Phase Plane Analysis of Congestion Control in DataCenter Ethernet Networks

Fengyuan Ren and Wanchun Jiang

Dept. of Computer Science and Technology, Tsinghua University, Beijing, 100084, ChinaEmail: renfy@tsinghua.edu.cn

AbstractEthernet has some attractive properties for networkconsolidation in the data center, but needs further enhancementto satisfy the additional requirements of unified network fabrics.Congestion management is introduced in Ethernet networks toavoid dropping packets due to congestion. The BCN (BackwardCongestion Notification) mechanism is a basic element of severalstandard drafts, and its stability underlies normal network oper-ations. Because the linear stability analysis method is incapableof handling the nonlinearity of the variable structure controlemployed by the BCN mechanism, some particular phenomenaare unexposed and the insights are insufficient. In this paper, wepropose the concept of strong stability of the queuing system tosatisfy the requirements of no dropped packets in the data center,and build a fluid-flow analytical model for the BCN congestioncontrol system. Considering the nonlinearity involved in the rateregulation laws, we classify the system into different categoriesaccording to the shapes of phase trajectories, and conduct anonlinear stability analysis using phase plane analysis techniqueson a case by case basis. The analysis details can provide acomprehensive understanding about the behaviors of the overallcongestion control system. Finally, we also deduce an explicitstability criterion presenting the parameters constraints for thestrongly stable BCN system, which can provide straightforwardguidelines for proper parameter settings.

Index TermsData Center Ethernet, Backward CongestionNotification, Strong stability and Phase Trajectory.

I. INTRODUCTION

In todays data centers, it is common to deploy an Ethernetnetwork for IP traffic, one or two storage area networks(SANs) for block mode Fibre Channel traffic, and an Infini-Band network for interprocess communication (IPC) traffic.As data centers grow in size and complexity, the effort tomanage different interconnect technologies for traffic frommultiple applications is becoming resource and cost inten-sive. Data center networking will become a new paradigmfor providing high-bandwidth and low-latency interconnectionfabrics to computing and storage applications. The deploymentof unified network fabrics in data centers is expected to bea methodical process. It offers many research challenges inthe areas of computing, storage, virtualization and networking.It also causes major shifts in the industry, and witnesses theconvergence of the different industries of computing, storageand networking.Ethernet is the most widespread wired local area network

technology because of its low cost, wide availability, simplic-ity, and broad compatibility. However, in the past its bandwidthlimitations kept it from being a candidate for unified fabric in

data centers. With recent advances in Ethernet speeds and thedevelopment of 40 and 100 Gbps, Ethernet has become anattractive choice for unified network fabrics in data centers[1]. However, Ethernet networks being a best effort datagramservice still need further enhancement to satisfy additionalrequirements, such as low latency, high reliability, and nopackets loss. In the IEEE standard body, several workinggroups are addressing these issues to ensure that Ethernet willbe equipped to meet data centers requirements. This enhancedEthernet is called Data Center Ethernet (DCE).

DCE is an architectural collection of Ethernet extensionsdesigned to improve Ethernet networking and management inthe data center, which has been well thought out to providethe next-generation infrastructure for data center networks bytaking advantage of classical Ethernets strengths and addingseveral crucial extensions, such as priority-flow control [?],enhanced transmission selection [?], shortest path bridging [?]and end-to-end congestion management [3].

Avoiding frame drops is mandatory for carrying nativestorage traffic, since storage traffic does not tolerate framedrops. SCSI is designed with the assumption of running overa reliable transport in which failures are rather rare. FibreChannel is the primary protocol used to carry storage traffic,and it avoids frame drops through a link flow control mech-anism based on credits called buffer-to-buffer flow control.Historically Ethernet has been a lossy network, since Ethernetswitches do not use any mechanism to signal to the sender thatthey are out of buffers [2]. A few years ago, IEEE 802.3 addeda PAUSE mechanism to Ethernet, which can be used to stopthe sender for a period of time. However it cannot properlyalleviate congestion and maintain high throughput becausethe congestion can roll back from switch to switch, affectingflows that do not contribute to the congestion, but happen toshare a link with flows that do. To deal with this problemand be competent in unified network fabrics in data centers,the link flow control is introduced in Ethernet networks toavoid packets dropping due to transient congestions. TheIEEE 802.1 standard committee has launched a new taskgroup to develop the end-to-end congestion control mechanismfor switched Ethernet networks [3], and four proposals arecurrently discussed.

