Applied Mathematics and Mechanics (English Edition) · Applied Mathematics and Mechanics (English...

Appl. Math. Mech. -Engl. Ed., 40(1), 25–48 (2019)

Applied Mathematics and Mechanics (English Edition)

https://doi.org/10.1007/s10483-019-2411-9

Suppression of oscillatory congestion via trunk link bandwidth

and control gain in star network∗

Sainan WANG, Shu ZHANG, Jian XU†

School of Aerospace Engineering and Applied Mechanics,

Tongji University, Shanghai 200092, China

(Received Jul. 7, 2018 / Revised Sept. 14, 2018)

Abstract The time delay-induced instability in an Internet congestion control modelis investigated. The star topology is considered, and the link bandwidth ratio and thecontrol gain are selected as the tunable parameters for congestion suppression. Thestability switch boundary is obtained by the eigenvalue analysis for the linearized systemaround the equilibrium. To investigate the oscillatory congestion when the equilibriumbecomes unstable, the center manifold reduction and the normal form theory are used tostudy the periodic oscillation induced by the delay. The theoretical analysis and numericalsimulation show that the ratio between bandwidths of the trunk link and the regular link,rather than these bandwidths themselves, is crucial for the stability of the congestioncontrol system. The present results demonstrate that it is not always effective to increasethe link bandwidth ratio for stabilizing the system, and for some certain delays, adjustingthe control gain is more efficient.

Key words Internet congestion control, Hopf bifurcation, delayed differential equation,stability, normal form

Chinese Library Classification O193, O137+.12010 Mathematics Subject Classification 34K20

1 Introduction

The smooth functioning of the Internet mainly relies on the transmission control proto-col/Internet protocol (TCP/IP). Congestion control is a central issue of TCP which aims atimproving the quality of Internet services. The congestion control algorithm regulates the enduser’s window size, which is defined as the maximum number of data packets during data trans-mission, as well as the queue size at the link or in the buffer of the router to avoid an overloadin the network system[1–2]. TCP-based congestion control has achieved admirable success sinceits early implementation[3–5] and has evolved into various TCP congestion control schemes andactive queue management (AQM) policies, such as TCP-Tahoe, TCP-Reno, TCP-NewReno for

∗ Citation: WANG, S. N., ZHANG, S., and XU, J. Suppression of oscillatory congestion via trunk linkbandwidth and control gain in star network. Applied Mathematics and Mechanics (English Edition),40(1), 25–48 (2019) https://doi.org/10.1007/s10483-019-2411-9

† Corresponding author, E-mail: [email protected] supported by the National Natural Science Foundation of China (Nos. 11572224, 11502168,11772229, and 11872277)

c©The Author(s) 2019

26 Sainan WANG, Shu ZHANG, and Jian XU

TCP congestion control, random early detection (RED), random exponential marking (REM),and active virtual queue (AVQ) for AQM policies[6–12]. Generally, these congestion controlschemes feature a moderate increase in the window size for low-level congestion and strongwindow control when the congestion indicator becomes remarkably significant, i.e., the additiveincreasing and multiplicative decreasing (AIMD) characteristic.

Hence, TCP-based congestion control plays a crucial role in current applications involvingthe Internet. However, the current congestion control scheme is not the ideal solution to thecongestion problem. For example, some TCP congestion control schemes have long been criti-cized for the oscillatory feature embedded in them[13]. According to the interpretation for theTCP-based congestion control from the operational research perspective[14], congestion will notoccur if data packets are transferred at a constant rate, pre-determined using the method ofoptimization. In other words, the stable steady state of the congestion control system rejectsthe occurrence of congestion. Conversely, the instability brings the risk of congestion, especiallyfor the synchronous oscillation case in which the network may collapse at the peak of oscillationof data transmission rates[15]. Besides, oscillatory rates of data transmission are harmful to thequality of service for specific Internet applications, such as video games, online chatting, andonline video watching. Therefore, significant effort has been made to study the stability of thecongestion control system, especially the factors that potentially lead to the oscillatory rates ofdata transmission[16–17].

The round trip time (RTT), or time delay, is among the most important parameters ofTCP-based congestion control systems. Unlike physical and algorithmic parameters such as linkbandwidth and control gain, the time delay is determined by neither hardware nor software.Therefore, significant uncertainty is introduced to the congestion control problem. Previousstudies have shown that the time delay may induce the oscillatory rates of data transmission,and consequently degrade the performance of the network[18–20]. Another key parameter is thebandwidth of the link, especially the trunk link, namely, the link at which data packet frommulti-sources stack. Such link is paramount in the investigation of congestion control. Beforethe data packets are injected into the trunk link, congestion is not likely to occur becausethe network resources serving these packets are distributive and thus are sufficient for packetstransmission. However, after these packets enter the trunk link, packets from various users willhave to compete for finite network resources, namely, the trunk link bandwidth. Therefore,the congestion may arise. As it is natural to ascribe the Internet congestion to the limitedbandwidth of the trunk link, one common intervention for frequently congested network is toincrease the trunk link bandwidth. However, such treatment may induce new problems. Forexample, it was argued that the excessively large buffer size, which acts similar to the linkbandwidth in the network, may cause unnecessary latency and consequently cause harm tothe network performance. A recent research has reported that the Braess’ paradox, whichstates that increasing the network bandwidth may unexpectedly induce congestion, may alsobe observed in the communication network[21]. Therefore, a careful investigation is necessary touncover the relation between the link bandwidth and the network performance for congestioncontrol. Moreover, the Internet congestion control system is present in the form of a feedbackcontrol system. It is well known that the control gain has significant influence on the stability ofsuch system[22–23]. Therefore, it is rational to seek the possibility of stabilizing the congestioncontrol system by selecting proper control gains.

