网络拥塞控制策略稳定性分析及应用
|
摘要
网络拥塞问题一直困扰着互联网(Internet)发展。基于源端的TCP拥塞控制算法和基于链路端的主动队列管理(AQM)算法相结合是解决拥塞问题的有效途径。
本文主要在综述TCP拥塞控制算法中FAST TCP协议参数选择、AQM算法中早期随机检测(RED)算法参数选择和基于滑模控制理论的AQM算法设计研究现状基础上,展开了如下研究:
(1)提出更为接近现实网络的FAST TCP拥塞控制模型,在对模型进行稳定性分析基础上得到了更为宽泛的FAST TCP协议参数设计指导方案。首先针对现有FAST TCP模型没有考虑窗口更新间隔参数,并忽略源端在估计拥塞信号时所采用指数平滑滤波稳定性因素的缺陷,改进其网络拥塞控制模型。其次将该模型在其平衡点附近进行线性化,进行Laplace变换后得到FAST TCP系统的开环传递函数。应用Routh判据对系统进行稳定性分析得到其稳定条件,然后提出了一种根据控制器增益参数、窗口更新周期和网络相关参数选择合适协议参数确保系统稳定的指导方案。最后理论计算和NS2仿真验证表明,和相关指导方案相比,该改进方案具有更宽泛的应用范围。
(2)提出包含TCP自同步特性的拥塞控制模型,并在稳定性分析基础上给出了更为宽泛的RED参数设置指导方案。TCP自同步特性就是TCP窗口在每接收到一个确认帧后才发送新的数据分组,这样TCP源端数据分组发送速率就不能简单的表示为窗口大小与往返时延的比值,而是由收到的确认帧的瞬时速率决定。本文首先针对现有TCP/RED拥塞控制模型忽略TCP自同步特性缺陷,提出新的包含TCP自同步特性的拥塞控制模型。然后在系统平衡点附近对模型进行线性化,再Laplace变换后得到整个TCP/RED系统的开环传递函数。在应用Nyquist稳定性对系统进行稳定分析后,得到比现有TCP/RED拥塞控制模型更为宽松的RED参数设置范围。最后NS2仿真验证了该模型分析RED参数设置范围的有效性。
(3)提出当系统参数扰动不满足匹配条件时基于滑模控制理论的AQM算法设计问题。首先改进了常用的网络拥塞控制模型,并在平衡点线性化后,采用还原算法进行延迟补偿,将该模型转化为无时滞的线性模型。在系统状态矩阵扰动不满足匹配但有界的条件下,采用积分滑模变结构控制算法,基于Lyapunov函数和线性矩阵不等式方法给出了滑模可到达和渐近稳定的可行条件,并根据该条件设计了鲁棒主动队列管理控制器。最后仿真结果验证了该控制器的有效性。
The development of Internet has been encumbered with the congestion problem. The combination of TCP congestion control algorithm based on the source side and Active Queue Management (AQM) algorithm based on the link has become a main approach to solve the congestion control problem.
Based on the corrent situation in review, which focuses in designing protocol parameter of FAST TCP in TCP congestion control algorithm, parameter selection of Random Early Detection (RED) algorithm in AQM algorithm, and the designing of AQM algorithm based on sliding mode control theory, the main research works and conclusions in this paper are as follows:
(1) A more accurate FAST TCP congestion control model is proposed, and a more broadly parameter design guidance is obtained based on the stability analsis of the new model.First of all, aiming at the existing FAST TCP model unconsidering the window update interval parameters exponential smoothing filter which the source uses to estimate congestion signal, this paper improves the network congestion control model. Second, the model is linerarized in the vicinity of the equilibrium point, and FAST TCP open-loop system transfer function is obtained after Laplace transforming. The system stability conditions is obtained with the help of Routh stability criterion, then a preferences guidance scheme was proposed to ensure the network stability according to the controller gain parameter and windows update interval and network related parameter. The theory computing and the NS2 simulation results demonstrate that this improved guidance scheme possesses much extensive application area in contrast to other related scheme.
