基于主动队列管理的拥塞控制机制的动态行为研究
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摘要
主动队列管理(AQM)算法的稳定性在拥塞控制中发挥着非常重要的作用,也就是说,通过链路的信息流的速率应该趋向于一种平衡,最好接近于链路的容量,而不是在剩余带宽和完全超载之间持续地振荡。但在较高带宽利用率和较小队列延迟情况下,还有一些亟待需要解决的问题,比如如何保持拥塞控制模型的稳定性以及如何推导出一些简单的判定稳定性的充分条件。另外,当TCP/AQM系统失稳时,系统将如何演化?这些问题已开始引起研究人员的关注,也正是本论文研究的动机。本文的主要贡献如下:
①提出了一个新的带多时滞的指数RED算法模型,在频域上获得了该算法模型稳定性的判据;
通过分析AIMD拥塞避免机制的本质特征并引入队列延迟和时变时滞,提出了一个与实际的网络运行特征相一致的新的基于指数RED算法的拥塞控制模型;通过使用自动反馈控制理论和广义Nyquist定理,在频域上推导并获得了一些判定系统稳定的充分条件,同时表明了系统的稳定性条件主要依赖于平衡往返时延和平衡丢弃概率。与原始的指数RED算法仿真对比表明,所提出的稳定性判据能有效地保持系统的局部稳定和好的性能(如高链路利用率和低队列延迟)。
②提出了带通信延迟的指数RED算法失稳时Hopf分岔存在的条件,进而给出了判定Hopf分岔的方向和分岔周期轨道的稳定性的公式;
通过分析线性化方程所对应的特征方程的特征值的分布范围,研究了带通信时滞的指数RED算法的Hopf分岔。利用时滞作为分岔参数,得出了通信时滞相关的线性稳定性判据,进而研究发现当时滞经过一个临界值时Hopf分岔就会发生,采用中心流形定理和规范形理论,确定了Hopf分岔的方向和分岔周期轨道的稳定性,最后仿真结果验证了理论分析的正确性。
③设计了一个动态时滞反馈控制器来控制二阶指数RED模型中的Hopf分岔,研究表明该控制器能够延迟Hopf分岔的发生;
将“washout滤波器”与控制器结构相结合,构建了一个新的动态时滞状态反馈控制器,使用该控制器可以增加通信延迟的临界值,保证系统在大通信延迟下具有稳定的数据发送速率。通过利用Hassard方法,得到了确定从平衡点分岔出的周期解的稳定性和方向的计算公式。仿真结果也表明所设计的控制器对于延迟Hopf分岔的发生是有效的。
④给出了一种新的基于LMI的带多个通信时滞的Primal-Dual算法的稳定性判据;
构造了一种适当的Lyapunov泛函,研究了带多个时滞的Primal-Dual算法的渐近稳定性问题。通过采用自由权值矩阵,克服了使用固定模型转换所带来的保守性,给出了基于LMI的时滞相关稳定性判据。对比仿真结果表明了所提出的稳定性判据具有更少的保守性,扩大了控制增益和可允许的通信时滞的稳定区间。
⑤研究了带不同区间时变通信时滞的二阶Primal-Dual算法的稳定性,给出了具有较少保守性的时滞相关稳定性判据;
采用一种新的方法来研究带多个区间时变时滞的Primal-Dual算法的局部稳定性。通过使用自由权值矩阵以及时滞区间分段方法,考虑时滞区间之间的相互关系,构造了一类新的Lyapunov-Krasoveskii泛函。研究表明所得出的时滞相关的稳定性判据与现有的结果相比具有较少的保守性。
⑥给出了带多个时滞的Primal-Dual算法中Hopf和余维2分岔存在的条件。
通过分析线性化方程的超越特征方程,研究了带多个时滞的Primal-Dual算法的平衡点的局部稳定性。通过使用广义Nyquist判据,得出了与时滞和系统参数相关的稳定性判据。进而通过选择某个时滞作为分岔参数,研究表明当时滞超过一个临界值时,极限环就会出现,Hopf分岔就会发生。进一步研究表明共振余维2分岔在该算法中也会出现。最后,仿真结果验证和说明了理论结果的正确性。
Stability of the AQM algorithm plays an important role in congestion control, i.e., the rate of information flow through its links should tend towards an equilibrium, preferably close to link capacity, rather than continually oscillating between the having bandwidth spare and the being completely overloaded. However, in the case of higher bandwidth utility and smaller queueing delay, there are some urgent problems to be solved, which are how to keep the stability of congestion control model, and how to derive some simple and sufficient stability conditions. In addition, how does TCP/AQM system evolve when the congestion control system loses its stability? This field also begins to draw much attention from researchers. These are also just the motivations of this thesis. The main contributions of this thesis are listed as follows:
①A novel exponential RED model with heterogeneous delays is introduced, and several stability criteria for the model in frequency domain are derived.
By analyzing the essential feature of the AIMD congestion avoidance mechanism and introducing queueing delay and time-varying delay, we develop a novel congestion control model in accordance with the characters of the real network operation. By use of automatic feedback control theory and the Generalized Nyquist theory, we derive and obtain the sufficient stability condition in frequency domain in the case of the round trip delay is dependent and independent of forward delay, respectively. It is shown that the stability condition is mainly dependent on the equilibrium round trip delay and the equilibrium drop probability. Through simulation and comparison with original exponential RED, the present stability criteria are effective to maintain both local stability and good performance, such as high utilization and low queue delay.
②Several existence conditions of Hopf bifurcation in a novel exponential RED model with communication delay are estiblished, and several explicit formulae for determining the direction of the Hopf bifurcation and stability of the periodic orbits are presented.
We investigate Hopf bifurcation in the exponential RED model with communication delay by analyzing the distributed ranges of eigenvalues of characteristic equation of the corresponding linearized equations. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay are also been derived, and furthermore, we find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, simulation results are verified and demonstrated the correctness of the theoretical results.
③A dynamic delayed state feedback controller for Hopf bifurcation in the exponential RED model is developed. Furthermore, it is shown that the controller can delay the onset of Hopf bifurcation.
The dynamic delayed state feedback controller is achieved by incorporating filters called“washout filters”into the controller architecture. With this controller, one can increase the critical value of the communication delay, and thus guarantee a stationary data sending rate for larger delay. Furthermore, the explicit formulae determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al’s approaches. Finally, numerical simulation results are presented to show that the dynamic delayed feedback controller is efficient in controlling Hopf bifurcation.
④Several LMI-based novel stability criteria of the primal-dual algorithm with heterogenous communication delays are proposed.
An appropriate type of Lyapunov functional is used to investigate the problem of asymptotic stability in the primal-dual algorithm with heterogeneous communication delays. Some LMI-based novel delay-dependent stability criteria are given by use of the free weighting matrices, which can overcome the conservativeness of method involving a fixed model transformation. Simulation results show that the stability criteria proposed in this thesis are less conservative in the sense that the larger range of control gains and communication delays can be accommodated.
⑤A second-order primal-dual algorithm with heterogenous interval time-varying communication delays is investigated, and several delay-dependent stability criteria with less conservativeness are proposed.
A new approach is proposed to investigate the asymptotic stability of the second-order primal-dual algorithm with heterogeneous interval time-varying delays, where a new classes of Lyapunov-Krasoveskii functionals are constructed by using the free weighting matrix technique and the delay interval division method as well as considering the interactions between the delay ranges. It is shown that the obtained delay-dependent stability criteria are less conservative than the existing results. Finally, numerical examples are given to demonstrate the effectiveness and merits of the proposed method.
⑥Several existence conditions of Hopf bifurcation and resonant codimension-two bifurcation in primal-dual algorithm with heterogeneous delays are obtained.
The local stability of the equilibrium solution of primal-dual algorithm with heterogeneous delays is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as a bifurcation parameter, it is shown that when the delays exceed a critical value a limit cycle emerges via a Hopf bifurcation. Resonant codimension-two bifurcation is also found in this model. Simulation results verify and demonstrate the correctness of the theoretical results.
