On the Scalability of Network Management Information for Inter-Domain Light-Path Assessment

13 Pages·0.548 MB·English

by  Liu G., Ji C., Chan V.

160 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
On the Scalability of Network Management
Information for Inter-Domain Light-Path Assessment
GuangleiLiu,StudentMember,IEEE,ChuanyiJi,Member,IEEE,andVincentW.S.Chan,Fellow,IEEE
Abstract—We investigate the necessary amount of network ofanetwork”,whentherearenowavelengthconvertersinthe
managementinformationforlight-pathassessmenttodynamically network. Wavelength converters can reduce state information
set up end-to-end light-paths across administrative domains in duetotheirabilitytorelaxthewavelengthcontinuityconstraint.
optical networks. Our focus is on the scalability of partial man-
However,it is expectedthat wavelengthconvertersremain ex-
agementinformation.Weposelight-pathassessmentasadecision
pensiveandarethususedmostlyontheboundariesofsub-net-
problem, and define the performance as the Bayes probability
of an erroneous decision. We then characterize the scalability of works [2].Therefore,generally completestate informationin-
management information as its growth rate with respect to the volvesthedetailedwavelengthoccupancywithinasubnet.Par-
total resources of the networkto achievea desired performance. tial information includes aggregated load and topology infor-
Scalability is achieved if the management information needed is
mationateachsubnet,andlocalstates,e.g.,thetotalnumberof
only a negligible fraction of the total network resources. Specif-
wavelengthsusedatwavelengthconverters.
ically, we consider in this work one type of partial information
that grows only logarithmically with the number of wavelengths Providingstateinformationisabasicfunctionalityofnetwork
supportedperlink.WederiveanupperboundfortheBayeserror management.Traditionalnetworkmanagementsystemsintend
intermsoftheblockingprobabilitywhenanewcallispresented toobtainascompletestateinformationaspossible[3].Butfu-
attheentranceofthenetwork.Weevaluatetheupperboundusing tureIP-WDMnetworksmayhavehundredsoflinkswitheach
bothindependentanddependentmodelsofwavelengthusagefor
link supporting hundreds of wavelengths [4]. This would re-
intra-andinter-domaincalls.Ourstudyshowsthatthereexistsa
sultinahugeamountofstateinformationfornetworkswithout
“thresholdeffect”:TheBayeserrordecreasestozeroexponentially
withrespecttotheloadwhentheloadiseitherbeloworabovea wavelength converters. For instance, let be the number of
thresholdvalue;andisnonnegligiblewhentheloadisinasmall linkswithineachsubnet, bethenumberofwavelengthssup-
duration around the threshold. This suggests that the partial in- portedperlinkateachsubnet,and bethenumberofsubnets.
formationconsideredcanindeedprovidethedesiredperformance,
The total amount of information about wavelength usage is in
andasmallpercentageoferroneousdecisionscanbetradedoffto
theorderof .When , and ,the
achievealargesavingintheamountofmanagementinformation.
number of states is about half a million. Storing and updating
IndexTerms—Bayesrule,blockingprobability,decisiontheory,
evenafractionofsuchalargenumberofstatesmayresultinan
light-pathassessment,managementinformation,scalability.
undesirably large amount of management traffic. Therefore, it
wouldbeprohibitivetomanagealargenetworkusingcomplete
I. INTRODUCTION stateinformation.
DYNAMICALLYassessingthequalityoflight-pathsisim- Usingpartialmanagementinformationisalsoarequirement
ofmulti-vendorservices.Alight-pathmaytraversemultiplead-
portanttomanyapplicationsinwavelength-routedoptical
ministrative domains (sub-networks) run by different service
networkssuchason-demandlight-pathprovisioning,protection
providers.Aserviceprovidermayprefertoexchangeonlymin-
andrestoration.Asthelight-pathqualityisacomplexmeasure
imalinformationwithothernetworkdomainsratherthanshare
[1], this work considers a simple quality, which is the wave-
the complete state information of its own. In fact, it has been
length availability on a candidate light-path. The assessment
the experience today in the Internet that network managers of
thenboilsdowntodetermineavailabilityofwavelengthsforin-
differentadministrativedomainsareextremelyreluctanttoand
comingcallrequestsbasedongivenmanagementinformation.
rarelysharedetailednetworkstateinformationoftheirsubnets
Completeorpartialnetworkmanagementinformationcanbe
withothers.Therefore,inter-domainsubnetsarelikeunknown
usedtoassessthewavelengthavailabilityonalight-path.Com-
networkcloudstoaserviceprovider[5].Light-pathassessment
plete information corresponds to the detailed states of wave-
mayhavetousepartialinformationonnetworkcloudssinceit
lengthusage,i.e.,“which wavelengthsareusedat which links
isinfeasibletoobtaincompletemanagementinformationacross
domainboundaries.
ManuscriptreceivedJanuary20,2003;revisedMarch1,2004;approvedby Therefore, a fundamental issue in light-path assessment is
IEEE/ACMTRANSACTIONSONNETWORKINGEditorA.Fumagalli.Thiswork
what performance can possibly be achieved given the partial
wassupportedinpartbytheNationalScienceFoundationunderECS0300605
andtheYamacrawCenterattheGeorgiaInstituteofTechnology. information.Specifically,therelatedquestionsare:(1)whatis
G.LiuandC.JiarewiththeSchoolofElectricalandComputerEngineering, the best performance of light-path assessments with the par-
GeorgiaInstituteofTechnology,Atlanta,GA30332-0250USA(e-mail:guan-
tial information? (2) What is the trade-off between the perfor-
[email protected];[email protected]).