As a general congestion management scheme, the BCNmechanism is adopted by several proposals. Its stability iscrucial for normal network operation. The theoretical stability

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analysis of BCN will definitely provide essential insights,which are beneficial for confirming a proper congestion controlalgorithm and accelerating the finalization of the specification.In [4], the inventors of BCN applied the stability criterion inthe classical linear control theory to analyze the BCN mecha-nism and provided the stable sufficient condition satisfied byparameters. The limitation of their work is that the overallcongestion control system governed by the BCN mechanismis intentionally divided into two linear subsystems to analyzetheir stability independently, and the transient behavior of theswitching process between two subsystems and its impacton the system stability are neglected, thus some importantphenomena and properties are likely unexplored. In this paper,we model BCN behavior as an autonomous second-ordersystem described by a set of nonlinear ordinary differentialequations using fluid-flow approximation technique. Remain-ing the nonlinearity involved in the variable structure ratecontrol of the BCN mechanism, and taking the switchingprocess into account, we conduct a nonlinear stability analysisapplying the phase plane method to provide a comprehensiveunderstanding about the behaviors of the overall congestioncontrol system, and deduce an explicit stability criterion.The remainder of this paper is organized as follows. The

background of congestion control in DCE networks and thebasic BCN mechanism are introduced in Section II. Subse-quently, the analytical model of the BCN congestion controlsystem is built, and the concept of strong stability is intro-duced. In Section IV, based on various phase trajectories ofthe BCN system, the stability is analyzed case by case anda stability criterion is deduced. Finally, the conclusions aredrawn in Section V.

II. BACKGROUNDA. Congestion Control in Ethernet

While numerous investigations have been made on con-gestion control in TCP/IP networks, Ethernet networks, evenafter 30 years of their invention, run without congestioncontrol in the link layer. To be competent for data centerapplications, it is crucial to introduce congestion managementin the link layer for Ethernet networks. The main tasks include:(1) Providing congestion detection and information feedbackto push congestion from the core towards the edge of thenetwork; (2) Supporting rate control at the edge switches toshape flows causing congestion. In addition, an appropriatecongestion management method for Ethernet networks shouldbe compatible with the current IEEE802.1/802.3 standards andalso work in harmony with the upper layer congestion controlscheme, such as TCP flow control.The IEEE 802.1Qau working group [3] is developing a new

specification for congestion management in Ethernet networks.Until now, the specification has not been finalized. Fourrepresentative proposals, however, have been discussed andinvestigated. The general Ethernet Congestion Management(ECM) framework employing the Backward Congestion No-tification (BCN) mechanism was proposed by D. Bergamasco[5]. The basic mechanism of BCN was initially used in frame

Edge Switch Core Switch

Regulator Samplingpackets

BCN message q(t)qsc q0qoff(t)

Fig. 1. Framework of BCN

relay networks called Backward Explicit Congestion Notifi-cation (BECN) [6]. Compared to FECN (Forward ExplicitCongestion Notification ) [6], the control messages in BCNare sent directly from switches back to sources when thecongestion happens. Due to early congestion indication, BCNis more responsive to congestion, which is attractive for superhigh-speed Ethernet networks, but needs switches to generateextra backward packets to carry control messages. Unliketraditional BECN whose control message only carries binarycongestion indication, the BCN mechanism also sends back thecurrent queue status, including queue length and variation, toguide the adjustment of sending rate (described in detail later).Raj Jain argued for the explicit feedback of allowed ratesand proposed the Forward Explicit Rate Advertising (FERA)mechanism [7], which is a variant of ERICA developed forexplicit flow control in ATM networks [8]. The researchersat IBM Zurich lab combined some ideas of BCN and FERAto develop a new scheme called Explicit Ethernet CongestionManagement (E2CM) [9]. The fourth proposal is quantizedcongestion notification (QCN) [10] in which the ECM queuefeedback is quantized to a few bits and the network onlyprovides negative feedback. Except for FERA, the otherthree proposals, including ECM, E2CM and QCN, follow theparadigm of BCN and send the queue dynamics back to thesources to instruct rate adjustment. In other words, the BCNmechanism is a cornerstone for most proposals of congestioncontrol in DCE networks. For the convenience of modelingand analysis, we firstly describe the main components of theBCN mechanism in the next subsection.

