Suppression of oscillatory congestion via trunk link bandwidth and control gain in star network

doi:10.1007/s10483-019-2411-9

Abstract

Abstract: The time delay-induced instability in an Internet congestion control model is investigated. The star topology is considered, and the link bandwidth ratio and the control gain are selected as the tunable parameters for congestion suppression. The stability switch boundary is obtained by the eigenvalue analysis for the linearized system around the equilibrium. To investigate the oscillatory congestion when the equilibrium becomes unstable, the center manifold reduction and the normal form theory are used to study the periodic oscillation induced by the delay. The theoretical analysis and numerical simulation show that the ratio between bandwidths of the trunk link and the regular link, rather than these bandwidths themselves, is crucial for the stability of the congestion control system. The present results demonstrate that it is not always effective to increase the link bandwidth ratio for stabilizing the system, and for some certain delays, adjusting the control gain is more efficient.

Key words: local strictly pseudocontractive, local strongly accretive, uniformly smooth Banach space, delayed differential equation, Internet congestion control, Hopf bifurcation, stability, normal form

2010 MSC Number:

34K20

Sainan WANG, Shu ZHANG, Jian XU. Suppression of oscillatory congestion via trunk link bandwidth and control gain in star network. Applied Mathematics and Mechanics (English Edition), 2019, 40(1): 25-48.

