American Institute of Mathematical Sciences

May  2017, 22(3): 1189-1206. doi: 10.3934/dcdsb.2017058

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

* The corresponding author

Received  September 2015 Revised  January 2016 Published  January 2017

Fund Project: The work is supported by the National Natural Science Foundation of China (11502168) the Fundamental Research Funds for the Central Universities, the Program for Young Excellent Talents at Tongji University (S.Z) and NSERC of Canada (203786 46310 2000)(Y.Y).

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.

Citation: Shu Zhang, Yuan Yuan. The Filippov equilibrium and sliding motion in an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 1189-1206. doi: 10.3934/dcdsb.2017058
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.

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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.
Configuration of marking function P
The topology of connection with single user group and $n$ links where $W(t)$ is the averaged window size of the user group and $L_i$ represents the $i$th link, $i=1, 2, \cdots, n$
The numerical simulation of (13) for (a) $q_1(t)$ and (b) $q_2(t)$ where $\tau_1=0.11$, $\tau_2=0.13$, $N=10$, $C_1=200$, $C_2=150$, $B_1=50$, $B_2=15$, $b_{1, 1}=b_{2, 1}=0.2$, $b_{1, 2}=b_{2, 2}=0.95$, $P_{1, max}=P_{2, max}=0.4$, $W(0)=0$, $q_1(0)=0, q_2(0)=0$. From the time history of $q_1(t)$ and $q_2(t)$, it is clear that the links are used simultaneously since $q_1(t)$ and $q_2(t)$ are not zero at any moment. This suggests that the two vector fields may be combined in some manner
The numerical continuation (a) by DDE-BIFTOOL [7] and simulation (b)-(e) by XPP-AUT [8] for (13) as $B_2=30$. Initial conditions: $W(0)=6$, $q_1(0)=11, q_2(0)=7$. (a) shows the distribution of the real and imaginary parts of the eigenvalues of $\dot{\mathbf{X}}(t)=\mathbf{K}(\mathbf{X}(t))$, $\lambda=\hat{\lambda}$ given by (16). The real parts of all the eigenvalues are negative and consequently the pseudo-equilibrium is stable in $\Sigma_s$ which is confirmed by the time history plots (b), (c) and (d). (e) shows that the dynamics of the system is restricted to $\Sigma$, in other words, $\Sigma_s$ is locally attractive
The numerical continuation (a) and simulation (b)-(e) for (13) when $B_2=15$. $W(0)=6$, $q_1(0)=11, q_2(0)=7$. Red dots in (a) represents the eigenvalue with positive real part. (a) shows the distribution of the eigenvalues of $\dot{\mathbf{X}}(t)=\mathbf{K}(\mathbf{X}(t))$. The maximum of the real parts of the eigenvalues is positive and consequently the pseudo-equilibrium is unstable in $\Sigma_s$ which is confirmed by the time history plots (b), (c) and (d). (e) shows that the dynamics of the system is restricted to $\Sigma$, implying the local attractivity of $\Sigma_s$
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