In this work, we study the oscillatory congestion induced by the time delay, as well as thecongestion suppression by adjusting the trunk link bandwidth and the control gain for a networkwith the star topology. The star network is a typical in the Internet. It is characterized by theexistence of a central hub that connects various hosts. Owing to such features, congestion mayoccur in the star network especially at the link that connects the central hub (typically, a router)with the outer Internet as such link is in fact a trunk link where traffic can be heavily congested.However, to our best knowledge, investigations related to the theoretical study of the congestion

Suppression of oscillatory congestion via trunk link bandwidth and control gain 27

control problem in the star network are scarce. To investigate such problems both qualitativelyand quantitatively, a mathematical model that describes the temporal evolution of congestioncontrol system is necessary. In Ref. [24], Kelly proposed a model for which the equilibriumis proved to be the global optimum of the corresponding source allocation problem over thenetwork. Hollot et al.[25] and Misra[26] formulated the mathematical model of TCP/AQMusing stochastic differential equations and studied the stability of the equilibrium of the model.Such a model includes the maximum amount of the technical details of the congestion controlalgorithm, and therefore attracts much attention from the researchers[27–31]. However, dueto the same reason, the model is highly complicated for further analysis from the theoreticalperspective. In this work, we use the model proposed by Kelly as the basic model since itcaptures the AIMD feature of TCP and meanwhile is well simplified.

To our best knowledge, few works are related to the joint influence of the delay, the trunklink bandwidth, and the control gain on the stability and dynamics of the congestion controlmodel. In this paper, we study the impact of time delay on the congestion control model as wellas the principle of selecting tunable parameters such as the link bandwidth ratio and controlgain. The remainder of the paper is organized as follows. The mathematical model of theInternet congestion control system and the location of the equilibrium are discussed in Section2. The stability of the equilibrium is investigated in Section 3. Based on the center manifoldreduction together with the normal form theory, the formulae for determining the propertiesof the Hopf bifurcating periodic solutions are derived in Section 4. In Section 5, we discussthe suppression of oscillatory congestion via tuning the trunk link bandwidth ratio and controlgain. Moreover, numerical results are provided to verify our theoretical analysis. Conclusionsare presented in Section 6.

2 Congestion control model and its equilibrium

In this section, we formulate the mathematical model of the congestion control algorithmfor the star network and study the equilibrium of the model.2.1 Mathematical model

In this paper, we focus on the Internet congestion control model with two sources and threelinks, as shown in Fig. 1.

Fig. 1 Topology of network under investigation

According to Fig. 1, the Internet congestion control model is described by{

.x1(t) = k1 (w1 − x1τ (p1(x1τ ) + p(x1τ + x2τ ))) ,.x2(t) = k2 (w2 − x2τ (p2(x2τ ) + p(x1τ + x2τ ))) ,

(1)

where x1(t) and x2(t) represent the sending rates of sources 1 and 2 at time t, respectively, and k1

and k2 are the positive control gains. w1 and w2 are the parameters of adjusting the equilibria,p1(·), p2(·), and p(·) are the congestion indication functions or marking functions, which areincreasing, nonnegative, and not identically zero. It is noteworthy that, if congestion does notoccur in the Internet,

.xi(t) = kiwi (i = 1, 2), implying that the rate of data transmission is


increasing unboundedly. When the Internet congestion occurs, the source receives a signal toreduce the data transmission rate. In (1), xiτ = xi(t− τ) (i = 1, 2), where τ represents the timedelay. We assume for simplicity that the time lags in the delayed terms in p1(·), p2(·), and p(·)are identical.

As in area networks, the network equipment and Internet speed are nearly the same for all

users. Herein, we consider the special case in which k1 = k2△= k, and w1 = w2

△= w. The

congestion indication functions are of the following form:

p1(x) = p2(x)△= p(x) =

θσ2x

θσ2x+ 2(C1 − x), p(x) =

θσ2x

θσ2x+ 2(C2 − x),

where C1 is the bandwidth of the regular link, namely, the link that is used by each sourcealone, and C2 is the bandwidth of the trunk link. According to the physical meaning of C1 andC2, the sending rates of the two sources should satisfy the following condition:

0 < x1(t) 6 C1, 0 < x2(t) 6 C1, 0 < x1(t) + x2(t) 6 C2. (2)

In the remainder of this paper, we assume θσ2 = 0.5 and w = 1 as in Ref. [22]. It is noteworthythat (1) describes the evolution of the congestion control system for the simplest star networkbecause the router on the right-hand side in Fig. 1 can be viewed as the hub.

To simplify the congestion control model and reduce the number of the system parameters,we introduce the following transformations that will not change the system qualitatively:

yi(t) =xi(t)

C1, W =

w

C1, C =

C1

C2, i = 1, 2. (3)

Then, (1) is transformed into{

.y1(t) = k (W − y1τ (p(y1τ ) + p(y1τ + y2τ ))) ,.y2(t) = k (W − y2τ (p(y2τ ) + p(y1τ + y2τ ))) ,

(4)

where p(y) = y4−3y

, and p(y) = y4C−3y

. In the following theoretical analysis, we consider (4)

instead of (1).The model investigated in the present work originates from the congestion control model of

a single user with a single link constructed by Kelly[24] as follows:

x(t) = k(w − xτp(xτ )),

where x(t), k, and w have the same meanings as their counterparts in the previous models.Again, xτ denotes the rate of data transmission delayed by τ . The congestion indicator, orequivalently, the penalty function p(·), is given by[24]

p(x) =θσ2x

θσ2x+ 2(C − x),

where θσ2 can be viewed as a parameter that regulates the strength of the punitive part inthe congestion control algorithm, and C represents the link bandwidth. Figure 2 shows theconfiguration of p(·) for different values of θσ2 when C = 5. It can be seen that for a smallrate of link occupancy, the network encourages the user to send packets because the penaltyfunction p(·) assumes a small value. When the link occupancy rate becomes large, the value ofp(·) approaches 1, implying the full punishment on the user. It should be noted that althoughthis model is derived based on the nonlinear optimization theory[24], it is usually interpreted asa mathematical abstraction of the aforementioned AIMD mechanism of the Internet congestioncontrol. More specifically, the right-hand side of the model above shows the balance betweenthe “additive increasing” and “multiplicative decreasing” of the TCP algorithm.


θσ

θσ

θσ

Fig. 2 Marking (penalty) function p(x) for C = 5 and different θσ2 (color online)

2.2 Location of equilibriumDenote the equilibrium of (4) by (y∗1 , y

∗2)T. Therefore,

{W = y∗1(p(y∗1) + p(y∗1 + y∗2)),

W = y∗2(p(y∗2) + p(y∗1 + y∗2)).(5)

First, we present a lemma to illustrate the symmetric property of the equilibrium.Lemma 1 Assume that the conditions guaranteeing 0 < y∗1 6 1, 0 < y∗2 6 1, and y∗1 +y∗2 6

C are satisfied. Then, y∗1 = y∗2 .Proof From (5), we obtain

(y∗1 − y∗2) (4(y∗1 + y∗2) − 3y∗1y∗2)

(4 − 3y∗1)(4 − 3y∗2)=

(y∗2 − y∗1)(y∗1 + y∗2)

4C − 3(y∗1 + y∗2).

