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Data Structures and Algorithms for Resource Scheduling in High Speed Networks

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Title:
Data Structures and Algorithms for Resource Scheduling in High Speed Networks
Creator:
Li, Yan
Place of Publication:
[Gainesville, Fla.]
Publisher:
University of Florida
Publication Date:
Language:
english
Physical Description:
1 online resource (147 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
Sahni, Sartaj
Committee Co-Chair:
Ranka, Sanjay
Committee Members:
Peir, Jih-Kwon
Newman, Richard E.
Ahuja, Ravindra K.
Graduation Date:
12/17/2010

Subjects

Subjects / Keywords:
Algorithms ( jstor )
Bandwidth ( jstor )
Data models ( jstor )
File transfers ( jstor )
Heuristics ( jstor )
Scheduling ( jstor )
Topology ( jstor )
Traffic loads ( jstor )
Wavelengths ( jstor )
Workloads ( jstor )
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
algorithm, bandwidth, data, multiple, optical, resource, scheduling, structure, twin, wavlength
Genre:
Electronic Thesis or Dissertation
born-digital ( sobekcm )
Computer Engineering thesis, Ph.D.

Notes

Abstract:
Large scale scientific applications require the collaboration of geographically distributed computational resources, that are connected by high speed and dedicated networks. These distributed computational resources, together with the network connecting them, define a heterogeneous computational environment. Finding an optimal resource allocation schedule is key to providing effective and reliable computation services in this environment. This dissertation focuses on resource scheduling problems for dedicated high speed networks. We formulate a series of scheduling problems according to various scheduling needs and performance metrics. We propose a set of data structures that characterize the temporal behavior of various resources. Based on these data structures, we propose algorithms for resource allocation and path computation for each formulated scheduling problem. We develop multi-path reservation algorithms for in-advance scheduling of large file transfers from multiple sources to multiple destinations. When the requests are processed one by one in online mode, a new max-flow based greedy algorithm and four variants that adapt the $k$-shortest paths and $k$-disjoint paths algorithms are proposed. Further, to find an earliest-finishing schedule for a batch of file transfers, a linear programming based algorithm is developed. We develop an extended network model that supports optical switches with or without wavelength converters. We customize some existing routing and resource allocation algorithms to the optical environment and study the impact of different wavelength conversion schemes for resource scheduling. We also present a novel wavelength assignment strategy that alleviates the need to keep track of the bandwidth allocation status of each wavelength. A Multiple Resource Reservation Model (MRRM) is presented for the case when multiple types of resources are scheduled together. This model enables the monitoring and scheduling of multiple heterogeneous and distributed resources. MRRM provides a unified representation for multiple types of distributed resources, and represents resource constraints such as compatibility and accessibility. Using MRRM, we solve the Multiple Resource First Slot (MRFS) problem based on a collection of algorithms that are customized for different request and resource types. We also consider the wavelength assignment problem in Time-domain Wavelength Interleaved Networks (TWIN). We propose a 2-step process to compute the wavelength assignment for a given set of the traffic demands. The goal of our scheduling algorithm is to find out the assignment that uses the minimum number of wavelengths. We show that this wavelength assignment problem is NP-Complete by reducing the Graph Coloring problem to it. Four greedy heuristics are presented to compute a near optimal solution within reasonable time. For each proposed algorithm, we conduct extensive experiments under various topologies and workloads to evaluate its performance and efficiency. ( en )
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2010.
Local:
Adviser: Sahni, Sartaj.
Local:
Co-adviser: Ranka, Sanjay.
Statement of Responsibility:
by Yan Li.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright Li, Yan. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
741565101 ( OCLC )
Classification:
LD1780 2010 ( lcc )

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DATASTRUCTURESANDALGORITHMSFORRESOURCESCHEDULINGINHIGHSPEEDNETWORKSByYANLIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2010

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c2010YanLi 2

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Idedicatemydissertationtomybeloveddaughter,Sarah,andmywife,LuChen. 3

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ACKNOWLEDGMENTS IwanttothankmyPhDadvisers:Dr.SartajSahniandDr.SanjayRanka.YouprovidedmethechancetofulllmydreamofbecomeaPhDinoneofthebestuniversitiesintheworld.Youtaughtmehowtoperformtherealacademicalresearches.Youinspiredmewithyourbrilliantmindandtrainedmyhowtoattackthosedifcultproblemsinmyway.Youbroughtmecondencetosolvetheproblemsbymyselfwhilepatientlysupportmebypointingouteventhesmallestaws.YoualsoprovidemethecomfortresearchenvironmentssothatIcandedicatedmyselftotheresearches.Yourdemeanors,yourkindnessesandyourwaysofthinkingwillbethebestexamplesformethroughallmylife.Iwanttothankmywife,Dr.LuChen.Youaretheonethatshareallmytroubles,myhappiness,mydesperationandmyhopes.Lookingbackalltheseyears,IcannotimaginehowIcangainwhatIhavetodaywithoutyou.Youtakecareofthefamily,youobtainedyourPhDtogetherwithme,andthemostamazingpart,youbroughtusourmostvaluabletreasure:littleSarah.Icannotexpectanymorefromyou.Iknowbehindallofthese,itisyournumerouseffortsandendlesspatients.Wordsarenotenoughtoexpressmysincerelyappreciationtoyou.However,Istillwanttosaythat,youarethebestgiftthatIhaveeverreceivedinallmylife.IwanttothankmyparentsinChina.Itisyouthattaughtmerightandwrong,theintegrityandproudandwhatarereallyvaluedforme;Itisyouthatstandallthelonelinessandmissingandencouragemepursuemyowndreaminothersideoftheocean.NowIhaveachieveyourexpectations.Now,IamaDoctorandafather.Iwillfollowtheexamplesyousettome.Iwilltrymybesttobringhappinessformyfamilymembers.Andforyoutwo,youwillalwaysbetheonesthatIlovethemost.Finally,IwanttothankallthepeoplesImeethere.ItisyouthatbringmeallthehappytimeinGainesville.Withoutyou,IwouldnotaccomplishmyPhDsosmoothand 4

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productive.Thesememorieswillforeverbeembeddeddeeplyinmymindandneverwillfadeaway. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 13 CHAPTER 1INTRODUCTION ................................... 15 1.1ProblemOverview ............................... 15 1.2OurContributions ................................ 17 1.3OrganizationOfThisdissertation ....................... 19 2TEMPORALNETWORKMODEL ......................... 20 2.1SlottedTime .................................. 20 2.2ContinuousTime ................................ 21 3SCHEDULINGINGENERALNETWORKS .................... 23 3.1ProblemDenition ............................... 23 3.2Single-PathScheduling ............................ 24 3.2.1ProblemDenitions ........................... 24 3.2.2PathComputationAlgorithms ..................... 25 3.2.2.1Fixedslot ........................... 25 3.2.2.2Maximumbandwidthinaslot ................ 28 3.2.2.3Maximumduration ...................... 28 3.2.2.4Firstslot ........................... 28 3.2.2.5Allslots ............................ 30 3.2.2.6Allpairsallslots ....................... 30 3.2.3PerformanceMetrics .......................... 31 3.2.3.1Spacecomplexity ...................... 31 3.2.3.2Timecomplexity ....................... 33 3.2.3.3Effectiveness ......................... 33 3.2.4Experiments ............................... 36 3.3Multiple-PathScheduling ........................... 42 3.3.1ProblemDenition ........................... 42 3.3.1.1Datastructures ........................ 43 3.3.2OptimalSolutionandN-BatchHeuristics ............... 44 3.3.2.1N-Batchheuristics ...................... 48 3.3.3OnlineSchedulingAlgorithms ..................... 49 3.3.3.1Greedyalgorithm ...................... 49 3.3.3.2Greedyschedulingwithnishtimeextension(GOS-E) .. 52 3.3.3.3K-Pathalgorithms ...................... 53 6

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3.3.4ExperimentalEvaluation ........................ 54 3.3.4.1Experimentalframework .................. 54 3.3.4.2Singlestarttimescheduling(SSTS) ............ 55 3.3.4.3Multiplestarttimescheduling(MSTS) ........... 58 3.3.4.4GOSv.s.GOS-E ...................... 61 3.4GeneralNetworkSchedulingAlgorithmsSummary ............. 63 4SCHEDULINGINOPTICALNETWORKS ..................... 65 4.1ProblemDenition ............................... 65 4.2SchedulinginFullWavelengthConversionNetwork ............ 67 4.2.1ProblemDenition ........................... 67 4.2.2RoutingAlgorithms ........................... 68 4.2.3WavelengthAssignmentAlgorithms .................. 71 4.2.4PerformanceEvaluation ........................ 73 4.2.5Experiments ............................... 74 4.2.5.1Simulationenvironment ................... 74 4.2.5.2Evaluatedalgorithms .................... 75 4.2.5.3Resultsandobservations .................. 76 4.3SchedulinginSparseWavelengthConversionNetwork ........... 77 4.3.1ProblemDescription .......................... 77 4.3.2ExtendedNetworkModel ....................... 79 4.3.3RoutingandWavelengthAssignmentAlgorithms .......... 81 4.3.3.1ExtendedBellman-Fordalgorithmforsparsewavelengthconversion .......................... 81 4.3.3.2k-Alternativepathalgorithm ................. 82 4.3.3.3Breakingthetiesinpathselection ............. 83 4.3.3.4Wavelengthassignment ................... 84 4.3.4ExperimentalEvaluation ........................ 85 4.3.4.1Experimentalframework .................. 85 4.3.4.2Slacktie-breakingscheme ................. 87 4.3.4.3Blockingprobability ..................... 89 4.3.4.4Requests'averagestarttime ................ 92 4.3.4.5Schedulingoverhead .................... 94 4.3.4.6Algorithmswitchingstrategy ................ 95 4.4OpticalNetworkSchedulingSummary .................... 97 5MULTIPLERESOURCESCHEDULING ...................... 99 5.1ProblemDenition ............................... 99 5.2ResourceModelandDataStructure ..................... 101 5.2.1ResourceModel:MRRM ....................... 101 5.2.2DataStructures ............................. 105 5.3ProblemDenition ............................... 106 5.4MultipleResourceSchedulingAlgorithm ................... 107 5.4.1WS)]TJ /F4 11.955 Tf 11.96 0 Td[(RCSchedulingAlgorithm .................... 108 7

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5.4.2WN)]TJ /F4 11.955 Tf 11.95 0 Td[(RCSchedulingAlgorithm .................... 110 5.4.3WS)]TJ /F4 11.955 Tf 11.96 0 Td[(RNSchedulingAlgorithm .................... 111 5.4.4WN)]TJ /F4 11.955 Tf 11.95 0 Td[(RNSchedulingAlgorithm .................... 113 5.5Evaluation .................................... 113 5.5.1EvaluationEnvironment ........................ 113 5.5.2ResultsandObservations ....................... 114 6SCHEDULINGINTIME-DOMAINWAVELENGTHINTERLEAEDNETWORKS 117 6.1ProblemDenition ............................... 117 6.2RelatedWork .................................. 119 6.3NetworkModelandProblemDenition .................... 121 6.4TreeConstruction ................................ 124 6.5Tree-WavelengthAssignment ......................... 126 6.5.1GenericFormoftheTree-WavelengthAssignmentProblem .... 126 6.5.2GreedyHeuristics ........................... 129 6.6Evaluation .................................... 132 6.6.1ExperimentalFramework ........................ 132 6.6.2EvaluationResults ........................... 133 7CONCLUSION .................................... 138 REFERENCES ....................................... 141 BIOGRAPHICALSKETCH ................................ 147 8

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LISTOFFIGURES Figure page 1-1TopologyofInternet2. ................................ 15 2-1Time-bandwidthlist ................................. 21 3-1ModiedDijkstra'salgorithm ............................ 29 3-2PseudocodeforextendedFloydalgorithm .................... 31 3-3Algorithmtimecomplexities ............................. 34 3-4Classicationofxed-slotalgorithms ........................ 35 3-5TopologiesofAbilenenetwork,MCInetwork,Burchard'snetworkandClusternetwork ........................................ 37 3-6AcceptanceratiovsrequestsnumberinAbileneMCIBurchardandClusternetwork ........................................ 39 3-7Acceptanceratiovsrequestsnumberinvariousrandomtopologies ....... 40 3-8AcceptanceratiovsmeanrequestsdurationinAbileneMCIBurchardandClusternetwork .................................... 40 3-9Acceptanceratiovsmeanrequestsdurationinvariousrandomtopologies ... 41 3-10AcceptanceratiovsmeanrequestsbandwidthinAbileneMCIBurchardandClusternetwork .................................... 41 3-11Acceptanceratiovsmeanrequestsbandwidthinvariousrandomtopologies .. 42 3-12Basicintervals .................................... 44 3-13GreedyonlineschedulingalgorithmGOS ..................... 51 3-14Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesinMCIusingSSTS. ..................................... 55 3-15Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesin100nodesrandomtopologyusingSSTS. ....................... 55 3-16Comparisonofdifferentalgorithms'MFTrandomTopologiesofdifferentsizeusingSSTS. ..................................... 56 3-17Comparisonofdifferentalgorithms'executiontimeondifferentnumbersoflesonMCInetworkusingSSTS. ......................... 57 3-18Comparisonofdifferentalgorithms'executiontimeondifferentnumbersofleson100nodesrandomtopologyusingSSTS. ................ 57 9

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3-19Comparisonofdifferentalgorithms'MFTexecutiontimeonrandomtopologieswithdifferentsizeusingSSTS. ........................... 58 3-20Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesinMCIusingMSTS. ..................................... 59 3-21Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesin100nodesrandomtopologyusingMSTS. ....................... 59 3-22Comparisonofdifferentalgorithms'MFTinrandomtopologiesofdifferentsizeusingMSTS. ..................................... 59 3-23Comparisonofdifferentalgorithms'executiontimeinMCInetworkusingMSTS. ............................................. 60 3-24Comparisonofdifferentalgorithms'executiontimein100nodesrandomtopologyusingMSTS. ..................................... 60 3-25Comparisonofdifferentalgorithms'executiontimeonrandomtopologieswithdifferentsizeusingMSTS. ............................. 60 3-26ComparisononnumberofMax-FlowsiscomputedbyGOS-EandGOSin100noderandomnetwork. ............................. 61 3-27ComparisononexecutiontimeusedbyGOS-EandGOSin100-noderandomnetwork. ....................................... 62 3-28ComparisononnumberofMax-FlowsiscomputedbyGOS-EandGOSin100noderandomnetwork. ............................. 62 3-29ComparisononexecutiontimeusedbyGOS-EandGOSin100-noderandomnetwork. ....................................... 62 4-1Arequesttablewith5requests ........................... 72 4-2ComparisonofwavelengthassignmentusingdifferentschemesforrequesttableofFigure 4-1 .................................. 73 4-3Timecomplexityofdifferentalgorithms ....................... 74 4-4NSFandGEANTnetwork .............................. 75 4-5Networkacceptanceratiovsnumberofrequests ................. 76 4-6Acceptanceratiovsrequestsnumberinvariousrandomtopologies ....... 77 4-7ExtendedNetworkModel .............................. 80 4-8ExtendedNetworkModel .............................. 84 4-9NetworkTopologies ................................. 86 10

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4-10Differenthvaluesfordifferenttopologies.NetworkTrafcLoad:=0.05 ... 88 4-11Benetofslacktie-breakingschemeinvarioustopologies.NetworkTrafcLoad:=0.05. ................................... 88 4-12BlockingProbabilityvs.WavelengthConverterRatioinvarioustopologywithlowtrafcload. .................................... 90 4-13BlockingProbabilityvs.WavelengthConverterRatioinvarioustopologywithhightrafcload. ................................... 90 4-14Totalresourceconsumptionina100-noderandomnetworkunderdifferentworkload. ....................................... 91 4-15AverageRequestStartTimevs.WavelengthConverterRatioinvarioustopologywithlowtrafcload. ................................. 92 4-16AverageRequestStartTimevs.WavelengthConverterRatioinvarioustopologywithhightrafcload. ................................ 93 4-17AveragecomputationtimeofEBF-SandKDP-S. ................. 94 4-18TheperformanceofalgorithmswitchingstrategyinSlowTrafcPatternSwitching. ............................................. 95 4-19TheperformancealgorithmswitchingstrategyinFastTrafcPatternSwitching. ............................................. 96 5-1GeneralModelofMRRM .............................. 101 5-2DetailedModelofMRRM .............................. 104 5-3WS)]TJ /F4 11.955 Tf 11.96 0 Td[(RCSchedulingAlgorithm .......................... 110 5-4ExtendedBellman-Fordalgorithm ......................... 112 5-5Ourco-schedulingalgorithms'performanceonacceptanceratio. ........ 115 5-6Ourco-schedulingalgorithms'performanceonconvergespeed. ........ 115 6-1AnexampleofTWINnetwork. ........................... 123 6-2AnexampleofTWINTreeConstruction. ..................... 124 6-3ThegreedyalgorithmforTree-Construction .................... 125 6-4ReductionfromGraph-Coloringproblemtotree-wavelengthassignmentproblem. ............................................. 128 6-5NetworkTopologies ................................. 133 11

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6-6TheperformancesofwavelengthassignmentheuristicsunderdifferentnumberofrequestsinMCInetwork. ............................ 134 6-7Theperformancesofwavelengthassignmentheuristicsunderdifferentnumberofrequestsin100-noderandomtopologies. ................... 134 6-8Theperformancesofwavelengthassignmentheuristicsinrandomnetworkswithvarioussizes. .................................. 135 6-9Thealgorithmrunningtimeofwavelengthassignmentheuristicsunderdifferentnumberofrequestsin100nodenetworks. .................... 136 6-10Thealgorithmrunningtimeofwavelengthassignmentheuristicsinrandomnetworkswithvarioussizes. ............................ 137 12

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDATASTRUCTURESANDALGORITHMSFORRESOURCESCHEDULINGINHIGHSPEEDNETWORKSByYanLiDecember2010Chair:SatarjSahniCochair:SanjayRankaMajor:ComputerEngineering Largescalescienticapplicationsrequirethecollaborationofgeographicallydistributedcomputationalresources,thatareconnectedbyhighspeedanddedicatednetworks.Thesedistributedcomputationalresources,togetherwiththenetworkconnectingthem,deneaheterogeneouscomputationalenvironment.Findinganoptimalresourceallocationscheduleiskeytoprovidingeffectiveandreliablecomputationservicesinthisenvironment.Thisdissertationfocusesonresourceschedulingproblemsfordedicatedhighspeednetworks.Weformulateaseriesofschedulingproblemsaccordingtovariousschedulingneedsandperformancemetrics.Weproposeasetofdatastructuresthatcharacterizethetemporalbehaviorofvariousresources.Basedonthesedatastructures,weproposealgorithmsforresourceallocationandpathcomputationforeachformulatedschedulingproblem. Wedevelopmulti-pathreservationalgorithmsforin-advanceschedulingoflargeletransfersfrommultiplesourcestomultipledestinations.Whentherequestsareprocessedonebyoneinonlinemode,anewmax-owbasedgreedyalgorithmandfourvariantsthatadaptthekshortestpathsandkdisjointpathsalgorithmsareproposed.Further,tondanearliest-nishingscheduleforabatchofletransfers,alinearprogrammingbasedalgorithmisdeveloped. 13

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Wedevelopanextendednetworkmodelthatsupportsopticalswitcheswithorwithoutwavelengthconverters.Wecustomizesomeexistingroutingandresourceallocationalgorithmstotheopticalenvironmentandstudytheimpactofdifferentwavelengthconversionschemesforresourcescheduling.Wealsopresentanovelwavelengthassignmentstrategythatalleviatestheneedtokeeptrackofthebandwidthallocationstatusofeachwavelength. AMultipleResourceReservationModel(MRRM)ispresentedforthecasewhenmultipletypesofresourcesarescheduledtogether.Thismodelenablesthemonitoringandschedulingofmultipleheterogeneousanddistributedresources.MRRMprovidesauniedrepresentationformultipletypesofdistributedresources,andrepresentsresourceconstraintssuchascompatibilityandaccessibility.UsingMRRM,wesolvetheMultipleResourceFirstSlot(MRFS)problembasedonacollectionofalgorithmsthatarecustomizedfordifferentrequestandresourcetypes. WealsoconsiderthewavelengthassignmentprobleminTime-domainWavelengthInterleavedNetworks(TWIN).Weproposea2-stepprocesstocomputethewavelengthassignmentforagivensetofthetrafcdemands.Thegoalofourschedulingalgorithmistondouttheassignmentthatusestheminimumnumberofwavelengths.WeshowthatthiswavelengthassignmentproblemisNP-CompletebyreducingtheGraphColoringproblemtoit.Fourgreedyheuristicsarepresentedtocomputeanearoptimalsolutionwithinreasonabletime. Foreachproposedalgorithm,weconductextensiveexperimentsundervarioustopologiesandworkloadstoevaluateitsperformanceandefciency. 14

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CHAPTER1INTRODUCTION 1.1ProblemOverview Manylarge-scalescienticandcommercialapplicationsproducelargeamountsofdata,oftheorderofterabytestopetabytes,whichmustbetransferredacrosswide-areanetworks.Forexample,forane-Scienceapplication,datasetsproducedonasupercomputerinLosAngelesmayneedtobestreamedtoaremotestoragecenterinHoustonforanalysis.TheresultsthenaresenttoAtlantaandvisualizedtheretoguidethenextroundtheexperiments.Whendataprovidersandconsumersaregeographicallydistributed,dedicatedconnectionsareneededtoeffectivelysupportavarietyofremotetasks[ 46 ].Morespecically,dedicatedbandwidthchannelsarecriticalinthesetaskstooffer(i)largecapacityformassivedatatransferoperations,and(ii)dynamicallystablebandwidthformonitoringandsteeringoperations.Itisimportantthatthesechannelsbeavailablewhenthedataisorwillbereadytobetransferred.Thus,theabilitytoreservesuchdedicatedbandwidthconnectionseitheron-demandorin-advanceiscriticaltobothclassesofoperations. Figure1-1. TopologyofInternet2. 15

