# American Institute of Mathematical Sciences

2011, 16(2): 653-668. doi: 10.3934/dcdsb.2011.16.653

## Time-varying delayed feedback control for an internet congestion control model

 1 School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai, 200092, China, China

Received  February 2010 Revised  November 2010 Published  June 2011

A proportionally-fair controller with time delay is considered to control Internet congestion. The time delay is chosen to be a controllable parameter. To represent the relation between the delay and congestion analytically, the method of multiple scales is employed to obtain the periodic solution arising from the Hopf bifurcation in the congestion control model. A new control method is proposed by perturbing the delay periodically. The strength of the perturbation is predicted analytically in order that the oscillation may disappear gradually. It implies that the proved control scheme may decrease the possibility of the congestion derived from the oscillation. The proposed control scheme is verified by the numerical simulation.
Citation: Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653
##### References:
 [1] T. Alpcan and T. Basar, Global stability analysis of an end-to-end congestion control scheme for general topology networks with delay,, in, (2003), 1092. [2] H. Brunner and S. Maset, Time transformations for delay differential equations,, Discrete Contin. Dyn. Syst. Ser A, 25 (2009), 751. doi: 10.3934/dcds.2009.25.751. [3] Z. Chen and P. Yu, Hopf bifurcation control for an Internet congestion model,, International Journal of Bifurcation and Chaos, 15 (2005), 2643. doi: 10.1142/S0218127405013587. [4] Y. Choi, Periodic delay effects on cutting dynamics,, Journal of Dynamics and Differential Equations, 17 (2005), 353. doi: 10.1007/s10884-005-3145-y. [5] S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations,, Nonlinear Dynamics, 30 (2002), 323. doi: 10.1023/A:1021220117746. [6] D. W. Ding, J. Zhu, X. S. Luo and Y. L. Liu, Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm,, Nonlinear Analysis: Real World Applications, 10 (2009), 2873. doi: 10.1016/j.nonrwa.2008.09.007. [7] S. Floyd and V. Jacobson, Random early detection gate-ways for congestion avoidance,, IEEE/ACM Transctions on Networks, 1 (1993), 397. [8] D. E. Gilsinn, Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter,, Nonlinear Dynamics, 30 (2002), 103. doi: 10.1023/A:1020455821894. [9] S. T. Guo, G. Feng, X. F. Liao and Q. Liu, Hopf bifurcation control in a congestion control model via dynamic delayed feedback,, Chaos, 18 (2008), 043104. doi: 10.1063/1.2998220. [10] S. T. Guo, X. F. Liao and C. D. Li, Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay,, Nonlinear Analysis: Real World Applications, 9 (2008), 1292. doi: 10.1016/j.nonrwa.2007.03.006. [11] S. T. Guo, X. F. Liao, Q. Liu and C. D. Li, Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay,, Nonlinear Analysis: Real World Applications, 9 (2008), 1768. doi: 10.1016/j.nonrwa.2007.05.014. [12] J. Hale, "Theory of Functional Differential Equations,", World Publishing Corporation, (2003). [13] V. Jacobson, Congestion avoidance and control,, ACM SIGCOMM Computer Communication Review, 