Robustness of Real and Virtual Queue-Based Active Queue Management Schemes

13 Pages·0.495 MB·English

by  Lakshmikantha A., Beck C., Srikant R.

IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005 81
Robustness of Real and Virtual Queue-Based Active
Queue Management Schemes
AshvinLakshmikantha,CarolynL.Beck,Member,IEEE,andR.Srikant,SeniorMember,IEEE
Abstract—In this paper, we evaluate the performance of both thevirtualqueueisalwayssmallerthanthecapacityofthereal
real and virtual queue-based marking schemes designed for use queue.
at routers in the Internet. Using fluid flow models, we show via
Weconsiderthefollowingcongestioncontrolmechanismsat
analysisandsimulationsthatVirtualQueue(VQ)-basedmarking
thesource:proportionallyfaircongestioncontrol(PFC)[8]and
schemesoutperformRealQueue(RQ)-basedmarkingschemesin
terms of robustness to disturbances and the ability to maintain minimumpotentialdelaycontrol[19].Duetothesimilarityto
lowqueueingdelays.Infact,weprovethatalinearizedmodelof TCPcongestionavoidancealgorithms,werefertotheminimum
RQ-based marking schemes exhibit a lack of robustness to con- potentialdelaycontrolschemeasaTCP-typecongestioncontrol
stantbutotherwiseunknownlevelsofdisturbances.Theanalytical
scheme[16].ThefollowingActiveQueueManagement(AQM)
results we present are applicable to combinations of proportion-
schemesareconsideredattherouter:RED(randomearlydetec-
ally fair and TCP-type congestion controllers at the source, and
RandomExponentialMarking(REM)and ProportionalControl tion) [5], REM (random early marking) [2], PC (proportional
(PC) schemes at the router. The behavior of Random Early Dis- control) [7] and PI (proportional-integral) control [7]. All of
card(RED)andProportional-Integral(PI)controlschemesatthe these schemes detect congestion based on the queue lengths
routerarealsostudiedviasimulations.
at the link, and, essentially, are distinguished by the specific
Index Terms—Active Queue Management, congestion control, functionusedtodeterminetheprobabilityofmarkingordrop-
fluid-flowanalysis. ping packets that is implemented at the router. We compare
implementations of each of these AQM schemes on the real
queuewithimplementationsonavirtualqueue.Wedemonstrate
I. INTRODUCTION
throughacombinationofanalysisandsimulationthat,byusing
INRECENTyears,therehasbeensignificantinterestinthe
avirtualqueueandreducingthelinkutilizationtoslightlyunder
designofalow-loss,low-delayInternet[9],[10],[13],[15],
100%(e.g.,to95%),wecanachieveasignificantimprovement
[17].Theprimaryenablingtechnologyforadvancingsuchade- inthe performanceof thenetwork. Weconsider twomaincri-
signisbasedontheuseofEarlyCongestionNotification(ECN)
teriaforassessingtheperformanceofthenetwork:
capabilityattherouters.Unlikethetraditionalcongestionnoti-
• Queueing delay: The queueing delay should be main-
ficationmechanismwherebyroutersdroppacketstosignalcon-
tainedatasmallfractionofthepropagationdelayinthe
gestion,withECN,routershavethecapabilitytomarkpackets
network.
toindicatecongestion.Markingreferstotheprocessofflipping
• Robustness: In most models of congestion control, it is
abitinthepacketheaderfromazerotoaonewhentherouterde-
typically assumed that all flows are long-lived and thus,
tectsincipientcongestion.Eachreceiverechoesthemarkstoits
steady-state stability analysis is reasonable. However, in
sourceandthesourceisexpectedtorespondtoeachmarkbyre-
a real network, there are many short-lived flows (popu-
ducingitstransmissionrate.Inthispaper,wefocusonthemech-
larlyknownaswebmice)andsourcesthatdonotrespond
anism by which marking is performed at the routers. Specifi-
tocongestionnotificationsuchasreal-timeflows,which
cally,wecompareschemeswherearoutermarkspacketsbased
wecollectivelyrefertoasdisturbances.Inthispaper,we
on the real queue length to schemes where the router marks
evaluatetherobustnessofoursystemwithrespecttothese
packetsbasedonthequeuelengthofavirtualqueue(VQ)[6].
unmodeleddisturbances.Specifically,weevaluatethedis-
A virtual queue is a fictitious queue, maintained at each link,
turbance traffic load that the system can tolerate while
withacapacitythatislessthantheactualcapacityofthelink.
maintainingsmallqueuelengths.