B. BCN mechanism

We shall only introduce those parts of the BCN mechanismwhich are relevant for our stability analysis, and omit concreteimplementation. The whole description can be seen in [5].BCN is a rate-based closed-loop feedback control mecha-

nism. Its framework and main components are shown in Fig.1.The technical goal of BCN is to hold the queue length of thecore switch at equilibrium point q0 so that the buffer is neitherover-utilized nor under-utilized. The core switches monitorthe instantaneous queue length q(t). If q(t) exceeds a highthreshold qsc > q0, then the network is regarded as severelycongested and the core switch sends out the PAUSE framesdefined in IEEE 802.3x to require all its uplink neighbors tostop forwarding packets. In addition, the core switches samplethe incoming packets with a deterministic probability pm,and also simply count the number of arrival and departurepackets to obtain the difference of instantaneous queue length

2121

0 47 95 111 127 143 175 207

DA SAEther

Type CPID FB

Fig. 2. Format of BCN message

q(t) in a sampling interval. When a packet is sampled, thecore switches determine the congestion level on the link bycomputing the key variable , which consists of a weightedsum of the instantaneous queue length offset and q(t) overthe last sampling interval:

(t) = [q0 q(t)] wq(t) (1)where w is a weight parameter. The core switches may send aBCN message to the source of the sampled packet if necessary.The BCN messages follow 802.1Q VLAN tag format to ensurethe coexistence and interoperability between BCN-aware andBCN-unaware switches. The key fields in BCN message areshown in Fig.2. Here, the 48-bits DA and SA fields containthe source address of the sampled packet and the address ofthe switch, respectively. EtherType tells the switches and endstations whether it is a BCN message. The CPID field is theID of the congestion point and its purpose is to univocallyidentify a congestion entity, which should at least include theMAC address of the switch interface. The FB field carries thekey measure (t). If (t) > 0, BCN message is positive, elsenegative. When a normal BCN message reaches the congestionreaction point, which refers to a rate regulator associated withthe source receiving the BCN message and is usually located innetwork interface card in the edge switch, the reaction pointadjusts its sending rate using a modified Additive Increaseand Multiplicative Decrease (AIMD) algorithm since it hasbeen proven to be stable, convergent and fair under commonnetwork environments [11]. AIMD is defined in BCN asfollows:

r(t) {

r(t) +GiRu(t) if (t) > 0r(t)[1 +Gd(t)] if (t) < 0

(2)

where r(t) is the rate of sources, Ru is the increase rateunit parameter, Gi and Gd are the rate additive increaseand multiplicative decrease gains, respectively. They need tobe appropriately set by the network manager. A congestionreaction point receiving a negative BCN message will associateitself with the congestion point identified by the correspondingCPID. Its subsequent packets will contain a rate regulator tag(RRT) to carry this CPID. If these packets are sampled bythe core switch, the value of CPID matches with the switchsCPID, a possible positive BCN message is generated and sentback the source of the sampled packets when q(t) < q0, orelse no BCN message is sent.

III. ANALYTICAL MODEL OF BCN

A. Assumption

In this section, we firstly build an analytical model ofthe BCN mechanism. Considering regular and symmetrical

network topologies in data centers, such as Monsoon [12], Fat-Tree [13] and D-Cell [14], and special traffic patterns drivenby the parallel reads/writes in cluster file systems, such asLustre [15] and Panasas [16], we assume that the sourcesare homogeneous, namely they have the same characteristics,follow the same routes, and experience the same round-trippropagation delays. In addition, the propagation delay in DCEnetworks is normally within the order of a few microsecondssince the diameter of the network is only a few hundredsmeters. Compared with the queuing delay in the order ofseveral tens or hundreds microseconds, the propagation delaycould be negligible. Furthermore, since links are assumed tobe of high capacity in DCE networks, the number of bitstream in the links is so large that it appears like a continuousflow fluid, the fluid-flow approximation extensively used innetwork modeling work (in [17] and [4] etc.) is assumed tobe practicable in our investigation.