(2) A new congestion control model including TCP self-synchronization is proposed, and a more broadly parameter design guidance is obtained based on the stability analsis of the new model. TCP self-synchronization is that TCP window doesn’t send a new packet to the link until receiving a confirmation frame,so packet sending rate in the source sind cannot simply expressed by the ratio of the window size and RTT, but by receiving the confirmation of the instantaneous speed frame. Aiming at the existing TCP/RED congestion control models ignoring TCP self-synchronization, this paper proposes a new congestion control model including TCP self-synchronization. Then the model is linerarized in the vicinity of the equilibrium point, and the open-loop system transfer function of TCP/RED is obtained after Laplace transforming. The system stability condition is obtained with the help of Nyquist stability criterion, then we receive a more relaxed RED parameters settings scope. At last, NS2 simulation is used to validate the effectiveness of the obtained RED parameters settings scope.
(3) AQM algorithm based on sliding mode control theory is proposed when the system parameter perturbation is unmatched. Firstly the model in common use is modified and linearized at the equilibrium. Then the no time-delay linearization model was obtained by the reduction algorithm. The feasible condition with reachable and asymptotic stability of sliding surface was given through the integral sliding mode control algorithm based on the Lyapunov function and LMI method under the condition of the mismatched but bounded perturbation of system state matrix, in which the robust AQM controller was designed. Finally, the effectiveness of the designed controller is validated by simulation.
本文主要在综述TCP拥塞控制算法中FAST TCP协议参数选择、AQM算法中早期随机检测(RED)算法参数选择和基于滑模控制理论的AQM算法设计研究现状基础上,展开了如下研究:
(1)提出更为接近现实网络的FAST TCP拥塞控制模型,在对模型进行稳定性分析基础上得到了更为宽泛的FAST TCP协议参数设计指导方案。首先针对现有FAST TCP模型没有考虑窗口更新间隔参数,并忽略源端在估计拥塞信号时所采用指数平滑滤波稳定性因素的缺陷,改进其网络拥塞控制模型。其次将该模型在其平衡点附近进行线性化,进行Laplace变换后得到FAST TCP系统的开环传递函数。应用Routh判据对系统进行稳定性分析得到其稳定条件,然后提出了一种根据控制器增益参数、窗口更新周期和网络相关参数选择合适协议参数确保系统稳定的指导方案。最后理论计算和NS2仿真验证表明,和相关指导方案相比,该改进方案具有更宽泛的应用范围。
(2)提出包含TCP自同步特性的拥塞控制模型,并在稳定性分析基础上给出了更为宽泛的RED参数设置指导方案。TCP自同步特性就是TCP窗口在每接收到一个确认帧后才发送新的数据分组,这样TCP源端数据分组发送速率就不能简单的表示为窗口大小与往返时延的比值,而是由收到的确认帧的瞬时速率决定。本文首先针对现有TCP/RED拥塞控制模型忽略TCP自同步特性缺陷,提出新的包含TCP自同步特性的拥塞控制模型。然后在系统平衡点附近对模型进行线性化,再Laplace变换后得到整个TCP/RED系统的开环传递函数。在应用Nyquist稳定性对系统进行稳定分析后,得到比现有TCP/RED拥塞控制模型更为宽松的RED参数设置范围。最后NS2仿真验证了该模型分析RED参数设置范围的有效性。
(3)提出当系统参数扰动不满足匹配条件时基于滑模控制理论的AQM算法设计问题。首先改进了常用的网络拥塞控制模型,并在平衡点线性化后,采用还原算法进行延迟补偿,将该模型转化为无时滞的线性模型。在系统状态矩阵扰动不满足匹配但有界的条件下,采用积分滑模变结构控制算法,基于Lyapunov函数和线性矩阵不等式方法给出了滑模可到达和渐近稳定的可行条件,并根据该条件设计了鲁棒主动队列管理控制器。最后仿真结果验证了该控制器的有效性。
The development of Internet has been encumbered with the congestion problem. The combination of TCP congestion control algorithm based on the source side and Active Queue Management (AQM) algorithm based on the link has become a main approach to solve the congestion control problem.