V. W. S. Chan is with the Laboratory for Information and Decision Sys- manceandtheamountofmanagementinformationmaintained
tems (LIDS), Electrical Engineering and Computer Science Department, forlight-pathassessments?
MassachusettsInstituteofTechnology,Cambridge,MA02139USA(e-mail:
We formulate the light-path assessment as a decision
[email protected]).
DigitalObjectIdentifier10.1109/TNET.2004.842436 problem, and define the performance as the probability of an
1063-6692/$20.00©2005IEEE
LIUetal.:ONTHESCALABILITYOFNETWORKMANAGEMENTINFORMATIONFORINTER-DOMAINLIGHT-PATHASSESSMENT 161
erroneous assessment. An error occurs when an assessment formation,whichistheBayesprobabilityoferror.Weshowthat
decision differs from the ground truth (in terms of availability the Bayes error is bounded by , where is
of wavelengths on a given path). The value of the error prob- the blocking probability of a light-path. This links the Bayes
ability measures the deviation from the optimal performance error with , a metric commonly used for WDM networks
(with zero error) when the complete information is available, [6]–[8]. The (Bayes) probability of error can then be investi-
and thus quantifies the sufficiency/insufficiency of the partial gatedthroughtheblockingprobabilitybasedondifferenttraffic
managementinformation. models.Wefirstadoptanindependentmodelthatcorresponds
Withalargeamountofmanagementinformation,agoodper- tolocalcalls.Wethenextendtheindependentmodeltoadepen-
formance,i.e.,asmallerrorprobability,couldbeachievedbut dent modelto include inter-domain calls. Oneimportant char-
at the cost of management complexity such as signaling and acteristicofthebestperformanceusingthepartialinformation
memoryoverhead.Withasmallamountofmanagementinfor- isa“thresholdeffect”,i.e.,thereexistsathresholdfortheload.
mation,theperformancemaydegradebutwithagainofman- Whentheloadisclosetothethresholdvalue,theblockingprob-
agementsimplicity.Thusatrade-offcanbemadebetweenthe abilitymakesasharptransitionfrom0to1.Thecorresponding
performanceandthenetworkmanagementinformation. probability of error remains close to zero for most of the load
Theamountofmanagementinformationneededvarieswith conditions.Thissuggeststhatthepartialinformationcouldpro-
respect to the size, and the resource of the network. The size videdesirableperformanceforlight-pathassessment.Hencethe
can be characterized by the number of links in a subnet and partialinformationisscalableandasmalllossinperformance
the number of subnets. The resource corresponds to the total maybetradedoffwithalargesavinginnetworkmanagement
numberofwavelengths,whichisrelatedtothenumberofusers information.
(flows)supportablebyanetwork.Futureopticalnetworksmay Thepaperisorganizedasfollows.SectionIIsummarizesthe
have hundreds of links, each of which supports hundreds of priorwork.SectionIIIprovidestheproblemformulation.Sec-
wavelengths. Therefore, the growth rate with respect to those tionIVpresentsBayesdecisiontheory,andanupperboundof
parametersisanimportantmeasureoftheamountofmanage- the best performance (the Bayes error) that can be achieved
ment information used. In particular, a desirable growth rate given the partial information. Sections V and VI investigate
shouldbeslowerthanthatofthetotalresourcetobemanaged thebestperformanceusinganindependentmodelandadepen-
ina network. dentmodelrespectively.SimulationresultsarepresentedinSec-
Combining the performance and the growth rate, we define tionVII.SectionVIIIconcludesthepaper.
the scalability of network management information for light-
pathassessment.Assumingthatagivenperformanceissatisfied,
II. RELATEDWORK
i.e., a small probability of error can be achieved; we consider
theneededmanagementinformationasscalableifitgrowsata VariousschemeshavebeenproposedformanagingIP-WDM
slowerratethanthetotalnetworkresource;andasnonscalable, networks based on different amount of management informa-
otherwise.Therefore,thescalabilityrequiresthattheamountof tion.Completestateinformationhasbeenusedtoestablishcon-
informationusedisonlyanegligiblefractionofthetotalwave- nections [9]. This approach, as discussed earlier, may not be
lengthresourceswithinthenetwork.Hencescalability/nonscal- feasible for dynamically setting up inter-domain connections
abilityprovidesasystematicwaytoinvestigatethetradeoffbe- for large networks. In contrast to using complete information,
tweenperformanceandthemanagementinformation. anothermethodistomanagesub-networksasseparateentities
Inthiswork,westudyonetypeof“strongly”scalableman- [10].Thecorrespondingperformance(i.e.,thecorrectnessofan
agementinformation,whichisonlylogarithmic in assessment) can be poor due to lack of information. An inter-
thenumberofwavelengthsupportedperlinkinthenetwork.We mediateapproachistousepartialinformation-exchangeamong
investigateasimplenetworkofbustopologytostudythescala- network domains [11]. The idea of using partial information
bilityofthepartialmanagementinformation.Wavelengthcon- is also investigated in other related research problems such as
vertersareonlylocatedattheboundariesof,butnotwithin,each network survivability [12]–[14], and wavelength routing [15].