Now, we show y∗1 = y∗2 . Conversely,

4(y∗1 + y∗2) − 3y∗1y∗2

(4 − 3y∗1)(4 − 3y∗2)= − y∗1 + y∗2

4C − 3(y∗1 + y∗2). (6)

From (2) and (3), we have 0 < y∗1 6 1, 0 < y∗2 6 1, and y∗1 + y∗2 6 C. It follows

4(y∗1 + y∗2) − 3y∗1y∗2

(4 − 3y∗1)(4 − 3y∗2)=y∗1

(1 − 3

8y∗2

)+ y∗2

(1 − 3

8y∗1

)

4(1 − 3

4y∗1

)(1 − 3

4y∗2

) > 0.

It is obvious that the right-hand side of (6) is negative, which brings a contradiction. Thus, weobtain y∗1 = y∗2 .

Let y∗ = y∗1 = y∗2 , and y∗ be the solution to the following equation:

6y3 + (9W − 4 − 2C)y2 − 6W (C + 2)y + 8WC = 0. (7)

Clearly, 0 < y∗ 6 1, and 2y∗ 6 C.

According to Ref. [32], we can obtain the discriminant of the roots of (7) as H(W ) =H1W

4 +H2W3 +H3W

2 +H4W , where H1 = −8 748C2 +34 992C−34 992, H2 = −11 664C3+23 328C2 + 46 656C − 93 312, H3 = −432C4 − 24 192C3 + 93 312C2 − 96 768C − 6 912, andH4 = −768C(C+2)3. Then, the following four cases are possible regarding the solutions to (7).

Case 1 If W = 0, and C = −2, then (7) has a triple root y = 0, which is contrary with thefact that y is positive.


Case 2 If H(W ) > 0, then (7) has one real root and one pair of conjugate complex roots.Case 3 If H(W ) < 0, then (7) has three different real roots.Case 4 If H(W ) = 0, then (7) has three real roots, two of which are repeated roots.H1, H2, H3, and H4 are plotted in Fig. 3. From Fig. 3, we can find that H1 and H2 are not

greater than zero (equal to zero when C = 2), while H3 and H4 are always less than zero. ForW > 0, we conclude that H(W ) < 0 always holds.

Fig. 3 Relations between (a) H1, (b) H2, (c) H3, and (d) H4 with C, respectively

Based on the analysis above, we can conclude that only Case 3 may occur. Moreover, thetwo roots of (7) are positive, and the third one is negative. However, it is difficult to derive aconcise expression for the solution to (7) in terms of W and C. Therefore, the numerical methodis introduced to study the relation between the equilibrium and parameters qualitatively.

According to the implicit function theorem, we have

∂y

∂C

∣∣∣y=y∗

=(y∗)2 + 3Wy∗ − 4W

9(y∗)2 + (9W − 4 − 2C)y∗ − 3W (C + 2). (8)

Through a simple deduction, we conclude that the numerator of (8) cannot be zero. The sameresults can be obtained for the denominator. That is to say, the sign of (8) is positive ornegative. We use W = 1

3 , i.e., C1 = 3 as an example to verify the sign of ∂y∂C

. It is clear from

Fig. 4 that ∂y∂C

∣∣y=y∗

> 0 for any C, i.e., y∗ increases with C. For convenience in the following

discussion, we define y∗∞ = limC→∞

y∗ and τ∗∞ as the critical value of the delay for the stability

switch when C → ∞. To obtain the limits of y∗, we set the right-hand side of (8) to be zero.


Then, we have y∗∞ =√

9W 2+16W−3W2 . This implies that if the bandwidth of the link used by

each source alone is incomparable to that of the trunk link, regardless of the size of C1, theequilibrium will not change significantly.

Fig. 4 Derivative of equilibrium with respect to C

Up to now, we have analyzed the number and location of the equilibria of the congestioncontrol model under consideration. Moreover, we have verified that the coordinates of theequilibrium will increase with the link bandwidth ratio C and approach their limits when C

approaches infinity.Next, we study the stability of the equilibrium as well as the nonlinear dynamics that arises

when the equilibrium becomes unstable.

3 Existence of Hopf bifurcation in Internet congestion control model

In this section, we consider the existence of bifurcating periodic solutions in the Internetcongestion control model.

Setting ui(t) = yi(t) − y∗ (i = 1, 2) and u(t) = (u1(t), u2(t))T, we obtain

.u(t) = Lu(t) +H(u(t)) + h.o.t., (9)

where

Lu(t) =

[a1u1τ + b1u2τ

a2u1τ + b2u2τ

],

H(u(t)) =

[d11u

21τ + d12u1τu2τ + d13u

22τ + e11u

31τ + e12u

21τu2τ + e13u1τu

22τ + e14u

32τ

d21u21τ + d22u1τu2τ + d23u

22τ + e21u

31τ + e22u

21τu2τ + e23u1τu

22τ + e24u

32τ

],

and h.o.t. stands for higher order terms. The coefficients ai, bi, dij , eik (i = 1, 2; j = 1, 2, 3; k =1, 2, 3, 4) and the derivatives of the congestion indication functions are listed in Appendix A.Due to the symmetry of the system, some of the coefficients are identical. For example, itshould be noted that a1 = b2, and a2 = b1. Further, 0 < y∗ 6 1, 2y∗ 6 C, and k > 0. Througha simple calculation, we can verify that all the coefficients in (9) are negative.

