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An Ordinary Differential Equation Framework for Stability Analysis of Networks with Finite Buffers

Xinyu Wu, Dan Wu, Eytan Modiano

TL;DR

An ordinary differential equation (ODE) model is proposed to capture the queuing dynamics and analyze the stability in buffered communication networks with general topology, and is useful for analyzing the effect of finite buffers on network stability.

Abstract

We consider the problem of network stability in finite-buffer systems. We observe that finite buffer may affect stability even in simplest network structure, and we propose an ordinary differential equation (ODE) model to capture the queuing dynamics and analyze the stability in buffered communication networks with general topology. For single-commodity systems, we propose a sufficient condition, which follows the fundamental idea of backpressure, for local transmission policies to stabilize the networks based on ODE stability theory. We further extend the condition to multi-commodity systems, with an additional restriction on the coupling level between different commodities, which can model networks with per-commodity buffers and shared buffers. The framework characterizes a set of policies that can stabilize buffered networks, and is useful for analyzing the effect of finite buffers on network stability.

An Ordinary Differential Equation Framework for Stability Analysis of Networks with Finite Buffers

TL;DR

An ordinary differential equation (ODE) model is proposed to capture the queuing dynamics and analyze the stability in buffered communication networks with general topology, and is useful for analyzing the effect of finite buffers on network stability.

Abstract

We consider the problem of network stability in finite-buffer systems. We observe that finite buffer may affect stability even in simplest network structure, and we propose an ordinary differential equation (ODE) model to capture the queuing dynamics and analyze the stability in buffered communication networks with general topology. For single-commodity systems, we propose a sufficient condition, which follows the fundamental idea of backpressure, for local transmission policies to stabilize the networks based on ODE stability theory. We further extend the condition to multi-commodity systems, with an additional restriction on the coupling level between different commodities, which can model networks with per-commodity buffers and shared buffers. The framework characterizes a set of policies that can stabilize buffered networks, and is useful for analyzing the effect of finite buffers on network stability.

Paper Structure

This paper contains 16 sections, 12 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Single-Commodity System
  3. Basic Setting
  4. Stability Analysis
  5. Local Result
  6. Global Result
  7. Existence of Equilibrium Point
  8. Multi-Commodity System
  9. Basic Setting
  10. Stability Analysis
  11. Existence of Equilibrium Point
  12. Shared buffer Case Study: Switched Networks
  13. Conclusion
  14. Proof of Theorem \ref{['prop:single1']}
  15. Proof of Lemma \ref{['lem:positive_vector_exist']}
  16. ...and 1 more sections

Key Result

Theorem 1

For a local policy, if there exists an equilibrium point $\mathbf{q}^*$, such that at $\mathbf{q}=\mathbf{q}^*$, then the ODE is asymptotically stable at $\mathbf{q}^*$.

Figures (2)

  • Figure 1: Finite buffer may affect stability result. On the right is an example of the queue dynamics in node $K$ following the backpressure policy. Due to its higher injection rate into the buffer of $K$, commodity 1 takes up higher ratio in node $K$ and squeezes out commodity 2 under backpressure, and in the final state we can show that average number of commodity 2 packets in the buffer of node $K$ is $1.5$, which arises from $1.5=(\mu_1/c_{1K})\times c_{2K}$, with details deferred to Section \ref{['subsec:shared-buffer']}. Therefore the actual throughput of commodity 2 is $1.5$, less than $\mu_2=3$, due to the finite buffer.
  • Figure 2: One-hop system with $C$ commodities

Theorems & Definitions (25)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Theorem 3
  • ...and 15 more