Themotivationformaintainingavirtualqueueisthatitprovides
WefirstconsiderthecaseofPFCwithREMattherouter.We
advance warning of network congestion, since the capacity of
present a necessary condition thatthe controller and the REM
parameters must satisfy to ensure local stability of the system
ManuscriptreceivedNovember12,2002;revisedJuly18,2003;approved underdelayedfeedback.Usingthiscondition,weprovethatit
byIEEE/ACMTRANSACTIONSONNETWORKINGEditorS.Low.Thisworkwas isnotpossibletomaintainlowqueueingdelaysinthepresence
supportedbytheAirForceOfficeofScientificResearchunderAFOSRGrant
F49620-01-1-0365 andby theDefenseAdvancedResearch ProjectsAgency of disturbances in an RQ-based system. We then show, alter-
underDARPAGrantF30602-00-2-0542. natively, that if REM is implemented in a VQ-based system,
A. Lakshmikantha and R. Srikant are with the Department of Electrical
both stability and small queueing delays can be achieved for
and Computer Engineering and Coordinated Sciences Laboratory, Uni-
versity of Illinois at Urbana-Champaign, Urbana, IL 61801 USA (e-mail: anyvalueoflinkutilizationlessthanone.Wefurthershowthat
[email protected];[email protected]) thesamephenomenonoccursforthefollowingcombinationsof
C.L.BeckiswiththeDepartmentofGeneralEngineering,UniversityofIlli-
source controllers and AQM schemes: PFC and PC, TCP and
noisatUrbana-Champaign,Urbana,IL61801USA(e-mail:[email protected])
DigitalObjectIdentifier10.1109/TNET.2004.842225 REM, and TCP and PC. Additional congestion control/AQM
1063-6692/$20.00©2005IEEE
82 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
scheme combinations are studied via simulations with results II. MODELS OF CONGESTION CONTROLLERS
summarizedinthefollowing. ANDAQMSCHEMES
1) PFC/RED, TCP/RED: The results that were established Weconsideradeterministicfluidflowmodelofasinglecon-
analytically for the PFC/PC and TCP/PC combinations gestion-controlledsourceaccessingasinglelink.Suchmodels
alsoholdinsimulationswhenREDisimplementedatthe havebeenusedwithmuchsuccessinmanypriorworks[4],[7],
router. [9],[10]–[13],[15],[17].Supposethatthetransmissionrateof
2) PFC/PI,TCP/PI:Inallofthepreviouscases,theVQ-based theuserattime isdenotedby ,thelinkcapacityisdenoted
controllers outperform the RQ-based controllers in the by ,andqueuelengthisdenotedby .Inadditiontothecon-
presence of constant but unknown disturbance levels. gestioncontrolledsource,wealsoassumethatafractionofthe
Note that in our analytic models, the disturbance levels linkcapacityisoccupiedbyshortflowsorunresponsiveflows.
are assumed to be constant, however this constant is Let denotethefractionofthelinkcapacitythatisusedbythese
unknown to the controller at the source and the router. disturbanceprocesses.Theevolutionofthequeueisthengov-
In contrast all our simulations indicate that the PI con- ernedbytheequation
trollerisrobusttodeterministicdisturbancesinbothRQ
and VQ-based systems. However, the results for the PI if
(1)
if .
controllerchangedramaticallyinthepresenceofrandom
disturbances, which are considered much more realistic
Similarlyifavirtualqueueismaintainedattherouter,thenthe
for modeling web mice in the Internet. Our simulations
evolutionofthevirtualqueuelengthisgivenby
indicatethatasthefractionofthetotalcapacityoccupied
by short-lived flows increases, the queue length in the if
(2)
RQ case increasesaccordingly; howeverwithVQ-based if
PIcontrolthequeuelengthremainssmall.
The main contribution of this paper is the demonstration of where isthedesiredlinkutilization.Notethattheonlyquan-
the fact that marking based on a virtual queue is more robust tity of interest in the virtual queue is its length, . Therefore,
than marking based on a real queue. In this work, a system is unlike the real queue, one does not need to know the detailed
said to be robust if it is locally stable and is able to maintain contentsofthevirtualqueue.Thus,thevirtualqueuecanbeim-
small queue lengths in the presence of disturbances. Further, plementedasa counterthatupdates accordingto(2).Also,
based on both our analytical and simulation results, we con- therateatwhichthequeuechangesisafunctionofthediffer-
cludethatwhenVQ-basedmarkingisused,thechoiceofwhich ence between the arrival and departure rates. In other words,
specific AQM scheme is implemented appears to be of mar- explicit knowledge of is not required to calculate the queue
ginal importance. However, for RQ-based marking, PI control length.