①提出了一个新的带多时滞的指数RED算法模型,在频域上获得了该算法模型稳定性的判据;
通过分析AIMD拥塞避免机制的本质特征并引入队列延迟和时变时滞,提出了一个与实际的网络运行特征相一致的新的基于指数RED算法的拥塞控制模型;通过使用自动反馈控制理论和广义Nyquist定理,在频域上推导并获得了一些判定系统稳定的充分条件,同时表明了系统的稳定性条件主要依赖于平衡往返时延和平衡丢弃概率。与原始的指数RED算法仿真对比表明,所提出的稳定性判据能有效地保持系统的局部稳定和好的性能(如高链路利用率和低队列延迟)。
②提出了带通信延迟的指数RED算法失稳时Hopf分岔存在的条件,进而给出了判定Hopf分岔的方向和分岔周期轨道的稳定性的公式;
通过分析线性化方程所对应的特征方程的特征值的分布范围,研究了带通信时滞的指数RED算法的Hopf分岔。利用时滞作为分岔参数,得出了通信时滞相关的线性稳定性判据,进而研究发现当时滞经过一个临界值时Hopf分岔就会发生,采用中心流形定理和规范形理论,确定了Hopf分岔的方向和分岔周期轨道的稳定性,最后仿真结果验证了理论分析的正确性。
③设计了一个动态时滞反馈控制器来控制二阶指数RED模型中的Hopf分岔,研究表明该控制器能够延迟Hopf分岔的发生;
将“washout滤波器”与控制器结构相结合,构建了一个新的动态时滞状态反馈控制器,使用该控制器可以增加通信延迟的临界值,保证系统在大通信延迟下具有稳定的数据发送速率。通过利用Hassard方法,得到了确定从平衡点分岔出的周期解的稳定性和方向的计算公式。仿真结果也表明所设计的控制器对于延迟Hopf分岔的发生是有效的。
④给出了一种新的基于LMI的带多个通信时滞的Primal-Dual算法的稳定性判据;
构造了一种适当的Lyapunov泛函,研究了带多个时滞的Primal-Dual算法的渐近稳定性问题。通过采用自由权值矩阵,克服了使用固定模型转换所带来的保守性,给出了基于LMI的时滞相关稳定性判据。对比仿真结果表明了所提出的稳定性判据具有更少的保守性,扩大了控制增益和可允许的通信时滞的稳定区间。
⑤研究了带不同区间时变通信时滞的二阶Primal-Dual算法的稳定性,给出了具有较少保守性的时滞相关稳定性判据;
采用一种新的方法来研究带多个区间时变时滞的Primal-Dual算法的局部稳定性。通过使用自由权值矩阵以及时滞区间分段方法,考虑时滞区间之间的相互关系,构造了一类新的Lyapunov-Krasoveskii泛函。研究表明所得出的时滞相关的稳定性判据与现有的结果相比具有较少的保守性。
⑥给出了带多个时滞的Primal-Dual算法中Hopf和余维2分岔存在的条件。
通过分析线性化方程的超越特征方程,研究了带多个时滞的Primal-Dual算法的平衡点的局部稳定性。通过使用广义Nyquist判据,得出了与时滞和系统参数相关的稳定性判据。进而通过选择某个时滞作为分岔参数,研究表明当时滞超过一个临界值时,极限环就会出现,Hopf分岔就会发生。进一步研究表明共振余维2分岔在该算法中也会出现。最后,仿真结果验证和说明了理论结果的正确性。
Stability of the AQM algorithm plays an important role in congestion control, i.e., the rate of information flow through its links should tend towards an equilibrium, preferably close to link capacity, rather than continually oscillating between the having bandwidth spare and the being completely overloaded. However, in the case of higher bandwidth utility and smaller queueing delay, there are some urgent problems to be solved, which are how to keep the stability of congestion control model, and how to derive some simple and sufficient stability conditions. In addition, how does TCP/AQM system evolve when the congestion control system loses its stability? This field also begins to draw much attention from researchers. These are also just the motivations of this thesis. The main contributions of this thesis are listed as follows:
①A novel exponential RED model with heterogeneous delays is introduced, and several stability criteria for the model in frequency domain are derived.
By analyzing the essential feature of the AIMD congestion avoidance mechanism and introducing queueing delay and time-varying delay, we develop a novel congestion control model in accordance with the characters of the real network operation. By use of automatic feedback control theory and the Generalized Nyquist theory, we derive and obtain the sufficient stability condition in frequency domain in the case of the round trip delay is dependent and independent of forward delay, respectively. It is shown that the stability condition is mainly dependent on the equilibrium round trip delay and the equilibrium drop probability. Through simulation and comparison with original exponential RED, the present stability criteria are effective to maintain both local stability and good performance, such as high utilization and low queue delay.
②Several existence conditions of Hopf bifurcation in a novel exponential RED model with communication delay are estiblished, and several explicit formulae for determining the direction of the Hopf bifurcation and stability of the periodic orbits are presented.
We investigate Hopf bifurcation in the exponential RED model with communication delay by analyzing the distributed ranges of eigenvalues of characteristic equation of the corresponding linearized equations. Using communication delay as the bifurcation parameter, linear stability criteria dependent on communication delay are also been derived, and furthermore, we find that the Hopf bifurcation occurs when the communication delay passes a sequence of critical values. The stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. Finally, simulation results are verified and demonstrated the correctness of the theoretical results.
③A dynamic delayed state feedback controller for Hopf bifurcation in the exponential RED model is developed. Furthermore, it is shown that the controller can delay the onset of Hopf bifurcation.
The dynamic delayed state feedback controller is achieved by incorporating filters called“washout filters”into the controller architecture. With this controller, one can increase the critical value of the communication delay, and thus guarantee a stationary data sending rate for larger delay. Furthermore, the explicit formulae determining the stability and the direction of periodic solutions bifurcating from the equilibrium are obtained by applying Hassard et al’s approaches. Finally, numerical simulation results are presented to show that the dynamic delayed feedback controller is efficient in controlling Hopf bifurcation.
④Several LMI-based novel stability criteria of the primal-dual algorithm with heterogenous communication delays are proposed.
An appropriate type of Lyapunov functional is used to investigate the problem of asymptotic stability in the primal-dual algorithm with heterogeneous communication delays. Some LMI-based novel delay-dependent stability criteria are given by use of the free weighting matrices, which can overcome the conservativeness of method involving a fixed model transformation. Simulation results show that the stability criteria proposed in this thesis are less conservative in the sense that the larger range of control gains and communication delays can be accommodated.
⑤A second-order primal-dual algorithm with heterogenous interval time-varying communication delays is investigated, and several delay-dependent stability criteria with less conservativeness are proposed.
A new approach is proposed to investigate the asymptotic stability of the second-order primal-dual algorithm with heterogeneous interval time-varying delays, where a new classes of Lyapunov-Krasoveskii functionals are constructed by using the free weighting matrix technique and the delay interval division method as well as considering the interactions between the delay ranges. It is shown that the obtained delay-dependent stability criteria are less conservative than the existing results. Finally, numerical examples are given to demonstrate the effectiveness and merits of the proposed method.
⑥Several existence conditions of Hopf bifurcation and resonant codimension-two bifurcation in primal-dual algorithm with heterogeneous delays are obtained.
The local stability of the equilibrium solution of primal-dual algorithm with heterogeneous delays is investigated by analyzing the corresponding transcendental characteristic equation of its linearized equation. Some general stability criteria involving the delays and the system parameters are derived by generalized Nyquist criteria. Furthermore, by choosing one of the delays as a bifurcation parameter, it is shown that when the delays exceed a critical value a limit cycle emerges via a Hopf bifurcation. Resonant codimension-two bifurcation is also found in this model. Simulation results verify and demonstrate the correctness of the theoretical results.
引文
[1] B. Braden, et al.. Recommendations on queue management and congestion avoidance in the Internet[J]. IETF RFC2309, 1998.
[2] S. Floyd, V. Jacobson. Random early detection gateways for congestion avoidance[J], IEEE/ACM Transactions on Networking, 1993, 1:397–413.
[3] T.J. Ott, T.V. Lakshman and L. Wong. SRED: Stabilized RED[J]. Proc. of INFOCOM’99, March 1999:1346–1355.
[4] W. Feng, K.G. Shin, D.D. Kandlur and D. Saha. The BLUE active queue management algorithms[J]. IEEE/ACM Transactions on Networking, 2002, 10(4):513–528.
[5] S. Floyd, R. Gummadi, S. Shenker. Adaptive RED: an algorithm for increasing the robustness of RED’s active queue management[EB/01], http://www.icir.org/floyd/papers/adaptiveRed.pdf, 2001.
[6] S. Athuraliya, D.E. Lapsley, S.H. Low.Random early marking for Internet congestion control[J] . Proceedings of the IEEE GLOBECOM, 1999, 3:1747–1752.
[7] S. Kunniyur, R. Srikant. Analysis and design of an adaptive virtual queue algorithm for active queue management[J]. Proceedings of the ACM SIGCOMM, San Diego, CA, August 2001:123–134.
[8] C.V. Hollot, V. Misra, D. Towsley and W. Gong. On designing improved controllers for AQM routers supporting TCP flows[J]. Proc. of INFOCOM’2001, April 2001:1726–1734.
[9] J. Aweya, M. Ouellette and D.Y.Montuno. A control theoretic approach to active queue management[J]. Computer Networks, 2001, 36(2/3) :203–235.
[10] M. Christiansen, K. Jaffey, D. Ott and D. Smith. Tuning RED for Web traffic[J], IEEE/ACMTransactions on Networking, 2001, 9(3):249–264.
[11] S. Ryu, C. Rump and C. Qiao. Advances in Internet congestion control[J]. IEEE Communications Surveys and Tutorials, 2003, 5(1):28–39.
[12] S. Ryu and C. Rump. Advances in Active Queue Management (AQM) Based TCP Congestion Control[J]. Telecommunication Systems, 2004, 25(4):317–351.
[13] S. Ryu and C. Rump. Design of a pro-active queue management[J]. The 11th IEEE International Conference on Networks, 2003:367- 372.
[14] J. Sun, G. Chen, K.T. Ko, S. Chan, M. Zukerman. PD-Controller: a new active queue management scheme[J]. Proc. IEEE Globecom., 2003, 6:3103–3107.