Based on the corrent situation in review, which focuses in designing protocol parameter of FAST TCP in TCP congestion control algorithm, parameter selection of Random Early Detection (RED) algorithm in AQM algorithm, and the designing of AQM algorithm based on sliding mode control theory, the main research works and conclusions in this paper are as follows:
(1) A more accurate FAST TCP congestion control model is proposed, and a more broadly parameter design guidance is obtained based on the stability analsis of the new model.First of all, aiming at the existing FAST TCP model unconsidering the window update interval parameters exponential smoothing filter which the source uses to estimate congestion signal, this paper improves the network congestion control model. Second, the model is linerarized in the vicinity of the equilibrium point, and FAST TCP open-loop system transfer function is obtained after Laplace transforming. The system stability conditions is obtained with the help of Routh stability criterion, then a preferences guidance scheme was proposed to ensure the network stability according to the controller gain parameter and windows update interval and network related parameter. The theory computing and the NS2 simulation results demonstrate that this improved guidance scheme possesses much extensive application area in contrast to other related scheme.
(2) A new congestion control model including TCP self-synchronization is proposed, and a more broadly parameter design guidance is obtained based on the stability analsis of the new model. TCP self-synchronization is that TCP window doesn’t send a new packet to the link until receiving a confirmation frame,so packet sending rate in the source sind cannot simply expressed by the ratio of the window size and RTT, but by receiving the confirmation of the instantaneous speed frame. Aiming at the existing TCP/RED congestion control models ignoring TCP self-synchronization, this paper proposes a new congestion control model including TCP self-synchronization. Then the model is linerarized in the vicinity of the equilibrium point, and the open-loop system transfer function of TCP/RED is obtained after Laplace transforming. The system stability condition is obtained with the help of Nyquist stability criterion, then we receive a more relaxed RED parameters settings scope. At last, NS2 simulation is used to validate the effectiveness of the obtained RED parameters settings scope.
(3) AQM algorithm based on sliding mode control theory is proposed when the system parameter perturbation is unmatched. Firstly the model in common use is modified and linearized at the equilibrium. Then the no time-delay linearization model was obtained by the reduction algorithm. The feasible condition with reachable and asymptotic stability of sliding surface was given through the integral sliding mode control algorithm based on the Lyapunov function and LMI method under the condition of the mismatched but bounded perturbation of system state matrix, in which the robust AQM controller was designed. Finally, the effectiveness of the designed controller is validated by simulation.
引文
[1] Jacobson V. Congestion avoidance and control [J]. IEEE/ACM Transactions on Networking, 1988, 6(3):314-329
[2] Stevens W R. TCP/IP illustrated volume 1: the protocols [M]. Addison-Wesley Press, 1994.
[3] Paxson V, Allman M, Dawson S, Fenner V J, et al. Known TCP implementation problems [DB/OL]. IETF, 1999. http://www.faqs.org/rfcs / rfc2525.
[4] V. Jacobson. Congestion Avoidance and Control[C], ACM Computer Communication Review 1988, 18(4): 314-329.
[5] W. Stevens.TCP Slow Start, Congestion Avoidance, Fast Retransmit, and Fast Recovery Algoritms [S], RFC 2001, 1997.
[6] S. Floyd, T. Henderson, The NewReno Modification to TCP’s Fast Recovery Algorithm[S], RFC 2582, 1999.
[7] M. Mathis, J.Mahdavi, S. Floyd. TCP Selective Acknowledgment Opions [S], RFC 2018, 1996.
[8] S. Athuraliya, V. H. Li, S. H. Low, Q. Yin. REM: Active Queue Management[J], IEEENetwork, 2001, 15(3):48-53.
[9] Jacobson V. Congestion avoidance and control [J]. ACM Computer Communication Review, 1988, 18(4): 314-329.
[10]罗万明,林闯,阎保平. TCP/IP拥塞控制研究[J].计算机学报. 2001, 24(1): 1-18.
[11]张淼.互联网端到端拥塞控制的研究[D].清华大学博士学位论文,2004.
[12] Jain R. Congestion control in computer networks: Issues and trends [J]. IEEE Network Magazine, 1999, 4(3): 24-30.
