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Periodic solutions, chaos and bi-stability in the state-dependent delayed homogeneous Additive Increase and Multiplicative Decrease/Random Early Detection congestion control systems
Mathematics and Computers in Simulation ( IF 4.4 ) Pub Date : 2021-04-01 , DOI: 10.1016/j.matcom.2020.06.001
Lijun Pei , Fanxin Wu

Abstract The combination of Additive Increase and Multiplicative Decrease (AIMD) congestion control and Random Early Detection (RED) queue as a whole congestion control system plays a key role in the overwhelming success of the Internet. Thus it is important to investigate the periodic oscillations and complicated dynamics of the state-dependent delayed homogeneous Additive Increase and Multiplicative Decrease/Random Early Detection (AIMD/RED) congestion control system and its modified version fully in this paper. Firstly employing the semi-analytical method called as the harmonic balance method with alternating frequency/time (HB-AFT) domain technique, the approximate analytical expressions of periodic solutions of the generalized homogeneous-flow Additive Increase and Multiplicative Decrease/Random Early Detection (AIMD/RED) system with state-dependent round-trip delay are considered. We compare them with the results of numerical simulations by WinPP, they agree very well with each other. It demonstrates that the method employed here is versatile, valid, simple and effective. Then to the end of improving its modeling and performance, we modify the above model by taking an easy approximate dropping function. Furthermore, for the modified delayed homogeneous system, the approximate analytical expressions of periodic solutions are obtained accurately, and some complex dynamics are also presented. Four kinds of bi-stability, i.e., the coexistence of chaos and Period-3 solution, that of Period-1 and Period-2 solutions, that of Period-2 and Period-2 solutions, that of Period-4 and Period-2 solutions are disclosed. And a route to chaos, i.e., Period Doubling bifurcation to chaos, and the window of Period-3 to chaos are also discovered. The periodic oscillation can reduce the link utilization, induce the TCP stream synchronization services and further congestion. Chaotic oscillation may result in collapse. Therefore, all complex dynamical phenomena found in this paper are harmful and should be avoided. The obtained results can be very helpful for the researchers to have a better understanding of the mechanism of the network congestion control system, and they can select the parameters properly to improve the network stability and performance.

中文翻译:

状态相关延迟齐次加法和乘法减少/随机早期检测拥塞控制系统中的周期解、混沌和双稳态

摘要 加法加乘减(AIMD)拥塞控制和随机早期检测(RED)队列作为一个整体拥塞控制系统的结合在互联网的压倒性成功中起着关键作用。因此,重要的是在本文中全面研究状态相关的延迟齐次加法和乘法减少/随机早期检测 (AIMD/RED) 拥塞控制系统及其修改版本的周期性振荡和复杂动态。首先采用称为交替频/时域技术的谐波平衡法(HB-AFT)的半解析方法,考虑了具有状态相关往返延迟的广义齐次流加法加乘减法/随机早期检测(AIMD/RED)系统周期解的近似解析表达式。我们将它们与 WinPP 的数值模拟结果进行比较,它们彼此非常吻合。它表明这里采用的方法是通用的、有效的、简单的和有效的。然后为了改进其建模和性能,我们通过采用简单的近似丢弃函数来修改上述模型。此外,对于修正的时滞齐次系统,精确地得到了周期解的近似解析表达式,并给出了一些复杂的动力学。四种双稳态,即混沌与Period-3解并存,披露了第 1 期和第 2 期解决方案、第 2 期和第 2 期解决方案、第 4 期和第 2 期解决方案的解决方案。并且还发现了一条通向混沌的途径,即周期倍增分岔到混沌,以及周期3到混沌的窗口。周期性振荡会降低链路利用率,诱发 TCP 流同步服务和进一步拥塞。混沌振荡可能导致崩溃。因此,本文中发现的所有复杂动力学现象都是有害的,应该避免。得到的结果可以非常有助于研究人员更好地理解网络拥塞控制系统的机制,他们可以适当地选择参数以提高网络稳定性和性能。披露了第 4 期和第 2 期解决方案的解决方案。并且还发现了一条通向混沌的途径,即周期倍增分岔到混沌,以及周期3到混沌的窗口。周期性振荡会降低链路利用率,诱发 TCP 流同步服务和进一步拥塞。混沌振荡可能导致崩溃。因此,本文中发现的所有复杂动力学现象都是有害的,应该避免。获得的结果对于研究人员来说非常有帮助,以更好地了解网络拥塞控制系统的机制,并且它们可以正确地选择参数以提高网络稳定性和性能。披露了第 4 期和第 2 期解决方案的解决方案。并且还发现了一条通向混沌的途径,即周期倍增分岔到混沌,以及周期3到混沌的窗口。周期性振荡会降低链路利用率,诱发 TCP 流同步服务和进一步拥塞。混沌振荡可能导致崩溃。因此,本文中发现的所有复杂动力学现象都是有害的,应该避免。得到的结果可以非常有助于研究人员更好地理解网络拥塞控制系统的机制,他们可以适当地选择参数以提高网络稳定性和性能。周期性振荡会降低链路利用率,诱发 TCP 流同步服务和进一步拥塞。混沌振荡可能导致崩溃。因此,本文中发现的所有复杂动力学现象都是有害的,应该避免。得到的结果可以非常有助于研究人员更好地理解网络拥塞控制系统的机制,他们可以适当地选择参数以提高网络稳定性和性能。周期性振荡会降低链路利用率,诱发 TCP 流同步服务和进一步拥塞。混沌振荡可能导致崩溃。因此,本文中发现的所有复杂动力学现象都是有害的,应该避免。得到的结果可以非常有助于研究人员更好地理解网络拥塞控制系统的机制,他们可以适当地选择参数以提高网络稳定性和性能。
更新日期:2021-04-01
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