[15] B. Wydrowski, M. Zukerman. GREEN: an active queue management algorithm for a selfmanaged internet[J]. Proc. ICC 4, 2002:2368–2372.
[16] E.C. Park, H. Lim, K.J. Park, C.H. Choi. Analysis and design of the virtual rate control algorithm for stabilizing queues in TCP networks[J]. Computer Networks, 2004, 44:17–41.
[17] C. Long, B. Zhao, X. Guan, J. Yang. The Yellow active queue management algorithm[J]. Computer Networks, 2005, 47:525–550.
[18] F. P. Kelly. Charging and rate control for elastic traffic[J]. Eur. Trans. Telecommunications, Jan. 1997, 8:33–37.
[19] F. Kelly, A. Maulloo, and D. Tan. Rate control in communication networks: Shadow prices, proportional fairness and stability[J]. J. Oper. Res. Soc., 1998, 49:237–252.
[20] J. Mo and J. Walrand. Fair end-to-end window-based congestion control[J]. IEEE/ACM Transactions on Networking, Oct. 2000, 8(5):556–567.
[21] R. J. La and V. Anantharam. Charge-sensitive TCP and rate control in the Internet[J]. Proc. IEEE INFOCOM, 2000:1166–1175.
[22] S. H. Low and D. E. Lapsley. Optimization flow control-I: Basic algorithm and convergence[J]. IEEE/ACM Transactions on Networking, Dec. 1999, 7(6):861–874.
[23] S. Kunniyur and R. Srikant. A time-scale decomposition approach to adaptive explicit congestion notification (ECN) marking[J]. IEEE Transactions on Automatic Control, Jun. 2002, 47(6):882–894.
[24] J.Wen and M. Arcak. A unifying passivity framework for network flow control[J]. Proc. IEEE INFOCOM, San Francisco, CA, Apr. 2003:1156–1166.
[25] Fernando Paganini. A global stability result in network flow control[J]. Systems & Control Letters, 5 July 2002, 46(3):165-172.
[26] Q. Yin and S. H. Low. Convergence of REM flow control at a single link[J]. IEEE Communication Letters, Mar. 2001, 5:119–121.
[27] G. Vinnicombe. On the stability of networks operating TCP-like congestion control[J]. Proc. 15th IFAC World Congr. Automatic Control, Barcelona, Spain, Jul. 2002.
[28] L. Massoulie. Stability of distributed congestion control with heterogeneous feedback delays[J]. IEEE Transactions on Automatic Control, Jun. 2002, 47(6):895–902.
[29] R. Johari and D. Tan. End-to-end congestion control for the Internet: Delays and stability[J]. IEEE/ACM Transactions on Networking, Dec. 2001, 9(6):818–832.
[30] A. Elwalid. Analysis of adaptive rate-based congestion control for highspeed wide-area networks[J]. Proc. IEEE Int. Conf. Communications (ICC), Seattle, WA, Jun. 1995, 3:1948–1953.
[31] S. Liu, T. Basar and R. Srikant. Controlling the Internet: A survey and some new results[J].Proc. 42nd IEEE Conf. Decision and Control, Maui, HI, Dec. 2003:3048–3057.
[32] Y.-P. Tian and H.-Y. Yang. Stability of the Internet congestion control with diverse delays[J]. Automatica, 2004, 40:1533–1541.
[33] Y.-P. Tian. Stability analysis and design of the second-order congestion control for networks with heterogeneous delays[J]. IEEE/ACM Transactions on Networking, Oct. 2005, 13(5):1082– 1093.
[34] Y.-P. Tian and G. Chen. Stability of the primal–dual algorithm for congestion control[J]. International Journal of Control , 2006, 79(6):662-676.
[35] Yu-Ping Tian. A general stability criterion for congestion control with diverse communication delays[J]. Automatica, 2005, 41:1255– 1262.
[36] Fengyuan Ren, Chuang Lin, Bo Wei. A robust active queue management algorithm in large delay networks[J]. Computer Communications, 2005, 28:485–493.
[37] Xiao Fan Wang, Guanrong Chen, King-Tim Ko. A stability theorem for Internet congestion control[J], Systems & Control Letters, 2002, 45:81–85.
[38] D. Bauso, L. Giarre and G. Neglia. AQM stability in multiple bottleneck networks[J]. IEEE International Conference on Communications,June 2004, 4(24):2267– 2271.
[39] M.L. Sichitiu, P.H. Bauer. Asymptotic stability of congestion control systems with multiple sources[J]. IEEE Transactions on Automatic Control, Feb. 2006, 51(2):292– 298.
[40] Cheng-Nian Long, Jing Wu, Xin-Ping Guan. Local stability of REM algorithm with time-varying delays[J]. IEEE Communications Letters, March 2003, 7(3):142– 144.
[41] Huijun Gao, James Lam, Changhong Wang and Xinping Guan. Further results on local stability of REM algorithm with time-varying delays[J]. IEEE Communications Letters, May 2005, 9(5):402– 404.
[42] Steven H. Low, Fernando Paganini, Jiantao Wang and John C. Doyle. Linear stability of TCP/RED and a scalable control[J]. Computer Networks, December 2003, 43(5):633-647.
[43] Zhu Li, N. Ansari. Local stability of a new adaptive queue management (AQM) scheme[J]. IEEE Communications Letters, June 2004, 8(6):406– 408
[44] Jinsheng Sun and Moshe Zukerman. RaQ: A robust active queue management scheme based on rate and queue length[J]. Computer Communications, June 2007, 30(8):1731-1741.
[45] I.C. Lestas and G. Vinnicombe. Scalable robust stability for nonsymmetric heterogeneous networks[J]. Automatica, April 2007, 43(4):714-723.
[46] Hamsa Balakrishnan, Nandita Dukkipati, Nick McKeown and Claire J.Tomlin. Stability Analysis of Explicit Congestion Control Protocols[J]. IEEE Communications Letters, October 2007, 11(10):823-825.
[47] Heng-Qing Ye. Stability of data networks under an optimization-based bandwidth allocation[J]. IEEE Transactions on Automatic Control, July 2003, 48(7):1238– 1242.
[48] Thomas Voice. Stability of Multi-Path Dual Congestion Control Algorithms[J]. IEEE/ACM Transactions on Networking, December2007, 15(6):1321-1239.
[49] Liansheng Tan, Wei Zhang, Gang Peng, Guanrong Chen. Stability of TCP/RED systems in AQM routers[J]. IEEE Transactions on Automatic Control, Aug. 2006, 51(8):1393– 1398.
[50]任丰原,林闯,任勇,山秀明.大时滞网络中的拥塞控制算法[J].软件学报,2003年03期.
[51]苏晓丽,郑明春,李锦涛,孟强.一种基于速率的组播拥塞控制算法及其性能分析[J].电子学报, 2004年02期.
[52]谭连生,熊乃学,杨燕.一种基于PGM的单速率组播拥塞控制方案[J].软件学报, 2004年10期.
[53]尹凤杰,井元伟,岳承君,王涛,潘伟.不确定输入延时网络系统的鲁棒拥塞控制[J].控制与决策, 2007年02期.
[54]许立波,吴国新,基于时序推断的拥塞控制策略的性能分析[J].计算机学报, 2007年09期.
[55] Xinbing Wang and Do Young Eun. Local and global stability of TCP-newReno/RED with many flows[J].Computer Communications, 8 March 2007, 30(5):1091-1105.
[56] F. P. Kelly. Models for a self-managed Internet[J]. Philosophic. Trans. Roy. Soc., 2000, A358: 2335–2348.
[57] V. Misra, W. Gong, and D. Towsley. A fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED[J]. Proc. ACM SIGCOMM, Stockholm, Sweden, Sep. 2000:151–160.
[58] S. Kunniyur and R. Srikant. Stable, scalable, fair congestion control and AQM schemes that achieve high utilization in the Internet[J]. IEEE Transactions on Automatic Control, Nov. 2003, 48(11):2024–2029.
[59] S. Kunniyur and R. Srikant. End-to-end congestion control: Utility functions, random losses and ECN marks[J]. IEEE/ACM Transactions on Networking, Oct. 2003, 11(10):689–702.
[60] F. Paganini, J. Doyle and S. Low. Scalable laws for stable network congestion control[J]. Proc. IEEE Conf. Decision Control, Dec. 2001:185–190.
[61] G. Vinnicombe. On the Stability of End-to-End Congestion Control for the Internet[R]. Univ. Cambridge Tech. Rep. CUED/F-INFENG/TR.398, 2001 [Online]. Available: http://www.eng.cam.ac.uk/~gv.
[62] S. Deb and R. Srikant. Global stability of congestion controllers for the Internet[J]. IEEETransations on Automatic Control, Jun. 2003, 48(6):1055–1060.
[63] C. V. Hollot and Y. Chait. Nonlinear stability analysis for a class of TCP/AQM schemes[J]. Proc. IEEE Conf. Decision Control, Dec. 2001:2309–2314.
[64] H. Yaiche, R. R. Mazumdar, and C. Rosenberg. A game-theoretic framework for bandwidth allocation and pricing in broadband networks[J]. IEEE/ACM Transactions on Networking, Oct. 2000, 8(5):667–678.