The linearized equation of (4) at the equilibrium (y∗, y∗)T is{ .u1(t) = a1u1τ + a2u2τ ,.u2(t) = a2u1τ + a1u2τ ,

(10)

and the corresponding characteristic equation of (10) is given by

λ2 − 2a1λe−λτ + (a2

1 − a22)e

−2λτ = 0. (11)


To obtain the explicit expression of the critical stability boundary, we assume that λ = iωis the root of (11), where ω > 0 is a real number. Then, substituting λ = iω into (11) andequating the real and imaginary parts to zero yield

{−2a1ω sin(ωτ) + (a2

1 − a22) cos(2ωτ) = ω2,

−2a1ω cos(ωτ) − (a21 − a2

2) sin(2ωτ) = 0,(12)

which can be rewritten as the following two equations:{−2a1ω sin(ωτ) + (a2

1 − a22) cos(2ωτ) = ω2,

cos(ωτ) = 0,(13)

or

−2a1ω sin(ωτ) + (a21 − a2

2) cos(2ωτ) = ω2,

sin(ωτ) = − a1ω

a21 − a2

2

.(14)

The following discussion will be divided into two cases.Case 1 From cos(ωτ) = 0, it follows that cos(2ωτ) = 2 cos(ωτ)2−1 = −1. If sin(ωτ) = −1,

substituting sin(ωτ) = cos(2ωτ) = −1 into the first equation of (13) yields ω2−2a1ω+(a21−a2

2) =0. It can be checked that a2

1 − a22 > 0. Obviously, the equation contains two negative roots

ω1 = a1 + a2 and ω2 = a1 − a2, which contracts the assumption that ω > 0. Likewise, for thecase sin(ωτ) = 1, we obtain

ω2 + 2a1ω + (a21 − a2

2) = 0. (15)

The roots of (15) are ω1 = −a1 + a2 and ω2 = −a1 − a2. According to sin(ωτ) = 1 and

cos(ωτ) = 0, we obtain ωτ = (4n+1)π2 (n = 0, 1, 2, · · · ). Thus, one has

τnl =(4n+ 1)π

2ωl

, n = 0, 1, 2, · · · , l = 1, 2.

Case 2 From sin(ωτ) = − a1ωa2

1−a2

2

, we have

cos(2ωτ) = 1 − 2 sin(ωτ)2 =(a2

1 − a22)

2 − 2a21ω

2

(a21 − a2

2)2

.

Substituting the formula above into the first equation of (14) yields ω =√a21 − a2

2. Becausesin(ωτ) ∈ [−1, 1], we can deduce −a2

2 > 0 from sin(ωτ) = − a1ωa2

1−a2

2

∈ [−1, 1], which is impossible

for any nonzero a2. Hence, this case cannot occur.Based on the discussion above, one can draw the conclusion that for the case sin(ωτ) = 1

and cos(ωτ) = 0, (11) contains a pair of purely imaginary roots ±iωl (l = 1, 2), where ω1 =

−a1 + a2, ω2 = −a1 − a2, and τnl = (4n+1)π2ωl

(n = 0, 1, 2, · · · , l = 1, 2).Now, we verify the second condition for the occurrence of Hopf bifurcation, i.e.,

dR(λ)dτ

∣∣τ=τnl

6= 0. The proof is detailed in Appendix A.

It is obvious that the roots of (11) depend continuously on the parameter τ . When τ = 0,we know that (11) contains two roots λ1 and λ2, which satisfy

λ1 + λ2 = 2a1 < 0, λ1λ2 = a21 − a2

2 > 0.

Therefore, both λ1 and λ2 contain negative real parts. Suppose that τ01 is the smallestpositive value so that (11) has a pair of purely imaginary roots. Because the roots of (11)


continuously depend on τ , it is clear that the roots of (11) contain negative real parts forτ ∈ [0, τ01]. It can be proved that (11) contains a pair of purely imaginary roots ±iω01 and allthe other roots have strictly negative real parts when τ = τ01. Assume the opposite, namely,λ = u + iv, where u > 0 is a root of (11) when τ = τ01. Since the roots of (11) continuouslydepend on τ , there exists τ ′ ∈ (0, τ01) so that (11) has purely imaginary roots at τ = τ ′, whichcontradicts the assumption that τ01 is the smallest such τ . Moreover, it follows from dλ

dτ> 0

that (11) has at least a pair of roots with positive real parts when τ > τ01.Up to now, we have studied the critical condition for the occurrence of Hopf bifurcation.

Figure 5 shows the stability switch boundary curve of the equilibrium of system (4) with k = 1,w = 1, and C1 = 3.

Fig. 5 Stability switch boundary of equilibrium of (4)

Based on Fig. 5, we claim that the system is free of congestion near the equilibrium forthe parameters selected from the shaded region. However, the mechanism that induces theinstability and the type of congestion that may arise after the stability switch remains unclear.The nonlinear analysis will be employed to study these issues in the next section.

4 Calculation of normal form on center manifold and stability of bifurcat-ing periodic orbit

Let C = C([−τ, 0] ,R2) and ut(θ) = u(t+ θ), where −τ 6 θ 6 0, and τ = τ∗ + µ. Then, (9)becomes

.u(t) = Lµ(ut) + F (µ, ut), (16)

where

Lµ(ut) =

[a1u1τ + a2u2τ

a2u1τ + a1u2τ

],

F (µ, ut) =

[d11u

21τ + d12u1τu2τ + d13u

22τ + e11u

31τ + e12u

21τu2τ + e13u1τu

22τ + e14u

32τ

d21u21τ + d22u1τu2τ + d23u

22τ + e21u

31τ + e22u

21τu2τ + e23u1τu

22τ + e24u

32τ

].

The expressions of the coefficients above are listed in (A1).According to the Riesz representation theorem, the function of bounded variation η(θ) exists

with θ ∈ [−τ, 0] such that

Lµ(φ) =

∫ 0

−τ

dη(θ, µ)φ(θ).


Define L0 =

[a1 a2

a2 a1

]. Then, η(θ, µ) = L0δ(θ + τ), where δ represents the Dirac delta

function.For φ ∈ C1([−τ, 0] ,R2), we define

A(µ)φ(θ) =

dφ

dθ, θ ∈ [−τ, 0) ,

∫ 0

−τdη(θ, µ)φ(θ), θ = 0,

R(µ)φ(θ) =

{0, θ ∈ [−τ, 0) ,F (µ, θ), θ = 0.

Then, (16) can be rewritten as

.ut = A(µ)ut +R(µ)ut. (17)

For ψ ∈ C∗([0, τ ] ,R2), the adjoint operator A∗ of A is defined as

A∗(µ)ψ(s) =

−dψ(s)

ds, s ∈ (0, τ ] ,

∫ 0

−τdηT(s, µ)ψ(−s), s = 0.