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Toprovidededicatednetworkconnectionserviceforthepreviousexample,themanagerofthehighspeednetworkthatconnectsthesesites(likeInternet2[ 29 ],Figure 1-1 )shouldaddressthefollowingissues: (a) Resourcemonitoringandmanagement.Toprovidearesourcereservationserviceonanetwork,itisfundamentaltoacknowledgethestatusofavailableresourcesandreservethemforacertainperiodoftime.Moresophisticatedresourcereservationmechanismscanonlybebuiltwiththesupportofthesetwobasicfunctionalities. (b) Dynamicandscalableresourcemodel.Asthelinkcapacityofanetworkissharedbydifferentrequestsfromtimetotime,adynamicresourcemodelthatrepresentsthechangingstatusofthenetwork'savailableresourcesisneeded.Plus,tobecapableoftheincreasingdemandsforthefutureapplications,thisresourcemodelisalsoexpectedtobescalablewiththenetworksizeandthenumberofrequests. (c) Efcientschedulingalgorithmsforvariousreservationrequests.Basedontheaboveresourcemodel,algorithmsareneededtondpathsthatfulllusers'requests.Userrequestsgenerallyspecifysourceanddestinationnodesaswellasbandwidthandduration.Whenthejobstarttimeistheprimaryconcern,thebandwidthschedulerneedtondtheearlieststarttimeforwhichafeasiblenetworkpathisavailable.Whenthejobrequiresalargeamountofbandwidth,multi-pathschedulingisneededtofullltheresourcerequirement.Whentherequiredpathisnotavailableduetolinkoutage,analterativepathneedtobecomputedandre-allocated.Hence,foreachtypeofrequest,weneedacorrespondingalgorithmtoeffectivelycomputethebestresourceallocationschedule. Inthisdissertation,wefocusonthelasttwoofthesethreeissuesandproposedatastructuresandalgorithmsforvariousresourceschedulingproblemsindedicatedhighspeednetworks. Theimportanceofdedicatedconnectioncapabilitieshasbeenrecognized,andseveralnetworkresearchprojectsarecurrentlyunderwaytodevelopresourceschedulingcapabilities.TheseincludeUserControlledLightPaths(UCLP)[ 61 ],UltraScienceNet(USN)[ 47 ],Circuit-switchedHigh-speedEnd-to-EndTransportArcHitecture(CHEETAH)[ 74 ],Enlightened[ 18 ],DynamicResourceAllocationviaGMPLSOpticalNetworks(DRAGON)[ 2 ],JapaneseGigabitNetworkII[ 31 ],BandwidthonDemand(BoD)onGeant2network[ 22 ],On-demandSecureCircuitsandAdvance 16

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ReservationSystem(OSCARS)[ 4 ]ofESnet,HybridOpticalandPacketInfrastructure(HOPI)[ 28 ],BandwidthBrokers[ 72 ]andothers.Inaddition,productionnetworksatthenationalandinternationalscalewithsuchcapabilitiesarebeingdeployedbyInternet2andLHCNet[ 37 ].Suchdeploymentsareexpectedtobeontheincreaseandproliferateintobothsharedandprivatenetworkinfrastructuresacrosstheglobeinthecomingyears. Bandwidthreservationsystemsoperateinoneoftwomodes: (a) Inon-demandscheduling,bandwidthisreservedforatimeperiodthatbeginsatthecurrenttime. (b) Inin-advancescheduling,bandwidthisreservedforatimeperiodthatbeginsatsomefuturetime. Inreality,on-demandschedulingisaspecialcaseofin-advancescheduling;thefuturetimeforeachschedulingrequestisseparatedfromthetimeatwhichtherequestismadebyatimeintervalofzero.Hence,inourresearch,alltheproblemsareformulatedforthein-advanceschedulingmodeandthesolutionsarenaturallyadaptedtotheon-demandcase. 1.2OurContributions Thecontributionsofthisdissertationarelistedbelow: (a) DynamicResourceModel.Torepresentthechangingresourceavailabilityinthenetwork,wedenedasetofdatastructuresthatcoupethetimeinformationtogetherwithresourceavailability.Wealsodiscussindetailthepros-and-consoftwodifferenttimerepresentations:continuoustimemodelanddiscretetimemodel. (b) SinglePathSchedulinginGeneralNetworks.Singlepathschedulingisthesimplestandmostcommonreservationmode,aseachrequestonlyasksforasinglepathinthenetwork.Forthistypeofscheduling,wehavecategorizedseveraldifferentproblemsbasedondifferentobjectives,includingxedslot,maximumbandwidthinslot,maximumduration,rstslot,allslotsandall-pairsall-slots.Foreachproblem,wehaveproposedseveralalgorithmsasitssolution.Someofthesealgorithmsareadaptedfromoriginalon-demandschedulingalgorithms,whileothersarerstproposedbyus.Ourexperimentsindicatethatforthosexed-slotproblem,theminimum-hopfeasiblepathalgorithmoftheseonlargenetworkswhileDAFPisbestonsmallnetworks.Forotherproblems,as 17

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noprioralgorithmshavebeenproposed,ourevaluationsshowonlytheowntheefciencyofourproposedalgorithms. (c) MultiplePathsSchedulinginGeneralNetworks.Itiswellknownthatusingmultiplepathsutilizestheavailablenetworkresourcesmoreeffectively[ 10 ].Toexplorethebenetsbroughtbymulti-pathrouting,theEarliestFinishTimeFileTransferProblem(EFTFTP)isproposed.Inthisproblem,leshavetobetransferredfrommultiplesourcestomultipledestinationsandtheobjectiveistominimizedtheoverallnishtimeofalltransfers.Tosolvethisproblem,twodifferentapproachesareproposedandcompared:onlinescheduling,inwhichtheletransfersarescheduledonebyone;andbatchscheduling,inwhichwescheduleallletransferstogether.Optimalsolutionsandheuristicsareprovidedforthesetwoapproach.Ourexperimentsshowthatonlineschedulingalgorithmgeneratescheduleswithamaximumnishtimeslightlylargerthanthoseobtainedbythebatchscheduling.However,onlineschedulingismoretimeefcientandprovidesbetteraveragenishtimes.Hence,onlineschedulingpresentsagoodbalanceamongmaximumnishtime,meannishtime,andcomputationtime. (d) SchedulinginOpticalNetworkswithFullWavelengthConversion.Manyhighspeednetworksarebasedonopticalinterconnectsandopticalswitches.Alightpathinanopticalnetworkhastotwoconstraints:wavelengthcontinuityconstraintandwavelengthsharingconstraint.Pathcomputationandresourceallocationalgorithmsforgeneralnetworksmaynotbesuitableforopticalnetworks.Hence,were-designourxed-slotandrst-slotalgorithmsfortheopticalenvironmentandconsiderthewavelengthassignmentproblem.OurresultsshowthattheExtendedBellmanFord(EBF)algorithmhasbetterperformancethatotheralgorithms.Forheterogeneousnetworks,ListSlidingWindow(LSW)algorithmalsoprovidedcomparablesolutions;whileforhomogeneousnetworksModiedSwitchPathFirst(MSPF)algorithmandModiedSwitchWavelengthFirst(MSWF)algorithmprovidecomparablesolutions.Wealsoshowedthatadeferredwavelengthassignmentstrategycanbeusedeffectivelyinconjunctionwithourroutingalgorithms. (e) SchedulinginOpticalNetworkswithSparseWavelengthConversion.Althoughfull-wavelengthconversionprovideagreatconvenienceforopticalnetworkscheduling,thehighcostsandtheaddedlatencyintroducedbywavelengthconvertersmakesparsewavelengthconversionanattractiveoptioninopticalnetworkdesign.Hence,weexaminetheimpactofsparsewavelengthconversionforresourceschedulingonopticalnetworks.Weproposeanewnetworkmodeltoemulatethefull-conversionalgorithmsinsparseconversionnetworks.Usingthismodel,weconductedextensiveexperimentstoassesstheimpactofwavelengthconvertersontheFirst-SlotRWAalgorithms'performance.Ourexperimentsindicatethatincreasingwavelengthconvertershaspositiveimpactonblockingperformance,butverylittleimpactontheavailabilityofearlierstarttimes.Wealsoshowthatfornetworksnolargerthanseveralhundredsnodes,deploywavelength 18

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convertersatmost60%ofnodesisenoughtoprovidesatisfyingperformance.Additionally,analgorithmswitchingstrategythatadaptstheschedulingalgorithmasthecurrentworkloadchangesisproposed.Whenthenetwork'strafcpatterndoesnotchangingdramatically,thisstrategyprovidesconsiderableperformanceimprovements. (f) MultipleResourcesScheduling.Toprovideacompleteserviceguaranteeforacomplexe-Scienceapplication,notonlynetworkconnections,butalsocomputationalresources,suchasCPUtimeanddiskspaceshouldbereserved.WeproposeaMultipleResourcesReservationModel(MRRM)thatrepresentsheterogeneousresourcesinauniedwaywhilekeepingtheirdiversity.WealsodenetheMultipleResourceFirstSlot(MRFS)problemanddivideitintofoursub-problems.ForeachsalientinstanceofMRFS,analgorithmispresentedtosolveitbasedonMRRM.Experimentsonaheterogeneouscomputernetworkshowthatouralgorithmsarescalablelinearlyintermsofnetworksizeandrequestratio. (g) SchedulinginTWIN.Time-domainWavelengthInterleavingNetwork(TWIN)isanovelopticalnetworkarchitecture.InTWIN,trafcfromdifferentsourcenodesmayshareonewavelengththroughTimeDimensionMultiplexing(TDM)inthenetworkiftheirhavethesamedestination.Thisnewfeaturemayimprovethelinkcapacityutilizationbyprovidingmuchnergranularityinopticalbandwidthreservationformulti-source-single-destinationtrafcs.Inthisdissertation,weconsiderthewavelengthassignmentproblemsinTWINnetworks.Severalalgorithmsforthisproblemareproposedandevaluated. 1.3OrganizationOfThisdissertation Therestofthisdissertationisorganizedasfollows.InChapter2,wepresentsthetemporalnetworkmodelandrelateddatastructures,whicharefundamentalforallfollowingchapters.InChapter3,ourresearchesonthegeneralnetworkmodelispresented,wherefractionalreservationofalink'scapacityisallow.Chapter4discussesbandwidththeschedulingprobleminthespeciccontextofopticalnetworks.Chapter5proposesanovelmultipleresourceschedulingmodelandcorrespondingschedulingalgorithms.Chapter6solvesthewavelengthassignmentprobleminTime-domainWavelengthInterleavedNetworks.WeconcludethisdissertationinChapter7. 19

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CHAPTER2TEMPORALNETWORKMODEL WeassumethatthenetworkisrepresentedasagraphG=(V,E).Eachnodeofthisgraphrepresentsadevicesuchasaswitchforlayers1-2andarouterforlayer3;andeachedgerepresentsalinksuchasSONETorEthernet.Whendevelopinganin-advancereservationsystemonemustdecideonarepresentationoftime.Theoptionsaretoeitherconsidertimeasdividedintoequalsizeslotsasisdonein[ 12 19 23 54 58 ]ortoconsidertimeasbeingcontinuousasin[ 46 48 52 73 ]. 2.1SlottedTime Explicitintheslottedmodelof[ 12 23 ]istheuseofanarraytostorelinkstatusforeachtimeslot.So,forexample,wemayuseatwo-dimensionalarraybsuchthatb[l,t]givesthebandwidthavailableonlinklinslott.Theuseofthismodeloftimehasseveralmeritsanddemerits.ThemeritsincludeitssimplicityandthefactthatthestatusofalinklinanyslottmaybedeterminedinO(1)time.Someofthedemeritsare: 1. Weneedtodecidethegranularityofatimeslot.Doesaslotrepresentaminute,anhour,aday,oraweekoftime?Thegranularityofatimeslotdeterminesthenumberoftimeslotsweneedtoprovisionforinourarrayb.So,iftheadvanceschedulerpermitsreservationstobemadeuptoayearinadvanceofthejobcompletiontime,thenumberoftimeslots(i.e.,sizeoftheseconddimensionofourarray),T,willneedtobe52incaseatimeslotrepresentsaweekand525,600incaseatimeslotis1minute(weassumea365-dayyear).Assumingthatittakes4bytestostoretheavailablebandwidthofalinkandanetworkwith1000links,thememoryrequiredforthelinkstatusarraybis208,000byteswhenweemploya1-weekgranularityandisabout2GBfora1-minutegranularity.Ontheotherhand,thepotentialtowastealotofresourcesishighwhenweusea1-weekgranularity.Thisisbecausetheschedulercanallocateonlyanintegralnumberofslotstoareservationrequest.So,ifataskneedsafractionalnumberofslots,theschedulermustroundtothenearestinteger.Arequestfor(say)1.1slotsresultsintheallocationof2slots.Witha1-weekgranularity,a1-minutetask,whichtiesupasource-destinationpathfor1-weekresultsina0.01%utilization! 2. Therun-timeofreservationalgorithmsisoftenafunctionofT,thetotalnumberofslots,or,thedurationofareservationrequestorboth.So,forexample,ittakesO()timetodeterminewhetherlinklhasbandwidthbavailableineachofslotst,,t+)]TJ /F6 11.955 Tf 12.39 0 Td[(1.Thisdeterminationistobemadetoverifythatlinklisavailable 20

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forabandwidthbreservationofdurationbeginningatslott.TondtherstslotduringwhichlinklhasabandwidthofatleastbavailabletakesO(T)time. 3. Theexacttimeofday/week/yearrepresentedbyslottcannotbexed.Inotherwordstheslotb[l,t]cannotbeassociatedwith(say)weektofayearwhenusingaslotgranularityof1week.Thisisbecausethereservationsystemneedstooperateessentiallyforeverusingthesamearray.Astimeadvancesfrom1weektothenextweneedtodroptheslotthatrepresentedtheelapsedweekandaddaslottorepresenttheweekthatisnowayearaway.Todothisefciently,wemustusetheslotsassociatedwitheachlinkinacircularmannerasisdoneinthecirculararray-representationofaqueue[ 51 ]. 2.2ContinuousTime Inthecontinuoustimemodeladoptedby[ 46 48 52 73 ],thestatusofeachlinklismaintainedusingatime-bandwidthlist(TBlist)TB[l]thatiscomprisedoftuplesoftheform(ti,bi),wheretiisatimeandbiisabandwidth.ThetuplesonaTBlistareinincreasingorderofti.If(ti,bi)isatupleofTB[l](otherthanthelastone),thenthebandwidthavailableonlinklfromtitoti+1isbi.When(ti,bi)isthelasttuple,abandwidthofbisavailablefromtito1.ConsiderthelinkshowninFigure 2-1 .Thegraphisapictorialrepresentationofthebandwidthavailableonthislinkasafunctionoftime.So,forexample,abandwidthof5isavailablefromtime0totime1andtheavailablebandwidthfrom2to3is4.ThecorrespondingTBlistis[(0,5),(1,2),(2,4),(3,5),(4,1),(5,5)]. Figure2-1. Time-bandwidthlist 21

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WenotethatTBlistsmaybeusedintheslottedmodeloftimeaswellwithtirepresentingaslotratherthanatime.Inthiscase,TB[4]=[(2,10),(9,5),(20,50)]wouldmeanthatlink4hasanavailablebandwidthof10inslots2through8,abandwidthof5inslots9through19,andabandwidthof50inslots20andbeyond;thelinkisnotavailableinslot1. EachTBlistmayberepresentedasanarraylinearlistusingdynamicarrayresizingasdescribedin[ 51 ]orasalinkedlist. Thedemeritsofthecontinuoustimemodelincludeitsrelativecomplexity(linearlistsaresomewhatmoredifculttohandlethanarrays)andthecomplexityofdeterminingthestatusofalinkatanygiventime.ThelattercanbedoneinO(logIjTB[l]j))timeusingabinarysearchincaseofanarraylinearlistandinO(jTB[l]j)timeincaseofalinkedTBlist.Someofthemeritsofthistimemodelare: 1. Thereisnoneedtopickatimegranularityortoplaceaboundonthelengthofthebookaheadperiod(i.e.,wedon'tneedtolimitourselvestoreservationsthatcompletewithinayear(say)). 2. Thememoryrequiredtorepresentalinkstate(i.e.,theTBlist)isafunctionofthetimevariationinlinkbandwidthavailabilityratherthantheschedulinghorizonT.So,forexample,alinkwithbandwidthcapacity100andnoreservationsisrepresentedbytheTBlist[(0,100)]irrespectiveofhowfaraheadonecanschedule.Ontheotherhand,iftheavailablebandwidthchangesattimes0,10,and40,theTBlistwillhave3tuples.Wenotethatthetimevariationinlinkbandwidthislooselyrelatedtothenumberoftasksscheduledonthatlink. 3. Therun-timeofreservationalgorithmsisnotafunctionoftheschedulinghorizon.Instead,itisafunctionofthesizeoftheTBlists,which,inturn,dependsonthenumberoftasksthathavebeenscheduled. Becauseofthecorrespondencebetweenaslotandtime,weoftenusethetwotermsinterchangeably. 22

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CHAPTER3SCHEDULINGINGENERALNETWORKS 3.1ProblemDenition Muchoftheresearchonbandwidthschedulinghavefocusedonreservingasinglepathforaspeciedbandwidthrequest.Foron-demandscheduling,itistypicallysupportedbyMultipleProtocolLabelSwitching(MPLS)[ 8 11 ]atlayer3andbyGeneralizedMPLS(GMPLS)[ 67 ]atlayers1and2.Algorithmsforon-demandschedulingaredescribedin[ 24 40 41 64 ],andimplementedbyCHEETAH,DRAGON,HOPI,UCLPandJGN.GeantII,OSCARS,USNandEnlightensupportin-advanceschedulingandalgorithmsforin-advanceschedulingaredescribedin[ 12 23 46 48 52 ],forexample. Ontheotherhand,itiswellknownthatusingmultiplepathscanutilizetheavailablenetworkresourcesmoreeffectively[ 10 ].Themulti-pathreservationproblemisformulatedin[ 10 ]asanetworkowproblemwiththeobjectiveofminimizinglinkcongestion.Algorithmsfordelay-constrainedletransferusingmultiplepathsareproposedin[ 49 ].Multi-pathletransferwithbothlinkutilizationconstraintsandpathlengthconstraintsisconsideredin[ 36 ].Amaximumconcurrentowformulationisusedin[ 45 ]tosolvethelargeletransferproblemwithxedstartandendtimes.Itsobjectiveistomaximizenetworkthroughput.[ 45 ]alsodevelopslinearprogrammingmodelstomaximizenetworkthroughputandproposestwoheuristicsformulti-pathrouting.Therstheuristic,k-ShortestPaths(KSP),usesthek-shortestpathsalgorithmof[ 69 ]tocomputeknotnecessarilydisjointpathsfromthesourcetothedestination.Theschedulingoftheletransferisrestrictedtothesekpaths.Thesecondheuristic,k-DisjointPaths(KDP),computeskdisjointpathsfromsourcetodestinationbyeliminatingthelinkscontainedinpreviouslycomputedpathsbeforecomputingthenextpath;eachpathcomputationgeneratestheshortestpathintheremainingnetwork. 23

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Inthischapter,weuseanetworkmodelinwhichallthenetworklinksallowfractionalbandwidthreservation.Inthiscase,wecanreserveanyamountofbandwidthonanylinkprovidedthereisenoughcapacityavailable.However,inthenextchapter,theintegerconstraintforbandwidthreservationisintroducedtoaccommodateforopticalnetworkconstraints. InSection 3.2 ,asetofin-advanceschedulingproblemsaredenedandcorrespondingalgorithmsareproposedandevaluated.InSection 3.3 ,amultiplepathschedulingproblemEarliestFinishTimeFileTransferProblem(EFTFTP)isproposedandsolvedbybothoptimalsolutionsandheuristics. 3.2Single-PathScheduling 3.2.1ProblemDenitions Thefollowingproblemsareofinterestinthecontextofin-advancescheduling. 1. FixedSlot:Reserveapathwithbandwidthbfromthesourcestothedestinationdfromtimetstarttotimetend. 2. MaximumBandwidthinSlot:Findthelargestbandwidthbsuchthatthereisabandwidthbpathfromthesourcestothedestinationdfromtimetstarttotimetend.Reservesuchapath. 3. MaximumDuration:Findthemaximumdurationsuchthatthereisabandwidthbpathfromthesourcestothedestinationdfromtimetstarttotimetstart+.Reservesuchapath. 4. FirstSlot:Findtheleasttforwhichthereisapathwithbandwidthbfromthesourcestothedestinationdfromtimettotimet+,whereisthedurationforwhichthepathisdesired.Reservesuchapath. 5. AllSlots:Findallrangesrsuchthatforeveryt2r,thereisabandwidthbpathfromthesourcestothedestinationdfromtimettotimet+.Reservesuchapathforauserselectedtinoneofthefoundranges. 6. AllPairsAllSlots:Foreverysource-destinationpair(s,d),ndallrangesrsuchthatforeveryt2r,thereisabandwidthbpathfromstodfromtimettotimet+. 24