[65] Lei Ying, G. E. Dullerud, R. Srikant. Global stability of Internet congestion controllers with heterogeneous delays[J]. IEEE/ACM Transactions on Networking, June 2006, 14(3):579– 590.
[66] T. Alpcan and T. Basar. A Globally Stable Adaptive Congestion Control Scheme for Internet-Style Networks With Delay[J]. IEEE/ACM Transactions on Networking, December 2005, 13(6):1261-1274.
[67] Z. Wang, F. Paganini. Boundedness and Global Stability of a Nonlinear Congestion Control With Delays[J]. IEEE Transactions on Automatic Control, Sept. 2006, 51(9):1514– 1519.
[68] Yueping Zhang, Seong-Ryong Kang and Dmitri Loguinov. Delay-Independent Stability and Performance of Distributed Congestion Control[J].IEEE/ACM Transactions on Networking, OCTOBER 2007, 15(5):838-851.
[69] Jinsheng Sun, Sammy Chan, King-Tim Ko, Guanrong Chen and Moshe Zukerman. Instability effects of two-way traffic in a TCP/AQM system[J]. Computer Communications, 2007, 30:2172–2179.
[70] M. Peet, S. Lall. On global stability of Internet congestion control[J]. 43rd IEEE Conference on Decision and Control, Dec. 2004, 1(14-17):1035 - 1041.
[71] Shao Liu, T. Basar, R. Srikant. Exponential-RED: a stabilizing AQM scheme for low- and high-speed TCP protocols[J]. IEEE/ACM Transactions on Networking, Oct. 2005, 13(5):1068– 1081.
[72] S.Floyd. Recommendation on using the‘gentle_’variant of RED[EB/02], http://www.icir.org/floyd/red/gent-le. html, 2000.
[73] W. Feng, D.D. Kandlur, D. Saha, K.G. Shin. A self-configuring RED gateway[J]. Proc. IEEE INFOCOMM,1999, 3:1320–1328.
[74] C. Diot, G. Iannaccone, M. May. Aggregate traffic performance with active queue management and drop from tail[J]. ACM SIGCOMM Comput Commun Rev, 2001, 31:4–13.
[75] Tomoya Eguchi, Hiroyuki Ohsaki, et al.. On Control Parameters Tuning for Active Queue Management Mechanisms using Multivariate Analysis[J]., 2003. Proceedings. Symposium on Applications and the Internet, 27-31 Jan. 2003:120– 127.
[76] Chengnian Long, Bin Zhao, Xinping Guan, Bo Yang. Stability analysis and filter design for LRED algorithm[J]. American Control Conference, 2005. Proceedings of the 2005, June 2005, 38-10:1847 - 1852.
[77] Hui Zhang, Mingjian Liu, et al.. Modeling TCP/RED :a Dynamical Approach[EB/03], http://www.ensc.sf-u.ca/~ljilja/papers/springer_indexed.pdf.
[78] E. Hashem. Analysis of random drop for gateway congestion control[R]. Report LCS TR-456, Laboratory for Computer Science, MIT, 1989:103.
[79] C. Wang, B. Li, Y. Thomas Hou, K. Sohraby, and Y. Lin. LRED: A robust active queue management scheme based on packet loss ratio[J]. Proceedings of IEEE INFOCOM, Hongkong, 2004.
[80] S.S. Kunniyur, R. Srikant. An adaptive virtual queue (AVQ) algorithm for active queue management[J]. IEEE/ACM Transactions on Networking, April 2004, 12(2):286– 299.
[81] C.Hollot, V. Mista, et al.. Analysis and design of controllers for AQM routers supporting TCP flows[J]. IEEE Transactions on Automatic Control, Jun.2002, 47:945-959.
[82] S. Athuraliya, S. Low, V. Li and Q. Yin. REM: Active queue management[J]. IEEE Network Magazine, May 2001, 15:48-53.
[83] Y. Gao and J.C. Hou. A state feedback control approach to stabilizing queues for ECN-enabled TCP flows[J]. In proceedings of IEEE INFOCOM, San Francisco, CA, 2003, 3:2301-2311.
[84] C.A. Deseor and Y. T. Yang. On the generalized Nyquist Stability Criterion[J]. IEEE Transactions on Automatic Control, 1980, 25:187-196.
[85] UCN/LBL/VINT. Network simulator-NS2[EB/04]. http://-mash.cs.berkelay.edu/ns.
[86] Kai Jiang, Xiaofan Wang and Yugeng Xi. Nonlinear analysis of RED– a comparative study[J]. Chaos, Solitons & Fractals, SEP 2004, 21 (5):1153-1162.
[87] K Cooke and Z. Grossman. Discrete delay, distributed delay and stability switches[J]. Journal of Mathematic Analysis and Application, 1982, 86:592–627.
[88] J. Hale. Theory of Functional Differential Equations[M], Springer, Berlin, 1977.
[89] B. Hassard, N. Kazarinoff, Y. Wan. Theory and Applications of Hopf Bifurcation[M], Cambridge University Press, Cambridge, 1981.
[90] C. Li, G. Chen, X. Liao, J. Yu. Hopf bifurcation in an Internet congestion control model[J]. Chaos, Solitons Fractals, 2004, 19 (4):853–862.
[91] L. Massoulie. Stability of distributed congestion control with heterogeneous feedback delays[J]. IEEE Transactions on Automatic Control, 2002, 47(6):895 - 902.
[92] F. Paganini, Z. Wang, J. Doyle, S. Low. A new TCP/AQM for stable operation in fastnetworks[J]. Proceedings of the IEEE INFOCOM, San Francisco, CA, April 2003:96–105.
[93] G. Raina. Local Bifurcation Analysis of Some Dual Congestion Control Algorithms[J]. IEEE Transactions on Automatic Control, 2005, 50(8):1135– 1146.
[94] P. Ranjan, R.J. La and E.H. Abed. Global stability conditions for rate control with arbitrary communication delays[J], IEEE/ACM Transactions on Networking, 2006, 14(1):94-107.
[95] F. Ren, C. Lin, B. Wei. A nonlinear control theoretic analysis to TCP–RED system[J]. Compututer Networks, 2005, 49 (4):580–592.
[96] H.-Y. Yang, Y.-P. Tian. Hopf bifurcation in REM algorithm with communication delay[J]. Chaos, Solitons Fractals, 2005, 25 (5):1093–1105.
[97] J. Nagle. Congestion control in IP/TCP internet works[R]. Request Comments RFC 896, Ian., 1984.
[98] S. Guo, X. Liao, C. Li and D. Yang. Stability analysis of a novel exponential-RED model with heterogeneous delays[J]. Computer Communications, 2007, 30(5):1058-1074.
[99] M. L. Sichitiu and P. H. Bauer. Asymptotic Stability of Congestion Systems With Multiple Sources[J]. IEEE Transactions on Automatic Control, 2006, 51(2):292-298.
[100] S. H. Low, F. Paganini, J.Wang, S. Adlakha and J. C. Doyle. Dynamics of TCP/RED and a scalable control[J]. Proc. IEEE Infocom, 2002.
[101] G. Raina and O. Heckmann. TCP: Local stability and Hopf bifurcation[J]. Performance Evaluation, 2007, 64(3):266-275.
[102] G. R. Chen, J. L. Moiola and H. O. Wang. Bifurcation control: Theories, methods and applications[J]. International Journal of Bifurcation and Chaos, 2005, 10(3):11-548 .
[103] J. Wang, L. Chen and X. Fei. Analysis and control of the bifurcation of Hodgkin–Huxley model[J]. Chaos, Solitons and Fractals, 2007, 31:247-256.
[104] E. H. Abed and J. H. Fu. Local feedback stabilization and bifurcation control: I. Hopf bifurcation[J]. System Control & Letters, 1986, 7:11-17.
[105] H. O. Wang and E. H. Abed. Bifurcation control of a chaotic system[J]. Automatica, 1995, 31(9):1213–1226.
[106] H. O. Wang and E. H. Abed. Robust control of period doubling bifurcations and implications for control of chaos[J]. In Proc. IEEE Int. Symp. Circ. Syst.,Orlando, USA , 1994:3287–3292.
[107] H. O. Wang, D. S. Chen and L. G. Bushnell. Dynamic feedback control of bifurcations. Latin American Applied Research, 2001, 31:219–225.
[108] G. Gu, X. Chen, A. Sparks and S. Banda. Bifurcation stabilization with local output feedback[J]. SIAM J. Contr. Optim., 1999, 37:934–956.
[109] P. Yu and G. R. Chen. Hopf bifurcation control using nonlinear feedback with polynomialfunctions[J]. International Journal of Bifurcation and Chaos, 2004, 14:1683–1704.
[110] N. Baba, A. Amann, E. Sch?ll and W. Just. Giant Improvement of Time-Delayed Feedback Control by Spatio-Temporal Filtering[J]. Physical Review Letters, 2002, 89, 074101.
[111] E. Sch?ll and H. G. Schuster. Handbook of Chaos Control[M]. 2nd edition, Wiley-VCH, Weinheim, 2007.
[112] K. Pyragas. Continuous control of chaos by self-controlling feedback[J]. Physics Letter A, 1992, 170(6):421-428.