B. Modeling

Considering the queue associated with the bottleneck link,and assuming that the queue length q(t) is continuous anddifferentiable and the propagation delay is negligible, we have:

dq(t)dt

=Ni=1

ri(t) C (3)

where ri(t) is the rate of the i-th source in the link, q(t)is the queue length, N is the number of active flows and Cdenotes the capacity of the bottleneck link. Because of thehomogeneity of sources in DCE networks, (3) can be writtenas

dq(t)dt

= N[ri(t) C

N

](4)

The difference of the queue lengthq(t) in a sampling intervalis

q(t) = tdq(t)dt

=1

pmC

dq(t)dt

(5)

where t is the average sampling period. Combining (1), (4)and (5), the feedback variable (t) in the BCN message canalso be rewritten as

(t) = {[q(t) q0] + wN

pmC

[ri(t) C

N

]}(6)

Referring to (2), we readily obtain

dri(t)dt

={

GiRu(t) (t) > 0Gd(t)ri(t) (t) < 0

(7)

So far, the dynamic of BCN congestion control system in DCEnetworks can be described by a set of first order ordinarynonlinear differential equations (4) and (7), which consistof a typical autonomous system. It is noted that equation(7) is characteristic of segmented nonlinearity. The switchingfunction (t) = 0 divides the whole state space into two parts.The behaviors in the different state spaces are determined bythe different differential equations.

2222

C. Understanding of BCN Stability

The BCN mechanism aims to limit the queue length withinbuffer size. Since the empty queue wastes the link resource andthe buffer overflow results in dropping packets, the rationalsolution is to regulate the queue length to the reference.Therefore, the stability of queue can be generally regardedas the stability of the whole system. In some investigationsof Internet congestion control, the stability criterions in theclassical linear control theory are straightforwardly used toevaluate system stability, such as [18] and [19]. The stabilityanalysis of the BCN system in [4] follows the same paradigm.After the BCN system is approximated as two linear isolatedsubsystems, the conditions for the stability of each of thetwo subsystems are derived separately using Nyquist stabilitycriterion. The sufficient condition for the stability of the overallBCN system is obtained through combining the conditionsfor stable subsystems stiffly. The conclusion can providesome insights, but its limitations are obvious. Because thenonlinearity of the variable structure control employed by theBCN mechanism is missed, some particular phenomena arenot opened out. For example, the stability criterion given in [4]cannot definitely explain why queue oscillations occur in BCNsystem. Moreover, the conclusion in [4] can only tell the truthof steady behaviors in BCN system. When taking the physicalconstraint of buffer into account, packet droppings due tothe transient overshoot of queue system cannot be properlydescribed by the analysis in [4]. Therefore, in this work, wewill use the phase plane analysis method in nonlinear controltheory to conduct a comprehensive and in-depth investigationon transient and steady behaviors in BCN system where thenonlinearity of the variable structure control and the physicalconstraint of buffer are held.The phase plane analysis is a graphic method to analysis

the transient phenomena of second order ordinary differentialequation. It can not only analyze the stability and oscillationsbut also give an explicit trajectory of the dynamics of the sys-tem, much better than displaying variables against time. Thedifferential equations (4) and (7) tell that the queue controlledby the BCN mechanism is a second order nonlinear system.Assuming the phase trajectory of the BCN queue system isthe curve l{q(t), q(t)} where q(t) = dq(t)dt , we illustrate therelationship between phase trajectories and queue stability.Some possible shapes of phase trajectories are shown in Fig.3, where the queue motion is restricted to the strip shaded areaon the phase plane because of the physical limitations of buffer(B denotes the buffer size). The system equilibrium state lies atpoint (q0, CN ), and the switching line divides the shaded areainto two parts, corresponding to rate decrease and increase,respectively. Apparently, the queue motions described by thephase curves l1 and l2 are unstable since they diverge from theequilibrium point. It is not difficult to obtain the similar con-clusion using the stability criterions in control theory, just likein [4]. Naturally, these stability criterions also suggest that thequeue motions described by the other phase curves is stable.Seemingly, the queue motions described by the phase curves

