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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: [email protected]
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
2010 International Conference on Distributed Computing
Systems
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2010 International Conference on Distributed Computing
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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
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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.
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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)
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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.
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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