schemesappeartoperformbetterthantheotherAQMschemes
A. CongestionControlSchemes
that we have considered. The intuitive reason for the robust-
nessofVQ-basedAQMschemesisasfollows:inaVQ-based Weconsidertwomechanismsfordeterminingthesourcerate
scheme,thequeuelengthisguaranteedtobesmallsincethear- dynamics:
rivalrateislessthanthelinkcapacity.Thus,thesystemparam- 1) PFC (Proportionally Fair Control): Let denote the
etershaveonlytobedesignedtomaintainstabilityinthepres- probability that a packet is marked at the link when the
enceofdisturbances.Ontheotherhand,withRQ-basedAQM queuelengthis .ThePFCmechanismisthengivenby
schemes,onehastofurtherdesignthesystemtomaintainsmall
queuelengths.Thus,inasense,VQ-basedschemeshaveanad- (3)
ditionaldegreeoffreedomthatallowsonetochoosethesystem
parameters to stabilize the system even in the presence of un- where the right-hand side of the above equation has to
knowndisturbances. beappropriatelymodifiedtoaccountforthefactthatthe
We note that adapting the virtual queue capacity to varying sourcerateisalwaysnonnegative.Here and arecon-
network traffic conditions is not considered in this paper. trollerparameters( isthecontrollergainand canbe
Typically,thisformofadaptationisperformedonatimescale thoughtofasthepricethatauseriswillingtopay[8]),and
that is slower than the dynamics of the congestion controllers denotes the feedback delay (also known as round-trip
[13]–[15],whereastheanalysisinthispaperisappropriatefor timeorRTT)inthenetwork.TheRTTisthesumoftwo
modelsatthecongestion-controltimescale. components:thepropagationdelay,whichwedenoteby
, and the queueing delay, given by . Note that
Thispaperisorganizedasfollows.InSectionII,wepresent
(3) should be modified by replacing with
the models for the various congestion controllers and AQM
whenmarkingisbasedonvirtualqueuelength.
schemesusedinthispaper.InSectionIII,wepresentthemain
2) TCP-typeControl:Thecongestionavoidancephaseofthe
analytical results. Section IV contains simulation results that
TCP-typecontrolmechanismcanbemodeledasfollows
strengthen the observations made in Section III, and provide
[16],[20]:
further insight into those combinations of controllers/AQM
schemes that are not addressed in Section III. Concluding
remarksareprovidedinSectionV. (4)
LAKSHMIKANTHAetal.:ROBUSTNESSOFREALANDVIRTUALQUEUE-BASEDACTIVEQUEUEMANAGEMENTSCHEMES 83
where , and , and are defined as be- RED is operating in the linear region between and
fore.TheTCPcongestioncontrolmechanismcanbein- .
terpreted as follows: when there is no congestion feed- 3) PI Control [7]: The primary motivation here is to elimi-
back, the source increases its transmission rate by nate or reduce possible limitations introduced by imple-
per unit time. For each congestion mark received, the mentingRED,namely:(a)thetradeoffbetweenresponse
transmission rate is reduced by a factor . Since timeandstabilityand(b)couplingofequilibriumqueue
istherateatwhichmarksare length and equilibrium loss probability values. The PI
received, we obtain the dynamics given in (4). For our controlschememarkspacketsataratethatisproportional
analysis, we consider the following general form of the toboththeinstantaneousqueuelengthandtheintegralof
controller: thequeue-lengthattime .Thisalgorithmcanthusberep-
resentedbythefollowingdifferentialequation:
(5)
(9)
Clearly,throughappropriatechoiceof , and ,(5)can
beputintheform(4).
where and arecontrollergainsthatmaybechosen
todeterminethesystembehavior,and isthedesired
B. AQMSchemes
queuelength.Notethat canbesettoanyvalue,thus
WeconsiderfourrecentlydevelopedAQMschemes([1],[5], the equilibrium marking probability and the equilibrium
[7]),whichwedescribebrieflyinthefollowing.Notethatthese queuelengthareeffectivelydecoupled.
AQM schemes are described as implemented in a real queue. 4) PC[7]:Theproportionalcontrolalgorithmmarkspackets
The same schemes can be implemented in a virtual queue, in with a probability that is directly proportional to the in-
whichcase shouldbereplacedby inwhatfollows. stantaneousqueuelength,thatis,
1) REM[1]:ThemarkingprobabilityfunctionusedbyREM
(10)
isgiven by
where isaproportionalityconstant.Notethattheuse
(6)
ofproportionalcontrolattheroutercapturesthebehavior
of RED when queue length averaging is not performed.
where representstheinstantaneousqueuelengthatthe
That is, the use of proportional control is essentially
buffer(whichmayberealorvirtual),and isaparameter
equivalent to using instantaneous queue length with the
thatdeterminestheresponsivenessoftheroutertoincip-
RED-markingprofile.
ientcongestion.Highervaluesof leadtohighermarking
probability,whichimpliestherouterismoreaggressivein
III. ANALYTICALRESULTS
respondingtocongestion.