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Thexed-slotproblemisreferredtoastheconnectionfeasibilityproblemin[ 23 ].Thesoonestcompletionproblemformulatedin[ 23 ]isreferredtoastherstavailabletransmissionperiodin[ 12 ];bothareidenticaltotherst-slotproblemstatedabove. Inadditiontothepathproblemsdenedabove,anin-advancereservationsystemmayimplementaggregatepathreservationalgorithmsinwhichtheavailablebandwidthonthereservedpathmaychangeduringthecourseofthereservationinterval.Theintegraloftheavailablebandwidthoverthereservationintervalistomeetaprespeciedrequirement.Aggregatedatareservationalgorithmsareusefulindatatransferapplicationswherewearenotconcernedwithtransferringdataatauniformratebutjustwithcompletingthetransferby(say)agivendeadline.[ 23 ]showsthattheaggregatepathreservationproblemisNP-hard. 3.2.2PathComputationAlgorithms WedescribeonlythealgorithmsneededtocomputethepathsforthevariousproblemsdescribedinSection 3.2.1 .TheactualschedulingorreservationofthefoundpathrequiresustoupdatetheTBlistsortheb-arrayentriesforthelinksonthepathaswellastosignaltheroutersonthepathatthereservedtime.Theformerisarelativelystraightforwardprocessandthelatterrequirestheuseofspecicsignalingprotocols. 3.2.2.1Fixedslot OftheproblemslistedinSection 3.2.1 ,thexed-slotproblemisthemoststudied.Thealgorithmsthathavebeenproposedaredescribedbelow. 1. FeasiblePath(FP):Alinkofthenetworkisfeasibleforxed-slotschedulingifftheavailablebandwidthonthelinkatalltimesintheinterval[tstart,tend]isb.Letpbeapathfromthesourcestothedestinationd.pisafeasiblepathiffitiscomprisedonlyoffeasiblelinks.InFPscheduling,afeasiblepathisreserved.Suchapathmaybefoundbyperformingasearch(depth-orbreadth-rst,forexample[ 51 ])onthesubgraphofG,calledthefeasiblesubgraph,obtainedfromGbyeliminatingalllinksthatarenotfeasible.FPschedulingisdonein[ 23 ]. 2. MinimumHopFeasiblePath(MHFP):Thenumberoflinksonapathisitshopcount.InMHFPscheduling,aminimum-hopfeasiblepathisreserved.Suchapathmaybefoundbyperformingabreadth-rstsearchonthefeasiblesubgraphof 25

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G.NoticethatFPschedulingisthesameasMHFPschedulingwhenthesearchmethodusedbyFPschedulingisbreadthrst.MHFPschedulingisusedinUSNandthepathcomputationalgorithmisformallystatedin[ 52 ]. 3. Widest/ShortestFeasiblePath(WSFP):Thisisanadaptationofthewidest-shortestmethodproposedin[ 24 ]foron-demandscheduling.Letpbeafeasiblepath.Letbminbbetheminimumbandwidthavailableonanylinkofpatanyinstantintheschedulinginterval[tstart,tend].InWSFPscheduling,weusetheminimum-hopfeasiblepaththathasthemaximumbminvalue.Tiesarebrokenarbitrarily.NoticethatWSFPschedulingisMHFPschedulingwithaspeciedtiebreaker.WSFPschedulingissuggestedin[ 12 ].AWSFPmaybefoundbyrunningamodiedBellman-FordalgorithmonthefeasiblesubgraphofG[ 24 ];theweightofalinkistheminimumbandwidthavailableonthatlinkduringtheinterval[tstart,tend]orbyrunningamodiedversionofDijkstra'sshortestpathalgorithmonthisfeasiblesubgraph[ 40 ].Inthelattercase,whenselectingthenextshortestpath,priorityisgiventonext-pathcandidateswithleasthopcountandtiesarebrokenbyusingthebminvalueofthepath. 4. Shortest/WidestFeasiblePath(SWFP):ThisisavariantofWSFPschedulingthatwasrstproposedforon-demandscheduling[ 64 ].Weselectafeasiblepaththathasmaximumbminvalue.Tiesarebrokenbyfavoringpathswithsmallerhopcounts.AnSWFPpathmaybefound[ 40 ]byrstrunningDijkstra'sshortestpathalgorithmmodiedtondapathwithmaximumbminandthendoingabreadth-rstsearchtondaminimum-hoppathwiththismaximumbminvalue;thebreadth-rstsearchignoreslinksthatviolatethisbminrequirement. 5. ShortestDistanceFeasiblePathAlgorithms(SDFP):Thesealgorithmsndashortestpath(thelengthofapathbeingthesumoftheweightsofthelinksonthatpath)inthefeasiblesubgraphofG.AnSDFPpathmaybefoundusingDijkstra'sshortestpathalgorithm. SDFPalgorithmsdifferintheirselectionofacostmetricforfeasiblelinks.SDFP-min(minimumSDFP)isanextensionoftheshortestdistancepathalgorithmforon-demandscheduling[ 41 ]tothecaseofin-advancescheduling.Theweightofafeasiblelinkisthereciprocaloftheminimumbandwidthavailableonthatlinkduringtheschedulinginterval[tstart,tend]. InSDFP-avg(averageSDFP),theweightofafeasiblelinkisthereciprocaloftheaverage(ratherthantheminimum)bandwidthavailableonthatlinkduringtheperiod[tstart,tend]. 6. DynamicAlternativeFeasiblePath(DAFP):Again,thisisanadaptationofthedynamicalternativepathalgorithmoriginallyproposedforon-demandscheduling[ 40 ].LethbethenumberofhopsintheMHFP.InDAFP,weuseawidestfeasiblepath(i.e.,onewithmaximumbminvalue)thathasnomorethanh+1hops.Suchapathmaybefound[ 40 ]byrestrictingtheBellman-FordalgorithmproposedforWSFPtousenopathwithmorethanh+1hops.WenotethatwhileDAFP 26

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guaranteestondafeasiblepathwheneversuchapathexists,thedynamicalternativepathalgorithmof[ 40 ]providesnosuchguarantee. 7. OSPFLikeAlgorithms:TheseareshortestpathalgorithmsthatworkonGorsomesubgraphofGotherthanthefeasiblesubgraph.Theydifferinhowthelinkweightsaredenedand/orinhowthesubgraphisdened.SincethesealgorithmsdonotworkonthefeasiblesubgraphofG,theymaygenerateaninfeasiblepathandsofailtoschedulearequestinsomecaseswhereoneoftheaforementionedfeasiblepathalgorithmssucceed.TheshortestpathmaybefoundusingDijkstra'salgorithm. InmostimplementationsoftheOSPFalgorithm,theweightofalinkisdenedtobethereciprocalofitsbandwidthcapacity.Notethatthisbandwidthisnotthelink'savailablebandwidthatanygiventimebutthelink'snominalunloadedbandwidth.So,theweightofa10Gblinkis1/10regardlessofthealreadyscheduledloadonthatlink.TheOSPFpathistheshortestpathinthegraphGusingtheselinkweights.IncasetheOSPFpathisnotfeasibleforxed-slotscheduling,thereservationrequestisdenied. IntheversionofOSPF-TEimplementedinOSCARS[ 4 ],youremovefromthenetworkgraphGthoselinksthatdonothaveanavailablebandwidththatisatleastbatthetimetheschedulingrequestisprocessed(notattimetstart);linkweightsareasforOSPF.Theshortestpathinthisreducedgraphisfoundandanattemptismadetoschedulethereservationrequestonthispath.AswithOSPF,theOSPF-TEpathmaynotbefeasible.NotethateventhoughtheOSPF-TEpathhasenoughbandwidthatthetimethepathiscomputed,itmaynothavesufcientbandwidthduringthereservationperiod[tstart,tend].ThefeasibilityoftheOSPF-TEpathisveried,inOSCARS,byusingadatabaseofpreviouslymadereservations.IncasetheOSPF-TEpathisinfeasible,theschedulingrequestisdenied. 8. kDynamicPaths(kDP):ThesealgorithmsareanextensionofOSPF-likealgorithms.RecognizingthatanOSPF-likealgorithmmayfailtondafeasiblepathinanetworkthathasafeasiblepath,kDPalgorithmsgenerateadditionalpathswiththehopethatoneoftheadditionalpathswillbefeasible.AnOSPF-likealgorithmgeneratesashortestpathandsucceedswhenthisshortestpathisfeasibleandfailsotherwise.InakDPalgorithm,whenthegeneratedpathisinfeasible,wereducethecurrentgraphbyremovingfromitlinksonthegeneratedinfeasiblepathwhoseavailablebandwidthduringthereservationinterval[tstart,tend]islessthanb.Wethenndtheshorteststodpathinthisreducedgraph.Thispathcomputationandgraphreductionprocessisrepeatedatmostktimes.Theprocessterminateswhentherstfeasiblepathisfoundorwhenkinfeasiblepathshavebeengenerated. kDP-OSPFandkDP-OSPF-TEarenaturalextensionsofOSPFandOSPF-TE.kDP-LOADisanadaptationofthealgorithmsusedin[ 58 ]forin-advanceschedulingofopticalnetworks.InkDP-LOAD,eachlinkisassignedaweightequaltothetotalloadalreadyscheduledonthatlink.Moreprecisely,alink'sweightistheaggregateallocatedbandwidthonthelinkbeginningatthecurrenttimeand 27

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goinguptothelatestscheduledtime.SomeotheradaptationsarekDPA-mininwhichthelinkweightisthereciprocaloftheminimumbandwidthavailableintheinterval[tstart,tend](asinSDFP-min)andkDP-avginwhichthelinkweightisthereciprocaloftheaveragebandwidthavailableinthisinterval(asinSDFP-avg). 9. kStaticPaths(kSP):Inthisalgorithm,wehaveuptokprecomputedpathsbetweeneverypairofsource-destinationvertices.Toscheduleapathbetweensandd,weexamine,insomeorder,theuptokprecomputedpathsforthepair(s,t)andselecttherstthatisfeasiblefortheinterval[tstart,tend].Ifnoneisfeasible,theschedulingrequestisdenied. 3.2.2.2Maximumbandwidthinaslot ThiscomputationisachievedbymodifyingDijkstra'sshortestpathalgorithm[ 51 ]asshowninFigure 3-1 .Here,b[u][v]istheminimumbandwidthavailableontheedge(u,v)duringthespeciedinterval/slotandbw[u]isthemaximumbandwidthalongpathsfromthesourcestovertexuundertheconstraintthatthesepathsgothroughonlythoseverticestowhichamaximumbandwidthpathhasbeenfoundalready.ThecomplexityofthealgorithmisO(n2)forageneralnvertexgraph.However,practicalnetworkgraphshaveO(n)edgesandthecomplexitybecomesO(nlogn)whenamaxheap(forexample)isusedtomaintainthebwvalues. 3.2.2.3Maximumduration Asnotedin[ 23 ],themaximumdurationproblemisverysimilartothewidestpathproblem,whichinturnisidenticaltothemaximumbandwidthproblem.Theweightofalinkissettothemaximumduration,beginningattstartforwhichthelinkhasabandwidthofbormoreavailable.Thewideststodpathinthisweightedgraphisthemaximumdurationpath.ThepathisfoundusingamodiedDijkstra'salgorithmasforthemaximumbandwidthinslotproblem. 3.2.2.4Firstslot Threedifferentalgorithmshavebeenproposedfortherstslotproblem[ 23 52 58 ]. 1. SlottedSlidingWindow(SSW):Theslidingwindowrstalgorithmproposedin[ 58 ]foropticalnetworksisavariationofthesoonestcompletionalgorithmproposedin[ 23 ].Boththesealgorithmstrytheslotststart,tstart+1,,inthisorder,tondtheleasttforwhichthegraphGhasafeasiblepath(i.e.,anstod 28

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MaxBandwidth(s,d,prev) { bw[i]=b[s][i],1in. prev[i]=s,1in. prev[s]=0. InitializeLtobealistwithallverticesotherthans. for(i=1,i
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3. ExtendedBellmanFord(EBF):Thisalgorithmforrstslotwasproposedin[ 52 ].First,weextendthenotionofanSTlistforalinktothatforapathinthenaturalway.Next,denest(k,u)tobetheunionoftheSTlistsforallpathsfromvertexstovertexuthathaveatmostkedgesonthem.Clearly,st(0,u)=;foru6=sandst(0,s)=[0,1].Also,st(1,u)=ST(s,u)foru6=sandst(1,s)=st(0,s).Fork>1(actuallyalsofork=1),weobtainthefollowingrecurrence st(k,u)=st(k)]TJ /F6 11.955 Tf 9.61 0 Td[(1,u)[f[vsuchthat(v,u)isanedgefst(k)]TJ /F6 11.955 Tf 9.6 0 Td[(1,v)\ST(v,u)gg(3) where[and\arelistunionandintersectionoperations.Forann-vertexgraph,st(n)]TJ /F6 11.955 Tf 12.7 0 Td[(1,d)givesthestarttimesofallpathsfromstodthathavebandwidthbavailableforaduration.Theavalueoftherst(a,b)pairinst(n)]TJ /F6 11.955 Tf 12.04 0 Td[(1,d)givesthedesiredrstslot. TheBellman-Fordalgorithm[ 51 ]maybeextendedtocomputest(n)]TJ /F6 11.955 Tf 12.47 0 Td[(1,d).Theextensionmerelyworkswithstlistsratherthanwithscalarsandisdescribedin[ 52 ]. 3.2.2.5Allslots Thereappearstobejustonealgorithmthathasbeenproposedfortheallslotsproblem.ThisistheextendedBellman-Fordalgorithm[ 52 ]tocomputest(n)]TJ /F6 11.955 Tf 12.16 0 Td[(1,d)(seeprecedingdiscussionofrstslotalgorithms).Asnotedabove,st(n)]TJ /F6 11.955 Tf 11.99 0 Td[(1,d)givesthestarttimesofallpathsfromstodthathavebandwidthbavailableforaduration. 3.2.2.6Allpairsallslots TheextendedBellman-Ford(EBF)algorithmofSection 3.2.2.4 computesst(u)=st(n)]TJ /F6 11.955 Tf 12.7 0 Td[(1,u)foragivensourcevertexsandalluinO(nel)time.st(u)givesthestarttimeofallavailableslotsofdurationdandbandwidthb.So,inO(nel),usingEBF,weareabletodetermineallavailableslotsfromstoeveryothervertexu(includingvertexd).Furthermore,todetermineallavailableslotsbetweenallpairsofvertices,wemayruntheEBFalgorithmforntimes,oncewitheachvertexasthesourcevertexs.So,thetimeneededtodetermineallslotsbetweenallpairsofverticesisO(n2el).AnalternativestrategytodetermineallavailableslotsbetweenallpairsofverticesistoextendFloyd'sall-pairsshortestpathalgorithm[ 51 ]asisdonein[ 47 ].Figure 3-2 givestheresultingextension.Here,st(u,v)istheSTlistforpathsfromutov.Initially,st(u,v)=ST(u,v).Ontermination,st(u,v)givesallpossiblestarttimesforpathsfromutov. 30

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algorithmExtendedFloyd() { for(intk=1;k
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memoryrequiredforlinkstatusinformationis,therefore,(eT),whereeisthenumberoflinksinthenetwork.Inthecontinuoustimemodel,whereTBlistsareusedtostorelinkstatus,weneedO(PijTB[i]j),wherejTB[i]jisthesizeoftheTBlistforlinki.AsnotedinChapter2,theslottedmodelisaspecialcaseofthecontinuousmodelandTBlistsmayalsobeusedforthislattermodel.Whenthisisdone,thesizeofeachTBlistisboundedbyTandtheTBlistrepresentationtakesO(eT)memory.ForalightlyloadedsysteminwhichthesizeofTBlistismuchlessthanitsmaximumpossiblesizeofT,theTB-listschemeusesmuchlessmemorythanisusedbytheslotted-arrayrepresentation.Infact,whennotasksarescheduled,theTB-listrepresentationuses(e)memoryvs.(T)memoryfortheslotted-arrayrepresentation.AttheotherextremewhereeachTBlisthasTpairs,theTB-listrepresentationtakesabouttwicethememorytakenbythearrayrepresentation(notethateachentryonaTBlistisapairwhileeacharrayentryisasingleton). Inadditiontothememoryrequiredtostorenetworkstatus,memoryisneededtorunthepathcomputationalgorithms.Algorithmsemployingagraphsearchstrategysuchasbreadth-rstsearchneedO(n)spacetokeeptrackofwhichverticeshaveorhavenotbeenvisitedsofar.Notethatwemaydeterminewhetherornotalinkisfeasiblewhilethebreadth-rstsearchalgorithmisrunningandsonoadditionalspaceisneededtomaintainthefeasiblesubgraphofG.ThosethatusesomeversionofDijkstra'sshortestpathalgorithm,needO(n)spaceforapriorityqueueandO(e)spaceforthelinkweightsthatareinuse. ForBellman-Fordextensions,weneedspaceforthelinkSTlistsandthepathstlists.ThenumberoflinkSTlistsise.AlthoughthereareO(n2)pathstlists,thecomputationofst(k,)canbedonein-place(i.e.,usingthesamespaceasusedbyst(k)]TJ /F6 11.955 Tf 12.86 0 Td[(1,)[ 52 ]).So,spaceforO(n)stlistsisneeded.Fortheslottedmodel,TisanupperboundonthesizeofanST/stlist.So,Bellman-FordextensionsrequireanadditionalO((n+e)T)memorytorun.Inlightlyloadedsystems,thesizeofanstlist 32

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ismuchlessthanTandcorrespondinglylessmemoryisneeded.Forthecontinuousmodel,thesizeofanST/stlistcouldbeaslargeasthenumberoftasksscheduledsofar.However,inrealisticapplications,weexpectthesizetobeconsiderablylessthantheTvalueforacorrespondingslottedsystem. TheFloydextensionusedintheall-pairsall-slotsproblemrequiresannnarrayofstlists.ThememoryforthisarraycorrespondstothatforO(n2)stlists.Boundsonthesizeofanstlistwerediscussedinthepreviousparagraph. 3.2.3.2Timecomplexity Figure 3-3 summarizesthetimecomplexityofeachofthealgorithmsdescribedinSection 3.2.2 .Weassumealsothatthenumberoflinkseisatleastequaltothenumberofnodesn.ThealgorithmsofSection 3.2.2 thatworkonthefeasiblesubgraphofGmaybeimplementedtoeitherbeginbyidentifyingthefeasiblelinksofGormaycheckeachlinkforfeasibilitywhenthelinkisrstexamined.Ineithercase,O(e)timeisspentonlinkfeasibilitycheckswhentheslotted-arraymodelisused.Inthecontinuousmodel,feasibilitycheckstakeO(L)time,whereListhesumofthelengthsoftheTBlists.Ineithercase,theremainderofthealgorithmtakesO(e)time. Whenthenumberofpreviouslyscheduledjobsissmall,theTBlistsalsoaresmallinsize.However,intheworstcase,thesizeofaTBlistmaybeT.So,forlightlyloadedsystems,weexpectthecontinuousversionofanalgorithmtooutperform(intermsofruntimeandmemory)thecorrespondingslotted-arrayversion.Thisisthetypicaltrade-offbetweensparseanddensedata-structurerepresentations.Additionally,theslotted-arrayversionsforthexedslotandmaxbandwidthproblemsareexpectedtooutperformthecorrespondingcontinuousversionswhentherequestedreservationdurationissmall. 3.2.3.3Effectiveness Therearetwoaspectstoeffectivenessguaranteesandutilization.Guaranteeshastodowithwhetherornottheschedulingalgorithmprovidesanyguaranteeonitsresult.Forexample,doesaxed-slotalgorithmguaranteetondafeasiblewhenever 33

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Problem Algorithm SlottedArray Continuous FixedSlot FP O(e) O(e+L) MHFP O(e) O(e+L) WSFP O(e+nlogn) O(e+L+nlogn) SWFP O(e+nlogn) O(e+L+nlogn) SDFP O(e+nlogn) O(e+L+nlogn) DAFP O(e+nlogn) O(e+L+nlogn) OSPF O(e+n(+logn)) O(e+L+nlogn) kDP O(ke+minfkn,eg+knlogn)) O(ke+L+knlogn) kSP O(minfkn,eg+kn) O(L+kn) MaxBandwidth Dijkstra O(e+nlogn) O(e+L+nlogn) MaxDuration Dijkstra O(eT+nlogn) O(e+L+nlogn) FirstSlot SSW-MHFP O(eT) )]TJ ET q .398 w 472.97 -190.78 m 472.97 -176.34 l S Q q .398 w 0 -205.23 m 0 -190.78 l S Q q .398 w 101.82 -205.23 m 101.82 -190.78 l S Q BT /F1 11.955 Tf 107.8 -200.9 Td[(LSW O((q+T)e) O(qe+L) EBF O(nel+eT) O(nel+L) AllSlots EBF O(nel+eT) O(nel+L) AllPairsAllSlots Floyd O(n3l+eT) O(n3l+L) l=sizeoflongeststlist q=numberofdifferentaisintheSTlists L=sumoflengthsofTBlists Figure3-3. Algorithmtimecomplexities suchapathexits?Doesarst-slotalgorithmactuallyndtheearliestfeasiblepath?Forthexed-slotproblem,allalgorithmsotherthantheOSPF-like,kdynamicpaths,andkstaticpathsalgorithmsprovideaguarantee.Fortheremainingin-advanceschedulingproblems,allalgorithmsdescribedinSection 3.2.2 ,otherthantherst-slotalgorithmof[ 58 ],provideaguarantee.Figure 3-4 givesapossiblehierarchicalclassicationofthexed-slotalgorithmsofSection 3.2.2.1 Sincetheschedulingalgorithmsworkinanonlinemode(i.e.,schedulingrequestsareprocessedintheordertheyarriveatthebandwidthmanagementsystemandadecisiononwhetherornottomaketherequestedreservationmadeonthebasisoflinkstatesatthetimethereservationrequestisprocessedwithoutregardtofuturerequests),thelinkstatusatthetimeadecisionismadeonthecurrentrequestbeingprocesseddependsondecisionsmadeinthepast.Thesepastdecisionsareafunctionofthepathcomputationalgorithm(s)inuse.Supposethatxed-slotreservationrequests 34