B0

q0

C/N

q(t)

)(tq

0)( =tSwitching Line

Rate Decrease Region

Rate Increase Region

Equilibrium Point

Fig. 3. Phase trajectories

l3 and l4 are stable since both of them eventually approach theequilibrium point. However, the movements along the dashedlines would never occur due to the physical limitations of thebuffer. The queue stays either full or empty, which implies thatthe incoming packets are dropped or the link is wasted. Thelinear stability analysis, which does not take the nonlinearityof physical limitation of buffer into account, is incapable ofopening up this phenomena. To overcome it, we introducea more meaningful definition of the stability with respect toqueue analysis in congestion control in DCE networks.

Definition 1: t0 > 0, when t > t0, if and only if 0 0b[y(t) + C][x(t) + ky(t)] (t) < 0

(8)Define functions f1(t) = y(t), f2(t) = a[x(t) + ky(t)], andf3(t) = b[y(t) + C][x(t) + ky(t)]. Since for i = 1, 2, 3 andany z = (z1, z2) = (x(t), y(t))

fi(t, z1) fi(t, z2) L z1 z2 Namely Lipschitz condition is satisfied, the set of nonlineardifferential equations (8) have a solution uniquely determinedby the initial value (x0, y0) [22]. Also since x = 0, y = 0 isunique solution of a set of equations{

f1(t) = 0f2(t) = 0

or{

f1(t) = 0f3(t) = 0

the origin is unique singular point of the nonlinear system (8).Lyapunov has shown that the stability of and the behavior ofthe trajectories in the neighborhood of a singular point can befound from a linearized version of nonlinear differential equa-tions about the singular point [23]. Expanding the equations(8) in a Taylor series at the singular point, i.e. the equilibriumpoint (0, 0), the equations of the first approximation are

dx(t)dt = y(t)

dy(t)dt =

{ ax(t) aky(t) (t) > 0bCx(t) bwpm y(t) (t) < 0

(9)

The eigenvalues of system (9) are roots of its characteristicequation:

2 +mi+ ni = 0 i {1, 2} (10)where m1 = ak and n1 = a as (t) > 0, or m2 = bwpm andn2 = bC as (t) < 0.Considering the physical meaning of the network parame-

ters, the coefficients mi and ni must be positive. According

to Routh-Hurwitz stability criterion [24], we can straightfor-wardly get the following proposition.Proposition 1: Both increasing and decreasing rate subsys-

tems are stable in the context of pure linear control theoryanalysis.Remarks: Without considering the nonlinear factors in the

BCN congestion management system, such as buffer limitationand variable structure control, the conclusion drawn fromseparate analysis to subsystems is impractical because thetransient behavior of switching between two subsystems ismissed.

B. Typical phase trajectories

If the physical limitations of the buffer and the transientbehavior of switching procedures are taken into account, weneed to carefully check the phase trajectories to judge whetherthe BCN system is strongly stable. The phase trajectories ofsystem (9) depend on its eigenvalues. Obviously,

1,2 =mi

m2i 4ni2

(11)

Case1: m2i 4ni < 0In this case, the eigenvalues are complex conjugates 1,2 = j, here = mi/2 and =

4ni m2i /2. The

solutions of a set of differential equations (9) are:{x(t) = Aet cos(t+ )y(t) = Aet cos(t+ )Aet sin(t+ ) (12)

here A =(2 + 2)[x(0)]2 2x(0)y(0) + [y(0)]2/,

= arctan y(0)x(0)x(0) , where x(0) and y(0) arethe initial values. A compact form of (12) is definedas {x(t), y(t)} = H{t|x(0), y(0)}. Accordingly, t =H1{x(t), y(t)|x(0), y(0)} denotes solving t from equations(12) when {x(t), y(t)} is known. The curves H {x(t), y(t)}are phase trajectories in this case .Multiplying both sides of the first equation in (12) by ,

then adding both sides of the second equation in (12), we have

y(t) x(t) = Aet sin(t+ ) (13)Combining (13) and the first equation in (12), we can yield