[113] K. Pyragas, V. Pyragas and H. Benner. Delayed feedback control of dynamical system at a subcritical Hopf bifurcation[J]. Physical Review E, 2004, 70, 056222.
[114] A. G. Balanov, N. B. Janson, and E. Sch?ll. Delayed feedback control of chaos: Bifurcation analysis[J]. Physical Review E, 2005, 71, 016222.
[115] A. Ahlborn and U. Parlitz. Stabilizing Unstable Steady States Using Multiple Delay Feedback Control[J]. Physical Review Letters, 2004, 93, 264101.
[116] M. G. Rosenblum and A. S. Pikovsky. Controlling Synchronization in an Ensemble of Globally Coupled Oscillators[J]. Physical Review Letters, 2004, 92, 114102.
[117] P. H?vel and E. Sch?ll. Control of unstable steady states by time-delayed feedback methods[J]. Physical Review E, 2005, 72, 046203.
[118] J. E. S. Socolar, D. W. Sukow and D. J. Gauthier. Stabilizing unstable periodic orbits in fast dynamical systems[J], Physical Review E, 1994, 50, 3245.
[119] K. Pyragas, V. Pyragas, I. Z. Kiss and J. L. Hudson. Stabilizing and Tracking Unknown Steady States of Dynamical Systems[J]. Physical Review Letters, 2002, 89, 244103.
[120] K. Pyragas, V. Pyragas, I. Z. Kiss and J. L. Hudson. Adaptive control of unknown unstable steady states of dynamical systems[J]. Physical Review E, 2004, 70, 026215.
[121] S. Schikora. All-optical noninvasive control of unstable steady states in a semiconductor laser[J]. Physical Review Letters, 2006, 97, 213902.
[122] V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum and M. Radziunas. Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation[J]. Physical Review E, 2006, 73, 046205.
[123] M. Feito, F. J. Cao. Time-delayed feedback control of a flashing ratchet[J]. Physical Review E, 2007, 76, 061113.
[124] H. Nakajima, On analytical properties of delayed feedback control of chaos[J]. Physics Letter A, 1997, 232:207 -210.
[125] B. Fiedler, V. Flunkert, M. Georgi, P. Hovel and E. Scholl. Refuting the Odd-Number Limitation of Time-Delayed Feedback Control[J]. Physical Review Letters, 2007, 98, 114101.
[126] W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hovel and E. Scholl. Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control[J]. Physical Review E, 2007, 76, 026210.
[127] Claire M. Postlethwaite and Mary Silber. Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control[J]. Physical Review E, 2007, 76, 056214.
[128] Thomas Dahms, Philipp H?vel and Eckehard Sch?ll. Control of unstable steady states by extended time-delayed feedback[J]. Physical Review E, 2007, 76, 056201.
[129] Serhiy Yanchuk, Matthias Wolfrum, Philipp H?vel and Eckehard Sch?ll. Control of unstable steady states by long delay feedback[J]. Physical Review E, 2006, 74, 026201.
[130] Z. Chen and P. Yu. Hopf bifurcation control for an Internet congestion model[J]. International Journal of Bifurcation and Chaos, 2005, 5(8):2643-2651.
[131] M. Xiao and J. Cao. Delayed feedback-based bifurcation control in an Internet congestion model[J]. Journal of Mathematical Analysis and Applications, 2007, 332(2):1010-1027 .
[132] H. O. Wang and Dong S.Chen. Bifurcation Control: Theory and Applications[A]. Springer, 2004, 293:229-248 .
[133] S. Guo, X. Liao and C. Li. Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay[J]. Nonlinear Analysis: Real World Applications, In Press, Corrected Proof, Available online 13 March 2007.
[134] D. S. Chen, H. O. Wang and G. Chen. Anti-Control of Hopf bifurcation[J]. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 2001, 48(6):661- 672.
[135] D. Fan and J. Wei. Hopf bifurcation analysis in a tri-neuron network with time delay[J]. Nonlinear Analysis: Real World Applications, In Press, Corrected Proof, Available online 2 October 2006.
[136] S. Guo, X. Liao, Q. Liu and C. Li. Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay[J]. Nonlinear Analysis: Real World Applications, In Press, Corrected Proof, Available online 3 June 2007.
[137] J. T. Wen and M. Arcak. A unifying passivity framework for network flow control[J]. IEEE Transactions on Automatic Control, 2004, 49(2):162-174.
[138] V. Jacobson. Congestion avoidance and control[J]. Proc. Symp. Communications Architectures and Protocols (SIGCOMM), Stanford, CA, Aug. 1988:314-329.
[139] R. Srikant. The Mathematics of Internet Congestion Control[M]. Cambridge, MA: Birkhauser, 2004.
[140] S. Det and R. Srikant. Global stability of congestion controller for the Internet[J]. IEEE Transactions on Automatic Control, 2003, 48(6):1055-1060.
[141] K. Kar, S. Sarkar and L. Tassiulas. A simple rate control algorithm for maximizing total user utility[J]. Proc. IEEE INFOCOM, Apr. 2001:133-141.
[142] D. Katabi, M. Handley and C. Rohrs. Congestion control for high bandwidth delay product networks[J]. Proc. ACM SIGCOMM, Aug. 2002:89-102.
[143] S. Kunniyur and R. Srikant. End-to-End congestion control schemes: Utility functions, random losses and ECN marks[J]. IEEE/ACM Transactions on Networking, 2003, 11(5):689-702.
[144] K. Gu, J. Chen, V. L. Kharitonov. Stability of time-delay system[M]. Springer-Verlag, New York, 2003.
[145] Y. He, M. Wu and J.-H. She. Delay-dependent stability criteria for linear system with multiple delays[J]. IEE proc.-Control Theory Appl., 2006, 153(4):447-452.
[146] Y. He, Q. G. Wang, L. H. Xie, et al.. Further improvement of free-weighting matrices technique for systems with time-varying delay[J]. IEEE Transations on. Automatic Control, 2007, 52(2):293-299.
[147] Y. He, Q. G. Wang, C. Lin, et al.. Delay-range-dependent stability for systems with time-varying delay[J], Automatica, 2007, 43(2):371-376.
[148] J. K. Hale and S. M. V. Lunel, Introduction to the Theory of Functional Differential Equations[M]. Springer, New York, 1991.
[149] S. Boyd, L. EI Ghaoui, E. Feron and V. Balakrishnam. Linear Matrix Inequalities in System and Control Theory[M]. SIAM, Philadelphia, PA, 1994.
[150] M. Wu, Y. He, J. H. She and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems[J]. Automatica, 2004, 40:1435-1439.
[151] M. Wu, Y. He and J. H. She. Delay-dependent Stabilization for Systems with Multiple Unknown Time-varying Delays[J]. International Journal of Control, Automation, and Systems, 2006, 4:682-688.
[152] L. Chen, X. Wang and Z. Han. Controlling chaos in Internet congestion control model[J].Chaos, Solitons & Fractals, 2004, 21:81-91.
[153] J. Hale and S. V. Lenel. Introduction to Functional Differential Equations[M]. Springer, NY, 1993.
[154] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Populations Dynamics[M]. Dordrecht, The Netherlands: Kluwer, 1992.
[155] K. L. Cooke and J. A. Yorke. Some equations modeling growth processes and gonorrhea epidemics[A]. Math. Biosci. 16:75-101.
[156] N. MacDonald. Biological Delay Systems: Linear Stability Theory, Cambridge UniversityPress, NY, 1989.
[157] G. Stepan. Retarded Dynamical System: Stability and Characteristic Functions[M]. Academic Press, NY, 1989.
[158] Z. Wang, T. Chu. Delay induced Hopf bifurcation in a simplified network congestion control model[J]. Chaos, Solitons and Fractals, 2006, 28:161–172.
[2] S. Floyd, V. Jacobson. Random early detection gateways for congestion avoidance[J], IEEE/ACM Transactions on Networking, 1993, 1:397–413.
[3] T.J. Ott, T.V. Lakshman and L. Wong. SRED: Stabilized RED[J]. Proc. of INFOCOM’99, March 1999:1346–1355.
[4] W. Feng, K.G. Shin, D.D. Kandlur and D. Saha. The BLUE active queue management algorithms[J]. IEEE/ACM Transactions on Networking, 2002, 10(4):513–528.
[5] S. Floyd, R. Gummadi, S. Shenker. Adaptive RED: an algorithm for increasing the robustness of RED’s active queue management[EB/01], http://www.icir.org/floyd/papers/adaptiveRed.pdf, 2001.
[6] S. Athuraliya, D.E. Lapsley, S.H. Low.Random early marking for Internet congestion control[J] . Proceedings of the IEEE GLOBECOM, 1999, 3:1747–1752.
[7] S. Kunniyur, R. Srikant. Analysis and design of an adaptive virtual queue algorithm for active queue management[J]. Proceedings of the ACM SIGCOMM, San Diego, CA, August 2001:123–134.
[8] C.V. Hollot, V. Misra, D. Towsley and W. Gong. On designing improved controllers for AQM routers supporting TCP flows[J]. Proc. of INFOCOM’2001, April 2001:1726–1734.
[9] J. Aweya, M. Ouellette and D.Y.Montuno. A control theoretic approach to active queue management[J]. Computer Networks, 2001, 36(2/3) :203–235.