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Figure3-4. Classicationofxed-slotalgorithms A,B,andCarriveatthebandwidthmanagementsysteminthisorder.RequestAmaybedeniedbyOSPF-TE(asOSPF-TEprovidesnoguarantee)andacceptedbyFP.Asaresult,thenetworkstatefollowingtheprocessingofrequestAisdifferentwhenthebandwidthmanagementsystemusesOSPF-TEforxed-slotreservationthanwhenitusesFP.Consequently,itisentirelypossiblethatOSPF-TEthenacceptsBandCwhileFPrejectsbothBandC.HencenetworkutilizationasmeasuredbythenumberofacceptedrequestsortotalnetworkbandwidththathasbeenscheduledmaybemoreusingOSPF-TEthatprovidesnoguaranteethanwhenusingFPthatprovidesaguarantee! Burchard[ 12 ]considerstwoutilizationmetricsrequestblockingratio(RBR)andbandwidthblockingratio(BBR),which,foranymeasurementinterval,aredenedas: RBR=numberofrejectedrequests totalnumberofrequests BBR=sumofbandwidth-durationproductsofrejectedrequests sumofbandwidth-durationproductsofallrequests Equivalently,wemayuserequestacceptance(RAR)andbandwidthacceptance(BAR)ratios,whicharedenedas: 35

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RAR=1)]TJ /F4 11.955 Tf 11.96 0 Td[(RBR BAR=1)]TJ /F4 11.955 Tf 11.96 0 Td[(BBR WenotethatBBRalsohasbeenusedinthecontextofon-demandscheduling(see[ 40 ],forexample). 3.2.4Experiments Foreachofthemaxbandwidth,maxduration,allslots,andallpairsallslotsproblems,onlyonealgorithmhasbeenproposedandsonorelativeeffectivenesscomparisonispossible.Fortherstslotproblem,therearethreealgorithmsSSW-MHFP,LSW,andEBF.All3guaranteetondtherstslotcorrectly.Hence,barringdifferencesresultingfromtheirpossibleimplementationusingdifferenttiebreakers,eachisjustaseffective.Ofcourse,therewillbesomedifferenceinthecomputer-timetakentoexecuteeachalgorithm(asindicatedinSection 3.2.3.2 ).Thereforeanexperimentalevaluationofeffectivenessisneededonlyforthevariousalgorithmsproposedforthexedslotproblem. WeprogrammedthevariousxedslotalgorithmsinC++andmeasuredtheeffectivenessofeachusingtheRARandBARmetrics.Althoughweexperimentedwithbothvariants(SDFP-minandSDFP-avg)ofSDFP,wereportonlytheresultsforSDFP-minastheretheresultsforbothvariantsarecomparable.ForOSPF,weprogrammedtheOSPF-TEvariantthatisusedinOSCARS[ 4 ].ThekDPvarianttestedbyusiskDP-LOADwithk=4.WeusedthisvariantasitisthevariantusedinEnlightened[ 18 ].Finally,forkSP,wesetkto4andused4shortestpaths. Fortestnetworks,weusedthetheAbilenenetwork[ 5 ],19-nodeMCInetworkandthe16-nodeclusternetworkof[ 40 ],the11-nodenetworkof[ 12 ],andseveralrandomlygeneratedtopologies.ThebackboneoftheAbilenenetworkusedbyushas11nodesasshowninFigure 6-5A andeachbackbonenodehasa5-nodestubnetworkattached 36

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toit.Thebandwidthofeachlinkis155Mbps.Thebandwidthofeachlinkinthenetworkof[ 12 ](Figure 3-5C )is100Mbps.Figures 3-5B and 3-5D givetheMCIandclustertopologiestogetherwiththelinkbandwidths.Therandomnetworkswetriedhad200,400,or800nodesandtheout-degreeofeachnodewasrandomlyselectedtobebetween3and7.Toensurenetworkconnectivity,therandomnetworkhasbidirectionallinksbetweennodesiandi+1forevery1i
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wererandomlyselectedusingauniformrandomnumbergenerator.ThetimeatwhichtherequestismadefollowedaPoissondistribution.Therequestedstarttimewassettobethetimeatwhichtherequestismadeplusarandomlygeneratedlag.Sincetheresultsarerelativelyinsensitivetothelagtime,wearbitrarilysetthemeanlagtimetobe100units. Forourexperiments,wehadthreecontrolparametersnumberofrequestsinthestudyinterval,meandurationofaschedulingrequest,andmeanbandwidthofarequest.Thestudyintervalwasarbitrarilysetto2000timeunits;thenumberofrequestsinthestudyintervalwassettooneofthevalues200,400,600,800,and1000fortherandomnetworksandtooneof100,200,300,400,and500fortheremainingnetworks;theallowablemeanrequestdurationswere200,400,600,800,and1000timeunits;andtheallowablemeanrequestbandwidthswere500,1000,1500,2000,and2500Mbpsfortherandomnetworksand10,30,50,70and90Mbpsfortheremainingnetworks.Foreachsettingofthe3controlparameters,weran10trials.Inthecaseofrandomnetworks,thenetworktopologywasrandomlyregeneratedforeachofthe10trials.Foreachtrial,wemeasuredtherequestacceptanceandbandwidthacceptanceratios(RARandBAR).Inreportingourresults,wecomputedtheaverageRARforallconductedexperimentswithagivennetworkandxedvalueforoneofthe3controlparameters.So,forexample,wecomputedtheaverageRARforthe250(5requestdurations*5requestbandwidths*10trials)experimentsdoneontheMCInetworkwiththenumberofrequestsinthestudyintervalbeing100. Sincetherelativeperformanceofthexed-slotalgorithmsisratherinsensitivetowhetherweusetheRARorBARmetric,wereportonlyontheRARresults.Figure 3-6 andFigure 3-7 givetheaverageacceptanceratiosforthexed-slotalgorithmsofSection 3.2.2.1 asafunctionofthenumberofrequestsinthestudyinterval.OnnetworkssuchastheMCI,Cluster,andBurchardnetworks,thathavearelativelysmallnumbernodes,thedynamicalternativefeasiblepathalgorithm(DAFP)givesbest 38

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performanceconsistentlyacrosstherangeofnumberofrequeststestedbyus.Theminimumhopfeasiblepathalgorithm(MHFP)isconsistentlysecondbest.However,onlargernetworkssuchastheAbilenenetwork,whichhas66nodes,andtherandomnetworksthathave200+nodes,MHFPisconsistentlysuperiortoDAFP.Forthetestedlargernetworks,MHFPisbestandDAFPissecondbest.Generally,thexed-slotalgorithmsOSPF,kDP,andkSPthatdonotguaranteetomakeareservationwhensuchareservationispossiblefaredworsethanthealgorithmsthatprovidesuchaguarantee.However,attimes,theperformanceofthebestnoguaranteealgorithmwasquiteclosetoorslightlybetterthanthatoftheworstguaranteealgorithm.Onournon-randomnetworks,OSPFconsistentlyhadtheworstperformance.However,onourrandomnetworks,kDPwasconsistentlyworstand,often,byquiteamargin.Asexpected,asthenetworkgetssaturated(i.e.,thenumberofrequestsinthestudyintervalincreases),theRARforallalgorithmsdeclinesandtherateofdeclineisaboutthesameforallalgorithms. AAbilene BMCI CBurchard DCluster Figure3-6. AcceptanceratiovsrequestsnumberinAbileneMCIBurchardandClusternetwork 39

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A200nodes B400nodes C800nodes Figure3-7. Acceptanceratiovsrequestsnumberinvariousrandomtopologies Figures 3-8 3-9 givetheaverageacceptanceratiosforthexed-slotalgorithmsofSection 3.2.2.1 asafunctionofthemeanrequestduration.Therelativeperformanceofthealgorithmsisthesameasforthecasewhenwexedthenumberofrequestsratherthanthemeanrequestduration. AAbilene BMCI CBurchard DCluster Figure3-8. AcceptanceratiovsmeanrequestsdurationinAbileneMCIBurchardandClusternetwork 40

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A200nodes B400nodes C800nodes Figure3-9. Acceptanceratiovsmeanrequestsdurationinvariousrandomtopologies Figures 3-10 3-11 givetheaverageacceptanceratiosasafunctionofthemeanrequestbandwidth.Therelativeperformanceofthealgorithmsisthesameasforthecasewhenwexedeitherthenumberofrequestsorthemeanrequestduration. AAbilene BMCI CBurchard DCluster Figure3-10. AcceptanceratiovsmeanrequestsbandwidthinAbileneMCIBurchardandClusternetwork 41

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A200nodes B400nodes C800nodes Figure3-11. Acceptanceratiovsmeanrequestsbandwidthinvariousrandomtopologies 3.3Multiple-PathScheduling 3.3.1ProblemDenition 2ConsiderascenarioofFusionExperiments[ 21 ],whichiscollaborationconsistingofresearchersindifferentEuropeancountries.Aftereachroundofexperiments,thesimulationdataaregeneratedindifferentsitesandprocessedbysupercomputersacrossthecontinent.Moreover,thedatatransferandprocessingtimeislimitedsincethenextroundofexperimentswillbeguidedbytheresultsfromthepreviousones.Thus,theletransfertimemaybecomeamajorbottlenecktoimprovetheexperiment'sefciency.Thedelayofanyletransferinthebatchwouldcausebiglosstothewholeproject.Inthispaper,wemodelsuchproblemastheBatched-FilePathSchedulingProblem(BFPSP),wherethegoalistominimizetheoveralltransfertimeofmultipleone-to-oneletransfers.Withoutlosinggenerality,weassumethatalltheletransferrequestsarepre-speciedbeforeschedulingstarts.Clearly,suchanalgorithmcanalsobeusedbybatchthenewlyarrivedrequestsatappropriateintervalswhentheyarrive.Inin-advancedscheduling[ 23 ],eachletransfermayhaveadifferentstarttime.Theirearlieststarttimecanbedenedbytheuseintheirrequests.Butactualstarttimeis 2Thissectionissubmittedthejournalofsupercomputingin2009andstillunderreview. 42

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decidedbytheschedulingalgorithmsatruntime.Inthispaper,weprovidebothoptimalandapproximatesolutionsforthisproblemandevaluatetheminmultiplescenarios. AllschedulingapproachescanbeconsideredasanvariationoftheBatchScheduling.Weassumethatallrequestsarecollectedasabatchinthescheduler;therequestsinabatcharescheduledasagroupwithcertainperiodicity.Obviously,ifalltheletransferrequestsarebatchedasonegroup,thesolutionisoptimal,whichisdenotedasAll-Batch.All-Batchisverytimeconsumingforlargebatchsizes.Also,itisnotrealisticasthearrivaltimeoftherequestsmaynotbethesame.WepresentanumberofheuristicsthathavemuchlowertimecomplexitythanusingAll-Batch,buthavesimilarperformanceandtheaddedbenetthatalltherequestsmaynotbeknownbeforehand.Theproposedapproach,N-Batch,groupsrequestsintobatchofconstantsizeNandscheduleseachbatchseparately.Forthespecialcaseofbatchsizeequalto1,theschedulingisequivalenttoOnlineScheduling.Forthiscase,wedeveloptwosetsofheuristics:GOSandk-Path. Wehavecomparedallthesealgorithmsforavarietyofscenariosandperformancemetrics.OursimulationsshowthatbothN-BatchandGOSprovidesschedulesthatarecomparableinqualitytousingAll-Batch,butrequiresignicantlylessschedulingtimethanusingAll-Batch.GOSiscomparabletoN-Batchbutrequiressignicantlylesscomputationtime.WealsoinvestigateGOS-Ealgorithmsthatminimizepathswitchingoverhead,whichisavariantofGOS.GOS-Eisknowntohavegoodperformancewhenpathswitchingoverheadcannotbeignored. 3.3.1.1Datastructures RecallthedatastructureTime-BandwidthList(TBList)introducedinChapter2.LetT=[T0,T1,],T0
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example,assumethesearetheonlytwolinksinthenetwork.TheTBlistfortherstlinkis[(0,5),(1,2),(2,5)]andthatforthesecondlinkis[(0,5),(1.5,3),(2,5)].Theglobaltimelistforourexampleis[0,1,1.5,2].Intheinterval[0,1),theavailablebandwidthonthetwolinksis5whereasintheinterval[1,1.5),therstlinkhasanavailablebandwidthof2whilethesecondlink'savailablebandwidthis5,andneitheroflinks'bandwidthchangeswithinthisbasicinterval. Figure3-12. Basicintervals Theintervals[T0,T1),[T1,T2),intheglobaltimelistarereferredasbasicintervals.Atanytimewithinacertainbasicinterval,eachedgehasaconstantamountofavailablebandwidth.Basicintervalsobtainedfromtheglobaltimelistcanbeorderedusingtherelationship[a,b)<[c,d)iffbc(notethatthebasicintervalsofaglobaltimelistaredisjointandthata
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Step1: Determinetheminimumnishtime,minFinishTime. Step2: Determinealetransferschedulethatachievesthisminimumnishtime. Tondouttheminimumnishtime,weconstructaglobaltimelistfromtheTBlistsofalllinksasbeforeandthenconstructthebasicintervalsfromthisglobaltimelist.Thebasicinterval[Ti,Ti+1)isreferredtosimplyasbasicintervali.Todeterminetheminimumnishtime,weusealinearprogramming(LP)modeltodetermine,foraspeciedbasicintervali,theminimumtimewithinthisbasicintervalbywhichitispossibletocompleteallletransfersinthegivenrequestsetF.ThisLPmodelwillhavenofeasiblesolutionforbasicintervalsiifitisn'tpossibletocompletetheletransferbytimeTi+1.Inthiscase,minFinishTimemustlieinabasicintervalq>i.SupposethevalueofLP'sobjectivefunctionftisavalidtimewithinthebasicintervali.ThenallthejobsinthebatchFcanbenishedbyft.Now,TiftTi+1.Ifft>Ti,minFinishTime=ft.However,whenft=Ti,itispossibletocompletetheletransfersinanintervalq
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minft (3) subjecttoXk:(l,k)2Efjlk(q))]TJ /F13 11.955 Tf 21.34 11.35 Td[(Xk:(k,l)2Efjkl(q)=08j2F,8l2V,l6=sj,l6=dj,0qi (3) iXq=0(Xk:(l,k)2Efjlk(q))]TJ /F13 11.955 Tf 21.34 11.36 Td[(Xk:(k,l)2Efjkl(q))=8><>:fjifl=sj)]TJ /F4 11.955 Tf 9.29 0 Td[(fjifl=dj8j2F (3) Xj2Ffjlk(q)blk(Tq)(Tq+1)]TJ /F4 11.955 Tf 11.96 0 Td[(Tq),8(l,k)2E,q
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transferamountsarenon-negativeinpermissiblebasicintervalsandEquation 3 ensuresthattheletransferamountsare0innon-permissiblebasicintervals. OnecanverifythateachsolutiontoEquation 3 through 3 denesavalidletransferscheduleforallrequestsinFandthatthenishtimeofthisscheduleisatmostft.Further,theinclusionofEquation 3 determinestheminimumnishtimeundertheconstraintthatnoletransfermaytakeplaceinintervalsq>i.Also,Equations 3 through 3 havenofeasiblesolutionifftheletransferscannotbescheduledsoastocompletebytimeTi+1. Asnotedabove,sincefractionalowisallowed,minFinishTimeispolynomiallysolvable[ 6 ].AbinarysearchoverthebasicintervalsisneededtodeterminetheintervalwhereminFinishTimeislocatedandalsoexactvalueofminFinishTime.ThisrequiresustosolveO(logB)LPs,whereBisthenumberofbasicintervals. Althoughthefjlk(q)sthatdetermineminFinishTimedenealetransferschedulethatachievesthisnishtime,thesefjlk(q)smaydeneatransferschedulethatincludescycles.Thatis,wehaveportionsofalebeingmovedfromnodeatonodebandbacktonodea,forexample,inthesamebasicinterval.Whilethesecyclicowsdonotnegativelyimpacttheoverallnishtime,theyaffectavailablebandwidthcapacityandsonegativelyimpactourabilitytoscheduleletransfersinfutureperiods. InStep2,weovercomethedecienciesoftheletransferscheduleobtainedfromStep1byusingaslightlydifferentLPformulationthatisgivenbyEquations 3 through 3 .Inthisformulation,weminimizesthesumofthefjlk(q)valuesacrossallbasicintervals.ThevalueU=minFinishTimecomputedinStep1isusedtolimittheletransfers'startandendtimes.WealsouseitodenotethebasicintervalforwhichTiminFinishTimeTi+1.ItisobviousthatthesolutiontoEquations 3 through 3 maycontainnocycle,oritcannotbeoptimal,sincewecanalwaysremovecyclesandproduceabettersolution. 47

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minXj2JX(l,k)2EiXq=0fjlk(q) (3) subjecttoXk:(l,k)2Efjlk(q))]TJ /F13 11.955 Tf 21.33 11.36 Td[(Xk:(k,l)2Efjkl(q)=08j2F,8l2V,l6=sj,l6=dj,0qi (3) iXq=0(Xk:(l,k)2Efjlk(q))]TJ /F13 11.955 Tf 21.34 11.36 Td[(Xk:(k,l)2Efjkl(q))=8><>:fjifl=sj)]TJ /F4 11.955 Tf 9.3 0 Td[(fjifl=dj8j2F (3) Xj2Ffjlk(q)blk(Tq)(Tq+1)]TJ /F4 11.955 Tf 11.95 0 Td[(Tq),8(l,k)2E,qi (3) fjlk(q)0,[Tq,Tq+1][Sj,U],8(l,k)2E,8j2F (3) fjlk(q)=0,[Tq,Tq+1]*[Sj,U],8(l,k)2E,8j2F (3) WenotethatwhiletheLPofEquations 3 through 3 issolvedO(logB)(Bisthenumberofbasicintervals)times,theLPofEquations 3 through 3 issolvedonlyonceasaminimum-costowproblem.UsingtheSuccessiveShortestPathalgorithm[ 6 ],thisowproblemcanbesolvedwithO(Elog(U)),whereEisnumberoflinksinthenetworkwhileUisthelargestamountofow.Thejustdescribedtwo-stepbatchschedulingalgorithmisreferredtoasalgorithmAll-Batch. 3.3.2.1N-Batchheuristics AswewillseeinSection 3.3.4 ,thecomputingtimerequiredtocomputetheoptimalscheduleusingalgorithmAll-Batchisveryhigh.OnewaytodecreasethecomputationtimeisthedividethesetofletransfersintosmallerbatchesofsizeNandprocessthemonebyone.WhenN>1,thecorrespondingheuristiciscalledN-Batch.ThesolutionforN-Batchisasfollows.Thebatchesareprocessedoneatatimesequentiallyinanincreasingorderofthebatch'scollectingtimes.Whencomputingtheoptimalscheduleforagivenbatch,thestarttimeforthatbatchisgivenbytheendtime(thetimeofthe 48

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lastscheduledjob)ofthepreviousbatch.Theoverallnishtimeisthenishtimeofthelastbatchscheduled. Asweusethegreedyapproachtoprocessalltherequests,oneofthekeyissuesistodecidethegreedyselectioncriterion.[ 9 ]suggestedthatLargestFileFirst(LFF)isareasonableandquiteeffectiveheuristictoselecttherequest(s)greedily.Thisapproachisbasedontheintuitionthatthelargerleswilltakemoretimetotransfer.WhenschedulingthelargestNlesrst,thelargerlesaregivenmorepriorityinthetheresourcecontention,whichresultsinapotentiallyearliernishtimeforthelargeletransfers.Sincetheselongtransfersactuallydeterminetheoverallnishtimeoften,thisheuristicisexpectedtoimprovetheoverallnishtime.ThisexpectationisborneoutinourexperimentaalevaluationandtheobservednishtimesareclosetothoseoftheoptimalsolutionsgeneratedbyAll-Batch. 3.3.3OnlineSchedulingAlgorithms Whenthebatchsizeequals1,theBFPSPturnsintoaninstanceofOnlineScheduling,whereallletransfersarescheduledonebyonewithoutusinganyknowledgeofthetransfersscheduledlaterinthesequence. Weproposesixonlineletransferschedulingalgorithms.InSections 3.3.3.1 and 3.3.3.2 ,wedescribetheGOSandGOS-Ealgorithms.Theremainingfouralgorithmsarevariationsofthek-PathheuristicsandaredescribedinSection 3.3.3.3 .Thegreedyalgorithm,GOS,employsnetworkowstominimizethenishtimeofeachsingleletransferbeingscheduled.GOS-Econsidersthepathswitchingoverhead.The4k-Pathheuristicsusethethek-shortestpathsork-disjointpathstocomputethescheduleonasmallernetworkthantheoriginalone.Theseadaptationsreducethecomplexityoftheschedulingalgorithm,butyieldlittleinmaximumnishtime. 3.3.3.1Greedyalgorithm Ourgreedyonlinealgorithm,GOS,schedulesaletransfer(si,di,fi,Si)byexaminingthebasicintervalsinthenetwork'scurrentglobaltimelistinincreasing 49

