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The Filippov equilibrium and sliding motion in an internet congestion control model

  • * The corresponding author

    * The corresponding author
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).
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  • 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.

    Mathematics Subject Classification: Primary:34K11, 34K18;Secondary:94C99.


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  • Figure 1.  Configuration of marking function P

    Figure 2.  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, $i=1, 2, \cdots, n$

    Figure 3.  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

    Figure 4.  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

    Figure 5.  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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