18 (1988), 314. doi: 10.1145/52325.52356. [14] K. Jiang, X. F. Wang and Y. G. Xi, Bifurcation analysis of an Internet congestion control model,, in, (2004), 590. [15] F. P. Kelly, Models for a self-managed Internet,, Philos Trans Roy Soc A, 358 (2000), 2335. doi: 10.1098/rsta.2000.0651. [16] F. P. Kelly, A. Maulloo and D. K. H. Tan, Rate control in communication networks: shadow prices, proportional fairness, and stability,, J. Oper. Res. Soc., 49 (1998), 237. [17] S. Kunniyur and R. Srikant, End-to-end congestion control: utility functions, random lossed and ECN marks,, IEEE/ACM Transactions on Networking, 7 (2003), 689. doi: 10.1109/TNET.2003.818183. [18] Y. Kuznetsov, "Elements of Applied Bifurcation Theory," 2nd, edition, (1997). [19] C. G. Li, G. R. Chen, X. F. Liao and J. B. Yu, Hopf bifurcation in an Internet congestion control model,, Chaos Solitons & Fractals, 19 (2004), 853. doi: 10.1016/S0960-0779(03)00269-8. [20] S. Liu, T. Basar and R. Srikant, Controlling the Internet: A survey and some new results,, in, (2003), 3048. [21] F. Liu, Z. H. Guan and H. O. Wang, Controlling bifurcations and chaos in TCP-UDP-RED,, Nonlinear Analysis: Real World Applications, 11 (2010), 1491. doi: 10.1016/j.nonrwa.2009.03.005. [22] F. Paganini, A global stability result in network flow control,, Systems & Control Letters, 46 (2002), 165. doi: 10.1016/S0167-6911(02)00123-8. [23] G. Raina, Local bifurcation analysis of some dual congestion control algorithms,, IEEE Transactions on Automatic Control, 50 (2005), 1135. doi: 10.1109/TAC.2005.852566. [24] G. Raina and O. Heckmann, TCP: Local stability and Hopf bifurcation,, Performance Evaluation, 64 (2007), 266. doi: 10.1016/j.peva.2006.05.005. [25] Shigeki Tsuji, Tetsushi Ueta, Hiroshi Kawakami and Kazuyuki Aihara, Bifurcation of burst response in an Amari-Hopfield Neuron pair with a periodic external forces,, Electrical Engineering in Japan, 146 (2004), 43. doi: 10.1002/eej.10217. [26] R. Srikant, "The Mathematics of Internet Congestion Control,", Birkhäuser, (2004). [27] X. F. Wang, G. R. Chen and King-Tim Ko, A stability theorem for Internet congestion control,, Systems & Control Letters, 45 (2002), 81. doi: 10.1016/S0167-6911(01)00165-7. [28] Z. F. Wang and T. G. Chu, Delay induced Hopf bifurcation in a simplified network congestion control model,, Chaos Solitons & Fractals, 28 (2006), 161. doi: 10.1016/j.chaos.2005.05.047. [29] M. Xiao and J. D. Cao, Delayed feedback-based bifurcation control in an Internet congestion model,, J. Math. Anal. Appl., 332 (2007), 1010. doi: 10.1016/j.jmaa.2006.10.062. [30] J. Xu and K. W. Chung, A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems,, Science in China Series E, 52 (2009), 698. doi: 10.1007/s11431-009-0052-1. [31] J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks,, SIAM J. Applied Dynamical Sysyems, 6 (2007), 29. doi: 10.1137/040614207. [32] H. Y. Yang and Y. P. Tian, Hopf bifurcation in REM algorithm with communication delay,, Chaos, 25 (2005), 1093. doi: 10.1016/j.chaos.2004.11.085. [33] H. Y. Yang and S. Y. Zhang, Hopf bifurcation of end-to-end network congestion control algorithm,, 2007 IEEE International Conference on Control and Automation, (2007).

