开发的数值方法和模拟都表明它也存在于里诺的情况下。在这个参数范围内,它转化为有限种群模型的双稳定性现象。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A mean-field analysis of short lived interacting TCP flows
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 suffers losses created when the total transmission rate of the flows exceeds the link rate, its transmission rate is decreased. The throughput obtained by a flow, 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 without 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 present in the Reno case too. It translates into a bi-stability phenomenon for the finite population model within this range of parameters.
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