Hopf bifurcation and oscillations in a communication network with heterogeneous delays

Abstract

In this paper, we investigate stability, bifurcation and oscillations arising in a single-link communication network model with a large number of heterogeneous users adopting a Transmission Control Protocol (TCP)-like rate control scheme with an Active Queue Management (AQM) router. In the system considered, different user delays are known and fixed but taken from a given distribution. It is shown that for any given distribution of delays, there exists a critical amount of feedback (due to AQM) at which the equilibrium loses stability and a limit cycling solution develops via a Hopf bifurcation. The nature (criticality) of the bifurcation is investigated with the aid of Lyapunov–Schmidt perturbation method. The results of the analysis are numerically verified and provide valuable insights into dynamics of the AQM control system.

Introduction

This paper is concerned with stability, bifurcation and oscillations arising in the model of a single-link communication network with heterogeneous users adopting a Transmission Control Protocol (TCP)-like rate control scheme and a single Active Queue Management (AQM) router. Each user is assumed to follow a TCP-like additive-increase multiplicative-decrease (AIMD) flow control scheme,

$${\dot{x}}_i\left(t\right)=\kappa [\frac{1}{d_i}-\beta x_i\left(t\right)x_i(t-\sqrt{d_i})p(t-\sqrt{d_i})]\text{,}$$

where $x_i\in {\mathbb{R}}^{+}\doteq [0,\infty )$ denotes the flow-rate for the $i$th-user, $0\le p\le 1$ is the observed rate of marking of its packets, $\sqrt{d_i}$ is the user specific delay and $\beta$ is a non-negative parameter. The delay parameters $d_i$ are assumed to be known and fixed but taken from a given distribution $g\left(d\right)$. We refer to Srikant (2004) for a detailed discussion and justification of the model considered in this paper.

The packet marking occurs at the link whose dynamics are next described. If the aggregate sending rate of users exceeds the capacity $C$ of the link, then the arriving packets are queued in the buffer $q$ of the link. The non-negative queue size evolves according to the ODE

$$\dot{q}\left(t\right)=\{\begin{array}{cc}\sum _{i=1}^M{x}_i\left(t\right)-C& \text{if }0<q<Q_{max}\text{,}\\ min(0,\sum _{i=1}^M{x}_i\left(t\right)-C)& \text{if }q=Q_{max}\text{,}\\ max(0,\sum _{i=1}^M{x}_i\left(t\right)-C)& \text{if }q=0\text{,}\end{array}$$

where we assume a maximum buffer size of $Q_{max}$ at which the queue saturates. Any incoming packet after this point is dropped; cf. Hollot, Misra, Towsley, and Gong (2002). $p(\cdot )$ in (1) is set by the AQM control and takes the general form $p=f\left(q\right)$, where $f$ is a given monotonic function of queue length with the range $[0,1]$.

In recent years, several studies have considered the issues of stability and bifurcation in simulations and models of communication networks (Firoiu and Borden, 2000, Raina et al., 2005, Ranjan et al., 2004, Veres and Boda, 2000). With simple fluid models, analytical methods including linear stability theory (Hollot et al., 2002), describing function methods (Han et al., 2005, Korkut, 2006), center manifold reduction (Ranjan & Abed, 2002), and normal form techniques (LI, Chen, Liao, & Yu, 2004) have been employed to investigate instability and bifurcation. Using these methods, a variety of bifurcation including Hopf bifurcation (Li et al., 2004, Raina, 2005, Ranjan and Abed, 2002), period doubling, and border-collision bifurcation (Ranjan and Abed, 2002, Ranjan et al., 2004) have been shown in models. Closely related to the topic of this paper is the work of Raina (2005), where local bifurcation results for a nonlinear delay differential equation model of communication network with a single discrete delay are described. Also related is Han et al. (2005) and Korkut (2006), where synchronization of TCP flows is studied by using a weakly coupled oscillator analogy.

The goal of this paper is to investigate the existence of an equilibrium and its bifurcation to non-equilibrium behavior vis-a-vis two important but complementary factors: (1) network delay and (2) network feedback. The heterogeneous user delays are modeled using distribution $g\left(d\right)$ with spread (or variance proportional to) $\gamma$, and the parameter $\beta$ is used to model the strength of feedback due to AQM queue. The stability and bifurcation results are described in the parameter space $(\beta ,\gamma )$.

The main analytical technique employed in this paper is to model user flow-rates $x_i\left(t\right)$ by functions $x(t,d_i)$ of the delay parameter $d_i$. For a large number of users, the delays are taken from a continuous distribution $g\left(d\right)$ and gives continuous approximation $x(t,d)$ of the user flow-rates. For modeling purposes, the communication network model is expressed in terms of such a continuous approximation. There are three kinds of analysis reported here: (1) analysis of the equilibrium solution, (2) analysis of stability properties of the equilibrium solution via linearization of the dynamic model, and (3) analysis of bifurcation and subsequent nonlinear oscillations via the method of Lyapunov–Schmidt.

The linear analysis entails consideration of both the discrete and the continuous spectrum in terms of the problem parameters. It is shown that the continuous spectrum always lies in the left half complex plane so stability can be analyzed by examining only the discrete spectrum. Numerical analysis of discrete spectrum shows that a complex eigenvalue pair crosses the imaginary axis as the feedback strength parameter $\beta$ is increased beyond a critical value. This suggests appearance of oscillations via a classical Hopf bifurcation and provides motivation for the nonlinear analysis.

The nonlinear analysis entails an analytic series expansion of the nonlinear oscillation solutions in terms of a small parameter $ϵ$. The unknown coefficients of the series are evaluated via a perturbation method based on Lyapunov–Schmidt reduction; cf., Ioose and Joseph (1980) and Kielhöfer (2003). The analysis leads to determination of the criticality of the Hopf bifurcation in terms of sign of a certain (Landau) coefficient. Validation results for both linear and the nonlinear analysis are presented using numerical simulations with the ODE model (1)–(2).

The remainder of the paper is organized as follows. Section 2 presents a summary of the numerical results with fluid model. These numerical results are explained with the aid of linear and nonlinear analysis presented in Sections 3 System equilibrium and stability, 4 Nonlinear analysis via a perturbation method, respectively. Section 5 presents an application of the methodology to investigate the effect of user delay distributions on instability in a single-link network. The conclusions appear in Section 6.