[y(t) x(t)]2 + [x(t)]2 = (A)2e2t (14)

tan (t+ ) = y(t) x(t)x(t)

(15)

Solving t from (15), then substituting it into (14), we have

[y(t)x(t)]2+[x(t)]2 = c1 exp{2

arctan

y(t) x(t)x(t)

}(16)

here c1 = (A)2 exp( 2 ). Let r(t) cos (t) = x(t) andr(t) sin (t) = x(t) y(t). Since tan (t) = y(t)x(t)x(t) =tan (t+ ), the form of (16) in polar coordinates should be{

r(t) =c1 exp

{ (t)

}(t) = t+

(17)

2424

x

y

Bq0 q0

(x1(0),y1(0))

(x2(0),y2(0)) maxxs(x2(0),y2(0))

minxs(x1(0),y1(0))

Fig. 4. Phase trajectories as m2i 4ni < 0

Thus, the phase trajectory H {x(t), y(t) | (x(0), y(0))}beginning at point (x(0), y(0)) actually is a logarithmic spiral.Since < 0 and > 0, r decreases but increases as timet increases, which implies that the motion along the phasetrajectory eventually approaches the origin. Therefore, thesingular point of the system is a stable focus. Fig.4 illustratestwo branches of these phase trajectories originating from dif-ferent initial points. Since y(t) = dx(t)dt , x(t) reaches its localextrema when y(t) = 0, namely x(t) crosses the horizontalaxis with its local extremum. Substituting y(t) = 0 into (12),we can obtain tan(t + ) = . Since the inverse tangentis a multivalued function, x(t) has infinite local extrema asshown in Fig.4. As y(0) > 0, the local extremum closest tothe initial point (x(0), y(0)) is the local maximum denoted bymaxsx{x(0), y(0)}, while y(0) < 0, it is the local minimumdenoted by minsx{x(0), y(0)}. Let t denote the time at whichthe system reaches the closest extremum. Since t 0, wehave

t =

{1 [tan

1 + tan

1 y(0)x(0)x(0) ] x(0)y(0) 0

1 [ + tan

1 + tan

1 y(0)x(0)x(0) ] x(0)y(0) < 0

(18)

where tan1(z) denote the principal value of the inversetangent. Substituting (18) and cos(t+ ) = /

2 + 2

into (12), we readily obtain

smax

x{x(0), y(0)} = A

2 + 2exp(t) (19)

sminx

{x(0), y(0)} = A2 + 2

exp(t) (20)

Case2: m2i 4ni > 0In this case, there are two different negative real eigenvalues.Assuming 1 < 2 < 0, the solutions of differential equations(9) are: {

x(t) = A1e1t +A2e2t

y(t) = A11e1t +A22e2t(21)

x

y

O Bq0 q0

(x1(0),y1(0))

(x2(0),y2(0))

maxxp(x2(0),y2(0))

minxp(x1(0),y1(0))

y=1x

y=2x

Fig. 5. Phase trajectories as m2i 4ni > 0

where

A1 =2x(0) y(0)

2 1 ; A2 =1x(0) y(0)

1 2A compact expression of (21) is defined as {x(t), y(t)} =

F{t|x(0), y(0)}. The curves F{x(t), y(t)} are phase trajec-tories in this case.

Multiplying both sides of the first equation in (21) by 1and 2, respectively, then subtracting the results from bothsides of the second equation in (21), we have:

y(t) 1x(t) = (2 1)A2e2t (22)y(t) 2x(t) = (1 2)A1e1t (23)

If (x(0), y(0)) satisfies with y(0)1x(0) = 0, then A2 = 0.From (22), we have

y(t) 1x(t) = 0 (24)If y(0) 2x(0) = 0, in the same way, we also have

y(t) 2x(t) = 0 (25)The phase trajectories defined (24) and (25) are particular.Both of them are straight lines. As A1 = 0 and A2 = 0, Com-bining (22) with (23), we make some algebraic operations, andthen yield