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order.TheexaminationbeginswiththebasicintervalthatincludesthetimeSi.Ineachexaminedinterval,wetransferasmuchoftheleasispossible.Thismaximumamountcanbedeterminedusingamax-owalgorithm(see[ 6 ],forexample).Theexaminationofbasicintervalsstopswhenallfibytesofthelehavebeenscheduled.Figure 3-13 givestheprocedureofourgreedyonlinealgorithm,GOS,toscheduletheithrequest.Inthespecicationofthisalgorithm,weconstructareducedgraphNfromGwithonlythelinkshavesomeavailablebandwidthincurrentbasicinterval.weusethetermmaxowlinkstodenotethoseedgesofNthathaveanon-zeroowinthemaxowsolutionforN.Also,notethatmaxFlowmaybezeroinsomebasicintervalsandcareneedstobetakenwhenprogrammingalgorithmGOStoavoidadividebyzeroerrorwhencomputingrfs=maxFlow. Theorem3.1. IfGhasapathfromsitodi,thenAlgorithmGOSschedulestheithletransferrequest(si,di,fi,Si)soastocompleteattheearliestpossibletime. Proof. Fromthefollowingfacts(a)Ghasapathfromsitodi,(b)theratedcapacityofeachlinkofGismorethan0,(c)thelastbasicintervaloftheglobaltimelistalwaysextendsto1,and(d)theavailablebandwidthofeachlinkisitsratedcapacityduringthislastbasicinterval,itfollowsthatthemaxowfromsitodiinthelastbasicintervalisnon-zeroandsotheremaininglesizerfscanalwaysbescheduledfortransferinthislastbasicinterval.Hence,GOSisabletoscheduleeveryletransferrequest. LetthenishtimeofaletransferscheduleconstructedbyGOSbeft.NotethatftisthevalueofmaxTimewhenGOSterminates.Weshow,bycontradiction,thatftistheearliestpossibletimeatwhichthisletransfercancomplete.Supposethereisanothertransferschedule,S,forthesamerequestthatcompletesthetransferbytimeft0
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GOS(i,G) { ConstructthecurrentglobaltimelistTfromthe TBlists; DeletefromTallTiSi; InsertSiand1intoTandrelabelthemembersofTin ascendingorderbeginningwiththelabelT0; rfs=fi;//remaininglesize j=0;//basicintervalindex; while(rfs>0) { LetNbethenetworkderivedfromGbyassigning toeachlinkacapacityequalstoitsavailable bandwidthinthebasicinterval[Tj,Tj+1), Removethelinkswith0capacityfromN; maxFlow=MaxowfromsitodiinN; maxTime=minfTj+rfs=maxFlow,Tj+1g; size=(maxTime)]TJ /F4 11.955 Tf 11.96 0 Td[(Tj)maxFlow; SchedulethetransferofsizebytesfromTjto maxTimeusingthemaxowlinks; UpdatetheTBlistsofthemaxowlinks; rfs)]TJ /F6 11.955 Tf 12.62 0 Td[(=size; j++; } } Figure3-13. GreedyonlineschedulingalgorithmGOS theGOSschedule.Thisisn'tpossiblesincetheGOSscheduletransfersthemaximumpossibleamountineachbasicintervalpriortoq.Ifq0=q,thensinceft0
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nisthenumberofverticesinthenetworkowgraph.Fornetworkswithfewedges,thesparsegraphnetworkowalgorithmofSleatorandTarjan(see[ 6 ],forexample)maybeused.ThecomplexityofthisalgorithmisO(nmlogn),wheremisthenumberoflinksinthenetwork.Whenschedulingtheithletransfer,thesizeoftheglobaltimelistisatmost2i,sinceeachpreviouslyscheduledrequestwillincreasethesizeofglobaltimelistbyatmost2:job'sstartandendtime.So,thecomplexityofGOSisO(n3i)whenthepush-relabelmaxowalgorithmisusedandO(nmilogn)whenthesparsegraphmaxowalgorithmisused.SincetypicalcomputernetworksaregenerallysparseandhaveonlyO(n)links,usingthesparsegraphmaxowalgorithmresultsinacomplexityofO(n2ilogn)forGOS. 3.3.3.2Greedyschedulingwithnishtimeextension(GOS-E) IntheGOSalgorithm,thenetworkowiscomputedforeachbasicinterval.Thisimpliesthat,foraletransferthatlastsnconsecutivebasicintervals,uptonestablishingandtearingdownoperationsonowpathwouldtakeplaceinthenetwork.Moreover,giventhefactthatmulti-pathsarerequiredforeachow,thepathswitchingoverheadwouldsignicantlyaffecttheGOS'sperformanceinpractice. Todecreasetheswitchingoverhead,GOS-Eisproposedtoreducethenumberofpathswitchingsbyreducingthetotalnumberofbasicintervalsinthenetwork.InGOS-E,wetriedtoextendthecurrentjob'snishtimetothetheendofnearestlaterbasicintervalti,iftiisnottoofarawayfromft,whichistheearliestnishtimecomputedfromGOS.Theextensioncanbedonebyeitherdirectlyover-reservingthebandwidthinthelastbasicintervalinvolvedintheletransferaccordingtotheoriginalreservationplanfromGOS,orreducetheamountofrequiredbandwidthinthelastintervaltocopewiththelongertransfertime,whichwillnotwastebandwidth.Asbandwidthresourcesarelimitedinourscenario,wetakethesecondapproach.Theextensionscopeshouldbelimitedtoacertainrangesothattheperformanceonasingleletransferisnotgreatlyaffected. 52

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WithGOS-E,weareabletoeliminatethesmallbasicintervalsbymergingthemintothepreviouslargeintervalsbyreducingitslinkbandwidth.Assmallintervalsgenerallyperformsmalleramountofletransfersthanlargeintervals,thismergingprocessactuallycostslittleadditionalnetworkthroughputbutprovidesthepotentialtoreducetheoverallpathswitchingoverhead.Intheevaluationsection,wewilltestthisheuristicandcomparewiththeoriginalGOSalgorithm,Also,therelationshipbetweentheextensionscopeandalgorithmperformanceisdiscussed. 3.3.3.3K-Pathalgorithms Anotherapproachtoacceleratethealgorithmistoreducetheproblemsize.WeincorporatetheideabehindKSP[ 69 ]andKDP[ 45 ]schedulingintoalgorithmGOSsoastocomputethemax-owinareducednetwork.IntheKSPandKDPadaptations,whenschedulingtherequest(si,di,fi,Si),welimitourresourceallocationstoasubgraphdenedbythekpathsfromsitodi.InthecaseoftheKDPadaptation,sincethekpathsaredisjoint,themaxowfromsitodiinanybasicintervaliseasilyseentobethesumoftheminimumavailablecapacityofalinkoneachofthekpaths.So,weavoidrunningacomplexnetworkowalgorithmtodeterminethemaxow.InthecaseoftheKSPadaptation,sincethepathsarenotdisjoint,westillneedtoruntheGOSalgorithmonthenetworkformedbythesekpaths.However,sincethesizeofthenetworkbeingconsideredissmaller,runtimeisreduced. ForboththeKSPandKDPadaptations,wedeneastaticandadynamicvariant.Inthestaticvariantthecostofalinkisdenedtobeitsratedcapacity(alternatively,someothernon-changingcostmaybeassigned)andthekpathsbetweeneverypairofnodes,whetherdisjointornot,arecomputedonceatthersttimearequestarrivesforthispairofnodesandusedirectlyfortheschedulingrequestbetweenthesamesource/destinationpairafterward.Inthedynamicvariant,linksareassignedacosteachtimeaschedulingrequestarrivesandthekshortestpathstousearecomputedusingthesenewlyassignedlinkcosts.Thecostassignedtoalinkinthedynamicvariantis 53

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proportionaltothefractionofitsratedcapacitythathasbeencommittedfromthecurrenttimetothenishtimeofthelastnishingletransfersofarscheduledinthenetwork.Inbothstaticanddynamicvariants,thelengthofapathisthesumofthelinkcosts.ThestaticanddynamicvariantsoftheKSPandKDPadaptationsofGOSarereferredtoasKSP-S,KSP-D,KDP-S,andKDP-D,respectively. 3.3.4ExperimentalEvaluation 3.3.4.1Experimentalframework Inthissection,wemeasuretheperformanceofthebatchschedulingalgorithminSection 3.3.2 andonlineschedulingalgorithmsinSection 3.3.3 .Forourexperiments,weusedtheMCInetworkandrandomtopologiesthatgeneratedatruntime. Filetransferrequestsweresyntheticallygenerated.Eachrequestisdescribedbythe4-tuple(sourcenode,destinationnode,lesize,requeststarttime).Thesourceanddestinationnodesforeachrequestwereselectedusingauniformrandomnumbergenerator.Thelesizeisalsouniformlydistributedbetween10GBand100GB.TheearliesttimeatwhichaletransfercanstartfollowedaPoissondistributionandtherequestarrivalrate(requestdensity)variedfrom0.05requests/timeunitto10requests/timeunit.Ourexperimentsstartedwithacleannetwork(i.e.,noexistingscheduledtransfers)andsimulatedthejobarrivalprocessfor100timeunits.So,forexample,witharequestdensityof5requests/timeunit,onerunofourexperimentwouldprocessapproximately500requests. Weusedthemaxnishtime(MFT),i.e.thetimewhenallletransfersinthesequencenishastheperformancemetric.Theexecutiontimeofanalgorithmismeasuredinseconds.ForGOS-E,theextensionscopeissettobeeither2%,5%or10%ofthecurrentletransfer'sduration.SupposethattheletransferJi,withstartandendtimeSiandEi,respectively,isbeingscheduled.SupposethatEi'snearestfuturebasicintervalendsatT.Then,theextensionofEitoTisperformedwhen(T)]TJ /F4 11.955 Tf 11.95 0 Td[(Ei)(Ei)]TJ /F4 11.955 Tf 11.95 0 Td[(Si)ExtensionScope. 54

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FortheKSPandKDPvariantsofGOS,wesetk,thenumberofpaths,to16.Thissettingisconsistentwiththeresultsof[ 45 ],whichshowthatnotmuchimprovementcanbeobtainedfromusingahigherk,andisalsoconsistentwithourownexperimentsonschedulingletransfersthatindicate16tobeagoodchoicefork. 3.3.4.2Singlestarttimescheduling(SSTS) AspecialcaseofBFPSPhasalllesreadyfortransferatthesametime,whichmeansallrequestsinthebatchhavethesameSi.Thisspeciaalcasearises,forexamp[le,whenallletranferrequestsoriginatefromasinglegroupofusers.Forthisexperiment,wesetSi=0foralltransferrequests.Thenumberoflestobescheduledvariesfrom200to1000.Thesizeoftherandomnetworkvariesfrom100nodesto500nodes. Figure3-14. Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesinMCIusingSSTS. Figure3-15. Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesin100nodesrandomtopologyusingSSTS. 55

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Figure3-16. Comparisonofdifferentalgorithms'MFTrandomTopologiesofdifferentsizeusingSSTS. Figures 3-14 and 3-15 showthemaximumnishtimeoftheschedulesforvariousnumberoflesontheMCInetworkanda100-noderandomnetwork,respectively.AllthealgorithmsproposedinSections 3.3.2 and 3.3.3 arecomparedexceptforGOS-E.ThemainobjectiveofGOS-Eistoreducethepathswitchingoverhead.Thisisnotaddressedbytheotheralgorithms.Figure 3-16 showshowtheschedulingresultschangeasnetworksizeincreasesandthenumberofrequestsisxedat400.Wemakethefollowingobservations. a. Batchschedulingperformsbetterthanonlineschedulinginallcases.Thelargerthebatchsize,thebettertherelativeperformanceofbatchscheduling.Inlargenetworkslikerandom-100,largerbatchsizehassignicantlylargerimprovementsontheoverallperformance;however,insmallnetworkslikeMCI,theimpactofbatchsizeisrelativelysmall. b. Whenbatchsizeis50,N-Batchperformsaswellastheoptimalbatchscheduleinmostcases.GOShasthebestperformanceamongonlineschedulingalgorithmsinallcases.Insmallnetworks,GOSperformsalmostaswellastheoptimalsolution.Inlargenetworks,theimprovementacheivedbyusingAll-Batchisusuallynomorethan5%overGOS. c. Amongthe4k-Pathheuristics,twoKDPheuristicsperformbetterthantheKSPheuristicsanddynamiclinkcostcanimprovetheperformance.Thiscanbepartlyattributedtothecongestionavoidancemechanismthatdynamiclinkcostprovided.Withoutdoubt,KDP-Dprovidesthebestperformanceamongthe4k-Pathheuristics.AlthoughnotcomparabletoGOSalgorithmforlargenetworks,theirperformancegapissmallenoughtobeacceptableforsmalltopologies. 56

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d. TheMFTincreasesasthenumberoflesincreases.Theamountofthisincreaseisfasterinsmallnetworksthaninlargeones,assmallnetworksarebecomecongestedandfullyloadedsooner.Ontheotherhand,MFTdecreasesasthenetworksizeincrease,asthenetworkprovidesmorebandwidthresourcesinaglobalview. Insummary,theAll-Batch'sadvantageoverN-BatchandGOSisnotobvious,especiallyinsmallnetworks.WhenthebatchsizeNreaches50,theperformancegapisrelativelysmallandcanbeignoredinmanypracticalscenarios. Figure3-17. Comparisonofdifferentalgorithms'executiontimeondifferentnumbersoflesonMCInetworkusingSSTS. Figure3-18. Comparisonofdifferentalgorithms'executiontimeondifferentnumbersofleson100nodesrandomtopologyusingSSTS. Weweremotivatedtoexploreonlineschedulingheuristicsbythedesiretoreducethecomputingtimerequiredbythescheduler.Ourheuristicssolveaneasiermax-owproblemratherthanthecomplexandtimeconsumingLPformulationsolvedbyAll-Batch.Figures 3-17 and 3-18 showtheexecutiontimefortheMCInetworkandfora 57

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randomnetworkwith100nodes,respectively.Thehorizontalaxisisthenumberoflestobescheduled.Theexecutiontimeismeasuredinseconds. Inallcases,theexecutiontimeofouronlinealgorithmsismuchlessthanthatofthebatchalgorithm.Insmallnetworks,theschedulingtimeisacceptableforallalgorithms.Lessthan2minutesaretakentoobtaintheoptimalscheduleinMCIfor1000lerequests;theonlinealgorithmstakeseveralseconds.Inlargenetworks,ouronlineheuristicsaredramaticallyfasterthanthebacthschedulingalgorithm. Figure3-19. Comparisonofdifferentalgorithms'MFTexecutiontimeonrandomtopologieswithdifferentsizeusingSSTS. Figure 3-19 showsthealgorithms'executiontimeforvariousnetworksizes.Duringthetest,wescheduled400jobsusingSSTS.Weobservedthat,althougheveryalgorithms'executiontimeincreaseswithnetworksize,thetimerequiredbyAll-BatchandN-Batchactuallyincreasemuchfasterthanthatrequiredbyouronlinealgorithms.Thisisduetothelowercomplexityoftheonlinealgorithms.Inthe500nodetopology,GOSonlytakesseveralminutes,butAll-Batchtakesabout2hourstocomputetheoptimalschedule,whichexceedstheactualletransfertimeinourexperiments. 3.3.4.3Multiplestarttimescheduling(MSTS) BFPSPactuallydoesnotrequirealltheletransferstostartatthesametime.Whenjobsstartatvarioustimes,thetotaltrafcloadisexpectedtobelessintensivethantheSSTScase,sincetheletransfersarelessoverlappedduetothedifferenceintheirstarttime. 58

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Figure3-20. Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesinMCIusingMSTS. Figure3-21. Comparisonofdifferentalgorithms'MFTfordifferentnumberoflesin100nodesrandomtopologyusingMSTS. Figures 3-20 to 3-25 givetheexperimentalresultsusingMSTS.Figures 3-20 3-21 and 3-22 showhowthemaximumnishtimechangeswithnumberofrequestsandnetworksize.Figures 3-23 3-24 and 3-25 showthealgorithmsscalabilitywithincreasingnumberofrequestsornetworksize.TheobservationsfromMSTSaresimilar Figure3-22. Comparisonofdifferentalgorithms'MFTinrandomtopologiesofdifferentsizeusingMSTS. 59

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Figure3-23. Comparisonofdifferentalgorithms'executiontimeinMCInetworkusingMSTS. Figure3-24. Comparisonofdifferentalgorithms'executiontimein100nodesrandomtopologyusingMSTS. tothoseforSSTS.ThebatchalgorithmsoutperformonlineonesintermsofMFTvalue,butrequiresignicantlylongercomputationtime.Thebatchalgorithmsalsoshowworsescalabilitywithbothnetworksizeandnumberofrequests.GOSperformsbetterthanallotheronlineheuristicsandprovidesverygoodscheduleswithverysmallexecutiontime. Figure3-25. Comparisonofdifferentalgorithms'executiontimeonrandomtopologieswithdifferentsizeusingMSTS. 60

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3.3.4.4GOSv.s.GOS-E Whenpathswitchingoverheadisconsidered,theGOS'spotentialtoswitchpathforeverybasicintervalcanpossiblybecomeitsmajordrawback.Thisproblemismoreseverewhentheleistransferredusingmultiplepaths.Inthissection,GOS-EisevaluatedandcomparedwiththeoriginalGOSalgorithm. Whenperformingthetest,oneoftheimportantissuesistosimulatethepathswitchingoverhead.Fromtheprevioussection,weknowthatpathswitchinghappensonlybetweentwoadjacentbasicintervals.Soweaddalagoftlinoursimulationbetweeneachbasicintervaltorepresentthedelayduetopathswitching.Inourexperiments,wesetthedelaytobe1secondforestablishingandtearingdownanopticalpath.Thatis,ifthescheduleforacertainletransferchangesitsroutesmtimesduringthetransfer,adelayof2msecondswillbeaddedtoitsnishtime. Figure3-26. ComparisononnumberofMax-FlowsiscomputedbyGOS-EandGOSin100noderandomnetwork. ThetestisperformedonGOSandthreeGOS-Evariantswithdifferentextensionscopes:2%,5%and10%,whichmeansGOS-Ecansearchforthenextexistingbasicintervalandextendthecurrentscheduleintherangeof2%,5%and10%,respectively.Figures 3-26 and 3-27 showtheMFTperformanceforallfouralgorithms,withandwithoutpathswitchingdelaycounted.Wecanseethatwhenthepathswitchingdelayisnotaccountedfor,theschedulesgeneratedbyGOS-Etakemoretimetocompletetheletransfers,butwhenthepathswitchingdelayisaccountedfor,GOS-Eactually 61

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Figure3-27. ComparisononexecutiontimeusedbyGOS-EandGOSin100-noderandomnetwork. Figure3-28. ComparisononnumberofMax-FlowsiscomputedbyGOS-EandGOSin100noderandomnetwork. outperformsGOS.WealsonoticethatalargerextensionscopedoesnotimplybetterMFT,asGOS-Ewith5%extensionscopegeneratesbetterschedulesthanwithanextensionscopeof10%.Althoughtheswitchingoverheadcanbereducedusingalarger Figure3-29. ComparisononexecutiontimeusedbyGOS-EandGOSin100-noderandomnetwork. 62

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extensionscope,theactualletransfertimeincreases,whichcompromisesthebenetoffewerpathswitchings. Figure 3-28 comparestheaverageroundsofmax-owcomputationforeachrequestinGOS-EwithGOS,whileFigure 3-29 comparestheirexecutiontimes.Theseresultsshowthatbacauseofthereductioninthenumberofmax-owcomputationrounds,theexecutiontimeforGOS-EislowerascomparedtotheGOSalgorithm.Also,theexecutiontimeisfurtherreducedwithlargerextensionscopesalbeitatapriceoflongerMFT.However,inlargenetworks,thisdegradationisconsiderablysmall.ThismakesGOS-Emoreattractiveforlargenetworks. 3.4GeneralNetworkSchedulingAlgorithmsSummary Inthischapter,wehavecategorizedthedifferentbandwidthalgorithmsthathavebeenproposedforin-advanceschedulingbyboththeproblembeingaddressed(xedslot,maximumbandwidthinslot,maximumduration,rstslot,allslots,all-pairsall-slots).Inaddition,wehaveproposedseveralnewalgorithms(SWFP,SDFP,DAFP,kDP,andkSP)forthexed-slotproblemthatareadaptationsofalgorithmsproposedearlierforon-demandscheduling.AlthoughtheDAFPalgorithmproposedbyusisanadaptationofthedynamicadaptivepathalgorithmproposedforon-demandschedulingin[ 40 ],DAFPguaranteestondafeasiblepathwheneversuchapathexistswhereasthedynamicadaptivepathalgorithmof[ 40 ]doesnotprovidesuchaguarantee.Wehaveconductedextensiveexperimentswiththevariousxed-slotalgorithmsforin-advancescheduling.Ourexperimentsindicatethattheminimum-hopfeasiblepathalgorithmproposedbyusin[ 52 ]isthebest(inthesenseofmaximizingnetworkutilization)oftheseonlargenetworks.Fornetworkswithasmallnumberofnodes(say20orless),theDAFPalgorithmproposedinthispaperisbest.Fromthestandpointofalgorithmiccomplexity,MHFPisconsiderablyfasterthanDAFP. Wehavealsodevelopedseveralmulti-pathreservationalgorithmsforin-advanceschedulingofsingleandmultipleletransfersinconnection-orientedopticalnetworks. 63

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Anoveltwo-stepsolution,All-Batch,hasbeendevelopedtocomputescheduleswithminimumnishtime(i.e.,optimalschedules).AnN-Batchheuristicwasdevelopedtoenablebatchschedulinginmorerealisticscenarios.Wealsoproposedanewmax-owbasedgreedyalgorithm(GOS)andfourvariantsofk-pathalgorithmstoreducecomputationtime.Theseheuristicsscheduleanindividualletransfertocompleteattheearliestpossibletime.ExtensivesimulationsusingbothrealworldnetworksandrandomtopologiesshowthatGOSpresentsagoodbalanceamongmaximumnishtime,meannishtime,andcomputationtime.Furtherreductionincomputationtimebysacricingmaximumnishtimemaybeobtainedusingourk-Pathvariants.Ofthese,KDP-Dworksbest.Whenpathswitchingoverheadisconsidered,GOS-Eprovidesgoodperformance. Inthefuture,itwouldbeusefultoexploreBATCHalgorithmsthatalsoincorporatethemeannishtimeintheoptimizationmetric.Also,whenswitchingtheletransferstothescenarioofOn-demandscheduling,thedelaycausedbythebatchingprocessmustalsobetakenintoaccount.Thereforestudyingthetradeoffbetweenthebatchsizeandthedelayincurredisofinterest. 64