[10] M. Christiansen, K. Jaffey, D. Ott and D. Smith. Tuning RED for Web traffic[J], IEEE/ACMTransactions on Networking, 2001, 9(3):249–264.
[11] S. Ryu, C. Rump and C. Qiao. Advances in Internet congestion control[J]. IEEE Communications Surveys and Tutorials, 2003, 5(1):28–39.
[12] S. Ryu and C. Rump. Advances in Active Queue Management (AQM) Based TCP Congestion Control[J]. Telecommunication Systems, 2004, 25(4):317–351.
[13] S. Ryu and C. Rump. Design of a pro-active queue management[J]. The 11th IEEE International Conference on Networks, 2003:367- 372.
[14] J. Sun, G. Chen, K.T. Ko, S. Chan, M. Zukerman. PD-Controller: a new active queue management scheme[J]. Proc. IEEE Globecom., 2003, 6:3103–3107.
[15] B. Wydrowski, M. Zukerman. GREEN: an active queue management algorithm for a selfmanaged internet[J]. Proc. ICC 4, 2002:2368–2372.
[16] E.C. Park, H. Lim, K.J. Park, C.H. Choi. Analysis and design of the virtual rate control algorithm for stabilizing queues in TCP networks[J]. Computer Networks, 2004, 44:17–41.
[17] C. Long, B. Zhao, X. Guan, J. Yang. The Yellow active queue management algorithm[J]. Computer Networks, 2005, 47:525–550.
[18] F. P. Kelly. Charging and rate control for elastic traffic[J]. Eur. Trans. Telecommunications, Jan. 1997, 8:33–37.
[19] F. Kelly, A. Maulloo, and D. Tan. Rate control in communication networks: Shadow prices, proportional fairness and stability[J]. J. Oper. Res. Soc., 1998, 49:237–252.
[20] J. Mo and J. Walrand. Fair end-to-end window-based congestion control[J]. IEEE/ACM Transactions on Networking, Oct. 2000, 8(5):556–567.
[21] R. J. La and V. Anantharam. Charge-sensitive TCP and rate control in the Internet[J]. Proc. IEEE INFOCOM, 2000:1166–1175.
[22] S. H. Low and D. E. Lapsley. Optimization flow control-I: Basic algorithm and convergence[J]. IEEE/ACM Transactions on Networking, Dec. 1999, 7(6):861–874.
[23] S. Kunniyur and R. Srikant. A time-scale decomposition approach to adaptive explicit congestion notification (ECN) marking[J]. IEEE Transactions on Automatic Control, Jun. 2002, 47(6):882–894.
[24] J.Wen and M. Arcak. A unifying passivity framework for network flow control[J]. Proc. IEEE INFOCOM, San Francisco, CA, Apr. 2003:1156–1166.
[25] Fernando Paganini. A global stability result in network flow control[J]. Systems & Control Letters, 5 July 2002, 46(3):165-172.
[26] Q. Yin and S. H. Low. Convergence of REM flow control at a single link[J]. IEEE Communication Letters, Mar. 2001, 5:119–121.
[27] G. Vinnicombe. On the stability of networks operating TCP-like congestion control[J]. Proc. 15th IFAC World Congr. Automatic Control, Barcelona, Spain, Jul. 2002.
[28] L. Massoulie. Stability of distributed congestion control with heterogeneous feedback delays[J]. IEEE Transactions on Automatic Control, Jun. 2002, 47(6):895–902.
[29] R. Johari and D. Tan. End-to-end congestion control for the Internet: Delays and stability[J]. IEEE/ACM Transactions on Networking, Dec. 2001, 9(6):818–832.
[30] A. Elwalid. Analysis of adaptive rate-based congestion control for highspeed wide-area networks[J]. Proc. IEEE Int. Conf. Communications (ICC), Seattle, WA, Jun. 1995, 3:1948–1953.
[31] S. Liu, T. Basar and R. Srikant. Controlling the Internet: A survey and some new results[J].Proc. 42nd IEEE Conf. Decision and Control, Maui, HI, Dec. 2003:3048–3057.
[32] Y.-P. Tian and H.-Y. Yang. Stability of the Internet congestion control with diverse delays[J]. Automatica, 2004, 40:1533–1541.
[33] Y.-P. Tian. Stability analysis and design of the second-order congestion control for networks with heterogeneous delays[J]. IEEE/ACM Transactions on Networking, Oct. 2005, 13(5):1082– 1093.
[34] Y.-P. Tian and G. Chen. Stability of the primal–dual algorithm for congestion control[J]. International Journal of Control , 2006, 79(6):662-676.
[35] Yu-Ping Tian. A general stability criterion for congestion control with diverse communication delays[J]. Automatica, 2005, 41:1255– 1262.
[36] Fengyuan Ren, Chuang Lin, Bo Wei. A robust active queue management algorithm in large delay networks[J]. Computer Communications, 2005, 28:485–493.
[37] Xiao Fan Wang, Guanrong Chen, King-Tim Ko. A stability theorem for Internet congestion control[J], Systems & Control Letters, 2002, 45:81–85.
[38] D. Bauso, L. Giarre and G. Neglia. AQM stability in multiple bottleneck networks[J]. IEEE International Conference on Communications,June 2004, 4(24):2267– 2271.
[39] M.L. Sichitiu, P.H. Bauer. Asymptotic stability of congestion control systems with multiple sources[J]. IEEE Transactions on Automatic Control, Feb. 2006, 51(2):292– 298.
[40] Cheng-Nian Long, Jing Wu, Xin-Ping Guan. Local stability of REM algorithm with time-varying delays[J]. IEEE Communications Letters, March 2003, 7(3):142– 144.
[41] Huijun Gao, James Lam, Changhong Wang and Xinping Guan. Further results on local stability of REM algorithm with time-varying delays[J]. IEEE Communications Letters, May 2005, 9(5):402– 404.
[42] Steven H. Low, Fernando Paganini, Jiantao Wang and John C. Doyle. Linear stability of TCP/RED and a scalable control[J]. Computer Networks, December 2003, 43(5):633-647.
[43] Zhu Li, N. Ansari. Local stability of a new adaptive queue management (AQM) scheme[J]. IEEE Communications Letters, June 2004, 8(6):406– 408
[44] Jinsheng Sun and Moshe Zukerman. RaQ: A robust active queue management scheme based on rate and queue length[J]. Computer Communications, June 2007, 30(8):1731-1741.
[45] I.C. Lestas and G. Vinnicombe. Scalable robust stability for nonsymmetric heterogeneous networks[J]. Automatica, April 2007, 43(4):714-723.
[46] Hamsa Balakrishnan, Nandita Dukkipati, Nick McKeown and Claire J.Tomlin. Stability Analysis of Explicit Congestion Control Protocols[J]. IEEE Communications Letters, October 2007, 11(10):823-825.
[47] Heng-Qing Ye. Stability of data networks under an optimization-based bandwidth allocation[J]. IEEE Transactions on Automatic Control, July 2003, 48(7):1238– 1242.
[48] Thomas Voice. Stability of Multi-Path Dual Congestion Control Algorithms[J]. IEEE/ACM Transactions on Networking, December2007, 15(6):1321-1239.
[49] Liansheng Tan, Wei Zhang, Gang Peng, Guanrong Chen. Stability of TCP/RED systems in AQM routers[J]. IEEE Transactions on Automatic Control, Aug. 2006, 51(8):1393– 1398.
[50]任丰原,林闯,任勇,山秀明.大时滞网络中的拥塞控制算法[J].软件学报,2003年03期.
[51]苏晓丽,郑明春,李锦涛,孟强.一种基于速率的组播拥塞控制算法及其性能分析[J].电子学报, 2004年02期.
[52]谭连生,熊乃学,杨燕.一种基于PGM的单速率组播拥塞控制方案[J].软件学报, 2004年10期.
[53]尹凤杰,井元伟,岳承君,王涛,潘伟.不确定输入延时网络系统的鲁棒拥塞控制[J].控制与决策, 2007年02期.
[54]许立波,吴国新,基于时序推断的拥塞控制策略的性能分析[J].计算机学报, 2007年09期.
[55] Xinbing Wang and Do Young Eun. Local and global stability of TCP-newReno/RED with many flows[J].Computer Communications, 8 March 2007, 30(5):1091-1105.
[56] F. P. Kelly. Models for a self-managed Internet[J]. Philosophic. Trans. Roy. Soc., 2000, A358: 2335–2348.
[57] V. Misra, W. Gong, and D. Towsley. A fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED[J]. Proc. ACM SIGCOMM, Stockholm, Sweden, Sep. 2000:151–160.
[58] S. Kunniyur and R. Srikant. Stable, scalable, fair congestion control and AQM schemes that achieve high utilization in the Internet[J]. IEEE Transactions on Automatic Control, Nov. 2003, 48(11):2024–2029.
[59] S. Kunniyur and R. Srikant. End-to-end congestion control: Utility functions, random losses and ECN marks[J]. IEEE/ACM Transactions on Networking, Oct. 2003, 11(10):689–702.
[60] F. Paganini, J. Doyle and S. Low. Scalable laws for stable network congestion control[J]. Proc. IEEE Conf. Decision Control, Dec. 2001:185–190.