show all references

##### References:
 [1] T. Alpcan and T. Basar, Global stability analysis of an end-to-end congestion control scheme for general topology networks with delay,, in, (2003), 1092. [2] H. Brunner and S. Maset, Time transformations for delay differential equations,, Discrete Contin. Dyn. Syst. Ser A, 25 (2009), 751. doi: 10.3934/dcds.2009.25.751. [3] Z. Chen and P. Yu, Hopf bifurcation control for an Internet congestion model,, International Journal of Bifurcation and Chaos, 15 (2005), 2643. doi: 10.1142/S0218127405013587. [4] Y. Choi, Periodic delay effects on cutting dynamics,, Journal of Dynamics and Differential Equations, 17 (2005), 353. doi: 10.1007/s10884-005-3145-y. [5] S. L. Das and A. Chatterjee, Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations,, Nonlinear Dynamics, 30 (2002), 323. doi: 10.1023/A:1021220117746. [6] D. W. Ding, J. Zhu, X. S. Luo and Y. L. Liu, Delay induced Hopf bifurcation in a dual model of Internet congestion control algorithm,, Nonlinear Analysis: Real World Applications, 10 (2009), 2873. doi: 10.1016/j.nonrwa.2008.09.007. [7] S. Floyd and V. Jacobson, Random early detection gate-ways for congestion avoidance,, IEEE/ACM Transctions on Networks, 1 (1993), 397. [8] D. E. Gilsinn, Estimating critical Hopf bifurcation parameters for a second-order delay differential equation with application to machine tool chatter,, Nonlinear Dynamics, 30 (2002), 103. doi: 10.1023/A:1020455821894. [9] S. T. Guo, G. Feng, X. F. Liao and Q. Liu, Hopf bifurcation control in a congestion control model via dynamic delayed feedback,, Chaos, 18 (2008), 043104. doi: 10.1063/1.2998220. [10] S. T. Guo, X. F. Liao and C. D. Li, Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay,, Nonlinear Analysis: Real World Applications, 9 (2008), 1292. doi: 10.1016/j.nonrwa.2007.03.006. [11] S. T. Guo, X. F. Liao, Q. Liu and C. D. Li, Necessary and sufficient conditions for Hopf bifurcation in exponential RED algorithm with communication delay,, Nonlinear Analysis: Real World Applications, 9 (2008), 1768. doi: 10.1016/j.nonrwa.2007.05.014. [12] J. Hale, "Theory of Functional Differential Equations,", World Publishing Corporation, (2003). [13] V. Jacobson, Congestion avoidance and control,, ACM SIGCOMM Computer Communication Review, 18 (1988), 314. doi: 10.1145/52325.52356. [14] K. Jiang, X. F. Wang and Y. G. Xi, Bifurcation analysis of an Internet congestion control model,, in, (2004), 590. [15] F. P. Kelly, Models for a self-managed Internet,, Philos Trans Roy Soc A, 358 (2000), 2335. doi: 10.1098/rsta.2000.0651. [16] F. P. Kelly, A. Maulloo and D. K. H. Tan, Rate control in communication networks: shadow prices, proportional fairness, and stability,, J. Oper. Res. Soc., 49 (1998), 237. [17] S. Kunniyur and R. Srikant, End-to-end congestion control: utility functions, random lossed and ECN marks,, IEEE/ACM Transactions on Networking, 7 (2003), 689. doi: 10.1109/TNET.2003.818183. [18] Y. Kuznetsov, "Elements of Applied Bifurcation Theory," 2nd, edition, (1997). [19] C. G. Li, G. R. Chen, X. F. Liao and J. B. Yu, Hopf bifurcation in an Internet congestion control model,, Chaos Solitons & Fractals, 19 (2004), 853. doi: 10.1016/S0960-0779(03)00269-8. [20] S. Liu, T. Basar and R. Srikant, Controlling the Internet: A survey and some new results,, in, (2003), 3048. [21] F. Liu, Z. H. Guan and H. O. Wang, Controlling bifurcations and chaos in TCP-UDP-RED,, Nonlinear Analysis: Real World Applications, 11 (2010), 1491. doi: 10.1016/j.nonrwa.2009.03.005. [22] F. Paganini, A global stability result in network flow control,, Systems & Control Letters, 46 (2002), 165. doi: 10.1016/S0167-6911(02)00123-8. [23] G. Raina, Local bifurcation analysis of some dual congestion control algorithms,, IEEE Transactions on Automatic Control, 50 (2005), 1135. doi: 10.1109/TAC.2005.852566. [24] G. Raina and O. Heckmann, TCP: Local stability and Hopf bifurcation,, Performance Evaluation, 64 (2007), 266. doi: 10.1016/j.peva.2006.05.005. [25] Shigeki Tsuji, Tetsushi Ueta, Hiroshi Kawakami and Kazuyuki Aihara, Bifurcation of burst response in an Amari-Hopfield Neuron pair with a periodic external forces,, Electrical Engineering in Japan, 146 (2004), 43. doi: 10.1002/eej.10217. [26] R. Srikant, "The Mathematics of Internet Congestion Control,", Birkhäuser, (2004). [27] X. F. Wang, G. R. Chen and King-Tim Ko, A stability theorem for Internet congestion control,, Systems & Control Letters, 45 (2002), 81. doi: 10.1016/S0167-6911(01)00165-7. [28] Z. F. Wang and T. G. Chu, Delay induced Hopf bifurcation in a simplified network congestion control model,, Chaos Solitons & Fractals, 28 (2006), 161. doi: 10.1016/j.chaos.2005.05.047. [29] M. Xiao and J. D. Cao, Delayed feedback-based bifurcation control in an Internet congestion model,, J. Math. Anal. Appl., 332 (2007), 1010. doi: 10.1016/j.jmaa.2006.10.062. [30] J. Xu and K. W. Chung, A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems,, Science in China Series E, 52 (2009), 698. doi: 10.1007/s11431-009-0052-1. [31] J. Xu, K. W. Chung and C. L. Chan, An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedbacks,, SIAM J. Applied Dynamical Sysyems, 6 (2007), 29. doi: 10.1137/040614207. [32] H. Y. Yang and Y. P. Tian, Hopf bifurcation in REM algorithm with communication delay,, Chaos, 25 (2005), 1093. doi: 10.1016/j.chaos.2004.11.085. [33] H. Y. Yang and S. Y. Zhang, Hopf bifurcation of end-to-end network congestion control algorithm,, 2007 IEEE International Conference on Control and Automation, (2007).
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