Dynamics of a Discrete Internet Congestion Control Model

We consider a discrete Internet model with a single link accessed by a single source, which responds to congestion signals from the network. Firstly, the stability of the equilibria of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark-Sacker bifurcations occur when the delay passes a sequence of critical values. Then, the explicit algorithm for determining the direction of the Neimark-Sacker bifurcations and the stability of the bifurcating periodic solutions is derived. Finally, some numerical simulations are given to verify the theoretical analysis.


Introduction
Congestion control algorithms and active queue management AQM for Internet have been the focus of intense research since the seminal work of Kelly et al. 1 .The primal algorithms they have introduced are based on explicit feedback functions, where the sum of user utilities is maximized within the capacity constraints of the links.In primal algorithms, the users adapt the source rates dynamically based on the congestion signal generated by the link, and the links select a static law to determine the link prices directly from the arrival rates at the links 2 .In 3 , a stability condition was provided for a single proportionally fair congestion controller with delayed feedback.Since then, this result was extended to networks in 4, 5 and for different classes of controllers in 6 .In this paper, we consider an Internet model with a single link and single source, which can be formulated as a congestion control system with feedback delay.The model is described by 3, 7, 8 x t k w − x t − τ p x t − τ , 1.1 where x t is the sending rate of the source at time t, k is a positive gain parameter, τ is the sum of forward and return delays, w > 0 is a target set point , and the congestion indication function p • is increasing, nonnegative, and not identically zero, which can be viewed as the probability that a packet at the resource receives a feedback congestion indication signal.System 1.1 corresponds to a rate control algorithm for the user, which can be used to adjust the sending rate x t so that the expected number of signals received by the user will tend to target w.A great deal of research has been devoted to the asymptotic behaviour, periodic solutions, global stability, and other properties of this model, for which, we refer to 3, 7-9 .
But considering the need of scientific computation and real-time simulation, our interest is focused on the behaviors of discrete dynamics system corresponding to 1.1 .Many authors considered the numerical approximation of a scalar delay differential equation by using different numerical methods, such as θ-method, Euler method, and Trapezoidal method see 10-15 .In this paper, we use the forward Euler scheme to make the discretization for system 1.1 .
Moreover, it is also of interest to find what will happen when the congestion control system loses stability.The purpose of this paper is to discuss this version as a discrete dynamical system by using Neimark-Sacker bifurcation theory of discrete systems.We not only investigate the stability of the fixed point and the existence of the Neimark-Sacker bifurcations, but also the stability and direction of the Neimark-Sacker bifurcation of the discrete system.
The paper is organized as follows: in Section 2, we analyze the distribution of the characteristic equation associated with the discrete model and obtain the existence of the local Neimark-Sacker bifurcation.In Section 3, the direction and stability of closed invariant curve from the Neimark-Sacker bifurcation of the discrete delay model are determined by using the theories of discrete systems in 16 .In the final section, some computer simulations are performed to illustrate the analytical results found.

Stability Analysis
Let u t x tτ .Then 1.1 can be rewritten as We consider sufficient small step size of the form h 1/m, where m ∈ Z .The Euler method applied to this equation yields the delay difference equation where u n is an approximate value to u nh .Let u * be a positive equilibrium of 2.2 , then we have u * p u * w.

2.3
Set y n u n − u * , then there follows: By introducing a new variable Y n y n , y n−1 , . . ., y n−m T , we can rewrite 2.4 in the form where F F 0 , F 1 , . . ., F m T , and

2.6
Clearly the origin is a fixed point of 2.5 , and the linear part of 2.5 is where The characteristic equation of A is given by It is well known that the stability of the zero equilibrium solution of 2.5 depends on the distribution of the zeros of the roots of 2.9 .In order to proof the existence of the local Neimark-Sacker bifurcation at fixed point, we need some lemmas as follows.

Lemma 2.1.
There exists a τ > 0 such that for 0 < τ < τ all roots of 2.9 have modulus less than one.
Consider the root λ τ such that λ 0 1.This root depends continuously on τ and is a differential function of τ.From 2.9 , we have

2.11
Since p • is increasing and nonnegative, we have So with the increasing of τ > 0, λ cannot cross λ 1.Consequently, all roots of 2.9 lie in the unit circle for sufficiently small positive τ > 0, and the existence of the τ follows.
A Neimark-Sacker bifurcation occurs when a complex conjugate pair of eigenvalues of A cross the unit circle as τ varies.We have to find values of τ such that there are roots on the unit circle.Denote the roots on the unit circle by e iω , ω ∈ −π, π .Since we are dealing with complex roots of a real polynomial, we only need to look for ω ∈ 0, π .For ω ∈ 0, π , e iω is a root of 2.9 if and only if e iω − 1 τkh p u * u * p u * e −imω 0.

2.13
Separating the real part and imaginary part from 2.

2.25
From 2.14 and 2.15 , we obtain that cos mω j 1 − cos ω j τ j kh p u * u * p u * , sin mω j sin ω j τ j kh p u * u * p u * .

2.26
It is easy to see that cos m 1 ω j cos mω j cos ω j − sin mω j sin ω j

2.27
Using 2.25 -2.27 , we have This completes the proof.
Proof.From 2.18 -2.20 , we have that 2.9 has toots e ±iω j if and only if τ τ j and ω ω j given in 2.19 and 2.20 .By Lemmas 2.1 and 2.2, we know that if τ ∈ 0, τ 0 , then all roots of 2.9 have modulus less than one; if τ τ 0 , then all roots of 2.9 except e ±iω j have modulus less than one; furthermore, by the Rouchs theorem Dieudonné 17, Theorem 9.17.4 , the statement on the number of eigenvalues with modulus more than one is as follows.
Lemma 2.3 immediately leads to stability of the zero solution of 2.4 , and equivalently, of the positive fixed point u u * of 2.2 .So we have the following results on stability and bifurcation in system 2.2 .

Direction and Stability of the Neimark-Sacker Bifurcation in Discrete Model
In the previous section, we obtain the conditions under which a family of periodic solutions bifurcate from the steady state at the critical value τ τ j , j 0, 1, 2, . . ., m − 1 /2 .Without loss of generality, denote the critical value τ τ j by τ * .For continuous systems, as pointed out in Hassard et al. 18 , it is interesting and important to determine the direction, stability, and the period of these periodic solutions bifurcating from the steady state.In this section, following the idea of Hassard et al., we shall study the direction, stability, and the period of the bifurcating periodic solution when τ τ 0 in the discrete Internet congestion control model.The method we used is based on the theories of discrete system by Kuznetsov 16 .Set τ τ 0 ν, ν ∈ R. Then ν 0 is a Neimark-Sacker bifurcation value for 2.4 .Rewrite 2.4 as 3.1 So system 2.5 is turned into where

3.4
Let q q τ 0 ∈ C m 1 be an eigenvector of A corresponding to e iω 0 , then Aq e iω 0 q, Aq e −iω 0 q.

3.7
If ξ is an eigenvalue of A, then Ap ξ ξp ξ .

3.10
Proof.Assign q * satisfies A T q * zq * with z e −iω 0 , then the following identities hold

Computer Simulation
In this section, we will confirm our theoretical analysis by numerical simulation.We choose p x x/ 20 − 3x as in 8 and set k 1.5, m 4, w 1.Then τ 0 1.0159 is the Neimark-Sacker bifurcation value.
Figures 1 and 2 are about delay difference equation 2.2 when step size h 0.25.