2) RED[5]:TheREDalgorithmemploysamarkingscheme Inthissectionweconsidersystemsdescribedbyvariouscom-
based on an average queue length, which we denote by binationsof(1)–(6)and(10).Ourprimaryfocusisananalysisof
.Theaveragequeuelengthatanytime iscomputed thebehaviorofbothrealandvirtualqueue-basedsystemswith
as a weighted sum of the current queue length and the proportionallyfaircontrolimplementedatthesourceandREM
previousvalueoftheaveragequeuelength.Themarking implementedat the router.Systems with TCP-type congestion
probabilityischosenasafunctionof asfollows: controlatthesource,andwithproportionalcontrolattherouter
• if ,then ; areevaluatedsimilarly.
• if ,then ;
A. PFCandREMWithRealQueueMarking
• if ,then ;
where and areuser-definedthresholdsand WefirststudyaPFC/REMsystemwithnouncontrolleddis-
is a constant. The averaging implemented by the RED turbancesaffectingthe link, thatis,weconsider the systemof
algorithmcanbemodeledbyalowpassfilter.Hence,as delaydifferentialequationsgivenin(1),(3),and(6)with .
in[7],weapproximatethisbehaviorasfollows: Webeginbylinearizing(1)and(3)aboutequilibrium,thus,we
studyonlylocalstability.Notethattheequilibriumpointof(1)
and(3),withREMimplementedattherouter,isgivenby
(7)
where denotestheLaplacetransformof .Inthe
timedomain(7)canberepresentedby (11)
(8) Clearly,thestabilityofthelinearizedsystemdoesnotguarantee
thatthesystemisgloballystable.However,thereisampleevi-
where is a function of the averaging parameter used dencetosuggestthatcontroldesignwiththegoalofstabilizing
in computing . We note that the above differential alinearizedversionofthesystemisaverygooddesigncriterion
equationisanapproximationbasedontheassumptionthat (seetheextensiveliteraturesurveyonthistopicin[18]).
84 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
We follow the usual approach and consider perturbations Now,from(17)notethatwhen , .Hence
around the equilibrium values of the source rate and queue
length,denotedby and ,respectively,i.e.,
Thereforegivenan ,thereexists suchthat
(12)
Linearizingaround yieldsthefollowinglineardelaydif-
(20)
ferentialequationfor :
(13) From(19)and(20)wethushave
where
(14)
Using the fact that for all , it is
and clear that by choosing and sufficiently small the stability
conditioncanbesatisfied.
(15) Recall that the equilibrium value of the queueing delay is
given by . Our goal is to ensure system stability while
In the above differential equation, denotes the equilibrium maintainingasmallqueueingdelay;specifically,wewouldlike
valueforthemarkingprobability(whichisgivenby ),and theequilibriumqueueingdelaytobeafractionofthepropaga-
denotesthefirstderivativeof evaluatedat . tiondelay.Inthe followingtheorem,weshowthatthisispos-
The following stability result may be derived from the sible only if the equilibrium marking probability is large (i.e.,
Nyquistcriterion,orasaspecialcaseofTheorem2.1in[3]. closeto1).
Lemma1: Thesystemdefinedby(13)–(15)isstableforall Theorem3: Supposethatthesystemparametersarechosen
,andisunstablefor ,where isgivenby suchthatthelinearizedformofthesystemgivenby(1),(3),and
(6)isstable,and forsome .Then, must
satisfythefollowinginequality:
(16)
and isthepositivesolutionoftheequation
(17) Thus, implies .
Proof: FromLemma1,forstabilitywerequire
Thestabilityconditiongiven inLemma1providesameans
fordeterminingstabilityintermsofthesystemdesignparame-
ters.Inparticular,ifPFCandREMareimplementedinanRQ
systemwithnodisturbances,thefollowingholds.
Lemma2: Given , and itisalwayspossibletochoose
where isasolutionoftheequation
and suchthatthesystemdescribedby(13)–(15)isstable.