[y(t) 2x(t)]2 = c2[y(t) 1x(t)]1 (26)where

c2 =[(1 2)A1]2[(2 1)A2]1 =

[y(0) 2x(0)]2[y(0) 1x(0)]1

Let u(t) = y(t) 1x(t) and v(t) = y(t) 2x(t), equation(26) is transformed into

v(t) = 2c2u(t)

12 (27)

Hence, the phase trajectory F{x(t), y(t) | (x(0), y(0))}originating from the initial point (x(0), y(0)), here y(0) =1,2x(0), is analogous to a parabola. Some phase trajectoriesoriginating from various initial points are depicted in Fig.5.Since the motion along the phase trajectory eventually ap-proaches the origin, the singular point is called a stable node.

2525

As y(t) = 0, x(t) arrives its global extremum denoted bymumpx{x(0), y(0)}:

mumpx{x(0), y(0)} = {(1)1 [y(0) 2x(0)]2(2)2 [y(0) 1x(0)]1

} 121

(28)mumpx{x(0), y(0)} is the maximum as y(0) > 0 or theminimum while y(0) < 0Case3: m2i 4ni = 0In this case, two real eigenvalues are identical, i.e. 1,2 = =mi2 .The solutions of equations (9)become:{

x(t) = (A3 +A4t)et

y(t) = (A3+A4 +A4t)et(29)

where A3 = x(0) and A4 = y(0) x(0). A compact formof (29) is defined as {x(t), y(t)} = L{t|x(0), y(0)}, and thecurves L {x(t), y(t)} describe the phase trajectories in thiscase.From (29), we can get

y(t) x(t) = A4et (30)If (x(0), y(0)) satisfies with y(0) x(0) = 0, then A4 = 0.Thus, the phase trajectory originating from (x(0), y(0)) is astraight line:

y(t) x(t) = 0 (31)From (29), we can also have

y(t)(A3 +A4t) = x(t)(A3 +A4 +A4t) (32)

Solving t from (32), and substituting it into (30)

y(t)x(t) = A4 exp{A4x(t)A3[y(t) x(t)]

A4[y(t) x(t)]}

(33)

Since < 0, limt tet = 0. Therefore, the movementalong the phase trajectory eventually approaches the origin.The singular point is a stable node. Although the solutionform of equations (9) in this case is different from that ofthe previous case, the phase trajectories L {x(t), y(t)} aresimilar toF{x(t), y(t)}. The only difference is that the formercontain one straight line, but the latter include two straightlines. Because of limited space, we omit the illustration of thephase trajectory in this case. As y(t) = 0, x(t) reaches itsunique extremum denoted by mumux{x(0), y(0)}:

mumux{x(0), y(0)} = A4

eA3+A4

A4 (34)

mumux{x(0), y(0)} is the maximum as y(0) > 0 or theminimum while y(0) < 0.

C. Stability analysis of BCN system

Next, according to the definition of strong stability, weuse the phase plane techniques to analyze the stability of theBCN system described by equations (9) with linear segmentednonlinear properties.Generally, the initial state of the BCN system is q(0) = 0

and ri(0) = , here is the initial rate of sources. Thisinitial state corresponds to the point (q0, N C) in the

t

x

q0

maxsx(x1d(0),y1d(0))

(b)

minsx(x2i (0),y2i (0))

t

y

(c)

x

y

Bq0 q0 (x1i (0),y1i (0))

(x1d(0),y1d(0)) T1i

I Region

D Region

(a)

(x2d(0),y2d(0))

(x2i (0),y2i (0))

(x3i (0),y3i (0))

x+ky=0

T1d

T2i

T2d

Fig. 6. Phase trajectory and dynamic behaviors as a < 4p2mC

2

w2and b 0, or n2 = bC as (t) < 0.As the characteristic equation (35) has two different nega-

tive real roots 1,2 =kni

(kni)24ni2 , and 2 +

1k =

(k2ni2)24(k2ni2)2k < 0, we have 1k > 2 > 1.

