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Generalized TCP-RED dynamical model for Internet congestion control

José M. Amigó, Guillem Duran, Angel Giménez, Oscar Martínez-Bonastre, José Valero

TL;DR

This work addresses Internet congestion control by extending the RED adaptive queue management framework to a beta-distributed drop probability with tunable parameters $(oldsymbol{eta})=(oldsymbol{eta})$. By formulating a discrete-time, one-dimensional dynamical system for the average queue size and proving global stability of the unique fixed point $q^*$ under monotonic and unimodal central dynamics, the authors derive practical tuning guidelines that expand stability domains beyond those of the original RED. Theoretical results on invariance of the central interval and avoidance of cycles, together with numerical simulations, demonstrate increased robustness to parameter variations and network conditions, suggesting a more stable adaptive congestion-control mechanism. The findings have direct implications for designing an AQM that maintains stable queueing behavior while accommodating real-time changes in network load and topology.

Abstract

Adaptive management of traffic congestion in the Internet is a complex problem that can gain useful insights from a dynamical approach. In this paper we propose and analyze a one-dimensional, discrete-time nonlinear model for Internet congestion control at the routers. Specifically, the states correspond to the average queue sizes of the incoming data packets and the dynamical core consists of a monotone or unimodal mapping with a unique fixed point. This model generalizes a previous one in that additional control parameters are introduced via the data packet drop probability with the objective of enhancing stability. To make the analysis more challenging, the original model was shown to exhibit the usual features of low-dimensional chaos with respect to several system and control parameters, e.g., positive Lyapunov exponents and Feigenbaum-like bifurcation diagrams. We concentrate first on the theoretical aspects that may promote the unique stationary state of the system to a global attractor, which in our case amounts to global stability. In a second step, those theoretical results are translated into stability domains for robust setting of the new control parameters in practical applications. Numerical simulations confirm that the new parameters make it possible to extend the stability domains, in comparison with previous results. Therefore, the present work may lead to an adaptive congestion control algorithm with a more stable performance than other algorithms currently in use.

Generalized TCP-RED dynamical model for Internet congestion control

TL;DR

This work addresses Internet congestion control by extending the RED adaptive queue management framework to a beta-distributed drop probability with tunable parameters . By formulating a discrete-time, one-dimensional dynamical system for the average queue size and proving global stability of the unique fixed point under monotonic and unimodal central dynamics, the authors derive practical tuning guidelines that expand stability domains beyond those of the original RED. Theoretical results on invariance of the central interval and avoidance of cycles, together with numerical simulations, demonstrate increased robustness to parameter variations and network conditions, suggesting a more stable adaptive congestion-control mechanism. The findings have direct implications for designing an AQM that maintains stable queueing behavior while accommodating real-time changes in network load and topology.

Abstract

Adaptive management of traffic congestion in the Internet is a complex problem that can gain useful insights from a dynamical approach. In this paper we propose and analyze a one-dimensional, discrete-time nonlinear model for Internet congestion control at the routers. Specifically, the states correspond to the average queue sizes of the incoming data packets and the dynamical core consists of a monotone or unimodal mapping with a unique fixed point. This model generalizes a previous one in that additional control parameters are introduced via the data packet drop probability with the objective of enhancing stability. To make the analysis more challenging, the original model was shown to exhibit the usual features of low-dimensional chaos with respect to several system and control parameters, e.g., positive Lyapunov exponents and Feigenbaum-like bifurcation diagrams. We concentrate first on the theoretical aspects that may promote the unique stationary state of the system to a global attractor, which in our case amounts to global stability. In a second step, those theoretical results are translated into stability domains for robust setting of the new control parameters in practical applications. Numerical simulations confirm that the new parameters make it possible to extend the stability domains, in comparison with previous results. Therefore, the present work may lead to an adaptive congestion control algorithm with a more stable performance than other algorithms currently in use.
Paper Structure (15 sections, 20 theorems, 101 equations, 6 figures)

This paper contains 15 sections, 20 theorems, 101 equations, 6 figures.

Table of Contents

  1. Introduction
  2. A RED dynamical model
  3. A generalized RED dynamical model
  4. Basic properties
  5. Local stability
  6. Examples
  7. Global stability
  8. Monotonic case
  9. Unimodal case
  10. Implementation of the model
  11. Numerical simulations
  12. Benchmarking RED vs generalized RED
  13. Robustness domains in the $(\alpha ,\beta )$-plane
  14. Discussion
  15. Conclusion and outlook

Key Result

Proposition 1

The constants $A_{1}$ and $A_{2}$ of the RED model are subject to the constraint

Figures (6)

  • Figure 1: Network topology (source Duran2018).
  • Figure 2: Graphs of the mapping $f$ for the parameter settings (\ref{['Par1']}), (\ref{['Par2']}), and $\alpha$, $\beta$ and $p_{\max }$ as given on the top of each panel: monotonic increasing (top left), unimodal continuous (top right), unimodal discontinuous (bottom left), and bimodal (bottom right). $A_{1}=2265.8$, $A_{2}=3852.0$, except for the discontinuous graph, in which case $A_{1}=4531.6$ ($p_{\max }=0.25$). The point $q^{\ast }$ (the fixed point of $f$, Section \ref{['sec-5']}) is shown for further reference.
  • Figure 3: Bifurcation diagram of the average queue length with respect to the parameter $A_{1}$ for the values of $\alpha$ and $\beta$ shown on the top of the panels. System parameters are given in \ref{['Par3']}. Other control parameters: $q_{\min }=500$, $q_{\max }=1500$ and $w=0.15$. The constant $A_{2}$ is $3852$ and $A_{1}$ ranges in the interval $[0,2500]$.
  • Figure 4: Bifurcation diagram of the average queue length with respect to the parameter $A_{2}$ for the values of $\alpha$ and $\beta$ shown on the top of the panels. System parameters are given in \ref{['Par4']}. Other control parameters: $q_{\min }=500$, $q_{\max }=1500$, $w=0.15$ and $p_{\max }=1$. The constant $A_{1}$ is $2265.8$ and $A_{2}$ ranges in the interval $[4000,7000]$.
  • Figure 5: Bifurcation diagram of the average queue length with respect to the parameter $w$ for the values of $\alpha$ and $\beta$ shown on the top of the panels. System parameters are given in \ref{['Par1']}. Other control parameters: $q_{\min }=500$, $q_{\max }=1500$ and $p_{\max }=1$. The parameter $w$ ranges in the interval $[0,1]$. The constant $A_{1}$ is $2265.8$ and $A_{2}$ is $3852$.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 6
  • Remark 7
  • Example 8
  • Example 9
  • Example 10
  • ...and 19 more