
- Previous Article
- DCDS-B Home
- This Issue
-
Next Article
Coexistence and competitive exclusion in an SIS model with standard incidence and diffusion
The Filippov equilibrium and sliding motion in an internet congestion control model
1. | School of Aerospace Engineering and Applied Mechanics, Tongji University, 1239 Siping Road, Shanghai 200092, China |
2. | Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's, Newfoundland, A1C5S7, Canada |
We consider an Internet congestion control system which is presented as a group of differential equations with time delay, modeling the random early detection (RED) algorithm. Although this model achieves success in many aspects, some basic problems are not clear. We provide the result on the existence of the equilibrium and the positivity and boundedness of the solution. Also, we implement the model by route switch mechanism, based on the minimum delay principle, to model the dynamic routing. For the simple network topology, we show that the Filippov solution exists under some restrictions on parameters. For the case with a single user group and two alternative links, we prove that the discontinuous boundary, or equivalently the sliding region, always exists and is locally attractive. This result implies that for some cases this type of routing may deviate from the purpose of the original design.
References:
[1] |
D. A. W. Barton, B. Krauskopf and R. E. Wilsona,
Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation, Dynam. Syst., 21 (2006), 289-311.
doi: 10.1080/14689360500539363. |
[2] |
D. Bertsekas,
Nonlinear Programming Athena Scientific, Belmont, MA, 1995. |
[3] |
Z. W. Cai, L. H. Huang, Guo and Z. Y. Chen,
On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33 (2012), 97-113.
doi: 10.1016/j.neunet.2012.04.009. |
[4] |
X. Chen, S. C. Wong, C. K. Tse and F. C. M. Lau,
Oscillation and period doubling in TCP/RED system, Int. J. of Bifurcation and Chaos, 18 (2008), 1459-1475.
doi: 10.1142/S0218127408021105. |
[5] |
K. Cooke and W. Huang,
On the problem of linearization for state-dependent delay differential equations, Proc. of AMS, 124 (1996), 1417-1426.
doi: 10.1090/S0002-9939-96-03437-5. |
[6] |
T. Dong, X. F. Liao and T. W. Huang,
Dynamics of a congestion control model in a wireless access network, Nonlinear Anal. RWA, 14 (2013), 671-683.
doi: 10.1016/j.nonrwa.2012.07.025. |
[7] |
K. Engelborghs, T. Luzyanina, G. Samaey, D. Roose and K. Verheyden, 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. |
[8] |
B. Ermentrout, XPPAUT5. 9-The differential equations tool, http://www.pitt.edu/~phase/, University of Pittsburgh, Pittsburgh, 2007. |
[9] |
A. F. Filippov,
Differential Equations with Discontinuous Right-Hand Side Mathematics and its Applications (Soviet Sereis), Kluwer Academic, Boston, MA, 1988. |
[10] |
M. Forti and P. Nistri,
Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Syst. I, 50 (2003), 1421-1435.
doi: 10.1109/TCSI.2003.818614. |
[11] |
M. Guardia, T. M. Seara and M. A. Teixeira,
Generic bifurcations of low dimension of planar Filippov systems, Journal of Differential equations, 250 (2011), 1967-2023.
doi: 10.1016/j.jde.2010.11.016. |
[12] |
C. V. Hollot, V. Misra, D. Towsley and W. B. Gong,
A control theoretic analysis of RED, In Proc. of IEEE Infocom., 3 (2006), 1510-1519.
doi: 10.1109/INFCOM.2001.916647. |
[13] |
V. Jacobson,
Congestion avoidance and control, Comput. Commun. Rev., 18 (1988), 314-329.
doi: 10.1145/52324.52356. |
[14] |
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), 237-252.
|
[15] |
P. Kowalczyk and M. Bernardo,
Two-parameter degenerate sliding bifurcations in Filippov systems, Physica D, 204 (2005), 204-229.
doi: 10.1016/j.physd.2005.04.013. |
[16] |
Y. A. Kuznetsov, S. Rinaldi and A. Gragnani,
One-parameter bifurcation in planar Filippov systems, Int. J. of Bifurcation and Chaos, 13 (2003), 2157-2188.