TrendMD

References

[1] JACOBSON, V. Congestion Avoidance and Control, Artech House, California (1988)
[2] ALLMAN, M., PAXSON, V., and Stevens, W. TCP congestion control. RFC 2581(1999)
[3] DEB, S. and SRIKANT, R. Global stability of congestion controllers for the Internet. IEEE Transactions on Automatic Control, 48(6), 1055-1060(2003)
[4] FLOYD, S. and FALL, K. Promoting the use of end-to-end congestion control in the Internet. IEEE/ACM Transactions on Networking, 7(4), 458-472(1999)
[5] HOE, J. C. Improving the start-up behavior of a congestion control scheme for TCP. ACM SIGCOMM Computer Communication Review, 26(4), 270-280(1996)
[6] WANG, Z. and CROWCROFT, J. Eliminating periodic packet losses in the 4.3-Tahoe BSD TCP congestion control algorithm. ACM SIGCOMM Computer Communication Review, 22(2), 9-16(1992)
[7] PADHYE, J., FIROIU, V., TOWSLEY, D. F., and KUROSE, J. F. Modeling TCP Reno performance:a simple model and its empirical validation. IEEE/ACM Transactions on Networking, 8(2), 133-145(2000)
[8] PARVEZ, N., MAHANTI, A., and WILLIAMSON, C. An analytic throughput model for TCP NewReno. IEEE/ACM Transactions on Networking, 18(2), 448-461(2010)
[9] PEI, L. J., MU, X. M., WANG, R. M., and YANG, J. P. Dynamics of the Intetnet TCP-RED congestion control system. Nonlinear Analysis:Real World Applications, 12(2), 947-955(2011)
[10] LAPSLEY, D. E. and LOW, S. Random early marking for Internet congestion control. Proceeding of Global Telecommunications Conference, IEEE, Brazil (1999)
[11] ZHAN, Z., ZHU, J., and XU, D. Stability analysis in an AVQ model of Internet congestion control algorithm. The Journal of China Universities of Posts and Telecommunications, 19(4), 22-28(2012)
[12] RYU, S., RUMP, C., and QIAO, C. Advances in active queue management (AQM) based TCP congestion control. Telecommunication Systems, 25(3/4), 317-351(2004)
[13] CHEN, X., WONG, S. C., TSE, C. K., and LAU, F. Oscillation and period doubling in TCP/RED systems:analysis and verification. International Journal of Bifurcation and Chaos, 18(5), 1459- 1475(2008)
[14] GIBBENS, R. J. and KELLY, F. P. Resource pricing and the evolution of congestion control. Automatica, 35(12), 1969-1985(1999)
[15] ZHANG, S., XU, J., and CHUNG, K. W. Desynchronization-based congestion suppression for a star-type Internet system with arbitrary dimension. Neurocomputing, 266, 42-55(2017)
[16] LOW, S. and PAGANINI, F. Internet congestion control. IEEE Control Systems, 22(1), 28-43(2002)
[17] PAGANINI, F., WANG, Z., DOYLE, J., and LOW, S. Congestion control for high performance, stability, and fairness in general networks. IEEE/ACM Transactions on Networking, 13(1), 43-56(2005)
[18] KATABI, D., HANDLEY, M., and ROHRS, C. Congestion control for high bandwidth-delay product networks. Proceedings of the 2002 Conference on Applications, Technologies, Architectures, and Protocols for Computer Communications, ACM, Pittsburgh (2002)
[19] JOHARI, R. and TAN, D. End-to-end congestion control for the Internet:delays and stability. IEEE/ACM Transactions on Networking, 9(6), 818-832(2001)
[20] DONG, T., LIAO, X. F., and HUANG, T. W. Dynamics of a congestion control model in a wireless access network. Nonlinear Analysis:Real World Applications, 14(1), 671-683(2013)
[21] MANFREDI, S., TUCCI, E. D., and LATORA, V. Mobility and congestion in dynamical multilayer networks with finite storage capacity. Physical Review Letters, 120(6), 068301(2018)
[22] LI, C. G., CHEN, G. R., and LIAO, X. F. Hopf bifurcation in an Internet congestion control model. Chaos, Solitons and Fractals, 19(4), 853-862(2004)
[23] CHEN, Z. and YU, P. Hopf bifurcation control for an internet congestion model. International Journal of Bifurcation and Chaos, 15(8), 2643-2651(2005)
[24] KELLY, F. P. Mathematical Modelling of the Internet, Springer, Berlin/Heidelberg (2001)
[25] HOLLOT, C. V., MISRA, V., TOWSLEY, D., and GONG, W. B. A control theoretic analysis of RED. Twentieth Annual Joint Conference of the IEEE Computer and Communications Society, IEEE, Anchorage (2001)
[26] MISRA, V., GONG, W. B., and TOWSLEY, D. Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED. Proceedings of the conference on Applications, Technologies, Architectures, and Protocols for Computer Communication, ACM, Stockholm (2000)
[27] SRIKANT, R. and YING, L. Communication Networks:An optimization, Control and Stochastic Networks Perspective, Cambridge University Press, New York (2014)
[28] STÉGER, J., VADERNA, P., and VATTAY, G. On the propagation of congestion waves in the Internet. Physica A:Statistical Mechanics and Its Applications, 360(1), 134-142(2001)
[29] XU, W. Y., CAO, J. D., and XIAO, M. Bifurcation analysis of a class of (n+1)-dimension Internet congestion control systems. International Journal of Bifurcation and Chaos, 25(2), 1-17(2015)
[30] ZHANG, S., XU, J., and CHUNG, K. W. Stability switch boundaries in an Internet congestion control model with diverse time delays. International Journal of Bifurcation and Chaos, 23, 1330016(2003)
[31] XU, C. J., TANG, X. H., and LIAO, M. X. Local hopf bifurcation and global existence of periodic solutions in TCP sysmtem. Applied Mathematics and Mechanics (English Edition), 31(6), 775-786(2010) https://doi.org/10.1007/s10483-010-1312-x
[32] WANG, Y., XHEN, J., YANG, Z. M., ZHANG, Z. K., ZHOU, T., and SUN, G. Q. Glaobal analysis of an SIS model with an infective vector on complex networks. Nonlinear Analysis:Real World Applications, 13(2), 543-557(2012)
[33] FARIA, T. and MAGALHAES, L. T. Normal forms for retarded functional differential equations and applications to Bogdanov-Takens singularity. Journal of Differential Equations, 122(2), 201-224(1995)
[34] ENGELBORGHS, K., LUZYANINA, T., SAMAEY, G., ROOSE, D., and VERHEYDEN, K. DDE-BIFTOOL v. 2.03:a Matlab package for bifurcation analysis of delay differential equations. http://twr.cs.kuleuven.be/research/software/delay/ddebiftool.shtml (2007)
[35] WANG, Q. and WANG, Z. H. An algorithm for the labeling stable regions of a class of timedelay systems with abscissa. Transactions of Nanjing University of Aeronanutics and Astronautics, 35(1), 94-100(2018)
[36] XU, Q. and WANG, Z. H. Exact stability test of neutral delay differential equations via a rough estimation of the testing integral. International Journal of Dynamics and Control, 2(1), 154-163(2014)
[37] XU, Q., STEPAN, G., and WANG, Z. H. Delay-dependent stability analysis by using delayindependent integral evaluation. Automatica, 70(8), 153-157(2016)