For ψ ∈ C∗([0, τ ] ,R2), and φ ∈ C([−τ, 0] ,R2), we define a bilinear form

〈ψ, φ〉 = ψT(0)φ(0) −∫ 0

−τ

∫ θ

0

ψT(ξ − θ)dη(θ)φ(ξ)dξ, (18)

where dη(θ) = dη(θ, 0), and C∗ is the dual space of C.Let q(θ) and q∗(θ) be such that A(0)q(θ) = iωq(θ), and A∗(0)q∗(θ) = −iωq∗(θ). In other

words, q(θ) and q∗(θ) are the eigenvectors of A(0) and A∗(0) associated with iω and −iω,respectively.

Obviously, q(θ) and q∗(θ) are of the form

q(θ) =

[1α

]eiωθ, q∗(s) = N

[1β

]eiωs,

where α, β, and N are given in Appendix B.4.1 Center manifold reduction

According to the work of Faria and Magalhaes[33], the phase space C can be decomposed byΛ = {iω,−iω} as C = P⊕Q, where P is the generalized eigenspace associated with Λ. Let Φ andΨ be the bases for P and P ∗ associated with the eigenvalues iω and −iω, respectively. Definem = dim(P ), and assume that B is an m×m matrix with the point spectrum σ(B) = Λ. For

the Internet congestion control model considered in this paper, we have m = 2, B =

[iω 00 −iω

],

and

Φ = (φ1, φ2), φ1 =

[1α

]eiωθ, φ2 =

[1α

]e−iωθ,

Ψ =

[ψ1

ψ2

], ψ1 = (N,Nβ)e−iωs, ψ2 = (N,Nβ)eiωs,

where N is the conjugation of N .


4.2 Normal form of (4)Recall µ = τ − τ∗. Therefore, the Hopf bifurcation occurs at µ = 0. Then, (17) can be

rewritten as

.u(t) = Lτ∗(ut) + F (ut, µ), (19)

where F (ut, µ) is the nonlinear term in (17). Enlarge the phase space C to the Banach spaceBC = C × R2, i.e., the space of uniformly continuous functions from [−τ, 0] to R2 with ajump discontinuity at zero. Using the decomposition ut = Φx(t) + y, with x(t) ∈ C2 andy ∈ Q1 = Q ∩ C leading to the decomposition BC = P ⊕ Fπ,Q ⊆ Fπ, where F denotes thekernel function. Then, (17) is decomposed as

x = Bx+ Ψ(0)F (Φx+ y, µ),

y = AQ1y + (I − π)X0F (Φx+ y, µ),(20)

where AQ1 : Q1 ⊆ Fπ is defined as

AQ1φ = φ+X0(Lτ∗φ− φ(0))

with

X0(θ) =

{0, θ ∈ [−τ, 0) ,

I, θ = 0

and

π : BC → P, π(φ +X0γ) = Φ ((Ψ, φ) + Ψ(0)γ) .

Expand the nonlinear terms of (20) into the Taylor series as follows:

Ψ(0)F (Φx+ y, µ) =1

2!f12 (x, y, µ) +

1

3!f13 (x, y, µ) + · · · ,

(I − π)X0F (Φx+ y, µ) =1

2!f22 (x, y, µ) +

1

3!f23 (x, y, µ) + · · · ,

(21)

where f1j (x, y, µ) and f2

j (x, y, µ) represent homogeneous polynomials in (x, y, µ) of degree j with

coefficients in C2 and Fπ, respectively. Therefore, (20) can be written as

x = Bx+∑j>2

1

j!f1

j (x, y, µ),

y = AQ1y +∑j>2

1

j!f2

j (x, y, µ).

(22)

The normal form of (17) on the center manifold at µ = 0 is written as

x = Bx+1

2!g12(x, 0, µ) +

1

3!g13(x, 0, µ) + h.o.t., (23)

where g12(x, 0, µ) and g1

3(x, 0, µ) are the second-order and third-order terms in (x, µ), respec-tively.

g1j (x, y, µ) = f1

j (x, y, µ) −(B,U1

j

)(x),

g2j (x, y, µ) = f2

j (x, y, µ) −(DxU

2j (x)Bx −AQ1(U2

j (x)))


with(B,U1

j

)(x) = DxU

1j (x)Bx − BU1

j (x). Here, U1j ∈ V 3

j (C2), and U2j ∈ V 3

j (Q1), where

V 3j (C2) represents the homogeneous polynomials of degree j in variables x1, x2, and µ, and

Q1 ≡ Q ∩C.

Let M1j be the operator from V 3

j (C2) to itself, which is defined by

(M1j h)(x, µ) = Dxh(x, µ)Bx −Bh(x, µ),

V 3j (C2) = G(M1

j ) ⊕ F (M1j ), g1

j (x, 0, µ) ∈ F (M1j ),

(24)

and

V 32 (C2) = span

{(x2

1

0

),

(x1x2

0

),

(x2

2

0

),

(x1µ

0

),

(x2µ

0

),

(µ2

0

),

(0x2

1

),

(0

x1x2

),

(0x2

2

),

(0x1µ

),

(0x2µ

),

(0µ2

)}.

Through a simple calculation, we have

G(M12 ) = span

{(x2

1

0

),

(x1x2

0

),

(x2

2

0

),

(x2µ

0

),

(µ2

0

),

(0x2

1

),

(0

x1x2

),

(0x2

2

),

(0x1µ

),

(0µ2

)}.

Then, we have

F (M12 ) = span

{(x1µ

0

),

(0x2µ

)}.

Similarly,

F (M13 ) = span

{(x2

1x2

0

),

(x1µ

2

0

),

(0

x1x22

),

(0

x2µ2

)}.

From the above results, we obtain

1

2!f12 (x, 0, µ) = Ψ(0)Lµ(Φx)

=

[N Nβ

N Nβ

](Lµ(φ1)x1 + Lµ(φ2)x2).

The second term on the right-hand side of the normal form is given by

1

2!g12(x, 0, µ) =

1

2!PF (M1

2)f

12 (x, 0, µ)

= PF (M1

2)

[N Nβ

N Nβ

](Lµ(φ1)x1 + Lµ(φ2)x2),

where

Lµ(φ1) = iµω

[1α

], Lµ(φ2) = −iµω

[1α

].