Thus, some theoretically feasible phase trajectories impossiblyappear in the actual BCN system. Subsequently, we discussthe stability of the BCN queue moving along six basic phasetrajectories case by case.Case1: a < 4p

2mC

2

w2 and b 4p2mC

2

w2and b 4p

2mC

2

w2 and b 1,2, thus the queue phasetrajectory must traverse the switching line x+ ky = 0 in thesecond quadrant. After traversing the switching line for thesecond time in the fourth quadrant, it gradually approachesthe equilibrium point and the straight line y = 2x, butnever traverse the latter because it is an asymptote of thephase trajectory. Substituting x1i (0) = q0, y1i (0) = 0 andy1d(0) = kx1d(0) into (26), we can get

y1d(0) = q0

[(k + 1/1)1

(k + 1/2)2

] 121

Following the same method in case 1, we also have

max2 {x(t)} = q0bC[(k+1/1)

1

(k+1/2)2

] 121 exp

{dd

[

+tan1 dd 1d]} (38)

2727

t x

q0 (b)

t

y

(c) (a)

x

y

Bq0 q0 (x1i (0),y1i (0))

(x1d(0),y1d(0))

T1i

D Region

I Region

T1d

x+ky=0 y=2x

y=1x

Fig. 9. Phase trajectory and dynamic behaviors as a < 4p2mC

2

w2and b >

4p2mC

w2

where dd =bkC

4bC(kbC)2 , and 1,2 =kaa2k24a

2 . Thus,

we have the proposition.Proposition 3: As a < 4p

2mC

2

w2 and b 4p2mCw2

Contrary to case 2, the phase trajectory is a logarithmic spiralin the rate increasing region, but a parabola in the decreasingregion. Fig.9 illustrates the representative behaviors of theBCN system in this case. After crossing the switching line,the phase trajectory moves directly to the equilibrium point.The queue motion in phase plane is limited in the secondquadrant because the line y = 2x is a asymptote for the phasetrajectory originating from the initial point (x1d(0), y

1d(0)) as

y1d(0) = 2x1d(0), which implies that the queue length doesnot overshoot the reference value q0 as shown in Fig.9 (b).Therefore, the BCN system is strongly stable forever in thiscase.Case4: a > 4p

2mC

2

w2 and b >4p2mCw2

In this case, the phase trajectory is a parabola in both rateincreasing and decreasing regions. The dynamic evolution ofsystem is shown in Fig.10. The strong stability can be alwaysguaranteed based on the same reasons in case 3.Case5: a = 4p

2mC

2

w2 or b =4p2mCw2

In this case, the switching line x+ ky = 0 is a special phasetrajectory due to 1,2 = 1k . Therefore, the phase trajectoryonly appears in the rate increasing region as a = 4p

2mC

2

w2 , orapproaches the equilibrium point along the switching line asb = 4p

2mCw2 . Naturally, the system is strongly stable.

The analysis in these three cases can be summarized as thefollowing proposition.Proposition 4: As b 4p2mCw2 or a = 4p

2mC

2

w2 , the BCNsystem is strongly stable.Summing up Proposition 2 to 4, we can draw a stability

criterion for the BCN congestion control system in DCEnetworks.Theorem 1: The sufficient condition for the strongly stable

BCN system is{1 +

RuGiNGdC

}q0 < B

t x

q0 (b)

t

y

(c) (a)

x

y

Bq0 q0 (x1i (0),y1i (0))

(x1d(0),y1d(0))

T1i

D Region

I Region

T1d

x+ky=0

y=i2x y=i1x

y=d1x y=d2x

Fig. 10. Phase trajectory and dynamic behaviors a > 4p2mC

2

w2and b >

4p2mC

w2

Proof: Since dd < 0 and (+tan1 d

d1d) > 0, from

(36), we have

max1{x(t)} < |x1d(0)|

kbC

Also x1d(0) = kA1i

4aa2k22 e

ak2 T 1i and A1i =2q0

a

4aa2k2 ,thus max1{x(t)} q0. In other words, as{1 +

abC

}q0 < B,

the conditions required by Proposition 2 can be satisfied.In case 2, 1,2 = ka

a2k24a2 < 0. Since 1 < 2