subnet.Thepartialinformationweconsiderincludes:(a)aggre- However, these works have a different focus, which is mostly
gatedinformationonnetworkloadandtopologywithinsubnets, ondevelopingapproachestomanagenetworksusingpartialin-
and (b) local state information at wavelength converters. The formation.Theymotivatethisworktoinvestigatethescalability
aggregated information serves as model parameters of wave- ofmanagementinformation.
lengthusage,andthelocalinformationcorrespondstorandom Probing methods have been proposed to obtain information
states or observations obtained locally at domain boundaries. from network clouds [16]. Probing, however, is intrusive, and
For a bus topology with available wavelengths at each link maybe impracticalfor inter-domainlight-path assessmentbe-
and subnets, the total amount of the partial information is causeofsecurityreasons.
.Thisisindeedmuchlessthanthetotalamountof Wavelengthconverters(opticalorelectronic)havebeencon-
resourcesavailabeinthenetwork .Therefore,thepartial sidered in designing WDM networks to improve wavelength
information will introduce much less management complexity utilization [17]. Sparsely-allocated wavelength converters are
thancompleteinformation. foundtobesufficienttoachieveadesiredutilizationgainsome-
Toevaluatetheachievableperformanceusingthepartialin- times[18].Theuseofwavelengthconvertershasalsobeencon-
formation,weconsidertheBayesdecisionrule.TheBayesrule jectured to result in simplified network management systems
resultsinthebestperformanceachievablegiventhepartialin- due to their ability to reduce the state information [17]. This
162 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
lengths at each link are used, i.e., the percentage of occupied
wavelengthsusedforinter-domainconnections.Forsimplicity
ofanalysis,weassumethateachsubnethasthesameaggregated
information.Thenwehave forall .
In practice, the aggregated information can be estimated
through measurements, which may deviate from true param-
Fig.1. Networkarchitecture. eters, and thus introduce additional information loss. For
simplicity, we consider aggregated parameters as accurate.
These parameters may also change with time but at a much
motivates us to consider a natural network architecture where
largertimescalethantheconnectiondynamics,andcouldthus
wavelengthconvertersarelocatedattheboundariesofsubnets
beregardedaslong-timeaverageandnearlystatic.
(administrativedomains).
The local information corresponds to the number of wave-
Priorinvestigationsinotherrelatedareasarealsobeneficialto
lengths used at the first hop of each subnet, which is readily
thisresearch.Inparticular,inaccurateoraggregatedinformation
available at the wavelength converters. Specifically, the local
hasbeeninvestigatedinthecontextofQoSroutingforIPnet-
informationcorrespondingtoobservations(states)atthewave-
works[19].Commonlyusedaggregatedinformationistopology
aggregation [20], [21] that can be regarded as a summarized lengthconvertersisgivenas where
characterizationofasubnet.Localinformationisconsideredin is the number of wavelengths used at the th wavelength con-
[22]for QoSroutinginIPnetworks.However,themainfocus verters,i.e.,thenumberofwavelengthsusedatthefirstlinkof
ofaforementionedworkisonmanagingexisting(IP)ratherthan domain . Such local information is changing with setup and
IP-WDMnetworks. teardownofconnections,andcanthusbeconsideredasrandom
Therefore,thetradeoffbetweenperformanceandtheamount variables.
ofmanagementinformationhasnotbeeninvestigatedquantita- The local information is informative due to the wavelength
tively.Inourpriorwork[23],[24],weformulatedtheproblem continuity constraint within a subnet. For instance, if nearly
ofnetworkmanagementinformationforlight-pathassessment all wavelengths are used at the first hop of a subnet, we can
based on independent and dependent models of wavelength inferthattheloadishighandtheremaynotbeanywavelength
usage.Thisworkextendsthepriorworktoamorecomprehen- available within the subnet to support an additional end-end
sivesetting.Weformallydefinethescalabilityofmanagement call.Likewise,theaggregatedinformationisinformativesince
information for light-path assessment, and use both analysis itcharacterizestheaverageloadinasubnet.Buttheaggregated
andsimulationstoinvestigatethescalabilityoftheinformation. and local information is incomplete in determining network
states,resultinginpossiblyerroneouslight-pathassessments.
III. PROBLEMFORMULATION
C. DecisionProblemandPerformance
A. NetworkArchitecture
Weposethelight-pathassessmentasadecisionproblem.A
Weconsiderassessingwavelengthavailabilityforanend-to-
decisionvariable isdefinedasfollows: ifthereisone
endcallrequestfromsourcebordernode todestinationborder
end-to-end wavelength continuous path across subnets for the
node asshowninFig.1.Wavelengthconvertersarelocatedat
connectionrequest;and otherwise.Theproblemoflight-
theboundariesofone-dimensionalsubnets,andthereare sub-
pathassessmentistodecideon giventhepartialinformation.
netsonagivenpath.Hereforsimplicity,weassumethatborder
Thenthe performanceof light-path assessment can be defined
nodes with wavelength converters connect two adjacent net-
astheprobabilityoferroneousdecisions.
workdomains.Eachsubnethas hopsandeachlinksupports
Definition 1: The probability of error is defined as the
wavelengths. Such network architecture, although simple,
probability that the assessment decision is different from the
captures the important characteristic of multi-domain network
groundtruth(intermsofavailabilityofwavelengthsonagiven
topologies.
path).