[13]秦凯运,杨煌普,谢剑英.面向带宽IP网络饿拥塞控制机制进展[J],通信学报, 2004, 25(11): 119-127.
[14] L.Tan, C. Yuan, M. Zukerman. FAST TCP: Fairness and queueing issues [J], IEEE Communications Letters, 2005, 9(8): 762-764.
[15] L. L. H. Andrew, L. Tan, T. Cui. Fairness comparison of FAST TCP and TCP Vegas [C], Proceedings of 19 ITC, Beijing, China, 2005, 1375-1384.
[16] J. Wang, A. Tang, S. H. Low. Local stability of FAST TCP [C]. Proceedings of 43th IEEE Conference on Decision and Control, Paradise Island, Bahamas, 2004, 1023-1028.
[17] J.Wang, D. X. Wei, S. H. Low. Modeling and stability of FAST TCP [C]. Proceedings of IEEE INFOCOM 2005, Miami, FL, USA, 2005, 938-948.
[18] J. Y. Choi, K. Koo, J. S. Lee. Global stability of FAST TCP in single-source network [C]. Proceedings of 4th IEEE Conference on Decision and Control, Seville, Spain, 2005, 1837-1841.
[19]罗万明,林闯,阎保平. TCP/IP拥塞控制研究[J].计算机学报, 2001, 24(1):1-18.
[20]王彬. TCP/IP网络拥塞控制策略研究[M].浙江大学博士学位论文, 2004.
[21] Lin D, Morris R. Dynamics of random early detection [C]. Proceedings of ACM SIGCOMM 1997, New York, USA, 1997. 127-138.
[22] Ott T J, Lakshman T V, Wong L H. SRED: Stabilized RED [C]. Proceedings of INFOCOM 1999, New York, USA, 1999. 1346-1355
[23] W. C. Feng, D. D. Kandlur. A self-configuring RED gateway [C]. Proceedings of INFOCOM 1999, New York, USA, 1999. 1320-1328.
[24] Hollot C V, Misra V. Analysis and design of controllers for AQM routers supporting TCP flows [J]. IEEE Trans. Automatic Control, 2002, 47(6): 945-959.
[25] Athuraliya S, Li V. REM: Active Queue Management [J]. IEEE Network Magazine, 2001, 15(3): 48-53.
[26] Feng W, Shin K, Kandlur D. The BLUE Active Queue Management Algorithms [J]. IEEE/ACM Trans. Networking, 2002, 10(4): 513-528.
[27] Kunniyur S S, Srikant R, An Adaptive Virtual Queue (AVQ) Algorithm for Active Queue Management [J]. IEEE/ACM Trans. Networking, 2004, 12(2): 286-299.
[28] Do Young Eun, Xinbing Wang. Achieving100%Throughput in TCP/AQM Under Aggressive Packet Marking With Small Buffer[J], IEEE TRANS on NETWORKING, 2008, 16(4): 945-957.
[29] M.Christiansen, K.Jeffay, D.Ott, etc. Tuning RED for web traffic [J]. IEEE/ACM Transactions on Networking, 2000, 30(4):139-150.
[30] C .V. Hollot, Vishal Misra, Don Towsley, etc. A Control Theoretic Analysis of RED[C].USA: IEEE INFOCOM, 3:1510-19.
[31]李金东,马东堂.基于RED算法的非线性拥塞控制[J],计算机工程,2008, 34(20): 91-92.
[32] Liansheng Tan, Wei Zhang, Gang Peng, etc. Stability of TCP/RED System in AQM Routers[J]. IEEE Transactions on Automatic Control, 2006, 51(8):1393-1398.
[33] Peng Yan,Yuan Gao.A Variable Structure Control Approach to Achive Queue Manage -ment for TCP with ECN[J].IEEE Trans on Control System Technology, 2005, 13(2): 203-212.
[34] F Ren, X Ying, Y Ren, et al. A robust active queue management algorithm based on sliding mode variable structure control Proceedings of IEEE INFOCOM 02, New York, USA, 2002:13-20.