doi: 10.1142/S0218127403007874. |
[17] |
J. Llibre, P. R. da Silva and M. A. Teixeira,
Regularization of discontinuous vector fields on R3 via singular perturbation, J. Dyn. Diff. Equat., 19 (2007), 309-331.
doi: 10.1007/s10884-006-9057-7. |
[18] |
A. Machina, R. Edwards and P. Driessche,
Singular dynamics in gene network models, SIAM J. Applied Dynamical Systems, 12 (2013), 95-125.
doi: 10.1137/120872747. |
[19] |
V. Misra, W. B. Gong and D. Towsley,
Fluid based analysis of a network of AQM routers supporting TCP flows with an application to RED, Proc. of ACM/SIGCOMM, 30 (2000), 151-160.
doi: 10.1145/347059.347421. |
[20] |
J. Nagle,
Congestion control in IP/TCP internetworks, Comput. Commun. Rev., 14 (1984), 11-17.
doi: 10.17487/rfc0896. |
[21] |
R. Srikant,
The Mathematics of Internet Congestion Control Birkhäuser, Boston, 2004. |
[22] |
Z. Wang and J. Crowcroft,
Analysis of shortest-path routing algorithms in a dynamic network environment, Comput. Commun. Rev., 22 (1992), 63-71.
doi: 10.1145/141800.141805. |
[23] |
S. Zhang, K. W. Chung and J. Xu, Stability switch boundaries in an Internet congestion control model with diverse time delays Int. J. of Bifurcation and Chaos 23 (2013), 1330016, 24 pp. |
[24] |
S. Zhang and J. Xu,
Time-varying delayed feedback control for an Internet congestion control model, Discrete Continuous Dynam. Systems -B, 16 (2011), 653-668.
doi: 10.3934/dcdsb.2011.16.653. |
[25] |
S. Zhang and J. Xu,
Quasiperiodic motion induced by heterogeneous delays in a simplified internet congestion control model, Nonlinear Anal. RWA, 14 (2013), 661-670.
doi: 10.1016/j.nonrwa.2012.07.024. |
[26] |
S. Zhang, J. Xu and K. W. Chung,
On the stability and multi-stability of a TCP/RED congestion control model with state-dependent delay and discontinuous marking function, Commun. Nonlinear. Sci. Numer. Simul., 22 (2015), 269-284.
doi: 10.1016/j.cnsns.2014.09.020. |
show all references
References:
[1] |
D. A. W. Barton, B. Krauskopf and R. E. Wilsona,
Periodic solutions and their bifurcations in a non-smooth second-order delay differential equation, Dynam. Syst., 21 (2006), 289-311.
doi: 10.1080/14689360500539363. |
[2] |
D. Bertsekas,
Nonlinear Programming Athena Scientific, Belmont, MA, 1995. |
[3] |
Z. W. Cai, L. H. Huang, Guo and Z. Y. Chen,
On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33 (2012), 97-113.
doi: 10.1016/j.neunet.2012.04.009. |
[4] |
X. Chen, S. C. Wong, C. K. Tse and F. C. M. Lau,
Oscillation and period doubling in TCP/RED system, Int. J. of Bifurcation and Chaos, 18 (2008), 1459-1475.
doi: 10.1142/S0218127408021105. |
[5] |
K. Cooke and W. Huang,
On the problem of linearization for state-dependent delay differential equations, Proc. of AMS, 124 (1996), 1417-1426.
doi: 10.1090/S0002-9939-96-03437-5. |
[6] |
T. Dong, X. F. Liao and T. W. Huang,
Dynamics of a congestion control model in a wireless access network, Nonlinear Anal. RWA, 14 (2013), 671-683.
doi: 10.1016/j.nonrwa.2012.07.025. |
[7] |
K. Engelborghs, T. Luzyanina, G. Samaey, D. Roose and K. Verheyden, 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. |
[8] |
B. Ermentrout, XPPAUT5. 9-The differential equations tool, http://www.pitt.edu/~phase/, University of Pittsburgh, Pittsburgh, 2007. |
[9] |
A. F. Filippov,
Differential Equations with Discontinuous Right-Hand Side Mathematics and its Applications (Soviet Sereis), Kluwer Academic, Boston, MA, 1988. |
[10] |
M. Forti and P. Nistri,
Global convergence of neural networks with discontinuous neuron activations, IEEE Trans. Circuits Syst. I, 50 (2003), 1421-1435.
