Abstract
We drive a scalar delay differential system to model the congestion of a wireless access network setting. The Hopf bifurcation of this system is investigated using the control and bifurcation theory; it is proved that there exists a critical value of delay for the stability. When the delay value passes through the critical value, the system loses its stability and a Hopf bifurcation occurs. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. Finally, some examples and numerical simulations are presented to show the feasibility of the theoretical results.
1. Introduction
Recently, the wireless access network has been wildly applied to various fields, especially to the Internment; therefore, it has received significant attention. The congestion control in wireless access network also plays a crucial role in the success of the wireless network technology.
The congestion and avoidance mechanism is a combination of the end-to-end TCP congestion control mechanism [1, 2] at the end hosts and the queue management mechanism at the routers. Because the congestion control algorithm is a highly complex dynamical model, many researchers have given much study to its dynamics and stability. In [3–5], the local stability in congestion control models is studied. In [6–9], the existence of Hopf bifurcation is analyzed in congestion control models.
For wired access network, the dynamic of window size is captured by the following equation [10]: where , , , and denote the TCP window size, TCP rate, round trap time at time of flow , and probability of packet mark at time , respectively.
However, there are seldom works which discuss the dynamical behaviors of the congestion control model in wireless access network such as stability and Hopf bifurcation. The observation provides us with the motivation to investigate the dynamical behaviors of the congestion control model in wireless access network.
In this paper, we consider the wireless access networks of only one bottleneck router and let TCP flows tracer the router. In the down link communication from the network to the sources, the marking probability is fed back to the sources. During channel fading, the source has failed to receive the marking probability. Therefore, we suppose that the drop probability is . In this case, the source will use the previous packet marking probability to reduce its window size, and also the window size is decreased by one by convention. Thus, we obtain The dynamic of queue length of the router is captured by the following equation [11]: where is the serving capacity of the link node and the function is the adjusted rate of the source based on the congestion rate from the link node, which is a decreasing and nonnegative derivative function.
Since and [12], we obtain
We assume that the is a constant and not time-varying and the queuing delay is neglected. So, we obtain the following congestion model in wireless network:
The paper is organized as follows. In Section 2, the stability of trivial solutions and the existence of Hopf bifurcation are discussed and the delay passes through the critical value, the system loses its stability, and a Hopf bifurcation occurs. In Section 3, based on the normal form theory and the center manifold theorem, we derive the formulas for determining the properties of the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. In Section 4, numerical simulations are given to justify the theoretical analysis. Finally, the conclusions appear in Section 5.
Since we focus on dynamical behavior analysis of the above model in the wireless access networks, we only need to choose the communication delay as the bifurcation parameter.
It is worth to point out that recent many works have been done for wired access network. For details, we refer to [13–16].
2. Stability of the System with Communication Delay
In this section, we assume that , is equal to , so (5) can be rewritten as follows: Let the equilibrium point of the system (6) be , which should satisfy and .
Hence, and we get(H1).
Remark 1. Consider , where
Let , . Linearizing the system (6) about the equilibrium point, we get
where
Then, the characteristic equation of the linearized equation (9) is
Note that the coefficients , , and depend on time delay , since is connected with . In order to apply the geometric criterion of Kuang [17, 18], we rewrite into
where
Lemma 2. If (H1) holds, then(a);(b) for all ;(c);(d) for each has at most a finite number of real zeros;(e)each positive root of is continuous and differentiable in whenever it exists.
Proof. (a) For ,
(b) Consider .
(c) From (13), we get
Hence, .
(d) From (13), we get
Hence, (d) holds.
(e) is continuous for and and differentiable in ; hence, implicit function theorem implies (e). This completes the proof of the theorem.
Supposing that and , we get
Hence,
Let , and then (18) can be rewritten as
Denote
Since , the equation has one positive root. We denote that the positive root is . Then, (18) has positive real root , where
For , let be defined by which combines with (18) and defines the following maps:
According to [18] and the above discussion, we have the following result.
Theorem 3. Assume that (H1) is satisfied, and then , , are a pair of simple and conjugate pure imaginary roots of the characteristic equation (11) if and only if for some . This pair of simple conjugate pure imaginary roots crosses the imaginary axis from left to right if and crosses the imaginary axis from right to left if , where
By the the expression of , , and , we know that they have singularity at . We can not gain the conclusion that the equilibrium by discussing roots of the characteristic equation . To our knowledge, this case is rarely considered by papers. But we can get the stability of the system (6) when by discussing the stability of the following auxiliary system: where
Then, the characteristic equation of the linearized equation (25) is
Definition 4. For simplicity, let
Lemma 5. The equilibrium of system (25) is locally asymptotically stable when .
Proof. When , (27) becomes Further, if(H2) and is satisfied, all roots of (29) have negative real parts by the Routh-Hurwitz criteria. So, when , the equilibrium point (0,0) of system (25) is locally asymptotically stable. This completes the proof of the lemma.
Let , where . Substituting it into (27) and separating the real and imaginary parts, we have
It follows from (30) that
Since , we have where
Since , we can rewrite (32) as where
Let ; then, (34) can be rewritten as
Denote
Since and , (36) has at the least one positive root. We define
Lemma 6. For cubic equation (36), the following cases need to be considered [19]:(a)if , then the equation has three distinct real roots;(b)if , then the equation has a multiple root and all its roots are real;(c)if , then the equation has one real root and two nonreal complex conjugate roots.