[61] G. Vinnicombe. On the Stability of End-to-End Congestion Control for the Internet[R]. Univ. Cambridge Tech. Rep. CUED/F-INFENG/TR.398, 2001 [Online]. Available: http://www.eng.cam.ac.uk/~gv.
[62] S. Deb and R. Srikant. Global stability of congestion controllers for the Internet[J]. IEEETransations on Automatic Control, Jun. 2003, 48(6):1055–1060.
[63] C. V. Hollot and Y. Chait. Nonlinear stability analysis for a class of TCP/AQM schemes[J]. Proc. IEEE Conf. Decision Control, Dec. 2001:2309–2314.
[64] H. Yaiche, R. R. Mazumdar, and C. Rosenberg. A game-theoretic framework for bandwidth allocation and pricing in broadband networks[J]. IEEE/ACM Transactions on Networking, Oct. 2000, 8(5):667–678.
[65] Lei Ying, G. E. Dullerud, R. Srikant. Global stability of Internet congestion controllers with heterogeneous delays[J]. IEEE/ACM Transactions on Networking, June 2006, 14(3):579– 590.
[66] T. Alpcan and T. Basar. A Globally Stable Adaptive Congestion Control Scheme for Internet-Style Networks With Delay[J]. IEEE/ACM Transactions on Networking, December 2005, 13(6):1261-1274.
[67] Z. Wang, F. Paganini. Boundedness and Global Stability of a Nonlinear Congestion Control With Delays[J]. IEEE Transactions on Automatic Control, Sept. 2006, 51(9):1514– 1519.
[68] Yueping Zhang, Seong-Ryong Kang and Dmitri Loguinov. Delay-Independent Stability and Performance of Distributed Congestion Control[J].IEEE/ACM Transactions on Networking, OCTOBER 2007, 15(5):838-851.
[69] Jinsheng Sun, Sammy Chan, King-Tim Ko, Guanrong Chen and Moshe Zukerman. Instability effects of two-way traffic in a TCP/AQM system[J]. Computer Communications, 2007, 30:2172–2179.
[70] M. Peet, S. Lall. On global stability of Internet congestion control[J]. 43rd IEEE Conference on Decision and Control, Dec. 2004, 1(14-17):1035 - 1041.
[71] Shao Liu, T. Basar, R. Srikant. Exponential-RED: a stabilizing AQM scheme for low- and high-speed TCP protocols[J]. IEEE/ACM Transactions on Networking, Oct. 2005, 13(5):1068– 1081.
[72] S.Floyd. Recommendation on using the‘gentle_’variant of RED[EB/02], http://www.icir.org/floyd/red/gent-le. html, 2000.
[73] W. Feng, D.D. Kandlur, D. Saha, K.G. Shin. A self-configuring RED gateway[J]. Proc. IEEE INFOCOMM,1999, 3:1320–1328.
[74] C. Diot, G. Iannaccone, M. May. Aggregate traffic performance with active queue management and drop from tail[J]. ACM SIGCOMM Comput Commun Rev, 2001, 31:4–13.
[75] Tomoya Eguchi, Hiroyuki Ohsaki, et al.. On Control Parameters Tuning for Active Queue Management Mechanisms using Multivariate Analysis[J]., 2003. Proceedings. Symposium on Applications and the Internet, 27-31 Jan. 2003:120– 127.
[76] Chengnian Long, Bin Zhao, Xinping Guan, Bo Yang. Stability analysis and filter design for LRED algorithm[J]. American Control Conference, 2005. Proceedings of the 2005, June 2005, 38-10:1847 - 1852.
[77] Hui Zhang, Mingjian Liu, et al.. Modeling TCP/RED :a Dynamical Approach[EB/03], http://www.ensc.sf-u.ca/~ljilja/papers/springer_indexed.pdf.
[78] E. Hashem. Analysis of random drop for gateway congestion control[R]. Report LCS TR-456, Laboratory for Computer Science, MIT, 1989:103.
[79] C. Wang, B. Li, Y. Thomas Hou, K. Sohraby, and Y. Lin. LRED: A robust active queue management scheme based on packet loss ratio[J]. Proceedings of IEEE INFOCOM, Hongkong, 2004.
[80] S.S. Kunniyur, R. Srikant. An adaptive virtual queue (AVQ) algorithm for active queue management[J]. IEEE/ACM Transactions on Networking, April 2004, 12(2):286– 299.
[81] C.Hollot, V. Mista, et al.. Analysis and design of controllers for AQM routers supporting TCP flows[J]. IEEE Transactions on Automatic Control, Jun.2002, 47:945-959.
[82] S. Athuraliya, S. Low, V. Li and Q. Yin. REM: Active queue management[J]. IEEE Network Magazine, May 2001, 15:48-53.
[83] Y. Gao and J.C. Hou. A state feedback control approach to stabilizing queues for ECN-enabled TCP flows[J]. In proceedings of IEEE INFOCOM, San Francisco, CA, 2003, 3:2301-2311.
[84] C.A. Deseor and Y. T. Yang. On the generalized Nyquist Stability Criterion[J]. IEEE Transactions on Automatic Control, 1980, 25:187-196.
[85] UCN/LBL/VINT. Network simulator-NS2[EB/04]. http://-mash.cs.berkelay.edu/ns.
[86] Kai Jiang, Xiaofan Wang and Yugeng Xi. Nonlinear analysis of RED– a comparative study[J]. Chaos, Solitons & Fractals, SEP 2004, 21 (5):1153-1162.
[87] K Cooke and Z. Grossman. Discrete delay, distributed delay and stability switches[J]. Journal of Mathematic Analysis and Application, 1982, 86:592–627.
[88] J. Hale. Theory of Functional Differential Equations[M], Springer, Berlin, 1977.
[89] B. Hassard, N. Kazarinoff, Y. Wan. Theory and Applications of Hopf Bifurcation[M], Cambridge University Press, Cambridge, 1981.
[90] C. Li, G. Chen, X. Liao, J. Yu. Hopf bifurcation in an Internet congestion control model[J]. Chaos, Solitons Fractals, 2004, 19 (4):853–862.
[91] L. Massoulie. Stability of distributed congestion control with heterogeneous feedback delays[J]. IEEE Transactions on Automatic Control, 2002, 47(6):895 - 902.
[92] F. Paganini, Z. Wang, J. Doyle, S. Low. A new TCP/AQM for stable operation in fastnetworks[J]. Proceedings of the IEEE INFOCOM, San Francisco, CA, April 2003:96–105.
[93] G. Raina. Local Bifurcation Analysis of Some Dual Congestion Control Algorithms[J]. IEEE Transactions on Automatic Control, 2005, 50(8):1135– 1146.
[94] P. Ranjan, R.J. La and E.H. Abed. Global stability conditions for rate control with arbitrary communication delays[J], IEEE/ACM Transactions on Networking, 2006, 14(1):94-107.
[95] F. Ren, C. Lin, B. Wei. A nonlinear control theoretic analysis to TCP–RED system[J]. Compututer Networks, 2005, 49 (4):580–592.
[96] H.-Y. Yang, Y.-P. Tian. Hopf bifurcation in REM algorithm with communication delay[J]. Chaos, Solitons Fractals, 2005, 25 (5):1093–1105.
[97] J. Nagle. Congestion control in IP/TCP internet works[R]. Request Comments RFC 896, Ian., 1984.
[98] S. Guo, X. Liao, C. Li and D. Yang. Stability analysis of a novel exponential-RED model with heterogeneous delays[J]. Computer Communications, 2007, 30(5):1058-1074.
[99] M. L. Sichitiu and P. H. Bauer. Asymptotic Stability of Congestion Systems With Multiple Sources[J]. IEEE Transactions on Automatic Control, 2006, 51(2):292-298.
[100] S. H. Low, F. Paganini, J.Wang, S. Adlakha and J. C. Doyle. Dynamics of TCP/RED and a scalable control[J]. Proc. IEEE Infocom, 2002.
[101] G. Raina and O. Heckmann. TCP: Local stability and Hopf bifurcation[J]. Performance Evaluation, 2007, 64(3):266-275.
[102] G. R. Chen, J. L. Moiola and H. O. Wang. Bifurcation control: Theories, methods and applications[J]. International Journal of Bifurcation and Chaos, 2005, 10(3):11-548 .
[103] J. Wang, L. Chen and X. Fei. Analysis and control of the bifurcation of Hodgkin–Huxley model[J]. Chaos, Solitons and Fractals, 2007, 31:247-256.
[104] E. H. Abed and J. H. Fu. Local feedback stabilization and bifurcation control: I. Hopf bifurcation[J]. System Control & Letters, 1986, 7:11-17.
[105] H. O. Wang and E. H. Abed. Bifurcation control of a chaotic system[J]. Automatica, 1995, 31(9):1213–1226.
[106] H. O. Wang and E. H. Abed. Robust control of period doubling bifurcations and implications for control of chaos[J]. In Proc. IEEE Int. Symp. Circ. Syst.,Orlando, USA , 1994:3287–3292.
[107] H. O. Wang, D. S. Chen and L. G. Bushnell. Dynamic feedback control of bifurcations. Latin American Applied Research, 2001, 31:219–225.
