The Rate Control Protocol (RCP) uses explicit feedback from routers to control network congestion. RCP estimates it’s fair rate from two forms of feedback: rate mismatch and queue size. An important design question that remains open in RCP is whether the presence of queue size feedback is helpful, given the presence of feedback from rate mismatch. The feedback from routers to end-systems is time-delayed, and may introduce instabilities and complex nonlinear dynamics. Delay dynamical systems are often modeled using delay differential equations to facilitate a mathematical analysis of their performance and dynamics. The RCP models with and without queue size feedback give rise to two distinct nonlinear delay differential equations. Earlier work on this design question was based on methods of linear systems theory. For further progress, it is quite natural to employ nonlinear techniques. In this study, we approach this design question using tools from control and bifurcation theory. The analytical results reveal that the removal of queue feedback could enhance both stability and convergence properties. Further, using Poincaré normal forms and center manifold theory, we investigate two nonlinear properties, namely, the type of the Hopf bifurcation and the asymptotic stability of the bifurcating limit cycles. We show that the presence of queue feedback in the RCP can lead to a sub-critical Hopf bifurcation, which would give rise either to the onset of large amplitude limit cycles or to unstable limit cycles. Whereas, in the absence of queue feedback, the Hopf bifurcation is always super-critical, and the bifurcating limit cycles are stable. The analysis is complemented with computations and some packet-level simulations as well. In terms of design, our study suggests that the presence of both forms of feedback may be detrimental to the performance of RCP.

速率控制协议（RCP）使用来自路由器的显式反馈来控制网络拥塞。RCP通过两种形式的反馈来估计它是合理的速率：速率不匹配和队列大小。在RCP中仍然存在的一个重要设计问题是，考虑到速率不匹配的反馈，队列大小反馈的存在是否有帮助。从路由器到终端系统的反馈是有时间延迟的，并且可能引入不稳定和复杂的非线性动力学。时滞动力系统通常使用时滞微分方程建模，以帮助对其性能和动力学进行数学分析。具有和不具有队列大小反馈的RCP模型产生两个截然不同的非线性延迟微分方程。有关此设计问题的早期工作是基于线性系统理论的方法。为了取得进一步的进展，采用非线性技术是很自然的。在这项研究中，我们使用控制和分歧理论中的工具来解决这个设计问题。分析结果表明，消除队列反馈可以增强稳定性和收敛性。进一步，利用庞加莱正态形式和中心流形理论，我们研究了两个非线性性质，即霍普夫分支的类型和分支极限环的渐近稳定性。我们表明，RCP中队列反馈的存在会导致次临界Hopf分叉，这可能会引起大幅度极限周期的开始或不稳定极限周期的出现。而在没有队列反馈的情况下，Hopf分叉总是超临界的，并且分叉极限周期是稳定的。该分析还辅以计算和一些数据包级仿真。在设计方面，我们的研究表明，两种形式的反馈的存在可能对RCP的性能有害。