Proof: TheconditionforstabilityfromLemma1is
(18) Since for ,anecessaryconditionforsta-
bilityisgivenby
Substitutingfrom(11)for gives
Alternatively,given anddenotingtheequilibriumqueueing
Wecanthusrewrite(18)as delayby ,theparameter mustsatisfy
(19) (21)
LAKSHMIKANTHAetal.:ROBUSTNESSOFREALANDVIRTUALQUEUE-BASEDACTIVEQUEUEMANAGEMENTSCHEMES 85
inorder forthe linearsystemtosatisfy the stabilitycondition. andmarkingprobabilities,itbecomesclearthattomaintainsta-
From(11)and(21)wethushave bility, mustvanishas .Toseethis,let denote
theequilibriummarkingprobabilitywhen ,i.e.,
orequivalently
Nownotethattheequilibriumthroughputatadisturbancelevel
of isgivenby
(22)
Ifweadditionallyrequirethat
Let denotetheequilibriummarkingprobabilityforadis-
turbancelevel .Since mustbelessthanorequaltoone,
we know
thentheinequalitygivenin(22)canbedirectlyrewrittenas
whichimplies
Remark 1: Note that for the main inequality of Theorem 3
Now, from Theorem 3, as we know , and
tohold,as ,wemusthave .Thisimpliesthatif
therefore .Inotherwords,ifthesystemisdesignedto
werequirealowqueueingdelay,thentheequilibriummarking
maintainbothstabilityandlowqueueingdelays,thenonlyvery
probabilitymustbeverycloseto1.Asanexample,supposewe
lowlevelsofdisturbancescanbehandledattherouter.
require the equilibrium queueing delay tobe lessthan 10% of
Building on the previous remark, we now show that if the
thepropagationdelay.Then,itiseasytoshowthat mustbe
system is designed to handle some “worst case” disturbance
greaterthan0.975.
level, ,theninstabilitymayresultwhenthesystemexperi-
WenowconsidertheeffectofdisturbancesonthePFC/REM
enceslowerlevelsofdisturbances.
real queue system,thatis,we considerthe systemrepresented
Theorem 4: Suppose that the parameters of the linearized
by (1), (3) and (6) with the disturbance parameter .
formofthesystemdescribedby(1),(3),and(6)aredesignedso
Againlinearizingthesystemabouttheequilibriumgives
thatthesystemmaintainsstabilityandlowqueueingdelays(i.e.,
)whenthedisturbancelevelis .Thengivenany
(23)
there existsan suchthatforall ,the
linearstabilityconditionisviolatedforall .
where,inthiscase,
Proof: Theequilibriummarkingprobabilitywhenthedis-
turbanceis isgivenby
and (24)
FromthestabilityconditiongiveninLemma1,wehave
For analysis purposes, we assume that is a constant whose
valueisunknownapriori.Thus,wehavealinearsystemwhose
parametersareunknown.Wewouldagainliketoensurestability
and also maintain low queueing delays, but now in the pres-
ence of disturbances. In particular, we are interested in deter-
miningtherangeofvaluesfor forwhichthesystemremains
stable when the original system is designed assuming . FromLemma2,weknowitisalwayspossibletochoose and
Inthefollowingtheorem,wedemonstratethatstabilitywillbe suchthatthesystemisstable.Hence,wecanwrite
maintainedonlywhen isverysmall,i.e.,designingthesystem
for low queueing delays results in poor disturbance rejection (25)
properties.
Remark2: Notethatifwedesigntheparametersofthelin-
where issomefixedconstant.Nowsupposethatthe
earizedsystemgivenby(23)and(24)tomaintainstabilityand
systemexperiencesadisturbanceof ,thenthesystem
lowqueueing delays(specifically )inthe absence
must satisfy
ofdisturbances,thenforany thereexistssomemaximum
disturbancelevel, ,suchthatforall thesystem
(26)
isstable.However,ifweconsiderequilibriumthroughputrates
86 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
Since the expression on the right hand side of (26) is an in- and isthepositivesolutionoftheequation
creasing function of , one can explicitly determine the value
of forwhichtherighthandsidesof(25)and(26)areequal.At
thisvalueof ,whichwerefertoas ,thestabilityconditionis
violated.Thatis, satisfies
As was shown in Lemma 2 for an RQ-based system, it can
alsoreadilybeshownthatforaVQ-basedsystem,given , ,
and ,itisalwayspossibletochoose and suchthatthe
Itfollowsthatforallvaluesof ,thestabilitycriterionis systemisstable.However,foraVQ-basedsystem,theequilib-
notmet.Solvingfor gives riumvalueoftherealqueueisalwayszero,sincetheequilibrium
arrival rate is bounded above by and hence is always less
thantheactuallinkcapacity.Asaresult,thequeueingdelayis
alwayszero.Thus,unliketheRQcase,placinganupperbound
onthequeueingdelayimposesnorestrictionsonthechoiceof
the system parameter values. As we will see, this allows us to
achieve better disturbance rejection properties in a VQ-based
system.
Thusitisclearthatas , andhence
WenowconsidertheeffectofdisturbancesinVQ-basedsys-
.Hencethereexistsan suchthatforall
tems. Specifically, consider the system defined by (2), (3) and
and thesystemisunstable.
(6) with . Note that the rate at which the queue length
Based on the preceding analysis we can conclude the
growsisgivenby
following.