Let bethedecisionregiononthemanagementinformation
B. PartialManagementInformation
for and bethedecisionregionfor .Inother
The partial information we consider consists of aggregated word,iftheobservation fallsin or .Thedecisionshould
informationandlocalstates.Theaggregatedinformationchar- be or respectively.Wethenhavetheprobability
acterizestheaveragebehaviorofeachnetworkdomainsothat of error
detailed network states within each subnet do not need to be
exchanged across domains. The aggregated information is de- (1)
noted as , where is the aggregated
information on subnet and . and characterizestheaverageperformancegiventhepartialin-
arethenumberofwavelengthsperlinkandthenumberoflinks formation.Thevalidityofsuchaperformancemeasurecanbe
atsubnet respectively. istheprobabilitythatawavelengthis understood through Fig. 2. When the complete information is
usedonalinkindomain ,whichistheloadinformationaggre- available,noerrorismadeinassessingwavelengthavailability,
gatedoveralldetailedstatesaboutwavelengthusagewithinthe and the performance is the best (i.e., zero error). When no in-
subnet. is a parameter related to wavelength usage. For ex- formation is available, decisions can only be made based on
ample, couldbetheparameterthatcharacterizeshowwave- random guessing, and the performance is the worst (i.e., 50%
LIUetal.:ONTHESCALABILITYOFNETWORKMANAGEMENTINFORMATIONFORINTER-DOMAINLIGHT-PATHASSESSMENT 163
Thepartialmanagementinformationconsideredinthiswork
satisfies
(3)
where isthenumberofbitsneededtostoretheaggregated
information, which is indexed with . is generally small,
and changes slowly with time. is the total number of
Fig.2. Performanceversusmanagementinformation.
bitsneededtocharacterizelocalstatesatonesubnet.Thenthe
amountofpartialinformationisintheorderof ,whichis
error).Thevalueof thusmeasuresthedeviationfromtheop- much less than that of the complete management information,
timalperformance(zeroerror)whenthecompleteinformation especiallywhenthenumberofwavelengthsislarge.Suchpartial
isavailable,andthusquantifiesthesufficiency/insufficiencyof informationcanbemaintainedeasilyevenforalargenetwork.
themanagementinformationavailable.Aquestioniswhetherit
ispossibletousepartialmanagementinformationatthecostof IV. OPTIMALPERFORMANCEUSINGBAYESRULE
asmallnumberofincorrectdecisions.
We now evaluate the best performance of the partial man-
agement informationtoseewhether it can providethe desired
D. ScalabilityofManagementInformation
performance.
We investigate this problem in the context of scalable net-
work management information. Intuitively, there are two im- A. BayesError
portant aspects of the scalability: The amount of management
With partial management information, assessment schemes
informationshouldbesufficientlylargetosatisfyagivenperfor-
basedonBayesdecisionrule[25]achievethebestperformance.
mance,i.e.,asmallprobabilityoferror.Meanwhile,theamount
Given a set of local states , the Bayes
ofmanagementinformationshouldbesmallenoughtokeepnet-
ruleistodecide
workmanagementsimple.Forinstance,itispreferredthatthe
managementinformationneededisjustanegligiblefractionof if ,
thetotalnetworkresource,i.e.,thetotalnumberofwavelengths otherwise,
supported in the network. Since the resource varies with re-
where and is a
spect to the size of a network and the number of wavelengths
posteriori probability given observation . The equality
supported per link, it would be meaningful to characterize the
corresponds
amountofmanagementinformationrequiredasitsgrowthrate
tothedecisionboundary,whichdividesthespace intotwo
withrespecttothosequantities.Combiningtheperformanceand
regions, todecide and todecide .TheBayes
the growth rate, we formally define the scalability of manage-
erroristheaverageprobabilityoferrorasgivenin(1).
mentinformationasfollows.
Definition2: Let betheamountofmanagementinforma-
B. CentralizedLight-PathAssessment
tionusedforlight-pathassessment.Let bethetotalamount
Such a Bayes rule essentially corresponds to an optimal
of wavelengthresources withinthe network. If grows at a
centralized assessment scheme. Imagine a fictitious central
slowerratethan withrespecttothenumberofwavelengths
manager, collecting partial information from all subnets. At
per link and the size of a network, and the corre-
a relatively larger time-scale than the flow dynamics, the
sponding performance of light-path assessment is acceptable
central manager could poll the aggregated information from
undermostloadconditions,thenetworkmanagementinforma-
each subnet. The central manager then could collect the local
tionisscalable;andnonscalableotherwise.
observation at a smaller time scale, and perform the Bayes
This definition essentially means that asymptotically (for a
ruletoassesswavelengthavailability.
largenetworkwithmanylinksandwavelengthseachlink),the
Thiscentralizedschemeisonlyconceptual,andusedinthis
scalablemanagementinformationisanegligiblefractionofthe
work for analysis rather than a practical solution. Centralized
total network resource. That is, when and
assessment may not be feasible for large optical networks be-
arelarge.
cause each subnet could belong to different administrativeen-
Consider the network shown in Fig. 1. The number of bits
tities.Thusadistributedlight-pathassessmentschememaybe
is used to quantify the management information. The detailed
anecessity.However,distributedassessmentschemesresultin
states within each subnet are “which wavelengths are used at
furtherinformationlossduetodecentralization.Thereforethere
whichlink”.Thetotalnumberofpossible(binary)statesis
is a need to understand the best performance achievableusing
foreachsubnet,and for subnets.Therefore,complete
thepartialinformation.Suchbestperformancewouldthenserve
informationsatisfies
asabasisforassessingtheperformanceofsub-optimalyetprac-
(2) ticalschemes.