Thus, we obtain

1

2!g12(x, 0, µ) =

[Lx1µ

Lx2µ

].


Similarly, we have

1

3!f13 =

1

3!f13 (x, 0, 0) = Ψ(0)F3(Φx, τ

∗)

=

[N Nβ

N Nβ

]×

[h1

h2

],

1

3!g13(x, 0, 0) =

1

3!PS f

13 (x, 0, 0)

=

[Kx2

1x2

Kx1x22

],

where

h1 = e11(x1 + x2)3 + e12(x1 + x2)

2(x1α+ x2α)

+ e13(x1 + x2)(x1α+ x2α)2 + e14(x1α+ x2α)3,

h2 = e14(x1 + x2)3 + e13(x1 + x2)

2(x1α+ x2α)

+ e12(x1 + x2)(x1α+ x2α)2 + e11(x1α+ x2α)3,

S = span

{(x2

1x2

0

),

(0

x1x22

)}.

L and K are shown in Appendix B.Thus, the normal form of (17) is given by

x = Bx+

[Lx1µ

Lx2µ

]+

[Kx2

1x2

Kx1x22

]+ O(|x|µ2 + |x|4). (25)

Introduce the following coordinate transformation:

x1 = p1 − ip2, x2 = p1 + ip2, p1 = ρ cos(ξ), p2 = ρ sin(ξ).

Then, (25) can be rewritten as

ρ = rµρ + sρ3 + O(µ2ρ+ |(ρ, µ)|4), ξ = −ω + O(|(ρ, µ)|), (26)

where r = R(L), and s = R(K).We should note that rs determines the direction of the Hopf bifurcation, i.e., if rs < 0 (> 0),

the Hopf bifurcation is supercritical (subcritical). s determines the stability of the bifurcatingperiodic solutions, i.e., the solutions are orbitally stable (unstable) if s < 0 (> 0). Now, we usean example to verify our theoretical results.4.3 Numerical simulation

In the following numerical simulation, we choose k = w = 1, C1 = 3, and C2 = 5 in theoriginal equation (1), resulting in W = 1

3 and C = 53 . According to (7), the equation about the

equilibrium (y∗, y∗)T is

6y3 − 13

3y2 − 22

3y +

40

9= 0. (27)

Through a direct calculation, the equilibrium is obtained as (y∗, y∗)T = (0.565 0, 0.565 0)T.Substituting W = 1

3 and C = 53 into (A1), we have a1 = −1.366 2, a2 = −0.350 8, and thus


ω∗ = 1.717 0, τ∗ = π2ω∗

= 0.914 8. From the theoretical results above, the normal form of (4)is given by

ρ = 0.777 8µρ− 5.473 9ρ3 + O(|(ρ, µ)|4), ξ = −1.717 0 + O(|(ρ, µ)|). (28)

According to the previous analysis, the equilibrium is stable if τ < 0.914 8 and unstable ifτ > 0.914 8. Therefore, a stable periodic solution bifurcates from the unstable equilibriumowing to a supercritical Hopf bifurcation. The amplitude of the periodic solution to (28) iscalculated as

ρ = 0.377 0√µ,

which will be compared with the numerical results in the following discussion. In the numericalsimulations, the values of the variables are listed as in Table 1 unless otherwise stated.

Table 1 Values of variables in numerical simulation

Variable k w C1 C2 x1(0) x2(0)

Value 1 1 3 5 0.3 0.15

From Figs. 6 and 7, we can find that the equilibrium is stable when τ < τ∗ and unstableif τ > τ∗. A stable periodic orbit bifurcates from the unstable equilibrium as predicted bythe theoretical analysis. The bifurcation diagram of (1) is shown in Fig. 8. As can be seenfrom Figs. 6 and 7 that (1) makes no qualitative difference with (4). Thus, we only plot thebifurcation diagram of (1) for simplicity.

Fig. 6 Time histories and phase portraits of Internet congestion control model with τ = 0.8 for (1)((a) and (b)) and (4) ((c) and (d))


Fig. 7 Time histories and phase portraits of Internet congestion control model with τ = 0.95 for (1)((a) and (b)) and (4) ((c) and (d))

Fig. 8 Bifurcation diagram of (1)

5 Suppression of oscillatory congestion via link bandwidth ratio and con-trol gain

As previously stated, the delay may deteriorate the system performance and even makethe system unstable. This section is devoted to the discussion on the selection of parameters,especially the suppression of the delay-induced oscillatory congestion via adjusting the tunableparameters, i.e., the link bandwidth ratio and the control gain. The idea is to select theseparameters appropriately so that the equilibrium is stabilized based on the study of Section 2.5.1 Stabilization of equilibrium through adjusting link bandwidth ratio C

From Section 2, the relationship between C and the critical delay τ∗ is given by


τ∗ =π(4 − 3y∗)2(2C − 3y∗)2

−4ky∗(27(y∗)3 − 36(C + 2)(y∗)2 + 6(C2 + 16C + 4)y∗ − 16C(C + 2)). (29)

We have obtained that the limit equilibrium is y∗∞ =√

9W 2+16W−3W2 and the corresponding

critical delay τ∗∞ = (3√

9W 2+16W−9W−8)2π

2k(√

9W 2+16W−3W )(9W+16−3√

9W 2+16W ). Recall that we study only the cases

of k = w = 1[22] and C1 = 3, implying that W = 13 , y

∗∞ = 0.758 3, and τ∗∞ = 1.076 7.

The stability switch boundary curve can be obtained based on the analysis in Section 2, asshown in Fig. 9. It can be seen from Fig. 9 that there is a vertical asymptote as C grows alongthe boundary curve, corresponding to τ∗∞ = 1.076 7. Based on Fig. 9, several observations thatare of practical importance can be obtained. First, increasing the link bandwidth ratio alwaysbenefits the stability of the equilibrium for small delays. For example, when τ = 0.9 (the dashedgreen curve in Fig. 9(a)), as C increases, the system parameters enter the region for which theequilibrium is stable. The green dashed line crosses the stability switch boundary for only once,implying that the equilibrium remains stable as C increases further. Next, for medium τ , thereexists an interval on which the equilibrium is stable. In other words, an excessively large orsmall C deteriorates the system stability for such cases. For instance, the dashed black lineintersects the stability switch boundary twice, implying that the equilibrium switches to beingunstable when C becomes sufficiently large, as shown in Fig. 9(c). Finally, as τ > 1.126 5 (thehorizontal coordinate of the rightmost point on the boundary curve), the equilibrium is alwaysunstable regardless of C. In other words, when τ exceeds some threshold, it is impossible tostabilize the congestion control system by adjusting the link bandwidth ratio, as revealed inFig. 9(d).