[35] Feng-jie Yin. Robust Stabilization of Uncertain Input Delay for Internet Congestion Control. Proceedings of the 2006 American Control Conference Minneapolis, USA, 2006:14-16.
[36] Ming Yan,Yuanwei Jing. Congestion Control over Internet with Uncertainties and Input Delay Based on Variable Structure Control Algorithm. Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, Harbin,China, 2007:5-8.
[37] Yuan-wei Jing,Ling He. Robust Stabilization of state and Input Delay for Active Queue Management Algorithm.Proceeding of the 2007 American Control conference Marriott Marquis Hotel at Times Square New York City USA, 2007:11-13.
[38] Ki Baek Kim. Design of Feedback Control Supporting TCP Based on the State-Space Approach IEEE TRANS on Automatic Control ,2006,51(7):1086-1098.
[39] Ao Tang, S.H.Low. An Accurate Link Model and Its Application to Stability Analysis of Fast TCP[C]. USA: IEEE INFOCOM, 2007, 27:161-169.
[40] David X.Wei, Cheng Jin, S. H. Low. FAST TCP: Motivation, Architecture, Algorithms, Performance [J]. IEEE TRANS on NETWORKING, 2006, 14(6): 1246-1259.
[41] JianTao Wang, David X.Wei, S.H.Low. Modeling and Stability of FAST TCP [C].in Proc.IEEE INFOCOM Miami, FL, USA, 2005:938-948.
[42] Ao Tang, S.H.Low. An Accurate Link Model and Its Application to Stability Analysis of Fast TCP [C]. In Proc. IEEE INFO COM, Anchorage, Alaska, USA, 2007:161-169.
[43] Krister Jacobsson,Lachlan L Steven H.Low. An Improved Link Model for Window Flow Control and Its Application to FAST TCP[J]. IEEE Transactions on Automatic Control, 2009, 54(3):551-564.
[44] Krister Jacobsson. Dynamic modeling of internet congestion control[PHD]. Royal Institute of Technology (KT H), Sweden, 2008.
[45]黄小猛,林闯,任丰源.高速传输协议研究进展[J].计算机学报.2006.29(11): 111-120.
[46] T.Cui, L.Andrew. FAST TCP simulator modle for NS-2. version1.1[Online]. Available: http:// www.cub inlab.ee.mu.oz.au/ns2fasttcp.
[47] Joon-Young Choi, K. Koo, S. H. Low. Global Exponential Stability of Fast TCP [C]. IEEE Conference on Decision on Control, 2006.
[48] Kyungmo Koo, Joon-Young Choi, Lee, J.S. Parameter Conditions for Global Stability of FAST TCP [J]. IEEE TRANS on Communications Letters, 2008, 12(2):155-157.
[49] Yun Jong Choi, Jeong Wan Ko, Sung Wook.Improved Global Stability Conditions of the Tuning Parameter in FAST TCP EEE TRANS on Communications Letters, 2009, 13(3):202-205.
[50] Liansheng Tan, Wei Zhang, Cao Yuan. On Parameter Tuning for FAST TCP[J]. IEEE TRANS on Communications Letters, 2007, 11(5):1-3.
[51] Hollot C V, Misra V, Towsley D. A control theoretic analysis of RED[C], Proceedings of IEEE INFOCOM, March 2003, 1510-1519,.
[2] Stevens W R. TCP/IP illustrated volume 1: the protocols [M]. Addison-Wesley Press, 1994.
[3] Paxson V, Allman M, Dawson S, Fenner V J, et al. Known TCP implementation problems [DB/OL]. IETF, 1999. http://www.faqs.org/rfcs / rfc2525.
[4] V. Jacobson. Congestion Avoidance and Control[C], ACM Computer Communication Review 1988, 18(4): 314-329.
[5] W. Stevens.TCP Slow Start, Congestion Avoidance, Fast Retransmit, and Fast Recovery Algoritms [S], RFC 2001, 1997.
[6] S. Floyd, T. Henderson, The NewReno Modification to TCP’s Fast Recovery Algorithm[S], RFC 2582, 1999.