C. BayesErrorandBlockingProbability
Clearly,itisnonscalabletousecompletemanagementinforma-
tionaccordingtoDefinition2eventhoughitwillalwaysresult Although the Bayes error characterizes the optimal perfor-
inzeroprobabilityoferror. mance, it is difficult to evaluate because the decision regions
164 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
Fig.3. Localcallsinindependentmodel.
and the corresponding probabilities are hard to obtain. There-
fore, we derive an upper bound for the Bayes error. Our goal
istorelatesuchaboundwithacommonlyusednetworkmea-
suresuchasblockingprobability.Sucharelationmayprovide
intuitiononhowerrordecisionsarerelatedtotheload and
wavelengthperlink ofeachsubnet.Forclarity,wedescribe
theblockingprobabilitybasedon[6].
Definition 3: The blocking probability is defined as the
probability that there does not exist a wavelength continuous
Fig.4. Load((cid:26))versusblockingprobability(P ):F =10;40;120,H=5,
pathineachnetworkdomaintosupportanincomingend-to-end L = 3.
inter-domainconnection.
ArelationbetweentheBayeserror andtheblockingprob- where .Thisexpressionmeansthatif wave-
ability canthenbederived. lengths are used at the first hop of subnet , one only needs
Theorem1: . todecidewhetherthereisawavelengthcontinuouspathatthe
TheproofofthetheoremisgiveninAppendixI.Intuitively, next hops from candidate wavelengths. Then
theupperbound canbeunderstoodasfol- is the probability that there is
lows.Considerthefollowingdecisionrule:Iftheblockingprob- a wavelength continuous path at the th subnet given , and
abilityofthenetworkis ,onecanrejectallconnection theproductistheprobabilitythattheconnectionrequestforan
requests.If ,onecansimplyacceptallconnectionre- end-to-endcalltobesupported.TheBayeserroris
quests. This decision rule will have
Since Bayes rule uses local observation as the additional (5)
information for light-path assessment in an optimal fashion; a
Equation(5)doesnothaveacloseform;andweturntoeval-
betterperformanceshouldbeachieved.Thatis,theBayeserror
uatetheupperboundof usingtheblockingprobabilityofthe
should be bounded by . The upper bound
independentmodel.
showsthattheprobabilityoferrorissmalliftheblockingprob-
abilityiscloseto1or0.
C. NumericalAnalysis
This theorem suggests an analytically feasible way to esti-
Undertheindependentassumption,theprobabilitythatthere
matetheBayeserror,whichisthroughtheblockingprobability.
isoneend-to-endwavelengthcontinuouspathcanbeobtained
Inaddition,theboundisindependentofaspecificmodelofthe
usingamodelin[6]:
blockingprobability.Theanalysiscanthenbeconductedusing
differentmodels.
(6)
V. INDEPENDENTMODEL where the sub-index means acceptation of a request based
A. IndependentModel on independent model. Therefore, the corresponding blocking
probabilityforanend-to-endcallis
We first assume independent wavelength usage on different
network links and among wavelengths. Such an assumption is (7)
equivalent to the case that all connections within the network
are local calls as shown in Fig. 3. Then the corresponding ag- Fig.4plotstheblockingprobability ,versustheload
gregatedinformationis where istheproba- for , , . One observation is that
bilitythatwavelengthisusedononelink.Thelocalobservation thereisathresholdeffecton .When isbelowthethreshold
is asdefinedinSectionIII.Duetothe value(e.g.,aboutat for ), remainsclose
independent assumption, all the ’s are independent random to 0. When is around the threshold value, increases to
variables. 1 rapidly with respect to . With a larger , the value of the
threshold increases, and the transition of from 0 to 1 gets
B. BayesError sharper.
Thisshowsthatundermostloadconditions,weeitherhavea
Undertheindependentmodel,theaposterioriprobabilityis
smalloralargeblockingprobability,bothofwhichresultina
smallprobabilityoferror.Therefore,basedonTheorem1,we
can conclude that, under most load conditions, the probability
of error for light-path assessment using partial information is
(4)
smallunderindependentmodel.Fig.5confirmsthisbyplotting
LIUetal.:ONTHESCALABILITYOFNETWORKMANAGEMENTINFORMATIONFORINTER-DOMAINLIGHT-PATHASSESSMENT 165
Fig.6. Inter-domaincallsandlocalcalls.
dent model in [6] to obtain a relatively accurate and tractable
dependentmodelforanalyzingtheprobabilityoferror.