doi: 10.1109/TCSI.2003.818614. |
[11] |
M. Guardia, T. M. Seara and M. A. Teixeira,
Generic bifurcations of low dimension of planar Filippov systems, Journal of Differential equations, 250 (2011), 1967-2023.
doi: 10.1016/j.jde.2010.11.016. |
[12] |
C. V. Hollot, V. Misra, D. Towsley and W. B. Gong,
A control theoretic analysis of RED, In Proc. of IEEE Infocom., 3 (2006), 1510-1519.
doi: 10.1109/INFCOM.2001.916647. |
[13] |
V. Jacobson,
Congestion avoidance and control, Comput. Commun. Rev., 18 (1988), 314-329.
doi: 10.1145/52324.52356. |
[14] |
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), 237-252.
|
[15] |
P. Kowalczyk and M. Bernardo,
Two-parameter degenerate sliding bifurcations in Filippov systems, Physica D, 204 (2005), 204-229.
doi: 10.1016/j.physd.2005.04.013. |
[16] |
Y. A. Kuznetsov, S. Rinaldi and A. Gragnani,
One-parameter bifurcation in planar Filippov systems, Int. J. of Bifurcation and Chaos, 13 (2003), 2157-2188.
doi: 10.1142/S0218127403007874. |
[17] |
J. Llibre, P. R. da Silva and M. A. Teixeira,
Regularization of discontinuous vector fields on R3 via singular perturbation, J. Dyn. Diff. Equat., 19 (2007), 309-331.
doi: 10.1007/s10884-006-9057-7. |
[18] |
A. Machina, R. Edwards and P. Driessche,
Singular dynamics in gene network models, SIAM J. Applied Dynamical Systems, 12 (2013), 95-125.
doi: 10.1137/120872747. |
[19] |
V. Misra, W. B. Gong and D. Towsley,
Fluid based analysis of a network of AQM routers supporting TCP flows with an application to RED, Proc. of ACM/SIGCOMM, 30 (2000), 151-160.
doi: 10.1145/347059.347421. |
[20] |
J. Nagle,
Congestion control in IP/TCP internetworks, Comput. Commun. Rev., 14 (1984), 11-17.
doi: 10.17487/rfc0896. |
[21] |
R. Srikant,
The Mathematics of Internet Congestion Control Birkhäuser, Boston, 2004. |
[22] |
Z. Wang and J. Crowcroft,
Analysis of shortest-path routing algorithms in a dynamic network environment, Comput. Commun. Rev., 22 (1992), 63-71.
doi: 10.1145/141800.141805. |
[23] |
S. Zhang, K. W. Chung and J. Xu, Stability switch boundaries in an Internet congestion control model with diverse time delays Int. J. of Bifurcation and Chaos 23 (2013), 1330016, 24 pp. |
[24] |
S. Zhang and J. Xu,
Time-varying delayed feedback control for an Internet congestion control model, Discrete Continuous Dynam. Systems -B, 16 (2011), 653-668.
doi: 10.3934/dcdsb.2011.16.653. |
[25] |
S. Zhang and J. Xu,
Quasiperiodic motion induced by heterogeneous delays in a simplified internet congestion control model, Nonlinear Anal. RWA, 14 (2013), 661-670.
doi: 10.1016/j.nonrwa.2012.07.024. |
[26] |
S. Zhang, J. Xu and K. W. Chung,
On the stability and multi-stability of a TCP/RED congestion control model with state-dependent delay and discontinuous marking function, Commun. Nonlinear. Sci. Numer. Simul., 22 (2015), 269-284.