[7] M. Mathis, J.Mahdavi, S. Floyd. TCP Selective Acknowledgment Opions [S], RFC 2018, 1996.
[8] S. Athuraliya, V. H. Li, S. H. Low, Q. Yin. REM: Active Queue Management[J], IEEENetwork, 2001, 15(3):48-53.
[9] Jacobson V. Congestion avoidance and control [J]. ACM Computer Communication Review, 1988, 18(4): 314-329.
[10]罗万明,林闯,阎保平. TCP/IP拥塞控制研究[J].计算机学报. 2001, 24(1): 1-18.
[11]张淼.互联网端到端拥塞控制的研究[D].清华大学博士学位论文,2004.
[12] Jain R. Congestion control in computer networks: Issues and trends [J]. IEEE Network Magazine, 1999, 4(3): 24-30.
[13]秦凯运,杨煌普,谢剑英.面向带宽IP网络饿拥塞控制机制进展[J],通信学报, 2004, 25(11): 119-127.
[14] L.Tan, C. Yuan, M. Zukerman. FAST TCP: Fairness and queueing issues [J], IEEE Communications Letters, 2005, 9(8): 762-764.
[15] L. L. H. Andrew, L. Tan, T. Cui. Fairness comparison of FAST TCP and TCP Vegas [C], Proceedings of 19 ITC, Beijing, China, 2005, 1375-1384.
[16] J. Wang, A. Tang, S. H. Low. Local stability of FAST TCP [C]. Proceedings of 43th IEEE Conference on Decision and Control, Paradise Island, Bahamas, 2004, 1023-1028.
[17] J.Wang, D. X. Wei, S. H. Low. Modeling and stability of FAST TCP [C]. Proceedings of IEEE INFOCOM 2005, Miami, FL, USA, 2005, 938-948.
[18] J. Y. Choi, K. Koo, J. S. Lee. Global stability of FAST TCP in single-source network [C]. Proceedings of 4th IEEE Conference on Decision and Control, Seville, Spain, 2005, 1837-1841.
[19]罗万明,林闯,阎保平. TCP/IP拥塞控制研究[J].计算机学报, 2001, 24(1):1-18.
[20]王彬. TCP/IP网络拥塞控制策略研究[M].浙江大学博士学位论文, 2004.
[21] Lin D, Morris R. Dynamics of random early detection [C]. Proceedings of ACM SIGCOMM 1997, New York, USA, 1997. 127-138.
[22] Ott T J, Lakshman T V, Wong L H. SRED: Stabilized RED [C]. Proceedings of INFOCOM 1999, New York, USA, 1999. 1346-1355
[23] W. C. Feng, D. D. Kandlur. A self-configuring RED gateway [C]. Proceedings of INFOCOM 1999, New York, USA, 1999. 1320-1328.
[24] Hollot C V, Misra V. Analysis and design of controllers for AQM routers supporting TCP flows [J]. IEEE Trans. Automatic Control, 2002, 47(6): 945-959.
[25] Athuraliya S, Li V. REM: Active Queue Management [J]. IEEE Network Magazine, 2001, 15(3): 48-53.
[26] Feng W, Shin K, Kandlur D. The BLUE Active Queue Management Algorithms [J]. IEEE/ACM Trans. Networking, 2002, 10(4): 513-528.
[27] Kunniyur S S, Srikant R, An Adaptive Virtual Queue (AVQ) Algorithm for Active Queue Management [J]. IEEE/ACM Trans. Networking, 2004, 12(2): 286-299.
[28] Do Young Eun, Xinbing Wang. Achieving100%Throughput in TCP/AQM Under Aggressive Packet Marking With Small Buffer[J], IEEE TRANS on NETWORKING, 2008, 16(4): 945-957.
[29] M.Christiansen, K.Jeffay, D.Ott, etc. Tuning RED for web traffic [J]. IEEE/ACM Transactions on Networking, 2000, 30(4):139-150.
[30] C .V. Hollot, Vishal Misra, Don Towsley, etc. A Control Theoretic Analysis of RED[C].USA: IEEE INFOCOM, 3:1510-19.