Fig. 9 Stabilization of (1) via adjusting link bandwidth ratio C based on stability switch boundaryin (a) for (b) τ = 0.9, (c) τ = 1.1, and (d) τ = 1.2 (color online)


5.2 Optimum C when equilibrium is stable

When the equilibrium is stable, the optimal C should be obtained so that the rate of datatransmission moves towards the equilibrium at the fastest speed. Hence, three examples areprovided as an illustration of the idea by plotting R(λ) versus C, where λ represents theeigenvalue with the largest real part among all eigenvalues.

Example 1 τ = 1.1 > τ∗∞. The equilibrium is stable for C ∈ (2.807 7, 13.962 5). As shownin Fig. 10(a), the optimal value of C is 4.41.

Example 2 τ = 1.076 7 = τ∗∞. The equilibrium is stable only if C > 2.485 0. According toFig. 10(b), C = 4.41 is the optimal value in the whole stable interval.

Example 3 τ = 0.9 < τ∗∞. The smallest C such that the equilibrium is stable is 1.625 1.As shown in Fig. 10(c), the optimal C is 4.41.

Fig. 10 Real parts of bλ for (a) τ = 1.1, (b) τ = 1.076 7, and (c) τ = 0.9

Based on the observations above, one can postulate that the optimal value of C is indepen-dent of τ when the equilibrium is stable. This assertion is validated by Figs. 11 and 12, in whichR(λ)−R(λτ ) instead of R(λ) itself, is plotted versus the Cτ -plane, where λτ is the eigenvalueof (10) evaluated at C = 4.41 for a given τ . It can be seen from Fig. 11 that the strongest“damping” is achieved at C = 4.41 for any C provided that the equilibrium is stable.

Figures 11 and 12 are plotted using DDE-BIFTOOL[34] and the integral evaluationmethod[35–37], respectively. First, we calculate the largest real part of eigenvalue for a spe-cific point (C, τ) = (3, 0.5) as −1.113 501 6 based on the method in Ref. [36]. The characteristicequation is given by f(λ) = λ2 − 2a1λe

−λτ + 2(a21 − a2

2)e−2λτ , where a1 and a2 are shown in


Fig. 11 Real part of bλ shifted by R(eλτ ) as C varies (plotted using DDE-BIFTOOL[34])

Fig. 12 (a) Real part of bλ shifted by R(eλτ ) as C varies, and (b) corresponding contour plot in Cτ -plane (color online). Both figures are plotted by integral evaluation method. Black line in(b) represents optimum C, i.e., C = 4.41

Appendix A. Through a direct calculation, we obtain

∫ 50

0

R(f ′(−1 + iω)

f(−1 + iω)

)dω ≈ 3.097 4, (30)

∫ 50

0

R(f ′(−2 + iω)

f(−2 + iω)

)dω ≈ −9.483 4. (31)

This implies that the largest real part of all eigenvalues is σ ∈ (−2,−1).Repeating the procedureabove, we can narrow the interval in which σ exists as (−1.2,−1). Then, the initial estimate ofσ is chosen as −1.1. Substituting λ = −1.1+ iω into f(λ), and equating the real and imaginaryparts of f(λ) with zero, respectively, σ can be solved by using the Newton-Raphson iterationmethod. Thus, we obtain the rightmost eigenvalue as −1.113 501 6+2.207 094 1i. Suppose thatthe step lengths of C and τ are both 0.01. Then, the largest real part of the eigenvalues fordifferent values of C and τ can be obtained as shown in Fig. 12. Obviously, Fig. 12 agrees wellwith Fig. 11.5.3 Stabilization of equilibrium through adjusting k

Figure 9(d) shows that it is impossible to stabilize the equilibrium by adjusting C for τ >1.126 5. In this part, we seek the possibility of stabilizing the equilibrium by adjusting thecontrol gain k. The stability switch boundary in the space of C, τ , and k is plotted in Fig. 13,in which the equilibrium of the system is stable for the parameters selected from the region


below the boundary surface. Figure 13 shows that the smaller the control gain k, the largerthe critical delay τ∗. In other words, lowering the control gain delays the onset of the Hopfbifurcation. For example, the equilibrium is unstable for k = w = 1, C1 = 3, C2 = 5, andτ = 0.95 > τ∗ = 0.914 8. If k is decreased to 0.8 while the other parameters remain unchanged,the equilibrium of (1) becomes stable, as shown in Fig. 14.

Fig. 13 (a) Stability switch boundary in space of C, τ , and k, where equilibrium of system is stablefor parameters selected from region below boundary surface, and (b) corresponding contourplot in Ck-plane

Fig. 14 Time histories of (1) for (a) k = 1 and (b) k = 0.8

6 Conclusions

In this paper, we study the oscillatory congestion induced by the time delay in an Internetcongestion control model, which describes the evolution of the congestion control algorithm fora star network. The link bandwidth ratio, defined as the ratio of bandwidth of the trunk linkand that of the common link, and the control gain are chosen as tunable parameters to suppressthe delay-induced oscillatory congestion. The stability switch boundary in the parameter spaceis obtained through the stability analysis for the equilibrium. The center manifold reductionand normal form theory are adopted to investigate the mechanism which leads to the oscillatory


congestion. Our analysis shows that, for the fixed control gain, the stability switch boundarycurve is not monotonic in the plane consisting of link bandwidth ratio and delay. Consequently,a complex relation is found between the delay and the interval of the link bandwidth ratio onwhich the oscillatory congestion can be suppressed. The length of such interval depends onthe delay. If the delay exceeds some threshold, the equilibrium cannot be stabilized throughadjusting the link bandwidth ratio. Besides, when the equilibrium is stable, there exists anoptimal value of the link bandwidth ratio such that the transient behavior of the congestioncontrol system dies out rapidly. Moreover, re-selecting the control gain may help stabilizethe equilibrium which cannot be stabilized by adjusting the link bandwidth ratio alone. Thenumerical simulations are in good agreement with the theoretical analysis.