To capture the dependence on traffic flows among subnets,
we assume that there are two types of calls supported by the
network.Onecorrespondstolocalcallsasassumedintheinde-
pendentmodel.Theothertypeofcallscorrespondstointer-do-
maincalls(Fig.6).Generally,inter-domaincallscanoriginate
and/or terminate anywhere at a network. But for simplicity of
analysis,weimposethefollowingassumptions:
Fig.5. Load((cid:26))versusupperboundofP :F =10;40;120,H=5,L=3.
(i) The inter-domain calls originate and exit only at edge
wavelengthconverters.
theupperboundof for , , .We
(ii) Ifawavelengthisnotusedforaninter-domaincallinone
can see that when the load is close to the threshold, the value
of increasestothemaximumvalueexponentially;and is subnet,itisusedforinter-domaincallinthenextsubnet
smallotherwise. withprobability .
(iii)Ifawavelengthisusedforoneinter-domaincallinone
D. SpecialCases subnet,thisinter-domaincallwillexitthecurrentsubnet
withprobability ,andwillcontinuetothenextsubnet
Toquantifythedecayrateoftheupperboundforlarge ,we
withprobability .
considerspecialcasesoflowandhighload,whichcorrespondto
(iv) If a wavelength is used for an inter-domain call in one
twopartsof belowandabovethethreshold.Wecanfindthat:
subnetandisreleasedattheedgeOXCofthissubnet,it
(i) Whentheloadislight,i.e., ,
isusedforinter-domaincallswithprobability inthe
nextsubnet.
(8)
(v) If an inter-domain call continues to the next subnet, it
(ii) Whentheloadisheavy,i.e., , willusethesamewavelength.
(vi) Ineachsubnet,awavelengthisusedforalocalcallina
(9)
linkwithprobability ,andforaninter-domaincallwith
probability .Theprobabilitythatawavelengthisused
Theseresultssuggestthattheperformancetrade-offisasmall
foreitheralocaloraninter-domaincallis .
probabilityoferrorthatdecreasesexponentiallywithrespectto
Thedependentmodelcapturesthelinkloadcorrelationacross
the number of wavelengths per link under low and high
subnetsduetointer-domaincalls,andisthusmoreaccuratethan
networkload.
theindependentmodel.Weareawarethatitislimitedtoassume
thattheinter-domaincallscanonlyenterorexitatthedomain
VI. DEPENDENTMODEL
boundaries.However,suchamodelprovidesunderstandingof
The above independent model fails to capture the inter-do- howinter-domaincallscontributetotheperformanceandman-
main calls, which extend beyond one subnet. In future optical agementinformationtrade-off.Amorerealisticmodelistobe
networks,asignificantpercentageofthetrafficmaybeinter-do- investigatedinsubsequentwork.
mainflowspassingthroughsubnets.Therefore,itisimportantto
taketheloadcorrelationamongsubnetsintoconsiderationwhen B. BayesError
estimating the performance. In this section, we investigate the
We begin evaluating the performance by considering the
probabilityoferrorbyconsideringbothintra-andinter-domain
probability of error. Again, we assume that all subnets have
calls.
identical aggregatedinformation. Under the dependent model,
the aggregated information is .
A. DependentModel
Local information is the same as that used for independent
Dependent models in a linear network have been investi-
model,whichis .Thentheaposteriori
gatedin[6]–[8].However,thestudyin[6]isrestrictedtohaving
probabilityusedinBayesruleis:
wavelength converters installed at each node, while the net-
workarchitectureasshowninFig.1iswithsparsely-allocated
wavelengthconverters.Moreaccuratedependentmodelsforthe
blockingprobabilityonsuchatopologycanbefoundin[7],[8]. (10)
However,bothmodelsarecomplex.Hereweextendthedepen-
166 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
where . isdefinedastheprobabilitythata
wavelength is used for local calls given that it is not used for
inter-domain calls. Such a posterior probability has a similar
formtothatoftheindependentcasein(4).
Theprobabilityoferroristhesameasin(5).Butduetointer-
domaincalls,thelocalobservations( ’s)atwavelengthcon-
vertersarenowdependentrandomvariables.TheBayeserroris
difficult to derive, we turn to study the upper bound based on
theblockingprobability .
C. BlockingProbability
To derive the blocking probability under the dependent
model, we define , which characterizes the per-
centage of occupied wavelengths used for inter-domain calls.
Then the independent model is just one special case of the
dependent model with . From assumptions in
SectionVI-A,wehave,
Fig.7. Load((cid:26))versusblockingprobability(P ):F =120,H =5,L=
(i) (wavelength is used for inter-domain call in 3,(cid:11) =0;0:6;0:9,P =0:2.
subnet is not used for inter-domain call in
subnet , where
(ii) (wavelength is used for inter-domain call in
subnet is used for inter-domain in subnet
. (16)
Therefore,
and
(11)
Itfollowsthat
(17)
(12)
Inserting(14)–(17)into(13), canbecomputedefficiently
Define ifthereisonewavelengthcontinuouspathwithin usingtheforwardpartoftheforward-backwardalgorithm[26].
subnet ;and otherwise.Thenadecision thatthere are
D. NumericalAnalysis
wavelengthsavailableforanend-to-endcall isequiva-
lentto forall . Theblockingprobabilitydoesnothaveaclose-formexpres-
Let bethenumberofinter-domainconnectionsinsubnet sion either, but can be evaluated numerically. Fig. 7 plots
.Thentheblockingprobabilityunderthedependentmodelcan versus for , , , ,
beexpressedasin(13),shownatthebottomofthepage,where .Itcouldbefoundthat hasasimilar“thresholdef-
fect” on the value of to that in the independent model. In
(14) addition,thethresholdisincreasingwith ,whichisdefinedas
the percentage of working wavelengths used for inter-domain
Let bethenumberofinter-domaincallsinthe thsubnet calls.This,intuitively,isbecauseofthefactthat,thedependence
thatcontinuetothenextsubnet.Wehave of wavelength usage introduced by inter-domain calls reduces
the blocking probability for a given load . When , the
dependentmodelisreducedtotheindependentmodel,andthe
thresholdhasthelowestvalue.