[31]李金东,马东堂.基于RED算法的非线性拥塞控制[J],计算机工程,2008, 34(20): 91-92.
[32] Liansheng Tan, Wei Zhang, Gang Peng, etc. Stability of TCP/RED System in AQM Routers[J]. IEEE Transactions on Automatic Control, 2006, 51(8):1393-1398.
[33] Peng Yan,Yuan Gao.A Variable Structure Control Approach to Achive Queue Manage -ment for TCP with ECN[J].IEEE Trans on Control System Technology, 2005, 13(2): 203-212.
[34] F Ren, X Ying, Y Ren, et al. A robust active queue management algorithm based on sliding mode variable structure control Proceedings of IEEE INFOCOM 02, New York, USA, 2002:13-20.
[35] Feng-jie Yin. Robust Stabilization of Uncertain Input Delay for Internet Congestion Control. Proceedings of the 2006 American Control Conference Minneapolis, USA, 2006:14-16.
[36] Ming Yan,Yuanwei Jing. Congestion Control over Internet with Uncertainties and Input Delay Based on Variable Structure Control Algorithm. Proceedings of the 2007 IEEE International Conference on Mechatronics and Automation, Harbin,China, 2007:5-8.
[37] Yuan-wei Jing,Ling He. Robust Stabilization of state and Input Delay for Active Queue Management Algorithm.Proceeding of the 2007 American Control conference Marriott Marquis Hotel at Times Square New York City USA, 2007:11-13.
[38] Ki Baek Kim. Design of Feedback Control Supporting TCP Based on the State-Space Approach IEEE TRANS on Automatic Control ,2006,51(7):1086-1098.
[39] Ao Tang, S.H.Low. An Accurate Link Model and Its Application to Stability Analysis of Fast TCP[C]. USA: IEEE INFOCOM, 2007, 27:161-169.
[40] David X.Wei, Cheng Jin, S. H. Low. FAST TCP: Motivation, Architecture, Algorithms, Performance [J]. IEEE TRANS on NETWORKING, 2006, 14(6): 1246-1259.
[41] JianTao Wang, David X.Wei, S.H.Low. Modeling and Stability of FAST TCP [C].in Proc.IEEE INFOCOM Miami, FL, USA, 2005:938-948.
[42] Ao Tang, S.H.Low. An Accurate Link Model and Its Application to Stability Analysis of Fast TCP [C]. In Proc. IEEE INFO COM, Anchorage, Alaska, USA, 2007:161-169.
[43] Krister Jacobsson,Lachlan L Steven H.Low. An Improved Link Model for Window Flow Control and Its Application to FAST TCP[J]. IEEE Transactions on Automatic Control, 2009, 54(3):551-564.
[44] Krister Jacobsson. Dynamic modeling of internet congestion control[PHD]. Royal Institute of Technology (KT H), Sweden, 2008.
[45]黄小猛,林闯,任丰源.高速传输协议研究进展[J].计算机学报.2006.29(11): 111-120.
[46] T.Cui, L.Andrew. FAST TCP simulator modle for NS-2. version1.1[Online]. Available: http:// www.cub inlab.ee.mu.oz.au/ns2fasttcp.
[47] Joon-Young Choi, K. Koo, S. H. Low. Global Exponential Stability of Fast TCP [C]. IEEE Conference on Decision on Control, 2006.
[48] Kyungmo Koo, Joon-Young Choi, Lee, J.S. Parameter Conditions for Global Stability of FAST TCP [J]. IEEE TRANS on Communications Letters, 2008, 12(2):155-157.
[49] Yun Jong Choi, Jeong Wan Ko, Sung Wook.Improved Global Stability Conditions of the Tuning Parameter in FAST TCP EEE TRANS on Communications Letters, 2009, 13(3):202-205.
[50] Liansheng Tan, Wei Zhang, Cao Yuan. On Parameter Tuning for FAST TCP[J]. IEEE TRANS on Communications Letters, 2007, 11(5):1-3.
[51] Hollot C V, Misra V, Towsley D. A control theoretic analysis of RED[C], Proceedings of IEEE INFOCOM, March 2003, 1510-1519,.