[108] G. Gu, X. Chen, A. Sparks and S. Banda. Bifurcation stabilization with local output feedback[J]. SIAM J. Contr. Optim., 1999, 37:934–956.
[109] P. Yu and G. R. Chen. Hopf bifurcation control using nonlinear feedback with polynomialfunctions[J]. International Journal of Bifurcation and Chaos, 2004, 14:1683–1704.
[110] N. Baba, A. Amann, E. Sch?ll and W. Just. Giant Improvement of Time-Delayed Feedback Control by Spatio-Temporal Filtering[J]. Physical Review Letters, 2002, 89, 074101.
[111] E. Sch?ll and H. G. Schuster. Handbook of Chaos Control[M]. 2nd edition, Wiley-VCH, Weinheim, 2007.
[112] K. Pyragas. Continuous control of chaos by self-controlling feedback[J]. Physics Letter A, 1992, 170(6):421-428.
[113] K. Pyragas, V. Pyragas and H. Benner. Delayed feedback control of dynamical system at a subcritical Hopf bifurcation[J]. Physical Review E, 2004, 70, 056222.
[114] A. G. Balanov, N. B. Janson, and E. Sch?ll. Delayed feedback control of chaos: Bifurcation analysis[J]. Physical Review E, 2005, 71, 016222.
[115] A. Ahlborn and U. Parlitz. Stabilizing Unstable Steady States Using Multiple Delay Feedback Control[J]. Physical Review Letters, 2004, 93, 264101.
[116] M. G. Rosenblum and A. S. Pikovsky. Controlling Synchronization in an Ensemble of Globally Coupled Oscillators[J]. Physical Review Letters, 2004, 92, 114102.
[117] P. H?vel and E. Sch?ll. Control of unstable steady states by time-delayed feedback methods[J]. Physical Review E, 2005, 72, 046203.
[118] J. E. S. Socolar, D. W. Sukow and D. J. Gauthier. Stabilizing unstable periodic orbits in fast dynamical systems[J], Physical Review E, 1994, 50, 3245.
[119] K. Pyragas, V. Pyragas, I. Z. Kiss and J. L. Hudson. Stabilizing and Tracking Unknown Steady States of Dynamical Systems[J]. Physical Review Letters, 2002, 89, 244103.
[120] K. Pyragas, V. Pyragas, I. Z. Kiss and J. L. Hudson. Adaptive control of unknown unstable steady states of dynamical systems[J]. Physical Review E, 2004, 70, 026215.
[121] S. Schikora. All-optical noninvasive control of unstable steady states in a semiconductor laser[J]. Physical Review Letters, 2006, 97, 213902.
[122] V. Z. Tronciu, H.-J. Wünsche, M. Wolfrum and M. Radziunas. Semiconductor laser under resonant feedback from a Fabry-Perot resonator: Stability of continuous-wave operation[J]. Physical Review E, 2006, 73, 046205.
[123] M. Feito, F. J. Cao. Time-delayed feedback control of a flashing ratchet[J]. Physical Review E, 2007, 76, 061113.
[124] H. Nakajima, On analytical properties of delayed feedback control of chaos[J]. Physics Letter A, 1997, 232:207 -210.
[125] B. Fiedler, V. Flunkert, M. Georgi, P. Hovel and E. Scholl. Refuting the Odd-Number Limitation of Time-Delayed Feedback Control[J]. Physical Review Letters, 2007, 98, 114101.
[126] W. Just, B. Fiedler, M. Georgi, V. Flunkert, P. Hovel and E. Scholl. Beyond the odd number limitation: A bifurcation analysis of time-delayed feedback control[J]. Physical Review E, 2007, 76, 026210.
[127] Claire M. Postlethwaite and Mary Silber. Stabilizing unstable periodic orbits in the Lorenz equations using time-delayed feedback control[J]. Physical Review E, 2007, 76, 056214.
[128] Thomas Dahms, Philipp H?vel and Eckehard Sch?ll. Control of unstable steady states by extended time-delayed feedback[J]. Physical Review E, 2007, 76, 056201.
[129] Serhiy Yanchuk, Matthias Wolfrum, Philipp H?vel and Eckehard Sch?ll. Control of unstable steady states by long delay feedback[J]. Physical Review E, 2006, 74, 026201.
[130] Z. Chen and P. Yu. Hopf bifurcation control for an Internet congestion model[J]. International Journal of Bifurcation and Chaos, 2005, 5(8):2643-2651.
[131] M. Xiao and J. Cao. Delayed feedback-based bifurcation control in an Internet congestion model[J]. Journal of Mathematical Analysis and Applications, 2007, 332(2):1010-1027 .
[132] H. O. Wang and Dong S.Chen. Bifurcation Control: Theory and Applications[A]. Springer, 2004, 293:229-248 .
[133] S. Guo, X. Liao and C. Li. Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay[J]. Nonlinear Analysis: Real World Applications, In Press, Corrected Proof, Available online 13 March 2007.
[134] D. S. Chen, H. O. Wang and G. Chen. Anti-Control of Hopf bifurcation[J]. IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, 2001, 48(6):661- 672.
[135] D. Fan and J. Wei. Hopf bifurcation analysis in a tri-neuron network with time delay[J]. Nonlinear Analysis: Real World Applications, In Press, Corrected Proof, Available online 2 October 2006.
[136] S. Guo, X. Liao, Q. Liu and C. Li. Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay[J]. Nonlinear Analysis: Real World Applications, In Press, Corrected Proof, Available online 3 June 2007.
[137] J. T. Wen and M. Arcak. A unifying passivity framework for network flow control[J]. IEEE Transactions on Automatic Control, 2004, 49(2):162-174.
[138] V. Jacobson. Congestion avoidance and control[J]. Proc. Symp. Communications Architectures and Protocols (SIGCOMM), Stanford, CA, Aug. 1988:314-329.
[139] R. Srikant. The Mathematics of Internet Congestion Control[M]. Cambridge, MA: Birkhauser, 2004.
[140] S. Det and R. Srikant. Global stability of congestion controller for the Internet[J]. IEEE Transactions on Automatic Control, 2003, 48(6):1055-1060.
[141] K. Kar, S. Sarkar and L. Tassiulas. A simple rate control algorithm for maximizing total user utility[J]. Proc. IEEE INFOCOM, Apr. 2001:133-141.
[142] D. Katabi, M. Handley and C. Rohrs. Congestion control for high bandwidth delay product networks[J]. Proc. ACM SIGCOMM, Aug. 2002:89-102.
[143] S. Kunniyur and R. Srikant. End-to-End congestion control schemes: Utility functions, random losses and ECN marks[J]. IEEE/ACM Transactions on Networking, 2003, 11(5):689-702.
[144] K. Gu, J. Chen, V. L. Kharitonov. Stability of time-delay system[M]. Springer-Verlag, New York, 2003.
[145] Y. He, M. Wu and J.-H. She. Delay-dependent stability criteria for linear system with multiple delays[J]. IEE proc.-Control Theory Appl., 2006, 153(4):447-452.
[146] Y. He, Q. G. Wang, L. H. Xie, et al.. Further improvement of free-weighting matrices technique for systems with time-varying delay[J]. IEEE Transations on. Automatic Control, 2007, 52(2):293-299.
[147] Y. He, Q. G. Wang, C. Lin, et al.. Delay-range-dependent stability for systems with time-varying delay[J], Automatica, 2007, 43(2):371-376.
[148] J. K. Hale and S. M. V. Lunel, Introduction to the Theory of Functional Differential Equations[M]. Springer, New York, 1991.
[149] S. Boyd, L. EI Ghaoui, E. Feron and V. Balakrishnam. Linear Matrix Inequalities in System and Control Theory[M]. SIAM, Philadelphia, PA, 1994.
[150] M. Wu, Y. He, J. H. She and G. P. Liu. Delay-dependent criteria for robust stability of time-varying delay systems[J]. Automatica, 2004, 40:1435-1439.
[151] M. Wu, Y. He and J. H. She. Delay-dependent Stabilization for Systems with Multiple Unknown Time-varying Delays[J]. International Journal of Control, Automation, and Systems, 2006, 4:682-688.
[152] L. Chen, X. Wang and Z. Han. Controlling chaos in Internet congestion control model[J].Chaos, Solitons & Fractals, 2004, 21:81-91.
[153] J. Hale and S. V. Lenel. Introduction to Functional Differential Equations[M]. Springer, NY, 1993.
[154] K. Gopalsamy. Stability and Oscillations in Delay Differential Equations of Populations Dynamics[M]. Dordrecht, The Netherlands: Kluwer, 1992.
[155] K. L. Cooke and J. A. Yorke. Some equations modeling growth processes and gonorrhea epidemics[A]. Math. Biosci. 16:75-101.
[156] N. MacDonald. Biological Delay Systems: Linear Stability Theory, Cambridge UniversityPress, NY, 1989.
[157] G. Stepan. Retarded Dynamical System: Stability and Characteristic Functions[M]. Academic Press, NY, 1989.
[158] Z. Wang, T. Chu. Delay induced Hopf bifurcation in a simplified network congestion control model[J]. Chaos, Solitons and Fractals, 2006, 28:161–172.