• To achieve low queueing delays in an RQ system with
PFC and REM, one must have very high equilibrium
marking probability ( ). Since is simply , a
andtheequilibriumrateatwhichtheusercansenddataisgiven
highvaluefor requiresthat mustbelarge.Inother
by
words, the QoS (quality of service) requirement that the
queueing delay remains small constrains the possible
valuesfor .
• Sincetheparameter isnowconstrainedduetotheQoS
Since ,wehave
requirement, the ability to freely choose the system pa-
rameters for the purpose of disturbance rejection is lim-
ited.Asaresult,thesystemexhibitspoordisturbancere-
jectionproperties.
• It is not possible to maintain stability, low queueing de-
orequivalently
lays and reject disturbances unless the disturbance level
isknownapriori.
B. PFCandREMWithVirtualQueueMarking
WenowconsidertheVQ-basedmarkingsystemrepresented Inthemaintheoremofthissectionweshowthatgivenaworst
by (2), (3), and (6) with . Linearizing this system about caseboundonthedisturbancelevel,itispossibletoensurethat
equilibrium gives the linear delay differential system repre- the system is stable for all levels of disturbance less than this
sentedby(13)–(15),butinthiscasetheequilibriumvaluesare upper bound. We begin by proving the following lemma re-
given by gardinguniformconvergenceofasequenceoffunctions,which
isrequiredinthemaintheorem.
Replacing by ,where ,weconsiderthefollowing
sequenceofreal-valuedfunctions:
where denotesthe design valuefor .A stabilitycondition (27)
fortheVQ-basedAQMschememaybedeterminedasintheRQ
casebyreplacing by in(16)and(17);thisisstatedinthe
Given a worst case bound on the disturbance, , where it
followinglemmaforconvenience.
isassumedthat ,thesequenceoffunctions
Lemma 5: The linearized form of the system described by
isdefinedinthedomain .Inthisdomain,
(2), (3) and (6) is stable for all values of and is
thesequenceoffunctionsconvergespointwiseto
unstablefor ,where isgivenby
LAKSHMIKANTHAetal.:ROBUSTNESSOFREALANDVIRTUALQUEUE-BASEDACTIVEQUEUEMANAGEMENTSCHEMES 87
Let denotethefirstderivativeof withrespectto .Recall for all . Letting and defining
that isthepositiverootoftheequationgiveninLemma5,that accordingly,wecanequivalentlystatethatgivenany there
is, exists suchthat
for all . The following inequality clearly
Notethatas , and . holds:
Lemma 6: The sequence of functions converges uni-
formlyto inthedomain .
Proof: Wefirstprovethatthesequenceoffunctions
and areuniformlybounded.Notethatforany ,thefunc-
tion has no singularities in . Further is clearly
Thusthereexistsa suchthat
boundedaboveby forall ,and hasnosingu-
laritiesin .Hence isuniformlybounded.Differentiating
withrespectto gives
(28)
orequivalently
where
where isdefinedin(29).Therefore,thereexists such
that
(29)
(30)
forall ,with asdefinedin(30).Notethat
(31) canbemadearbitrarilysmall.Further,as , for
some .Henceitfollowsthatthereexistsa suchthat
and
forall .Thatis,thereexistsa such
that isadecreasingfunctionof ,implyingthatif ,
then
Here, denotecontinuousboundedfunctionsof givenby
Thusthesystemisstableforall .
Remark3: BasedonTheorem7,weknowthataVQsystem
It is straightforward to show that the sequence is also withPFCandREM isstableforall .Thus,ifa
uniformly bounded. This implies that the family of functions smallvalueischosenfor ,wecanrejectlargelevelsofdistur-
is equicontinuous. A direct application of the Arzela– bances.
Ascolitheorem(see,forexample,[21])givestheresult.
Theorem 7: Considerthe linearizedform of the system de- C. TCPCongestionControlandREM
scribedby(2),(3)and(6)with .Let denotethedesign WenowconsiderTCP-typecongestioncontrolimplemented
valueof .Givenanupperbound onthedisturbancelevel, atthesource.AsbothPFCandTCPcontrolmechanismshave
whereitisassumedthat ,thereexist and similardelaydifferentialequationrepresentations,theanalysis
suchthatthesystemisstableforany . for TCP-type congestion controlproceeds ina manner similar
Proof: Fix andconsiderthesequenceoffunctions to that for PFC. The system is now described by (5) and (6),
defined in (27). By Lemma 6, we know that given any withtheequilibriumgivenby
thereexistsan suchthatforall wehave
and
88 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
when RQ-based marking is used. In the case of VQ-based levels of disturbances. That is, a direct analog of Theorem 7
schemes, this equilibrium point is modified by replacing the holdsaswell.