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CHAPTER4SCHEDULINGINOPTICALNETWORKS 4.1ProblemDenition Dedicatedconnectionsareneededtoeffectivelysupportavarietyofgeographicallydistributedapplicationtasks.Manyhighspeednetworksthatprovidededicatedconnectionsarebasedonopticalinterconnectsandopticalswitches.Forthesenetworks,thebandwidthalongagivenlinkcanbedecomposedintomultiplewavelengths.ThebandwidthschedulingandpathcomputationprobleminthecontextofopticalnetworksisusuallycalledRWA(RoutingandWavelengthAssignment)[ 71 ].AlgorithmsforRWAmayhavetoadheretooneormoreofthefollowingconstraints: 1. Wavelengthcontinuityconstraint:Thisconstraintforcesthatasinglelightpathmustoccupythesamewavelengththroughoutallthelinksthatitspans.Thisconstraintisnotrequiredwhenanopticalnetworkisequippedwithwavelengthconverters.Whensuchconvertersarepresent,thenetworkiscalledwavelengthconvertiblenetwork.Thealgorithmspresentedinthispaperassumethateitherwavelengthconversionisavailableatallswitchesornotavailableatanyswitch. 2. Wavelengthsharingconstraint:Formanydeployments,itismosteffectivetoconsiderthebandwidthonalinkasconsistingofintegermultiplesofwavelengthandasinglewavelengthasaunitforassignmenti.e.,onewavelengthisoccupiedbyonlyonereservationatacertainpointoftime.Thealgorithmsinthispaperassumethatthisconstraintneedstobesatised.ItisworthnotingthattechniquesbasedonTimeDivisionMultiplexing(TDM)/WavelengthDivisionMultiplexing(WDM)[ 73 ]allowfordecomposingthebandwidthonawavelength. TheEnlightenedproject[ 18 ]hasdevelopedseveralroutingalgorithmsforopticalnetworksassumingwavelengthconvertibilityandnowavelengthsharingconstraints.TheyaredevelopedusingtheFlexibleAdvanceReservationModel(FARM)[ 58 ]thattriestoreducetheblockingprobabilityofrequestsbyassigningaschedulingwindowforeachrequest[ 27 ].ThealgorithmsdevelopedintheEnlightenedprojectcanbetermedaskDynamicPaths(kDP)algorithmsaccordingtotheclassicationof[ 32 ].Thesealgorithmsdonotguaranteedtondafeasiblepathwheneversuchapathispresent. Thewavelengthassignmentproblemisarelativelyorthogonalproblemfromtheroutingproblemandmanyheuristicshavebeendevelopedforitssolution[ 71 ].Forour 65

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experimentalcomparisons,weusethewavelengthassignmentpolicyasproposedforthatgivenroutingalgorithm. InSection 4.2 ,weextendedtheExtendedBellman-Ford(EBF)algorithmtoincorporatethewavelengthsharingandwavelengthcontinuityconstraints.WealsoproposemodiedversionsofthealgorithmsintheEnlightenedsystemforcontinuoustimemodel.ThesealgorithmsarecalledModiedSwitchPathFirst(MSPF)andModiedSwitchWindowFirst(MSWF).MSPFtriestondtheearliestpathwithintheschedulingwindow,whileMSWFtriestondtheshortestpathwithintheschedulingwindow.Moreover,adeferredwavelengthassignmentstrategyispresented.Thisstrategyonlycountsthenumberofwavelengthsthatareusedonalink.Theactualassignmentofthewavelengthisdonewhentherequestisactuallyfullled. Althoughthedesignandimplementationofafull-conversionschedulerisrelativelystraightforward,thehighcostandtheaddedlatencyintroducedbywavelengthconvertersmakesparse-conversionmoreattractiveinpractice.ExistingresearchonOn-Demandschedulinghasshownthatwavelengthconvertershavethepotentialtoimproveblockingperformancesignicantlyandthatitisnecessaryonlyforarelativelysmallfractionofthenodestohaveawavelengthconvertertoachieveblockingperformancecomparabletothatoffullwavelengthconversion[ 14 35 56 ].Sinceon-demandschedulingreservesapathforaxedtimeslot,thisresultappliesdirectlytotheFixed-Slotscenario. InSection 4.3 ,wepresentanewnetworkmodelthatthatcanemulatetheexistingfull-conversionalgorithmswhenonlyasubsetofnodeshaveawavelengthconverter.WedemonstratetheutilityofthisapproachusingtheExtendedBellman-Ford(EBF)andk-AlternativePath(k-Path)algorithms.Weevaluatethealgorithmson3performancesmetrics:blockingprobability,averagestarttimeandschedulingoverhead.Blockingprobability,whichmeasurestheratioofblockedrequeststothenumberofscheduledrequests,istheprimarymetricusedtoevaluateaschedulingalgorithm. 66

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Averagestarttime,whichpresentshowearlytherequestedlightpathisavailable,isofspecialimportanceintheFirst-Slotscenario.Schedulingoverhead,whichcomparesalgorithmsaccordingtotheircomputationcosts,isanimportantmetricforthealgorithmspracticality. 4.2SchedulinginFullWavelengthConversionNetwork 4.2.1ProblemDenition AnopticalnetworktopologyisrepresentedasagraphG=(V,E,W)whereVisthesetofnodes,EisthesetoflinksandWisthesetofwavelengthssupportedbyeachlink.Anin-advancereservationrequestforalightpathcanbemadebetweenanytwonodesonG.AlgorithmsforRWAmayhavetoadheretothewavelengthcontinuityandwavelengthsharingconstraintsasdescribedintheintroduction.WeusetheFlexibleAdvanceReservationModel(FARM).Thismodeltriestoreducetheblockingprobabilityofrequestsbyassigningaschedulingwindowforeachrequest[ 27 ]. Inthissection,weaddressthefollowingqueries: 1. Findtheleasttimetwithintheschedulingwindowforwhichthereisapathwithanavailablewavelengthfromthesourcestothedestinationdfromtimettotimet+durandreservesuchapath.ThisisavariationoftheFirst-SlotProblemin[ 52 ]. 2. Findtheshortestpathwithanavailablebandwidthfromasourcestoadestinationdwithintheschedulingwindowandreservesuchapath. EachrequestRforasinglelightpathisdenedasfollows:R=[s,d,dur,start,end],wheresisthesourcenodeofthelightpath,disthedestinationnodeofthelightpath,duristhereservationduration,andstartandendarethestarttimeandendtimeoftheschedulingwindowrespectively.Theschedulingwindowmustbelargerthanthereservationdurationd.Theschedulermustcheckifapathisavailableduringanypossibleintervalintheschedulingwindow.Inslottedtimemodel,theintervalswillbe[start+t,start+t+dur]wheret=0,1,2,....,end)]TJ /F4 11.955 Tf 12.03 0 Td[(start)]TJ /F4 11.955 Tf 12.03 0 Td[(dur.Inthecontinuoustimemodel,adiscreteslidingapproachmaymissintermediatestarttimes. 67

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Thealgorithmspresentedin[ 58 ]donotguaranteetondafeasiblesolutionevenifasolutionispresent.Thealgorithmsdevelopedinthissectionprovidesuchguarantees.Wecomparethetimerequirementsandtheeffectivenessofthesealgorithmsinthefollowingsections.Theroutingandthewavelengthassignmentportionsofthesealgorithmsarepresentedinseparatesubsections. 4.2.2RoutingAlgorithms Thegoaloftherstqueryistondthepathwiththeleaststarttimeduringaschedulingwindow,whilethegoalofthesecondqueryistondthepathwithshortestdistance/hopduringaschedulingwindow.Wedeveloptwoalgorithms-ModiedSwitchPathFirst(MSPF)andListSlidingWindow(LSW)-foransweringtherstquery.Wealsodeveloptwoalgorithms-ModiedSlideWindowFirstAlgorithm(MSWF)andExtendedBellman-FordAlgorithm(EBF)foransweringthesecondquery.Thesearedescribedindetailintherestofthissection. Thealgorithmsdevelopedin[ 58 ]canbetermedaskDynamicPaths(kDP)algorithmaccordingtotheclassicationof[ 32 ].Thesealgorithmscheckkdynamicpathsbasedonthecurrentnetworkstatusandtestthemfortheirfeasibility.Ifmorethanonepathsarefoundtobefeasible,tiesarebrokenappropriately.Thesealgorithmsdonotguaranteetondafeasiblesolutionevenifasolutionispresent.Experimentalresults[ 58 ]showedthatwhenthelinkcostsaredynamicallyupdatedtoincorporatecurrentallocationandprovideLoadBalancing(LB),theblockingrateissignicantlyreduced.TheLoadBalancing(LB)scheme[ 58 ]assignsthecostofalinkwithnoreservationtobeequalto1.Thiscostisincrementedasadditionalreservationsareassignedtothelink.ThetwoversionsbasedonloadbalancingusingthedifferentapproachestochoosebetweenmultiplepathsarecalledSwitchPathFirst(LB-SPF)andSwitchWindowFirst(LB-SWF)algorithm.LB-SPFtriestondthepathstartingearliestwithintheschedulingwindow,whileLB-SWFtriestondtheshortestpathwithintheschedulingwindow.Wehaveextendedthesealgorithmsforthecontinuoustime 68

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modeltoprovidecomparisonswithouralgorithmsandarecalledMSPFandMSWFrespectively. ModiedSwitchPathFirstAlgorithm(MSPF) .Fortheslottedtimemodel,thisalgorithmstartsatthebeginningoftheschedulingwindow.ItcomputestheshortestpathusingDijkstra'salgorithm.Ifthispathisfeasible,itassignsanavailablewavelengthalongthepath;else,itdeletesthebusylinksonthepathandrecomputestheshortestpath.Foragivenxedreservationinterval[tstart,tstart+duration],thisshortestpathcomputationhastoberepeatedatmostktimes.Ifnofeasiblepathisfound,thereservationintervalisincrementedbyoneslotuntiltheinterval'sendtimeequalstheschedulingwindow'sendtime.Toadaptthisapproachtothecontinuoustimemodel,weadvancetothenextstarttimeintheSTlistinsteadofslidingthereservationintervalbyonetimeslot.Themodiedalgorithmisasfollows: Step1: SortallavaluesintheSTlistofeachlinkinthenetwork.Foreachaiinthesortedlist,repeatSteps2and3atmostktimes. Step2: Computetheshortestpathbasedoncurrentlinkcosts.Verifyfeasibilityforreservationintervalstartingattimeai. Step3: Ifthepathisfeasible,stopthealgorithmandreturnthispath;elsedeleteallthebusylinksonthepath. ModiedSwitchWindowFirstAlgorithm(MSWF) .TheoriginalSWFalgorithmissimilartoSPFexceptthatitslidesthereservationintervalbeforeswitchingthepath.Thisrequiresacheckonallthepossiblereservationintervalswithintheschedulingwindowforonepathbeforecheckingthenextpath.Forthecontinuoustimemodel,modiedalgorithmscanbeimplementedinseveralways.Onepossiblevariationisasfollows: TheintersectionofalltheSTlistsonapathcanbeusedtoderivethefeasibilityofagivenpath.Ifthepathisfeasible,anearlieststarttimecanbeeasilyderivedbyusing 69

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thesmallestnumberintheintersectedlist.Thealgorithmattemptsatmostkdifferentpathsandcheckallthepossiblestarttimeforeachpathasfollows: Step1: ComputetheshortestpathanditassociatedSTlists.VerifythepathbyintersectingalltheseSTlists. Step2: IfintersectedSTlistisnotempty,returnthispathwithrstpossiblestarttime,elseremovethebusiestlinkduringthewindow. Step3: Repeatstep1and2forktimes.Ifnoneofthepathisfeasible,rejecttherequest. Ifthenetworkisequippedwithwavelengthconverters,onesingleSTlistcanbemanagedperonelink.Fornetworkswithnowavelengthconverters,asinglepathhastobetreatedasmultiplesub-pathscorrespondingtoeachwavelengthandrepeattheprocessoverallsub-paths. TheMSPFandMSWFalgorithmsarenotguaranteedtondasolutionevenifoneexists.Theyonlytrykdynamicpaths,wherekisarbitrarilyspeciedinadvance.Inthefollowing,wedescribetwoalgorithmsListSlidingWindow(LSW)andExtendedBellmanFord(EBF)thatareguaranteedtondafeasiblepathifoneexists. Wealsomodifytwoalgorithmsfrompreviouschaptertoprovideaguaranteeonndingafeasiblepathifoneexists: 1. ListSlidingWindow(LSW)TheLSWalgorithm[ 32 ]ndsthepathwiththeleaststarttime.ThisalgorithmsortsalltheavaluesinSTlistsofeachlinkinthenetwork.Itdeterminesthesmallestaiforwhichafeasiblepathisavailablebyscanningthislistsequentially.Thesearchforafeasiblepathusesabreadth-rstsearch[ 32 ].Theexecutiontimecanbeimprovedbysavingandutilizingthebreadthrstsearchcomputationsfromthepriorstarttimes.Forexample,thebreadthrstsearchforaicanusethesearchcomputationsrequiredforthebreadthrstsearchofai)]TJ /F9 7.97 Tf 6.58 0 Td[(1utilizingthefollowingobservation:Thebreadth-rstsearchmustscantheSTlistofeachwavelengthoneachlinkthatistraversedduringthesearch,thisscanmaybeginwherethemostrecentscanofthislist(fromthebreadth-rstsearchforanearlierai)wascompleted. 2. ExtendedBellman-Ford(EBF)Thisalgorithmndstheminimumhoppathwithinaschedulingwindow.ItwasoriginallyproposedfortheFirstSlotandAll-AvailableSlotsproblemsin[ 52 ].Wedevelopamodiedversionoftheoriginal 70

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EBFalgorithmsothatitcanstopassoonastheSTlistatthedestinationnodeisnotempty.ItalsoextendsthenotionofanSTlistforalinktoincorporatelightpaths.Theresultingstarttimecorrespondstothesolutionwithminimumhoppath. 4.2.3WavelengthAssignmentAlgorithms Onceapathisfound,thewavelengthassignmentalgorithmisapplied.Thisalgorithmisrelativelyorthogonaltotheroutingalgorithmalthoughitsharesthecurrentreservationinformationwithit.Whennowavelengthconversionisallowed,weextendedeachalgorithmbyapplyingtheroutinetoeachwavelengthsequentiallyi.e.aFirst-Fitwavelengthassignmentschemeisused. Whenwavelengthconversionisallowed,exibleheuristicscanbedesignedsincewecanchooseanyavailablewavelengthinalink.Severalmetricssuchasmin-leadingormin-trailinggap[ 58 ]canbeusedinadditiontopath-widemetrics.Topickupthemostappropriatewavelengthamongpossiblewavelengths,theMin-Leading-Gapstrategybasesitsdecisionontheleadinggapbetweenthenewandpreviousreservations.WeusetheMin-Leading-Gapstrategy[ 58 ]fortheMSPFandMSWFalgorithm;wealwayschoosetheonethatproducestheminimumleadinggapwiththeassignmentofthecurrenttask.Thisstrategywasshowntohavethebestperformancein[ 58 ]forlinkutilizationandacceptanceratio. ForLSWandEBFalgorithms,weproposeanewmechanismcalleddeferredwavelengthassignment.Thismechanismdefersassigningaspecicwavelengthatreservationtime.Adeferredstrategyonlycountsthenumberofwavelengthsthatareusedonalink.Theactualassignmentofthewavelengthisdoneatthetimeoftherequestisactuallyfullled.Adeferredwavelengthstrategycanbeshowntoalwaysguaranteeafeasiblesolutionaslongasthetotalamountofreservedbandwidthdoesnotexceedthetotalcapacityofbandwidthofalinkataspecicpointoftime.Thisalleviatestheneedtokeeptrackofbandwidthallocationstatusofeachwavelength.Only,acountneedstobemaintainedforeachlink. 71

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ReqID Source Destination StartTime EndTime R1 n0 n1 t0 t2 R2 n0 n1 t4 t6 R3 n0 n1 t0 t3 R4 n0 n1 t2 t5 R5 n0 n1 t2 t5 Figure4-1. Arequesttablewith5requests AnalgorithmbasedontheleftedgealgorithmforchannelassignmentintheVLSIroutingcontext[ 26 ]canbeusedforthispurpose.Although,thisalgorithmwasforbatchassignment,itmaybeadaptedtoourcontextasbelow: 1. Assignjobssolongasnolinkhasmorethankjobsassignedatanytime,wherekisthenumberofwavelengths. 2. Atcurrenttimet,assigneachjobthatbeginsattimettoanyoneoftheavailablewavelengthonthepathreservedforthisjob;thewavelengthisthesameforeachlinkonthereservedpath.Consideranylinke.Ifthelinkhasqjobsscheduledtostartatt,itcanhaveatmostk)]TJ /F4 11.955 Tf 12.12 0 Td[(qjobscontinuingfrombeforet.Hencethereareatleastqwavelengthsavailableonefromtto1. Thus,itcanbeshownthatifthereservednumberofwavelengthsdoesnotexceedthemaximumnumberofwavelengthsofalink,allrequestscanbeaccommodatedwithdeferredwavelengthassignment. Figure 4-1 showsanexampleofarrivingrequests;thenetworkhasonlytwonodesn0andn1thatareconnectedtoeachotherbyalinkwith3wavelengths(1,2and3).Therequestsaresortedinascendingorderofarrivaltime,i.e.,R1arrivedbeforeR2.Apictorialcomparisonbetweenmin-leadinggapanddeferredwavelengthassignmentisshowninFigure 4-2 .Themin-leadinggapwavelengthassignmentschemeschedulesrequestsinarst-come-rst-servefashion,i.e.,R1!R2!R3!R4!R5.ThisresultsinasituationthatR2isscheduledon1,R3on2,R4on3andnallyR5failstobescheduled.Usingadeferredwavelengthassignment,allrequestsareacceptedsincethelinkiscomputedtobeavailablefortheperiod[t2,t5].Duringthisperiodthenumberofallocatedwavelengthsislessthanorequalto2.Thedeferredwavelengthassignment 72

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scheme(Figure 4-2 (b))schedulesrequestsaccordingtostarttimesofacceptedrequests.Att0,R1andR3areactivated,andscheduledon1and2respectively.R4andR5arescheduledon1and3att2.Finallyatt4,thereisstillroomat2forR2andR2isscheduled. Figure4-2. ComparisonofwavelengthassignmentusingdifferentschemesforrequesttableofFigure 4-1 4.2.4PerformanceEvaluation Inadditiontothetraditionalmetricsofspaceandtimecomplexity,theeffectivenessofanin-advanceschedulingalgorithminaccommodatingreservationrequestsiscritical.Thespacecomplexityneedstobereasonable.Thatis,thespacerequirementshouldnotexceedtheavailablememoryonthecomputeronwhichthebandwidthmanagementsystemistorun.Thetimecomplexityisimportantasthisinuencestheresponsetimeofthebandwidthmanagementsystemand,inturn,determineshowmanyreservationrequeststhissystemcanprocessperunittime.Schedulingeffectivenessis,ofcourse,criticalasrevenueisgeneratedonlyfromtasksthatareactuallyscheduled. Figure 4-3 summarizesthetimecomplexityofeachofthealgorithmsdescribedinSection 3.2.2 .Ifdeferredwavelengthassignmentisusedforawavelengthassignmentalgorithmcombinedwithacertainroutingalgorithmonawavelengthconvertiblenetwork,allabovementionedcomplexitieswillbesmallerbythefactorofWsinceitonlycountsthenumberofwavelengthsthatareusedonalink. 73

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Problem Algorithm SlottedArray Continuous FirstSlot SPF O(k(nlogn+Wdn)) O(qk(nlogn+L)) LSW O(ed) O(q(e+L))) AnySlot SWF O(k(nlogn+Wdn)) O(qk(nlogn+L)) EBF O(nel+ew) O(nel+L) d=durationofarequest,k=numberofpathstotry l=sizeoflongeststlistwithinaschedulingwindow q=numberofdifferentaisintheSTlistswithinaperiod L=sumoflengthsofTBlists,=endofschedulingwindow-d w=sizeofschedulingwindow Figure4-3. Timecomplexityofdifferentalgorithms 4.2.5Experiments Inthissection,werstbrieypresentoursimulationenvironmentincludingthenetworktopologiesusedandtherequestgenerationprocess.Wethenpresentthekeyvariationsthatwereimplementedandcompared.Thisisfollowedbyourexperimentalresultsandobservations. 4.2.5.1Simulationenvironment Fortestnetworks,weusedthe24-nodeNSFnetwork(Figure 4-4A )and33-nodeGEANTnetwork(Figure 4-4B )of[ 58 ],the19-nodeMCInetworkandthe16-nodeclusternetworkof[ 40 ],the11-nodenetworkof[ 12 ],theAbilenenetwork[ 5 ],andseveralrandomlygeneratedtopologies.Althoughmanyofthesenetworks(exceptNSFandGEANT)donotuseopticalinterconnects,weconvertedthesenetworksintoopticalnetworksbysettingthenumberofwavelengthsbasedontheoriginalbandwidthoflinks;Forexample,theMCInetworkthathasbandwidthsrangingfrom45Mbpsto310Mbpswasconvertedtoacorrespondingopticalnetworkbydividingthebandwidthofeachlinkby5. Therandomnetworksweusedforoursimulationshad200,400,or800nodes.Theout-degreeofeachnodewasrandomlyselectedtobebetween3and5.Toensurenetworkconnectivity,therandomnetworkhadbidirectionallinksbetweennodesiandi+1forevery1i
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ANSF BGEANT Figure4-4. NSFandGEANTnetwork Wegeneratedasyntheticsetofreservationrequestsinthesamewayasinpreviouschapters.Foreachtrial,wemeasuredtherequestacceptanceandbandwidthacceptanceratiosfor10timesandpresentstheaveragevalueoftheresults. 4.2.5.2Evaluatedalgorithms Thereareseveralvariantsforeachbasicalgorithmthatisdescribedintheprevioussectionbasedonthepropertiesofnetwork(convertersorlackofconverters)andwhetherornotadeferredstrategywasusedforwavelengthassignment.Forexample,thethreevariantofModiedSwitchPathFirst(MSPF)areasfollows: 1. MSPFw/converterandw/wavelengthassignment, 2. MSPFw/converteranddeferredwavelengthassignment,and 3. andMSPFw/oconverter SimilarvariationscanalsobederivedforModiedSwitchWindowFirst(MSWF)algorithms. ListSlidingWindow(LSW)andExtendedBellman-Ford(EBF)algorithmshavetwovariantsdependingonthepresenceorabsenceofwavelengthconvertersintheopticalnetwork.Forthesealgorithms,deferredwavelengthassignmentisassumedforthenetworkswithoutconverters.Forexample,thevariationsforLSWarelabeledasLSWw/ConvandLSWw/oConvrespectively.Weprogrammedallthereasonablevariantsfor 75

