Skip to main content
SpringerLink
Log in

Linear stability and Hopf bifurcation in an exponential RED algorithm model

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

We consider a novel congestion control model, i.e., the exponential RED algorithm with a single link and single source. Using the gain parameter of the system instead of the delay as the bifurcation parameter, the linear stability and Hopf bifurcation are investigated, and the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. Numerical simulations are carried out to illustrate our theoretical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Kelly, F.P., Maulloo, A.K., Tan, D.K.H.: Rate control in communication networks: shadow prices, proportional fairness and stability. J. Oper. Res. Soc. 49, 237–252 (1998)

    Article  MATH  Google Scholar 

  2. Jacobson, V.: Congestion avoidance and control. ACM SIGCOMM Comput. Commun. Rev. 18(4), 314–329 (1988)

    Article  Google Scholar 

  3. Braden, B., Clark, D., Crowcroft, J., Davie, B., Deering, S., Estrin, D., Floyd, S., Jacobson, V., Minshall, G., Partridge, C., Peterson, L., Ramakrishnan, K., Shenker, S., Wroclawski, J., Zhang, L.: Recommendations on queue management and congestion avoidance in the Internet. IETF RFC2309 (1998)

  4. Liu, S., Başar, T., Srikant, R.: Exponential-RED: a stabilizing AQM scheme for low- and high-speed TCP protocols. IEEE/ACM Trans. Netw. 13(5), 1068–1081 (2005)

    Article  Google Scholar 

  5. Athuraliya, S., Low, S.H., Yin, Q.: REM: active queue management. IEEE Netw. 15, 48–53 (2001)

    Article  Google Scholar 

  6. Yin, Q., Low, S.H.: On stability of REM algorithm with uniform delay. GLOBECOM’02 IEEE 3, 2649–2653 (2002)

    Google Scholar 

  7. Long, C.-N., Wu, J., Guan, X.-P.: Local stability of REM algorithm with time-varying delays. IEEE Commun. Lett. 7(3), 142–144 (2003)

    Article  Google Scholar 

  8. Tian, Y.-P.: Stability analysis and design of the second-order congestion control for networks with heterogeneous delays. IEEE/ACM Trans. Netw. 13(5), 1082–1093 (2005)

    Article  Google Scholar 

  9. Sichitiu, M.L., Bauer, P.H.: Asymptotic stability of congestion control systems with multiple sources. IEEE Trans. Autom. Control 51(2), 292–298 (2006)

    Article  MathSciNet  Google Scholar 

  10. Low, S.H., Paganini, F., Wang, J., Doyle, J.C.: Linear stability of TCP/RED and a scalable control. Comput. Netw. 43, 633–647 (2003)

    Article  MATH  Google Scholar 

  11. Ranjan, P., Abed, E.H., La, R.J.: Nonlinear instabilities in TCP-RED. IEEE Trans. Netw. 12(6), 1079–1092 (2004)

    Article  Google Scholar 

  12. Wang, X., Eun, D.Y.: Local and global stability of TCP-newReno/RED with many flows. Comput. Commun. 30(5), 1091–1105 (2007)

    Article  Google Scholar 

  13. Yang, H.-Y., Tian, Y.-P.: Hopf bifurcation in REM algorithm with communication delay. Chaos Solitons Fractals 25(5), 1093–1105 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kelly, F., Raina, G., Voice, T.: Stability and fairness of explicit congestion control with small buffers. ACM SIGCOMM Comput. Commun. Rev. 38(3), 51–62 (2008)

    Article  Google Scholar 

  15. Michiels, W., Niculescu, S.I.: Stability analysis of a fluid flow model for TCP like behavior. Int. J. Bifur. Chaos 15, 2277–2282 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kunniyur, S., Srikant, R.: Analysis and design of adaptive virtual queue algorithm for active queue management. ACM SIGCOMM Comput. Commun. Rev. 31(4), 123–134 (2001)

    Article  Google Scholar 

  17. Li, C., Chen, G., Liao, X., Yu, J.: Hopf bifurcation in an Internet congestion control model. Chaos Solitons Fractals 19(4), 853–862 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, Z., Chu, T.: Delay induced Hopf bifurcation in a simplified network congestion control model. Chaos Solitons Fractals 28(1), 161–172 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Guo, S., Liao, X., Li, C.: Stability and Hopf bifurcation analysis in a novel congestion control model with communication delay. Nonlinear Anal. Real World Appl. 9, 1292–1309 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Raina, G.: Local bifurcation analysis of some dual congestion control algorithms. IEEE Trans. Autom. Control 50(8), 1135–1146 (2005)

    Article  MathSciNet  Google Scholar 

  21. Ding, D., Zhu, J., Luo, X.: Hopf bifurcation analysis in a fluid flow model of internet congestion control algorithm. Nonlinear Anal. Real World Appl. 10, 824–839 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xiao, M., Cao, J.: Delayed feedback-based bifurcation control in an Internet congestion model. J. Math. Anal. Appl. 332, 1010–1027 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Cao, J., Xiao, M.: Stability and Hopf bifurcation in a simplified BAM neural network with two time delays. IEEE Trans. Neural Netw. 18, 416–430 (2007)

    Article  Google Scholar 

  24. Yu, W., Cao, J., Chen, G.: Stability and Hopf bifurcation of a general delayed recurrent neural network. IEEE Trans. Neural Netw. 19, 845–854 (2008)

    Article  Google Scholar 

  25. Beretta, E., Kuang, Y.: Geometric stability switch criteria in delay differential systems with delay dependent parameters. SIAM J. Math. Anal. 33, 1144–1165 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Hassard, B., Kazarinoff, N., Wan, Y.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  27. Dieudonné, N.: Foundations of Modern Analysis. Academic Press, New York (1960)

    MATH  Google Scholar 

  28. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New York (1993)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. College of Mathematics and Econometrics, Hunan University, Changsha, Hunan, 410082, People’s Republic of China

    Haijun Hu & Lihong Huang

Authors
  1. Haijun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lihong Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lihong Huang.

Additional information

Research supported by 973 Program (2009CB326202), National Natural Science Foundation of China (10771055, 60835004) and Key Program of Hunan Basic Research for Applications (2008FJ2008).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hu, H., Huang, L. Linear stability and Hopf bifurcation in an exponential RED algorithm model. Nonlinear Dyn 59, 463–475 (2010). https://doi.org/10.1007/s11071-009-9553-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-009-9553-5

Keywords

Search

Navigation