(15) Fig.8plots versus for , , ,
, .Wecanfindthatthethresholdisincreasing
(13)
LIUetal.:ONTHESCALABILITYOFNETWORKMANAGEMENTINFORMATIONFORINTER-DOMAINLIGHT-PATHASSESSMENT 167
large (and is small), local states at wavelength converters,
are joint Gaussian random variables
withprobability [27].SuchaGaussiandistribu-
tion can be completely characterized by the means, variances,
and covariances of ’s. Specifically, all ’s are random
variableswiththesamemean andvariance ,where
(18)
and
(19)
Thecovariance between and for character-
izesthedependencebetweentwosubnets,where
Fig.8. Load((cid:26))versusblockingprobability(P ):F =20;40;120,H = (20)
5,L=3,(cid:11)=0:6,P =0:2.
Such dependence can be further characterized through parti-
tioning and intodifferentcomponents
(21)
and
(22)
where
number of wavelengths occupied by local calls at the
firsthopofthe thsubnet;
number of wavelengths occupied by local calls at the
firsthopofthe thsubnet;
number of wavelengths in the th subnet occupied by
inter-domaincalls;
number of wavelengths in the th subnet occupied by
Fig.9. Load((cid:26))versusupperboundofP :F = 120,H = 5,L = 3, inter-domaincalls.
(cid:11) =0;0:6;0:9,P =0:2.
Define as the correlation coefficient between
and . Then and have a bivariate normal dis-
withthenumberofwavelengths .Thisisduetothefactthat
tribution: . Since
themorewavelengths,thesmallertheblockingprobabilityfor
form a Gaussian Markov Chain, the joint
agivenload.Thesharpnessofthetransitionalsoincreaseswith
probabilitydistributionof is
respectto ,suggestinganasymptoticbehavioroftheblocking
probabilityforalarge .
(23)
Fig.9plotstheupperboundfortheprobabilityoferrorfrom
Fig.7usingTheorem1. Itshowsthatthe valueof is small
undermostloadconditions. where
E. SpecialCases
A question rises why the threshold effect persists for both
independentanddependentmodels.Weinvestigatethisquestion
byconsidering special caseswhen the number ofwavelengths
is large, and all the sub-networks are weakly-connected ( is
large).Undertheseconditions,analyticalformoftheblocking
probabilitycanbederived.
1) Gaussian Approximation: An important step to obtain ... ... ...
a close form expression for the blocking probability is to
approximate the joint probability of the local states ( ’s) at ... ... ... ...
wavelengthconverters.Whenthenumberofwavelengths is
168 IEEE/ACMTRANSACTIONSONNETWORKING,VOL.13,NO.1,FEBRUARY2005
with where is the nonblocking probability of the decoupled
subnets as given in (25), and is proportional to (see
for or , AppendixIIIfordetails).
Wecanfindthat:
otherwise,
(i) When , all inter-domain calls last one subnet.
Hence all the sub-networks are decoupled, and we have
, .
(ii)When is large (e.g., ), a small percentage
of the inter-domain calls last more than one subnets.
Itcanbeshownthat
Hence the sub-networks are weakly-connected, and
.Here the nonblocking probability is
justthatofthedecoupledsub-networksplusasmallper-
(24) turbation. Thus we can expect a threshold effect occurs
forall .(Detailedderivationscanbefound under the weakly-connected sub-networks. The analysis
inAppendixII).Twoobservationscanbemadehere: herefurtherexplainswhythethresholdeffectpersistsfor
bothindependentanddependentmodel.
(i) is monotonically decreasing when is increasing.
Specifically, when , all the inter-domain calls
supportedbyanetworkdomainexitatthecurrentnetwork VII. SIMULATIONRESULTS
domain . When all the inter-domain
For more realistic scenarios with dynamic call arrivals and
callsareend-to-endconnectionstraversingallthenetwork
departures,theBayesianapproachweusewouldbeapplicable
domains .Notethatwealways
conceptually.However,theexactaposteriorprobabilitywould
have .
berathercomplex.Hence,aquestioniswhetherornotthestatic
(ii) ismonotonicallyincreasingwithrespectto .where
modelweusecouldresultinagoodapproximation.Inthissec-
isthepercentageofinter-domaincalls.Forinstance,when
tion, we investigate this issue through simulation of light-path
,i.e.,allthecallssupportedbythenetworkarelocal
assessment for dynamic call patterns. Of particular interest is
calls,wehave .When i.e.,allthecallsare
theperformanceoftheanalyticalboundon ,whichisderived
inter-domaincalls,wehave .
usingthestaticmodel,inadynamicsetting.