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eachbasicalgorithmsinC++andmeasuredtheireffectiveness.Wealsostudiedtheimpactofusingconvertersontheeffectiveness. 4.2.5.3Resultsandobservations Figures 4-5 and 4-6 providetheaverageacceptanceratiosforthealgorithmsasafunctionofthenumberofrequestsinthestudyintervalfortwonetworktopologies.Theaverageacceptanceratiosforthesealgorithmsasafunctionofthemeanrequestdurationforthevariousnetworktopologiesweresimilar. ANSF BGEANT CBurchard DCLUSTER EAbilene FMCI Figure4-5. Networkacceptanceratiovsnumberofrequests Ourexperimentalresultsshowthefollowing: 76

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A200nodes B400nodes C800nodes Figure4-6. Acceptanceratiovsrequestsnumberinvariousrandomtopologies 1. EBFconsistentlyoutperformedallotheralgorithms.Fortheremainingalgorithms,therelativeperformancevariesbasedonnetworkproperties.Forhomogeneousnetworktopologies(NSF,GEANT,BurchardandAbilene)(resultsnotpresentedduetospacelimitations)thathavesamenumberofwavelengthsonallthelinks,MSWFperformsbetterthanMSPFandLSW.However,fornon-homogeneousnetworktopologies(MCI,clusterandrandomnetworks),LSWhasbetterperformancethanMSWFandMSPF. 2. Fornon-homogeneoustopologies,thealgorithmsthatdonotguaranteetondfeasiblereservations(MSPFandMSWF)faredworsethanthealgorithmsthatprovidesuchaguarantee.However,attimes,theperformanceofthebestnoguaranteealgorithmwasquiteclosetoorslightlybetterthanthatoftheworstguaranteealgorithm.Asthenetworkgotsaturated(i.e.,thenumberofrequestsinthestudyintervalincreases),theRARforallalgorithmsdeclinedandtherateofdeclineforthenoguaranteealgorithmwasfoundtobehigherthanguaranteealgorithm.algorithms. 3. Therelativeperformanceofthealgorithmsdoesnotchangesignicantlywithmeanrequestduration.However,theperformancegapgrowslargerasthemeanrequestdurationdecreases.Theperformancedifferenceusingdeferredwavelengthassignmentmethodornotusingitwasnotsignicant. 4. Theuseofwavelengthconvertersgenerallyledtobetterperformance.Thus,theadditionalexibilitythatwavelengthconvertersprovideinanetworkisworthwhile. 4.3SchedulinginSparseWavelengthConversionNetwork 4.3.1ProblemDescription Inopticalnetworkscheduling,theprimaryconcernisroutingandwavelengthassignment(RWA).Wavelengthdivisionmultiplexing(WDM)allowsmultiplelightpathsfromdifferentuserstoshareoneopticalbersimultaneously.Normally,afeasible 77

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lighpathintheopticalnetworkhastofulllthewavelengthcontinuityconstraint,whichforcesasinglelightpathtooccupythesamewavelengththroughoutallthelinksthatitspans.However,thisconstraintisrelaxedinanopticalnetworkthatisequippedwithwavelengthconverters.Thesignalsreceivedbyawavelengthconvertermaybetransmittedonadifferentwavelengthinthenexthop.Wheneverynodeinthenetworkisequippedwithawavelengthconverter,thenetworksupportsfullwavelengthconversion.Whenonlysomeofthenetworknodeshaveawavelengthconverter,thenetworksupportssparsewavelengthconversion. Theimpactofwavelengthconvertersinall-opticalroutinghasbewidelystudied.[ 56 ]showedthatforcertaintopologiesandxed-pathrouting,sparsewavelengthconversionisalmostaseffectiveasfullwavelengthconversion.[ 14 16 ]haveinvestigatedtheperformanceofthek-AlternativePathsalgorithmsinthepresenceofwavelengthconverters.Theirfocusistheblockingprobabilityofsparsewavelengthconversionforonon-demandscheduling.[ 33 ]considersin-advanceschedulingusingacontinuoustimemodelandevaluatesthevariousalgorithms'blockingperformanceforfullwavelengthconversionandlowworkload. Inthissection,ourprimarygoalistostudytheimpactofwavelengthconvertersonFirst-SlotschedulingandtoanalyzethetemporalbehavioroftypicalRWAalgorithmsinthecontextofsparsewavelengthconversion.Extensiveexperimentalresultsarepresentedinthispaper.Accordingtoourtestresults,increasingthefractionofnodeswithwavelengthconvertersisofgreatervalueforblockingperformanceinrelativelylowtrafccasesthaninthehighworkloadcases.However,theaveragestarttimesarealmostunaffectedbywavelengthconvertersratioexceptforthemarginallyimprovementinsmalltopologies. Anotherkeyconsiderationhereistoexploretheadvantagesanddisadvantagesofthe2schedulingstrategiesrepresentedbyEBFandk-Pathrespectively.Intuitively,alwaysacceptingarequestwheneverthereisafeasiblelightpathinthenetwork,which 78

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isdonebyEBF,shouldprovidebetterperformancethanlimitingthesearchforafeasiblepathtoasmallsetofcandidatepathsasisdoneink-Path.However,ourresultsshowthatthisstatementisonlytruewhentheoverallworkloadissmallcomparedtonetworkcapacity.EBFoftenschedulesarequestonalongerpaththanusedbythek-Path.So,whentheworkloadishigh,theadditionalresourcesutilizedbyEBFtoacceptarequestnegativelyimpactstheacceptanceoffuturerequests.Ourresultsclearlydemonstratethesetradeoffs:EBFperformsbetterwhennetworkcapacityisamplefortherequestedworkload,butk-Pathoutperformswhenthetrafcscongestthenetwork.Thisobservationleadsustoproposeahybridapproachthatautomaticallyswitchesbetweenthetwoalgorithmsbasedoncurrentnetworktrafc. Wealsostudydifferenttie-breakingapproacheswhenmultiplepathsarefeasibleandtheimpactofdifferenttie-breakingschemesonoverallperformanceinthepresenceofsparsewavelengthconversion.Aslacktie-breakingschemeisproposedanditsperformancerelativetootherwidelyusedstrategiesisanalyzed. 4.3.2ExtendedNetworkModel ThetopologyofaopticalnetworkwithsparseconversionisrepresentedasagraphG=(V,E,W)whereVisthesetofopticalswitchesorrouters,EisthesetofopticallinksandWisthenumberofwavelengthssupportedbyeachlink.EachnodeninVisassociatedwithabooleanfunctionF(n),whichistrueifandonlyifthenodeisequippedwithwavelengthconverter.Toemulatethefull-conversionalgorithms,werstconverttheabovegraphintoanewgraphG0=(V0,E0).TomapnodesetGtoG0,foranoden2V,ifnequipsawavelengthconverter(i.e.F(n)istrue),acorrespondingnoden0willbeinsertedintoV0.IfF(n)isfalse,Wpseudo-nodes(n01,n02,...,n0w)willbeinserted.WisthenumberofwavelengthsdenedinG.Foralinkl2E,ifeconnects2nodeswithconverters,itwillbemappedtoalinkl0inE0;else,lwillbemappedintoWpseudo-links(l01,l02,...,l0w)andeachpseudo-linkstandsforaspecicwavelengththatcarriedinl.In 79

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theextendedmodel,l0iisincidentton0iifflisincidenttonintheoriginalgraphandl0iandn0iistheirithpseudocopyrespectively. Figure 4-7 showsaexampleofa4-noderingtopology.Thenetworkcontains2wavelengthconverters,nodesAandB.Eachlinkcarries2wavelengths.Itsextendedpresentation,shownontherightside,contains6nodesand7links.Eachnon-converternodesaresplitinto2pseudo-nodes(nodesfC1,C2gfornodeCandfD1,D2gfornodeD).Eachopticallinkthatisincidenttoanon-converternodeissplitinto2pseudo-linksandeachpseudo-linkstandsforoneindividualwavelength(link(B,C1)and(B,C2)forlink(B,C)forexample).Twopseudo-nodesthatareadjacentintheextendedmodelmustfulll2conditions:1)theircorrespondingnodesareadjacentintheoriginalgraph,and2)theyareeitherconverternodesortheirwavelengthindexmatches. Figure4-7. ExtendedNetworkModel TheextendedgraphG0isequivalenttotheoriginalgraphG.EveryfeasiblelightpathinGhasacorrespondingpathinG0andviceversa.Thewavelengthcontinuityconstraintisalsopreservedintheextendedmodel.IfnodenhasnowavelengthconverterinG,everycorrespondingpseudo-noden0iinG0isincidentbyandonlybythepseudo-linksofwavelengthindexi.Hence,wecandirectlyapplytheRWAalgorithmsthatoriginallydesignedforfull-conversion/no-conversionnetworkstothesparseconversionscenariowithlittleadaptation. Whenschedulingarequest,ifthesourceordestinationnodesareextendedtomultiplepseudo-nodes,theRWAalgorithmneedstocheckeverycorrespondingpseudo-source-destinationpairs.SoatmostW2roundsoftheoriginalalgorithmare 80

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neededintheextendedmodel,whereWisthemaximumnumberofwavelengthcarriedbyalinkinthenetwork.Also,duringthemodelextension,thegraphsizecanincreasebyatmostWtimesforitslinkandnodenumber.Therefore,analgorithm'scomputationtimeontheextendedmodelG0isboundedbyaconstantratio(apolynomialofW)ofthecomputationtimeforthatalgorithminfull-conversionscenarios,whichrunsontheoriginalnetworkG.So,theadaptedalgorithmhasthesameasymptoticcomplexityasitsoriginalversion. Inthispaper,weemployExtendedBellman-Fordalgorithmandk-AlternativepathalgorithmasourRWAalgorithms.ThedetailofthealgorithmswillbepresentedinSection 4.3.3 4.3.3RoutingandWavelengthAssignmentAlgorithms TheRWAalgorithmforFirst-Slotproblemusuallycontains3steps: 1. Identifytheearlieststarttime. 2. Findtheshortestpaththatprovidessuchstarttime. 3. Assignthewavelength. Asthewavelengthassignmentareindependentfromtherst2steps,mostRWAalgorithmshandlethisstepseparately.Inourpaper,EBFandk-pathalgorithmproceedtherst2steps.AwavelengthassignmentstrategycalledLeastConversionAssignmentisexplainedinSection 4.3.3.4 4.3.3.1ExtendedBellman-Fordalgorithmforsparsewavelengthconversion ExtendedBellman-Fordalgorithm[ 52 ]appliestheBellman-Fordshortestpathalgorithm[ 17 ]totheSTlistsonlinksthatmayconnectsourceanddestination.Thekeystepsofthealgorithmareasfollows: 1. Letst(k,u)representtheunionoftheSTlistsforalllightpathsfromvertexstovertexuthathaveatmostkedges.Clearly,st(0,u)=;foru6=sandst(0,s)=[0,1].Also,st(1,u)=ST(s,u)foru6=sandst(1,s)=st(0,s).Fork>=1,the 81

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followingrecurrencecanbederived: st(k,u)=st(k)]TJ /F6 11.955 Tf 11.95 0 Td[(1,u)[f[v,(v,u)2Efst(k)]TJ /F6 11.955 Tf 11.96 0 Td[(1,v)\ST(v,u)gg 2. Constructthelistst(n)]TJ /F6 11.955 Tf 12.44 0 Td[(1,d),whichgivesthestarttimesofallpathsfromstodthathavebandwidthBWavailableforadurationDur.Ifst(n)]TJ /F6 11.955 Tf 12.05 0 Td[(1,d)isnotempty,ainitsrst(a,b)pairistheearlieststarttimeforthiscurrentsource/destinationpair,denotedasesti. 3. Forallpossiblesource/destinationpairs,ndtheminimumestiastheearlieststarttimefortherequestandthecorrespondingsource/destinationpair(si,di)arerecorded. 4. Removethelinksthatcannotprovidetherequestedbandwidthattheearliestnishtime. 5. RunBreath-FirstSearch[ 17 ]ontheextendedmodelandndashortestroutefromthesitodi,maptheroutebacktotheoriginalgraph. ThecomplexityofintersectionandunionoperationislineartothelengthofthecurrentSTList.Foreachiterationofconstructingst(k,d)forst(k)]TJ /F6 11.955 Tf 12.33 0 Td[(1,d),weneedstocomputetheSTListforeachlinkandeachcomputationtakesO(L)time,whereListhelengthofthelongeststlist.SincetheconstructioniteratesatmostN)]TJ /F6 11.955 Tf 12.23 0 Td[(1times,thecomplexityoftheextendedBellman-FordalgorithmisO(NEL),whereNandEisthenumberofnodesandlinksinthegraph. 4.3.3.2k-Alternativepathalgorithm Thek-AlternativepathalgorithmisextendedfromtheshortestpathalgorithmsroutingusedbyOSPFinInternetrouting.RecognizingthatanOSPF-likealgorithmmayfailtondafeasiblepathinanetworkthathasafeasiblepath,k-Pathalgorithmgeneratesadditionaldisjointpathswiththehopethatoneoftheadditionalpathswillbefeasible.ByconstructingSTListoneachpath,itiseasytodecidewhichpathprovidesearlieststarttime.Iftheallgeneratedpathisinfeasible,therequestisreject. Thekpathscanbeeitherxedordynamicallygenerated.k-Fixedpatharecomputedforeachnodepairbeforetherstschedulingbeginsaccordingtosomegivenlinkcosts.Thek-dynamicpatharecomputedforeveryrequestaccordingto 82

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currentnetworkstatus.Usually,themostcongestlinkwillhavelargecostsoastobeavoidinthepathcomputation.[ 45 ]haveshownthek-Dynamicpathscanprovideamuchlargernetworkthroughputthank-Fixedpaths,especiallywhenthetrafcloadisrelativelylargeforthenetwork'scapacity.So,inthispaper,weadoptthethisk-DynamicpathsalgorithmandcallitKDP. 4.3.3.3Breakingthetiesinpathselection Whenmultiplepathscansatisedauser'srequest,atie-breakingschemeisneededtoselectoneofthemastheactualscheduledpath.Themoststraightforwardschemebreakstiesbasedonrst-t(FF)strategyorShortestPath(SP)strategy.First-tstrategyterminatesthepathselectiononceasucceedpathisfound.Shortest-pathstrategychosethepathwithminimumhopnumber.Bothstrategiesarewidelyusedinnon-conversionandfull-conversionscenarios,butusingthemdirectlyinSparsewavelengthconversionneedscarefulconsiderationasthesepathsarecomputedonextendedmodels. Intheextendedmodel,theshortestpathdoesnotnecessarilycorrespondstotheshortestpathintheoriginalgraphforsamesource/destinationpairs.Figure 4-8 givesanexampleofa4-noderingwithonly2converters,nodesAandC.Eachlinkconsists2wavelengths.Atsomeconjuncture,theavailablewavelengthsareshowninextendedgraphontherightside.Forarequestof1wavelengthofcapacityfromnodeBtoA.Twocandidatepathsareavailable:B1!C!D1!AandB2!A.Assumethatbothpathprovidesthesamestarttime,TheFirst-Fittie-breakingschemewillchoosethepair(B1,A)assourceanddestinationwhenthenode-pairsarecheckedinlexicalorderandtheShortestPathtiebreakingschemewillchoosethe3-hoppathasitistheoneavailableB1toA. Atie-breakingschemethatchoosetheshortestpathbyexaminingallsource-destinationpairs,ratherthantherstsucceedpair,wouldsolvetheproblem.However,consideringonlythepathlengthwouldnotbeenoughinthecontextofsparseconversion.In 83

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Figure4-8. ExtendedNetworkModel networkswithoutwavelengthconversion,shorterpathismorelikelyhavecommonwavelengths,thereforelesslikelytoblocktherequests.However,thepresentofwavelengthconvertersreducesthecorrelationbetweenpathlengthandpathcapacitiesduetotheeliminationofthewavelengthcontinuityconstraint.Hence,intermsofloadbalancing,choosinglongerpathwithhighercapacitymayalsoreducethepotentialcongestionbyalleviatethetrafcintheshortpaths.Inthispaper,insteadofusingFirst-FitstrategyinoriginalEBFandKDPalgorithm,weemployaslacktie-breakingschemethatselectsthepaththatatmosthhopslongerthantheshortestpathbuthavethemostfreewavelengthsamongallthepathsthathavefewerorequalhopcounts.WecallthesevariantEBF-SandKDP-S. Intuitively,hshouldnotbetoolargeorthebenetofloadbalancingwouldbecanceledbythewasteoflinkcapacitiesonlongerpaths.InSection 4.3.4 ,wewillcomparingtheFirst-FitschemewithSlackschemeandperformanumericanalysisonthechoiceofhindifferentscenarios. 4.3.3.4Wavelengthassignment Withthepresenceofwavelengthconverters,wavelengthassignmentbecomeslessimportantinopticalrouting.However,aswavelengthconversioncontributesaconsiderabledelayintheopticaltransmission[ 16 ],aproperwavelengthassignmentwouldpotentiallyreducesuchoverheads. Forthoselinksthatconnecttoanon-converternodes,thewavelengthassignmentisquitestraightforward,asthespecicwavelengthhasbeenalreadyidentiedwith 84

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thelightpathduringthepathselectionprocess.However,forthoselinksconnecttwowavelengthconverters,weapplytheleastconversionassignmentratherthantherst-talgorithmusedinmostRWAalgorithms.Inleastconversionassignment,thelinkthatconnects2converterswillusethesamewavelengtheitherasitsprevioushoporasitnexthopunlessneitherwavelengthsisnotavailable.Forinstance,letl2Econnectstwowavelengthconverters,wibethewavelengthassignedforthepreviouslinkonthepathandfwngbethesetofcommonwavelengthofbothlandl'snextlink.Ifwiisavailableonl,weassignthiswitothelightpath,elseweassignanywavelengthinfwngiffwng6=;.Ifneitherisavailable,arandomavailablewavelengthinassigned.Thisstrategycanalwaysguaranteeafeasiblesolutionwhileavoidingtheunnecessaryconversions. 4.3.4ExperimentalEvaluation 4.3.4.1Experimentalframework Inthissection,wemeasuretheperformanceoftheschedulingalgorithmsdescribedinSection 4.3.3 andanalyzehowwavelengthconvertersaffectsthealgorithms'performanceinvariousscenarios.Wecomparethetie-breakingschemesinSection 4.3.3.3 toshowtheeffectivenessofourslackstrategy;WeanalyzetheperformanceofRWAalgorithmsfor3metrics:blockingprobability,averagestarttimeandexecutiontime.Inourexperiments,blockingprobabilityismeasuredbyratioofrejectedrequestscomparingtothetotalsubmittedrequest.Averagestarttimeismeasuredbytheaveragedelaybetweenthereservationwindow'sstarttimeforajobanditsactualstarttime.Executiontimemeasurestheconvergencespeedofeachalgorithmandhowitvarieswithnetworksizeandworkloads.Wealsoproposedaself-adaptivealgorithmswitchingstrategythatdynamicallychoosethesuitablealgorithmaccordingtothecurrentworkload.Theperformanceofthisswitchstrategyistestedinvariousscenarios. Tosimulateae-Sciencebackbone,weusea10-noderingtopology,a25-nodemesh-torustopology,arealwork11-nodeAbilenenetwork(Figure 4-9 )andseveral 85

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randomlygeneratedtopologies.Eachlinkisassumedtocarry10wavelengths.Forrandomlygeneratedtopologies,wesettheout-degreeofeachnodetoberandomintegersbetween3and7.Toensurenetworkconnectivity,therandomnetworkhasbidirectionallinksbetweennodesiandi+1forevery1i
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willcomparethealgorithmsperformanceunderdifferentWLCRatiotoplottheimpactofwavelengthconverters.Althoughasimplestrategyisappliedhere,ourexpectationisthataperformancecomparisonbetweendifferentapproachesshouldbeapplicableevenwhenamoresophisticatedplacementisused. Filetransferrequestsaresyntheticallygenerated.Eachrequestisdescribedbythe6-tuple(si,di,BWi,Duri,STi,ETi).Thesourceanddestinationnodesforeachrequestwereselectedusingauniformrandomnumbergeneratorsothattheworkloadisdistributeduniformlyamongdifferentnodepairs.Withoutlossofgenerality,weassumeeachrequestasksforacapacityofonly1wavelength.Thedurationisuniformlydistributedwithinarangeof100to500timeunits.ThearrivaloftherequestfollowsaPoissondistributionwithrate.Thereservationwindowstartsatsometimeaftertherequests'arrival.Astheresultsarerelativelyinsensitiveforthislag,wearbitrarilychoseitas100timeunits.Thelengthofthewindowisrandomlyselectfrom2to4timesoftherequestduration. WeassumethattherequestsarriveinaPoissonprocessforeachsource/destinationpairwithanarrivalrate.Followingtheexperimentalsettingin[ 56 ],ispickedintherangefrom0.01requests/timeunitto0.1requests/timeunitforeachnodepair.So,forexample,withaarrivalrateof0.05requests/timeunit,onerunofourexperimentona100noderandomtopologywouldprocessapproximately5105requestsduringthetestswhichlasts1000timeunits.Allourexperimentassumethatwestartwithnoloadi.e.,noexistingscheduledtransfers. 4.3.4.2Slacktie-breakingscheme RecallthatinSection 4.3.3.3 ,toselectthebestcandidatepath,weconsiderallthepathsthatareatmosthhopslongerthantheshortestpath.Hence,toevaluatetheperformanceoftheseheuristics,wemustrstdecidethevalueoftheh.Figure 4-10 explorehowh'svalueinuencetheblockingperformanceofEBF)]TJ /F4 11.955 Tf 12.53 0 Td[(Sinvarioustopologies.Inthesmalltopologylike8-nodering,h=1providesthebestperformance, 87