Without loss of generality, we assume that (36) has three positive roots: , , and . Since and , we have
Thus, we know that
Denote
Lemma 7. Assume that are simple roots of (27) when .
Proof. Since , we obtain
Substituting , into (42), by using (30), we can obtain
Similarly, we can get
This completes the proof of the lemma.
Hence, is a simple pair of purely imaginary roots of (27) with .
Lemma 8. Let be the root of (27) satisfying , ; the following transversality condition holds:
Proof. By equation (27) with respect to and applying the implicit function theorem, we get
Since , we obtain
where
By again using (30), we can obtain
Hence, . This completes the proof of the lemma.
From the above discussion about the system (25), we have the following result.
Theorem 9. When , the equilibrium point of system (25) is locally asymptotically stable.
Further, if(H3)is satisfied, we will get the following lemma.
Lemma 10. The equilibrium point of system (6) is locally asymptotically stable when .
Proof. Since Lemma 5 and hypotheses (H3), the equilibrium point of system (25) is locally asymptotically stable when . When and , system (9) and system (25) are the same system. So, there are no roots of with nonnegative real parts and the equilibrium point of system (6) is locally asymptotically stable when . This completes the proof of the lemma.
According to [18] and the above discussion, we have following the result.
Theorem 11. Assume that (H1), (H2), and (H3) hold, if the function has positive zeros in ; the equilibrium of system (6) is asymptotically stable for all and becomes unstable for staying in some right neighborhood of ; hence, system (6) undergoes Hopf bifurcation when .
3. Direction and Stability of the Hopf Bifurcation
In this section, we will study the direction of Hopf bifurcation and the stability of bifurcating periodic solution of system (6) at . The approach employed here is the normal form method and center manifold theorem introduced by Hassard [20]. More precisely, we will compute the reduced system on the center manifold with the pair of conjugate complex, purely imaginary solutions of the characteristic equation (11). By this reduction, we can determine the Hopf bifurcation direction, that is, to answer the question of whether the bifurcation branch of periodic solution exists locally for supercritical bifurcation or subcritical bifurcation.
Let , , , , , and , so that system (6) is transformed into an FDE in as with where Then, is a one parameter family of bounded linear operator in . By the Riesz representation theorem, there exists a function of bounded variation for such that
In fact, we can choose where is Dirac delta function. For , the infinitesimal generator is defined by
Further, let and then system (50) is equivalent to where for .
The adjoint operator of is defined by and a bilinear form where and are row vector space.
Let ; by the discussion in the previous section, we know that are common eigenvalues of and . We need to compute the eigenvector of and corresponding to and , respectively. Suppose that and are the eigenvector of and corresponding to and , respectively; then, we have
Then, we have the following lemma.
Lemma 12. Consider where
Proof. From (55), we can rewrite (60) as
Based on (53) and (64), we have
For , we have
We can choose and get
Similar to the proof of (64)–(68), we can obtain
Now, we can calculate as
On the other hand, since , we have
Therefore, . This completes the proof of the lemma.
In the remainder of this section, by using the same notations as in Hassard [20], we first compute the coordinates to describe the center manifold at , which is a locally invariant, attracting two-dimensional manifold in . Let be the solution of (57) when . Define and, then on the center manifold , we have and then and are local coordinates for center manifold in the direction of and . Note that is real if is real and we only deal with real solutions . It is easy to see that where
So, we can get where
Since and , we have
From the definition of , we have where
Since , we have
Comparing the coefficients of the above equation with those in (75), we have
In order to get the expression of , we need to compute and . Now, we determine the coefficients in (73). By (72) and (57), we have
From (73), (74), (83), and (84), we obtain
From (75) and (83), for , we have Comparing the coefficients of the above equation with those in (84), it follows that
When , we have Comparing the coefficients of the above equation with those in (84) gives that
From (85) and the definition of , we have
Hence, and, by a similar method, we get where and are both two-dimensional vectors. In the following, we will find out and . From the definition of and (85), we can obtain
Notice that Hence, we can obtain
Similarly, we have
Thus, we can get
From (97), we can obtain
Based on the above analysis, we can determine and from (91) and (92). Furthermore, in (82) can be expressed by the parameters and delay. Thus, we can compute the following quantities: which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value and we have the following result.
Theorem 13. In (99), the following results hold:(i)the sign of determines the directions of the Hopf bifurcation: if , then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for ;(ii)the sign of determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if ;(iii)the sign of determines the period of the bifurcating periodic solutions: the period increases (decreases) if .
4. Numerical Simulation Examples
In this section, we use the formulas obtained in Sections 2 and 3 to verify the existence of the Hopf bifurcation and calculate the Hopf bifurcation value and the direction of the Hopf bifurcation of system (6) with , , and .
From (7), we have
By calculation, we obtain that , , and . It follows from (3.32) that
These calculations prove that the system equilibrium is asymptotically stable when by computer simulation (see Figures 1, 2, and 3; ). When passes through the critical value , loses its stability and a Hopf bifurcation occurs (see Figures 4, 5, and 6; ).
5. Conclusion
A delayed model of congestion control was analyzed in this paper. Based on our theoretical analysis and numerical simulation, we can find that there exists a critical value for this delay and the whole system is stable when the delay of the system is less than this critical value. By using the time delay as a bifurcation parameter, we have shown that a Hopf bifurcation occurs when this parameter passes through a critical value, which means that the wireless access system will be congested, even collapsed, when the communication delay becomes large.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.