2) Weakly-ConnectedSub-Networks: When ,allsub-
networksarecompletelydecoupled,i.e.,eachinter-domaincall
A. SimulationSetup
lastsforonesubnet .Thenonblockingprobabilityof
We simulate light-path assessment in a network of bus
decoupledsubnetsis
topologywiththreenetworkdomains.Eachnetworkdomainis
assumed to have five hops. Connection requests are assumed
(25)
to obey a Poisson Process with unit exponential holding time.
Define as the arrival rate of connection requests for local
where is the probability that a wavelength is used for
calls at each link, and as the arrival rate of connection
local calls given that it is not used for inter-domain calls,
requests for inter-domain calls from domain to domain .
. For the nonblocking probability of the
Note that the connections between two border nodes of do-
independentmodel,wehavein(6)
main are considered as inter-domain calls from domain
to domain . Let the total arrival rate to the network be ,
then .Furthermore,followingthe
assumptionsinSectionVI,wehave
Equation (25) bears a similar form to (6), and thus it can be
shownthatthereexistsathresholdeffectintheblockingproba-
bilityfordecoupledsubnetssimilartothatfortheindependent
model. (27)
Ofparticularinterestiswhenallthesub-networksareweakly
connected.When issmall,allsub-networksareweakly-con-
nected,i.e.,asmallpercentageofthecallsareinter-domaincalls
( issmall),and/orinter-domaincallsexitatcurrentsubnetwith
alargeprobability( islarge).Forweaklyconnectedsub-net- (28)
works,weobtainthefollowingtheoremthroughTaylorExpan-
sion.
Theorem 2: For weakly-connected sub-networks, i.e., is
and
smalland/or islarge,thenonblockingprobabilityofthede-
pendentmodelcanbeexpressedas
(29)
(26)
LIUetal.:ONTHESCALABILITYOFNETWORKMANAGEMENTINFORMATIONFORINTER-DOMAINLIGHT-PATHASSESSMENT 169
Fig.10. AnalyticalboundandsimulatedP :F = 40,H = 5,L = 3, Fig.11. AnalyticalboundandsimulatedP :F = 80,H = 5,L = 3,
(cid:11) = 0:6,P = 0:2. (cid:11) = 0:6,P = 0:2.
canbeconsideredasthetotalarrivalrateforinter-domain The simulation result confirms the threshold effect that is
connectionsateachnetworkdomain.Solving(27)–(29),wecan predicted by the analytical model and shows the good perfor-
obtain the arrival rates for connection requests with different manceoftheanalyticalbound.Wecanfindthat,usingBayesian
sourcesanddestinationsasfollows: approach based on only aggregated information and the static
model, is neglible under most load conditions; and in-
creases to its peak exponentially when the load is close to the
threshold.Furthermore,thestaticmodelpredictsthethreshold
oftheloadaccurately.ForFig.10, .
Fig. 11 shows the simulation results for , ,
and .ItalsoconfirmsthattheBayesianapproachcould
give us a small except when the load is in a small region
close to the threshold. When the load is close to the threshold
( for Fig. 11), increases to its peak ex-
(30) ponentially. Because of its static nature, the dependent model
used in simulation cannot capture the instantaneous blocking
where . probabilityofthenetworkcarryingdynamictrafficwith100%
For a specific network load we adjust the total traffic ar- accuracy.Therefore,theprobabilityoferrorexceeds0.5when
rivalratetothenetwork tobe eitherhighorlow,sothatthe theloadisatthethreshold.However,simulationresultsconfirm
probabilitythatawavelengthisusedinthenetworkremainsap- thatitispossibletoachieveaprobabilityoferrorcloseto0using
proximately . onlyaggregatedinformationundermostloadconditions.
Thesimulatorisbasedondiscreteeventsimulation.Foreach
simulation, 10 runs are performed where each run consists of VIII. CONCLUSION
100000 end-to-end connection requests. Four decisions may
Inthiswork,wehaveinvestigatednetworkmanagementin-
resultfromlight-pathassessmentusingthepartialinformation
formation for light-path assessment across administrative do-
discussedpreviously:(i)correctacceptation(CA);(ii)incorrect
mains(subnets).Ourfocushasbeenonstudyingthescalability
acceptation(IA);(iii)correctrejection(CR);(iv)incorrectrejec-
of management information, which includes aggregated infor-
tion(IR).Theprobabilityoferrorisobtainedasthepercentage
mationofeachsubnet,andlocalinformationfromwavelength
ofIAandIRoutofallthedecisions.
convertersonnetworkboundaries.
We have formulated the problem based on decision theory,
B. SimulationResults
and defined the performance of using partial management in-
Fig.10depictstheprobabilityoferrorforlight-pathassess- formation through the Bayes probability of error. A bound in
mentusingaggregatedinformationwith , ,and termsofblockingprobabilityisderivedtoestimatesuchaper-
. The reasons for choosing these parameters are: (1) formance. We then defined the scalability of management in-
morewavelengthswouldbeusedforinter-domaincallsthanfor formation as the growth rate with respect to network size and
localcalls;and(2)alargepercentageofinter-domaincallssup- resourcewhenadesiredperformanceisachieved.
portedbyonenetworkdomainwouldbecallspassingthrough A scalable case has been studied where the partial manage-
thatdomain. mentinformation grows onlylogarithmically with the number