link capacity with the virtual link capacity . We first
completetheanalysisforanRQ-basedsystem.Linearizingthe D. ProportionalControlBasedMarking
TCP/REMrealqueuesystemgives ConsideranRQsystemwithPFCimplementedatthesource
andPCatthe router;thatis,considerthesystemdescribedby
(32) (1),(3),and(10).FromLemma1,itfollowsthatthesystemis
stable if
where
As in the preceding analyses, based on the inequality
, a necessary condition for stability is given
by
(33)
and
orequivalently
(34)
(35)
As before, is the equilibrium marking probability and
giventhat,inthiscase, , ,and
is the first derivativeof evaluatedat . Thecounterpart of
.
Lemma1forTCPcontrolisstatedinthefollowing.
Ifthemaximumdesiredqueueingdelayisdenotedby ,then
Lemma8: Thesystemdefinedby(32)–(34)isstableforall
the equilibrium queue length is bounded above by . From
andisunstablefor ,where
(35)wethenhave
whichleadsdirectlytotherelation
and
ThisinequalityimpliesthatifPFC-PCisimplementedinanRQ
system,regardlessofthevalueschosenforthesystemparame-
We now state a stability condition that is derived under the
ters,thequeueingdelayisalwaysgreaterthanthepropagation
usualQoSconstraintthatthequeueingdelaysremainsmall.The
delay.AsimilaranalysiscanbeusedtoshowthatifTCP-type
proofforthisresultcloselyfollowsthatofTheorem3,thuswe
congestioncontrolisimplemented,thefollowingholds:
omit ithere.
Theorem9: Supposethatthesystemparametersarechosen
suchthatthelinearizedformofthesystemgivenby(1),(5)and
(6)isstable,and forsome .Then must
satisfythefollowinginequality: Thussmallqueueing delaysare notattainablefor RQsystems
whenthemarkingprofileisdeterminedbyaPCscheme.
Alternatively,completingananalysissimilartoSectionIII-B
revealsthatwithaVQ-basedPCscheme,itispossibletomain-
tain stability and small queueing delays using either PFC or
Thus, implies . TCP-typecongestioncontrol,eveninthepresenceofnontrivial
Thisresultimpliesthattoachievealowqueueingdelayone disturbancelevels.
must haveahigh equilibriummarking probability.Continuing
asinthePFCcase,wecanshowthatinanRQsystemwithTCP E. MultipleUsersWithIdenticalRTT
and REM it is not possible to simultaneously ensure stability WenotethattheanalysisforREMandPCcanbereadilyex-
andsmallqueueingdelayinthepresenceofdisturbances.That tendedtothecasewheretherearemanysourceswithidentical
is,adirectanalogofTheorem4holdsforthiscase. RTTsaccessingalink.Toseethis,considerthecaseofasingle
Similar to the analysis in Section III-B, it further can be linkaccessedby TCPsources.Thecongestioncontrolequa-
shown that in a VQ system with TCP-type congestion control tionofthe sourceisgivenby
and REM implemented at the router,it is possible tomaintain
bothstabilityandlowqueueingdelaysinthepresenceoflarge
LAKSHMIKANTHAetal.:ROBUSTNESSOFREALANDVIRTUALQUEUE-BASEDACTIVEQUEUEMANAGEMENTSCHEMES 89
andthequeuedynamicsaregivenby
with the appropriate modification to the right-hand side when
thequeuelengthhitszero.Now,considerthefollowingchange
ofvariables:
Further, suppose that REM is used at the link, with the REM
parameter ,i.e.,
Fig.1. EvolutionoftheQueueingdelaywithPFCatthesource,RQ-based
Then, defining , it is easy to see that the linearized
REMattherouterandaconstantdisturbanceof3.1%oflinkcapacity.
dynamicsfor arethesameasthatofasingleuserandallour
previousresultsapply.Tocompletetheproof,onehastoshow
that the stability of impliesthe stabilityof ;this can be
easilydoneandweomittheproofhere.
In the case of the proportional control AQM, the parameter
hastobescaledby togetthecorrespondingresult.These
scalings can also be applied to proportionally fair source con-
trollersaswell,toshowthatthe -linkresultcanbeobtained
bysuitablyrescalingthesystemtoputitinthesingle-userform.
IV. SIMULATIONS
In this section, we demonstrate the robustness of VQ-based
congestion controllers via simulations. We study REM, PI
and RED marking schemes at the router, along with PFC and
TCP-type congestion controllers at the source. For all virtual
queues, the target utilization is set to 0.95. The capacity
is chosen to be 10000 packets/s. (The bandwidth associated Fig.2. EvolutionoftheQueueingdelaywithPFCatsource,VQ-basedREM
with this capacity is approximately equal to the bandwidth of attherouter.