doi: 10.1016/j.cnsns.2014.09.020. |





[1] |
Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 |
[2] |
Zuowei Cai, Jianhua Huang, Liu Yang, Lihong Huang. Periodicity and stabilization control of the delayed Filippov system with perturbation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1439-1467. doi: 10.3934/dcdsb.2019235 |
[3] |
Bertrand Maury, Aude Roudneff-Chupin, Filippo Santambrogio, Juliette Venel. Handling congestion in crowd motion modeling. Networks and Heterogeneous Media, 2011, 6 (3) : 485-519. doi: 10.3934/nhm.2011.6.485 |
[4] |
Jian-Wu Xue, Xiao-Kun Xu, Feng Zhang. Big data dynamic compressive sensing system architecture and optimization algorithm for internet of things. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1401-1414. doi: 10.3934/dcdss.2015.8.1401 |
[5] |
Soliman A. A. Hamdallah, Sanyi Tang. Stability and bifurcation analysis of Filippov food chain system with food chain control strategy. Discrete and Continuous Dynamical Systems - B, 2020, 25 (5) : 1631-1647. doi: 10.3934/dcdsb.2019244 |
[6] |
Ramasamy Kavikumar, Boomipalagan Kaviarasan, Yong-Gwon Lee, Oh-Min Kwon, Rathinasamy Sakthivel, Seong-Gon Choi. Robust dynamic sliding mode control design for interval type-2 fuzzy systems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1839-1858. doi: 10.3934/dcdss.2022014 |
[7] |
Carles Bonet-Revés, Tere M-Seara. Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3545-3601. doi: 10.3934/dcds.2016.36.3545 |
[8] |
Hao Sun, Shihua Li, Xuming Wang. Output feedback based sliding mode control for fuel quantity actuator system using a reduced-order GPIO. Discrete and Continuous Dynamical Systems - S, 2021, 14 (4) : 1447-1464. doi: 10.3934/dcdss.2020375 |
[9] |
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control and Related Fields, 2021, 11 (4) : 905-934. doi: 10.3934/mcrf.2020051 |
[10] |
Carl. T. Kelley, Liqun Qi, Xiaojiao Tong, Hongxia Yin. Finding a stable solution of a system of nonlinear equations arising from dynamic systems. Journal of Industrial and Management Optimization, 2011, 7 (2) : 497-521. doi: 10.3934/jimo.2011.7.497 |
[11] |
D. J. W. Simpson, R. Kuske. Stochastically perturbed sliding motion in piecewise-smooth systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2889-2913. doi: 10.3934/dcdsb.2014.19.2889 |
[12] |
Cecilia Cavaterra, Denis Enăchescu, Gabriela Marinoschi. Sliding mode control of the Hodgkin–Huxley mathematical model. Evolution Equations and Control Theory, 2019, 8 (4) : 883-902. doi: 10.3934/eect.2019043 |
[13] |
Shuren Liu, Qiying Hu, Yifan Xu. Optimal inventory control with fixed ordering cost for selling by internet auctions. Journal of Industrial and Management Optimization, 2012, 8 (1) : 19-40. doi: 10.3934/jimo.2012.8.19 |
[14] |
Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Fractional optimal control problems on a star graph: Optimality system and numerical solution. Mathematical Control and Related Fields, 2021, 11 (1) : 189-209. doi: 10.3934/mcrf.2020033 |
[15] |
Zhi-Xue Zhao, Mapundi K. Banda, Bao-Zhu Guo. Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2533-2554. doi: 10.3934/dcdsb.2020021 |
[16] |
Hernán Cendra, María Etchechoury, Sebastián J. Ferraro. Impulsive control of a symmetric ball rolling without sliding or spinning. Journal of Geometric Mechanics, 2010, 2 (4) : 321-342. doi: 10.3934/jgm.2010.2.321 |
[17] |
Antonia Katzouraki, Tania Stathaki. Intelligent traffic control on internet-like topologies - integration of graph principles to the classic Runge--Kutta method. Conference Publications, 2009, 2009 (Special) : 404-415. doi: 10.3934/proc.2009.2009.404 |
[18] |
Tao Jiang, Liwei Liu. Analysis of a batch service multi-server polling system with dynamic service control. Journal of Industrial and Management Optimization, 2018, 14 (2) : 743-757. doi: 10.3934/jimo.2017073 |
[19] |
Mingyong Lai, Hongming Yang, Songping Yang, Junhua Zhao, Yan Xu. Cyber-physical logistics system-based vehicle routing optimization. Journal of Industrial and Management Optimization, 2014, 10 (3) : 701-715. doi: 10.3934/jimo.2014.10.701 |
[20] |
Chenyin Wang, Yaodong Ni, Xiangfeng Yang. The inventory replenishment policy in an uncertain production-inventory-routing system. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021196 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]