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Appendix A

In this appendix, we want to give the proof process of transversality condition and some formulaein Section 4. First, we give the coefficients in (9),

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

a1 = −k(p(y∗) + y∗p′(y∗) + ep(2y∗) + y∗ep′(2y∗)),

b1 = −ky∗ep′(2y∗),

a1 = b2, a2 = b1,

d11 = −k“p′(y∗) +

p′′(y∗)

2y∗ + ep′(2y∗) +

ep′′(2y∗)

2y∗

”,

d12 = −k(ep′(2y∗) + y∗ep′′(2y∗)),

d13 = −kep′′(2y∗)

2y∗,

d21 = d13, d22 = d12, d23 = d11,

e11 = −k“p′′(y∗)

2+

p′′′(y∗)

6y∗ +

ep′′(2y∗)

2+

ep′′′(2y∗)

6y∗

”,

e12 = −k“

ep′′(2y∗) +ep′′′(2y∗)

2

”,

e13 = −k“ ep′′(2y∗)

2+

ep′′′(2y∗)

2y∗

”,

e14 = −ep′′′(2y∗)

6y∗,

e21 = e14, e22 = e13, e23 = e12, e24 = e11,

(A1)

8>><>>:

p′(y) =4

(4 − 3y)2, p′′(y) =

24

(4 − 3y)3, p′′′(y) =

216

(4 − 3y)4,

ep′(y) =4C

(4C − 3y)2, ep′′(y) =

24C

(4C − 3y)3, ep′′′(y) =

216C

(4C − 3y)4.

(A2)

The proof process of transversality condition is presented as follows.Proof Taking the derivative of λ with respect to τ in (11), we get

dλ(τ )

dτ=

λ(a21 − a2

2)e−2λτ − a1λ

2e−λτ

λ − a1e−λτ + a1λτe−λτ − (a21 − a2

2)τe−2λτ.

For the sake of simplicity, ω and τ are denoted by ωl and τnl, respectively. Then,

dλ

dτ=

ω(a21 − a2

2)e−2iωτ i + a1ω

2e−iωτ

ωi − a1e−iωτ + a1ωτe−iωτ i − (a21 − a2

2)τe−2iωτ

,A1 + B1i

A2 + B2i,

where

A1 = ω(a21 − a

22) sin(2ωτ ) + a1ω

2 cos(ωτ ),

B1 = ω(a21 − a

22) cos(2ωτ ) − a1ω

2 sin(ωτ ),

A2 = −a1 cos(ωτ ) + a1ωτ sin(ωτ ) − τ (a21 − a

22) cos(2ωτ ),

B2 = ω + a1 sin(ωτ ) + a1ωτ cos(ωτ ) + τ (a21 − a

22) sin(2ωτ ).


Let

Q =A22 + B

22

=(−a1 cos(ωτ ) + a1ωτ sin(ωτ ) − τ (a21 − a

22) cos(2ωτ ))2

+ (ω + a1 sin(ωτ ) + a1ωτ cos(ωτ ) + τ (a21 − a

22) sin(2ωτ ))2.

Then,

QR“dλ

dτ

”=Re((A1 + B1i)(A2 − B2i))

=A1A2 + B1B2

=(ω(a21 − a

22) sin(2ωτ ) + a1ω

2 cos(ωτ ))(−a1 cos(ωτ ) + a1ωτ sin(ωτ )

− τ (a21 − a

22) cos(2ωτ )) + (ω(a2

1 − a22) cos(2ωτ ) − a1ω

2 sin(ωτ ))

· (ω + a1 sin(ωτ ) + a1ωτ cos(ωτ ) + τ (a21 − a

22) sin(2ωτ )).

Thus, we have

QR“dλ

dτ

”˛˛τ=τnl

= (−ω(a21 − a

22) − a1ω

2)(ω + a1)

= −ω((a21 − a

22)ω + (a2

1 − a22)a1 + a1ω

2 + a21ω)

= −ω((a21 − a

22)ω + (a2

1 − a22)a1 − 2a

21ω − a1(a

21 − a

22) + a

21ω)

= a22ω

2

> 0.

This completes the proof.

Appendix B

According to the definition of q(θ) at θ = 0, one obtains

Z 0

−τ

»a1 a2

a2 a1

–δ(θ + τ )

»1α

–eiωθdθ = iω

»1α

–

⇒

»a1 a2

a2 a1

– »e−iωτ

αe−iωτ

–= iω

»1α

–.

Thus, we get the expression of α,

α =a2

iωeiωτ − a1.

Similarly,

Z 0

−τ

»a1 a2

a2 a1

–δ(s + τ )

»1β

–eiωsds = −iω

»1β

–

⇒

»a1 a2

a2 a1

– »eiωτ

βeiωτ

–= −iω

»1β

–,

and

β =−a2e

iωτ

a1eiωτω.


In order to obtain N , let 〈q∗, q〉 = 1.

〈q∗, q〉 = (q∗)T(0)q(0) −

Z 0

−τ

Z θ

0

(q∗)T(ξ − θ)dη(θ)q(ξ)dξ

= N(1 + αβ) −

Z 0

−τ

Z θ

0

N(1, β)e−iω(ξ−θ)

»a1 a2

a2 a1

–δ(θ + τ )dθ

»1α

–eiωξdξ

= N(1 + αβ) −

Z 0

−τ

Z θ

0

N(1, β)eiωθ

»a1 a2

a2 a1

–δ(θ + τ )

»1α

–dθdξ

= N(1 + αβ) + Nτe−iωτ (a1 + a2α + a2β + a1αβ).

Then,

N =1

1 + αβ + τe−iωτ(a1 + a2α + a2β + a1αβ).

L and K in Section 4 are

L = iωN(1 + αβ),

K =N((e12 + e13β)(2α + α) + (e13 + e12β)(α2 + 2αα)

+ 3(e14 + e11β)α2α + 3(e11 + e14β)).

Applied Mathematics and Mechanics (English Edition) · Applied Mathematics and Mechanics (English...

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