100Mb/slink, assumingthateachpacketis 1000bytes long).
Thenumberofusersis1000.Theround-trippropagationdelay we chose , and . We introduce
for all users is taken to be 40 ms. Although the analysis in a constant disturbance of 3.1% of the link capacity. Our ana-
the earlier sections was carried out treating the disturbance as lytical results in Section III-A indicate that the linear system
an unknown constant, in these simulations random flows are is unstable for this disturbance. Fig. 1 shows that, in the orig-
taken into account. Whenever the system is simulated in the inal nonlinear system, the queue length indeed becomes very
presence of random flows, we take these random flows to be large. In the case of VQ-based REM, we chose and
i.i.d. Bernoulli random variables with the total mean flow rate .Notethatsuchlowvaluesof cannotbechosen
equal to 25% of the link capacity. We choose the number of inthecaseofRQ-basedREMsincetheresultingqueueingde-
random flows in the network to be 100. In all the simulations layswillbetoolarge.InFig.2,theperformanceofVQ-based
reportedinthepaper,theinitialconditionsforthesourcerates REMwithnodisturbanceisshown.Todemonstraterobustness,
andqueuelengthsweretakentobeequaltozero.However,we aconstantdisturbanceof80%ofthelinkcapacityisintroduced
have also conducted other simulations, not shown here, with withoutchanginganydesignparameters;theresultsareshown
initial conditions taking values up to three times the equilib- inFig.3.ThisdemonstratesthatVQ-basedREMisabletore-
rium values of the different state variables. The performance jectveryhighlevelsofdisturbance.
of the various controllers remains the same, and appearsto be NextweconsidertheimpactofrandomdisturbancesonRQ
independentoftheinitialconditions. andVQ-basedversionsofREM.Sinceouranalysisisbasedon
adeterministicanalysis,whenrandomdisturbanceswereintro-
A. Experiment1:REM
duced,thesystem parametersforRQ-basedschemes werede-
We first consider an RQ-based REM scheme at the router, signedassumingthesystemwassubjectedtoaconstantdistur-
withPFCatthesource.Forthisexperiment,thevaluesof , bance equal to the mean. Accordingly, we set and
and arechosensothatthesystemisstableandthequeueing . The VQ-based REM parameters were chosen
delay is less than 10% of the propagation delay. Specifically assumingnodisturbances.Fig.4showsthatthequeueingdelay
90 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
Fig.3. EvolutionoftheQueueingdelaywithPFCatthesource,VQ-based Fig.5. EvolutionoftheQueueingdelaywithTCPatthesource,RQ-based
REMattherouterandaconstantdisturbanceof80%oflinkcapacity. REMattherouterandaconstantdisturbanceof1.3%oflinkcapacity.
Fig.6. EvolutionoftheQueueingdelaywithTCPatthesourceandVQ-based
REMattherouter.
Fig.4. ComparisonbetweenPFC/RQ-basedREMandPFC/VQ-basedREM
withrandomflowsattherouteramountingto25%oflinkcapacity.
performanceofVQ-basedREMissignificantlysuperiortothat
ofRQ-basedREMdespitethefactthatVQ-basedREMparam-
eterswerenotchosentohandleanydisturbancelevel.
SimilarexperimentswereconductedwiththeTCP-typecon-
gestioncontroller(5)andbothRQandVQ-basedREM.Inthe
case of RQ-based REM we set , and .
When dealing with VQ-based REM we set ,
and .Again,aswithPFC,asmallconstant
disturbance(inthiscase1.3%)causesthequeuetobecomevery
large for RQ-based REM, as shown in Fig. 5. With VQ-based
REM,thequeueingdelayiszerowithorwithoutdisturbances
(Figs.6and7).Whenrandomflowsareintroducedintherouter,
fortheRQ-basedREMwechose and .
Fig.7. EvolutionoftheQueueingdelaywithTCPatthesource,VQ-based
From Fig. 8 it is clear that VQ-based REM performs signifi- REMattherouterandaconstantdisturbanceof50%oflinkcapacity.
cantlybetterthanRQ-basedREM.
parametersforRED.However,furtherexperimentswereusedto
B. Experiment2:RED
ensurethatthebestvalueswerechosenforRED.Accordingly,
Aspreviouslynoted,REDcanbethoughtofasproportional weset tobe100and tobe5 whenaTCP-typecon-
controlwiththeaveragingbeingperformedattherouter.Thus, gestioncontrollerisusedatthesource,andweset tobe100
theparametersforPCwereusedasaguidelineforchoosingthe and tobe7 whenPFCisusedatthesource.Further,