# American Institute of Mathematical Sciences

September  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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] S. Floyd and V. Jacobson, Random early detection gate-ways for congestion avoidance,, IEEE/ACM Transctions on Networks, 1 (1993), 397.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. Hale, "Theory of Functional Differential Equations,", World Publishing Corporation, (2003).   Google Scholar [13] V. Jacobson, Congestion avoidance and control,, ACM SIGCOMM Computer Communication Review, 18 (1988), 314.  doi: 10.1145/52325.52356.  Google Scholar [14] K. Jiang, X. F. Wang and Y. G. Xi, Bifurcation analysis of an Internet congestion control model,, in, (2004), 590.   Google Scholar [15] F. P. Kelly, Models for a self-managed Internet,, Philos Trans Roy Soc A, 358 (2000), 2335.  doi: 10.1098/rsta.2000.0651.  Google Scholar [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.   Google Scholar [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.  Google Scholar [18] Y. Kuznetsov, "Elements of Applied Bifurcation Theory," 2nd, edition, (1997).   Google Scholar [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.  Google Scholar [20] S. Liu, T. Basar and R. Srikant, Controlling the Internet: A survey and some new results,, in, (2003), 3048.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] R. Srikant, "The Mathematics of Internet Congestion Control,", Birkhäuser, (2004).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar

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.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] S. Floyd and V. Jacobson, Random early detection gate-ways for congestion avoidance,, IEEE/ACM Transctions on Networks, 1 (1993), 397.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [12] J. Hale, "Theory of Functional Differential Equations,", World Publishing Corporation, (2003).   Google Scholar [13] V. Jacobson, Congestion avoidance and control,, ACM SIGCOMM Computer Communication Review, 18 (1988), 314.  doi: 10.1145/52325.52356.  Google Scholar [14] K. Jiang, X. F. Wang and Y. G. Xi, Bifurcation analysis of an Internet congestion control model,, in, (2004), 590.   Google Scholar [15] F. P. Kelly, Models for a self-managed Internet,, Philos Trans Roy Soc A, 358 (2000), 2335.  doi: 10.1098/rsta.2000.0651.  Google Scholar [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.   Google Scholar [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.  Google Scholar [18] Y. Kuznetsov, "Elements of Applied Bifurcation Theory," 2nd, edition, (1997).   Google Scholar [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.  Google Scholar [20] S. Liu, T. Basar and R. Srikant, Controlling the Internet: A survey and some new results,, in, (2003), 3048.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] R. Srikant, "The Mathematics of Internet Congestion Control,", Birkhäuser, (2004).   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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).   Google Scholar
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