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ARing BRandom Figure4-10. Differenthvaluesfordifferenttopologies.NetworkTrafcLoad:=0.05 whileinthe100-noderandomnetwork,theh=2caseismarginallybetterthanh=1.Also,largehvalueslike3or4actuallydeterioratetheperformanceasthelinkcapacityiswastedbychoosinglongpaths.Inourtest,h=1isalsothebestchoiceforMesh-TorusandAbilene.Similarresults(notpresentedhere)werealsoobservedforKDP)]TJ /F4 11.955 Tf 12.47 0 Td[(S.Inthefollowingtests,wechoseh=1inforRing,AbileneandMesh-Torustopologiesandh=2forrandomnetwork. ARing B100-nodeRandom Figure4-11. Benetofslacktie-breakingschemeinvarioustopologies.NetworkTrafcLoad:=0.05. Figure 4-11 depictsthewavelengthconverters'impactonblockingperformancebyvaryingthewavelengthconverterratioforvarioustopologies.EBF-SandKDP-SusethesimpleFirst-Fittie-breakingschemebutEBF-SandKDP-Semployslackscheme.WeobservethatthealgorithmswithslackschemeworkmuchbetterthanthealgorithmsusingFFschemeinallcases.Thus,choosinglongerpathinpresenceofexcess 88

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capacitybenetstheblockingperformanceforsparsewavelengthconversion.AnotherobservationisthatwhenWLCRatioequals20%,theblockingprobabilityofFirst-Fitalgorithmsisworsethannothavinganyconverters.Thisphenomenoncanbeexplainedasfollows:whennowavelengthconverterisavailable,manyrequestsarerejectedduetolackofcontinuouswavelength,buttheacceptedrequestaremoreevenlydistributedamongallwavelengthsandlonglightpathsarelesslikelytobeestablished.Whenasmallnumberofnodesareequippingwithconverters,thetrafcscheduledbyFirst-Fitstrategyaremorelikelytousethoselongpathsbetweenwithwavelengthsoflowerindex,asshowninSection 4.3.3.3 .Theadditionalbenetfromhaving20%convertersisnotenoughtocoverthedegradationduetothewastedcapacityoflongpaths.Althoughthisdegradationcanbecompensatedbyeitherinsertingmoreconverters,asplottedinFigure 4-11 ,usingtheslackschemeisobviouslymoreeffectiveandeconomical.WealsonoticethattheimprovementbroughtbytheslackschemeinRingandAbilenetopologyisnotasmuchasinmesh-torusandrandomtopology.Thisisconsistentwiththeconclusionin[ 56 ],whichstatesthatwavelengthconversioncanhelpmoreinthetopologieswithmoredivergenceandconnectivity,asmorevariantsinthepathsareavailable. InthetestforbothFigure 4-17 andFigure 4-11 ,thenetworktrafcloadissettoamoderatedegree:=0.05.However,similarresultsarealsoobservedunderdifferentworkloads. 4.3.4.3Blockingprobability Inthissection,theblockingperformanceofEBF-SandKDP-SareevaluatedandcomparedtheFixed-ShortestPathroutingalgorithm.ForFixed-ShortestPathrouting,notie-breakisspeciedasonlyonepathisavailable.However,asFirst-Fitwavelengthassignmentisapplied,Fixed-ShortestPathroutingisdenotedasSP-FFinourdiagrams.Figure 4-12 andFigure 4-13 depictthehowtheblockingprobabilitieschangedwithwavelengthconverterratioinRing,Mesh-Torus,AbileneandRandom-100topology. 89

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ARing B100-nodeRandom Figure4-12. BlockingProbabilityvs.WavelengthConverterRatioinvarioustopologywithlowtrafcload. Figure 4-12 presenttheresultsobtainedwhenthenetwork'strafcloadisrelativelylow:2[0.01,0.05],whileFigure 4-13 presenttheresultwhenworkloadisrelativelyhigh:2(0.05,0.1]. ARing B100-nodeRandom Figure4-13. BlockingProbabilityvs.WavelengthConverterRatioinvarioustopologywithhightrafcload. FromFigure 4-12 andFigure 4-13 ,wenotethatIncreasingthewavelengthconverterratiocandecreasetheblockingprobabilitiesforallalgorithms.However,theimprovementisalsodependentonthenetwork'strafcload.Whennetworktrafcloadisrelativelylow,EBFonlyneedsabout40%ofwavelengthconverterstoprovideasatisfactoryblockingperformance,buttheblockingprobabilitiesofKDPandSPdecreasesmoregraduallyastheincreaseofWLCRatio.Whentrafcloadishigh,increasingthewavelengthconvertershasonlymarginalimprovementonallalgorithms.Thiscanbeexplainedasfollows.EBFexploresthenetworkmorethoroughlyforan 90

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availablepaththanKDPandSP.Withasmallamountofconverters,EBFisabletosatisfythetrafcdemandsbutKDPandSPcannot.However,whentrafcloadishigh,majorityofblockingoccursduetolackoflinkcapacitiesbutnottheavailabilityofcontinuouswavelengths.Hence,increasingwavelengthconvertershaslittleimpactinheavyloads. Wealsonotethatinalltopologies.EBF-SandKDP-SalgorithmscanachieveamuchsmallerblockingprobabilitycomparingtoSP-FFalgorithm.InsimpletopologieslikeRing(Figure 4-12A and 4-13A ),EBF-SandKDP-SleadsSP-FFforabout2-5%onblockingprobability,whileinthosemorecomplextopologieslikerandomnetwork(Figure 4-12B and 4-13B ),theadvantagesaredoubled.Thisisconsistentwiththeresultsin[ 14 ]thatshortest-pathroutingislesslikelytobeimprovedduetothesmallnumberofalternativelightpaths. ALowWorkload BHighWorkload Figure4-14. Totalresourceconsumptionina100-noderandomnetworkunderdifferentworkload. AnotherimportantobservationisthatEBFoutperformsKDPinlowtrafcworkloads(Figure 4-12 ),butKDPleadsEBFinhightrafcworkloads(Figure 4-13 ).Thiscanbeexplainedasfollows:EBFtriestondanypossiblepathforcurrentrequestifitexists,butKDPonlytestsnomorethankpaths.Whenthelong-termtrafcloadislow,thenetwork'scapacityisampletoaccommodatemostrequests,theEBFthatactsgreedilywouldacceptmorerequeststhanKDP.However,whentrafcloadishigh,limitingtheroutestoonlythoseshortpathsandrejectsomelongpathswoulddenitely 91

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benetthescheduleoffuturerequests.Inthatcase,themoreconservativeKDPwouldprovideabetterperformanceinthelongrun. ThisobservationcanalsobesupportedbythefactthatEBFactuallyconsumesnetworkbandwidthfasterthanKDPdoes.Figure 4-14 depictsthetotalamountsoflinkcapacitiesthatEBFandKDPconsumeunderdifferentworkloads.ThetotalresourceamountsaredenedasPp2RLP(Dur(p)length(p)),whereRLPisthesetofallestablishedlightpaths,Dur(p)islightpathp'sdurationandlength(p)isp'shopcount.Weseethat,forthesamerequestset,EBFconsumesmorelinkcapacitiesthanKDPinlowworkloadcases,asKDPrejectsthoselong-pathrequestswhileEBFacceptsthem.Whentheworkloadishigh,KDP,whichrejectssomeearlylong-pathrequests,substantiallyacceptsmorerequestsinthelongrun.Therefore,thetotalamountsoflinkcapacitiesbothalgorithmsconsumesarealmostequalinhighworkload. 4.3.4.4Requests'averagestarttime ARing B100-nodeRandom Figure4-15. AverageRequestStartTimevs.WavelengthConverterRatioinvarioustopologywithlowtrafcload. Intheall-opticalroutingarea,blockingprobabilityisalwaystheprimaryconcern,butforthisspecialcaseofFirst-Slotscheduling,theavailabilityofearlierstarttimemayalsobeanimportantmetrictoevaluateascheduler'sperformance.Figure 4-15 andFigure 4-16 presenttheinuenceofwavelengthconverterratioontherequests'averagestarttime.Similarwiththeprevioussection,Figure 4-15 showsthestarttimeperformanceinthelowtrafccaseandFigure 4-16 showsthetestresultinhighworkloadcases. 92

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ARing B100-nodeRandom Figure4-16. AverageRequestStartTimevs.WavelengthConverterRatioinvarioustopologywithhightrafcload. Toexcludetheimpactofblockedrequestsontheaveragerequeststarttime,wesetthereservationwindowforeachrequesttoalargeenoughtimeintervalsuchthateveryrequestwillbeacceptedatsometimewithinthewindow.Fromtheabovegures,wecanobservethatincreasingwavelengthconvertershaspositiveimpactsontherequests'starttime,buttheimprovementsarenotasobviousastheimpactonblockingperformance,especiallyforEBF-SandKDP-S.InRingtopology,theimprovementfrom0%WLCratioto100%WLCratioisonlyabout5%inlowtrafcloadcase,and15%inhightrafcloadcases.Inrandomtopology,theimprovementsarealmostnegligible. WealsonotethatEBF-SandKDP-SleadtheaveragestarttimeoverSP-FFinallcase.TheadvantagesarelargerinrandomtopologythaninRingtopologyandtheyincreaseastheworkloadincrease.ThisshowsthatEBF-SandKDP-ShavetheabilitytoscheduletherequestsinamoreparallelwaythanSP-FF.Whenthenetworkcapacitiesareamplefortherequests,EBF-SandKDP-Sprovidemuchfasterschedules.Whenthetrafccongeststhenetwork,thoseforthcomingrequestshavetowaitforpreviousrequeststonishduetolackofnetworkcapacity,whichreduceEBF-SandKDP-S'sadvantages. Similartotheresultsinprevioussection,EBF-SagainoutperformedKDP-Sinlowworkloadcasesonaveragestarttime,andKDP-Sgainsbetterperformanceforconservativereservationstrategyintheheavyworkload.Whenthenetworks'capacityis 93

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relativeamplecomparingtotherequestedworkload,EBF-ShastheabilitytostartthejobearlierthanKDP-S,buttheirperformancesareverycloseinlargetopologies.Whentheworknetworkisundercongestionbyhighrequestrates,KDP-Sleadstheaveragestarttimeinalltopologiesanditsadvantagearemoreobservableinlargenetworks. 4.3.4.5Schedulingoverhead Inthissection,wediscussabouttheschedulingoverheadofourRWAalgorithms.Figure 4-17 presentscomparisonsontheschedulingoverheadofEBF-SandKDP-SwithvariousWLCratioandnetworksize.TheseresultsshowthatEBF-Sisabout2-5timesslowerthanKDP-SanditscomputationtimegrowsfasterthanKDP-Swiththeincreaseofnetworksize.TheexecutiontimedecreaseswiththeincreaseinWLCratio,asthesizeofextendednetworkissmaller.However,evenfora400-nodetopology,EBF-Scanschedulearequestaveragelywithinseveralseconds.Thisshouldbeacceptableinmostscenarios. ACTvs.WLCRatio BCTvs.NetworkSize Figure4-17. AveragecomputationtimeofEBF-SandKDP-S. Insummary,equippingthenetworkwithsomewavelengthconvertersdoimprovetheblockingperformance,butaddingmoreconverteraftercertainthresholddoesnotbringmorebenets.Meanwhile,thetrafcloadandnetworktopologyalsocastgreatinuenceontheblockingprobability.Fortheaveragestarttime,wefoundthattheeffectofwavelengthconverterratioisminorforbothEBF-SandKDP-S,butthetrafcloadandnetworktopologyhavemoreevidentinuencesonthismetric.Comparingtwoalgorithms,EBF-Sgenerallyperformsbetterinthelowtrafccases,butKDP-Sisa 94

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betterchoicewhenthetrafcloadisheavy.KDP-SisfasterthanEBF-Sinallcase,butinevenintheworstcase,EBF-Scanstillscheduletherequestsinanacceptablespeed. 4.3.4.6Algorithmswitchingstrategy WendoutinSection 4.3.4.3 andSection 4.3.4.5 thatthegreedyapproachEBF-Shasbetterperformancewhenthetrafcloadiscomparablylight,whiletheconservativeapproachKDP-Sworksmoreeffectivelyinthehighworkloadcase.Inthissection,weproposeaself-adaptivealgorithmswitchingstrategythatautomaticallychoosesEBF-SorKDP-Saccordingtocurrenttrafcloadinthenetwork. Themainideaoftheswitchstrategyisasfollow.Thetimedomainisdividedintoequal-lengthtimeslots.Theschedulerrunsbothalgorithmsimultaneouslyoneachrequest,thechoiceofwhoseresulttoapplyincurrenttimeslotismadeatthebeginningofthisslot,assumingcurrenttrafchasthesimilarpatternasinlastslot.Theperformancesofthecandidatealgorithmscanbeevaluatedbyeithertheirblockingperformance,oraveragestarttimeoftherequestsscheduledinthelastslot,oracombinationofthetwo.Comparingthestatisticalperformanceofbothalgorithmsinthelastslot,thealgorithmthatperformedbetterinthelastslottakeseffectinthecurrentslot.Ifitdoesnotperformasgoodastheotheroneincurrentslot,theschedulewillswitchtoitsalternativeatthebeginningofthenextslot. ABlockingPerformance BStartTimePerformance Figure4-18. TheperformanceofalgorithmswitchingstrategyinSlowTrafcPatternSwitching. 95

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ABlockingPerformance BStartTimePerformance Figure4-19. TheperformancealgorithmswitchingstrategyinFastTrafcPatternSwitching. Toevaluatetheperformanceofouralgorithmswitchstrategy,wedesigned2scenarios:slowtrafcpatternswitching(STPS)andfasttrafcpatternswitching(FTPS).Inourtest,weassumetherequestarrivalrateisrandomlyselectedfromtherange[0.01,0.1]andthelengthofeachtimeslotis10timeunits.InSTPSscenario,thearrivalratechangesitsvaluewith50%probabilityevery100timeunites.InFTPSscenario,thearrivalratechangesitsrangewith50%probabilityevery10timeunits. Figure 4-18 andFigure 4-19 depictstheblockingandstarttimeperformanceofourswitchstrategyinbothscenariosina100-noderandomtopology.DS-SSalgorithmstandsfortheswitchstrategyworkedintheSTPSscenario,whileDS-FSalgorithmstandsfortheswitchstrategyworkedintheFTPSscenario.TheresultsshowthatthehybridalgorithmworkedinSTPSscenario,DS-SS,havethebestperformanceonbothmetrics,asthehistoryinformationfromlastslotpredictthecurrenttrafcpatternquiteaccurately.Ontheotherhand,theperformanceofDS-FSalgorithmwhichworkedinFTPSscenariodegradeddramaticallyduetotheincreasingprobabilityofinaccuratepredictionsinFTPSscenario.Insomecases,DS-FSevenprovidestheworstperformance.So,whenthetrafcpatternremainstaticorchangedinfrequently,ouralgorithmswitchstrategycanprovideaprettygoodperformancebysyntheticthemeritsofbothalgorithms.However,whenthetrafcpatternischangingfrequently,theswitchstrategydoesnotguaranteeanyimprovementcomparingtoeitherEBF-SorKDP-S. 96

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4.4OpticalNetworkSchedulingSummary Inthischapter,weconductedextensivesimulationstoevaluatetheperformanceofalgorithmsforavarietyofrequestpatternsandnetworktopologies.OurresultsshowthatExtendedBellmanFord(EBF)algorithmhasconsistentlybetterperformancethatotheralgorithms.Fornon-homogonousnetworks,LSWalsoprovidedcomparablesolutions;whileforhomogeneousnetworksMSPFandMSWFprovidecomparablesolutions.Oursimulationswereperformedforpresenceorabsenceofconvertersthatcanbeusedtoconvertfromagivenwavelengthtoanotherwavelength.Ourexperimentalresultsshowedthattheuseofwavelengthconvertersgenerallyledtobetterperformance.Thus,theadditionalexibilitythatwavelengthconvertersprovideinanetworkisworthwhile.Wealsoshowedthatadeferredwavelengthassignmentstrategycanbeeffectivelyusedinconjunctionwiththeroutingalgorithms.Adeferredstrategyonlycountsthenumberofwavelengthsthatareusedonalink.Since,theactualassignmentofthewavelengthisdoneatthetimeoftherequestfulllment,thisalleviatestheneedofkeepingtrackofbandwidthallocationstatusofeachwavelength.Adeferredwavelengthstrategyalwaysguaranteestondafeasiblesolutionaslongasthetotalamountofreservedbandwidthdoesnotexceedthetotalcapacityofbandwidthofalinkataspecicpointoftime. Inthischapter,wealsoexaminedtheimpactofsparsewavelengthconversiononFirst-Slotscheduling.Weproposedanewnetworkmodeltoemulatethefull-conversionalgorithmsinsparseconversionnetworks.Usingthismodel,weconductedextensiveexperimentstoassesstheimpactofwavelengthconvertersonFirst-SlotRWAalgorithms'performance.Thisassessmentusedthreemetrics:blockingprobability,averagestarttimeandschedulingoverhead.Ourexperimentshaveindicatethatincreasingwavelengthconvertershaspositiveimpactonblockingperformance,butverylittleimpactontheavailabilityofearlierstarttimes.Meanwhile,asmostimprovementsareachievedbyhavingnomorethan60%nodeswithconverters,deployingtoomany 97

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wavelengthconvertersmaynotbeworththeadditionalcost.WealsoproposedaSlacktie-breakingschemewhenmultiplefeasiblepathsareavailable.Thistie-breakingschemeisshowntohavemuchbetterperformancethanthetraditionalFirst-FitorShortest-Pathtie-breakingschemes.ThecomparisonsbetweenEBFandKDPalsoleadtotheconclusionthatacceptingrequestsgreedilywouldprovidebetterperformancesinlowtrafcloadcase,butrejectingsomerequestswithlongpathcanbeasuperiorstrategywhenworkloadishigh.Analgorithmswitchingstrategythatadaptstheschedulingalgorithmasthecurrentworkloadchangesisproposed.Whenthenetworktrafcpatternchangesslowly,thisstrategyhasconsiderableadvantageoverstaticalgorithms.Overall,ourresultsshowthataddingasmallnumberofwavelengthconvertersmayhavealimitpositiveimpactonFirst-Slotscheduling.However,thisimpactshouldbecarefullyweightedagainsttheadditionalcost. 98

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CHAPTER5MULTIPLERESOURCESCHEDULING 5.1ProblemDenition Manycomplexe-Scienceapplicationsrequirealargeamountofschedulableresourceswhicharesubjectedtodynamicchanges.Theabilitytoreservevarioustypesofcomputationalresourcessuchasbandwidthchannels,CPU,memoryanddiskspacehasbecomeakeyrequirementforoveralleffectivenessforthee-Sciencecommunity.Tomeetthisneed,weproposeaframeworkforconductingadvancereservations,admissioncontrol,andschedulingofnetworkservicerequestsalongwithotherresourcessuchasCPUs,memory,diskspace,andsoftwarelicenses.Thisframeworkissegregatedfromtheschedulingofnetworkresourcessuchasnetworkbandwidth.Forexample,LSF-HPC[ 44 ]andMaui[ 42 ],whicharepopularhigh-performancecomputingschedulersforclustersandsupercomputers,donotschedulenetworkresources,althoughtheyareabletodoadvancereservationofresourcessuchasCPUs,memory,diskspace,andsoftwarelicenses[3].ThesameistrueforCondor(UniversityofWisconsin'shighthroughputscheduler),PBS(portablebatchsystem-theparallelschedulerfortheIBMSP2)andVMWareESX.Ontheotherhand,networkbandwidthmanagementsystemssuchasthoseforUltraScienceNet(USN)andESnetdonotschedulecomputerresources.TheSharcsystem[ 63 ]modeledbothnetworkandCPUresourcesintheclustersasuniedresourcesblocks,buttheconstrainsamongresources,suchasnetworktopology,resourcecompatibilitiesarenotconsidered. Weenvisionanenvironmentinwhichacomputationalnetworkcontainshundredsofnodes,andcomputationalresourceswithdifferentplatforms.Whenmultipleresourcesarereserved,theirtopologies,dependencies,andcompatibilitiesmustbehandled.Thepurposeofourpaperistodevelopaco-schedulerthatsimultaneouslyschedules 99

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multipletypesofresourceswithanetworkfocusbasedonaourMultipleResourcesReservationModel(MRRM).WithMRRM,wecan 1. givemultipleheterogeneousresourcesauniedpresentation,butstillkeeptheirdiversity; 2. efcientlypresentthevarioustypesofconstraintsamongdifferentresources,suchascompatibility,accessibility,andassignability;aswellas 3. fordifferenttypeofuser'requests,themodelcanbeexibleenoughtoadjusttheresourcepresentationforbetterschedulingefciency. WedeneaMultipleResourceFirstSlotproblem(MRFS)withtheobjectiveofdeterminingtheearliesttimethatcanbeusedtoreserveallresourcesrequiredtocomputeagivenrequest.WedividetheMRFSproblemintofoursub-problems,basedon(a)whethertherequestconsistsofmultiplesubtasksthatcanbeassignedtodifferentresources,and(b)whethereachresourcecanbeassignedindependently. Insystemandarchitectureeld,manypaperalsofocusedonresourcereservation.[ 20 ]proposedtheGARAframeworkthatallowmultipleresourcestobemonitoredandreservedinaClustersystem.InGARA,allresourcesareconsiderasuniedblockssothatonesinglealgorithmisenoughtohandleallresources.However,GARAisonlyaframeworkforresourcemanagement,Nospecicalgorithmismentioned.[ 62 ]modeleddifferentresourcesasqueues.Onerequestisallowedtoenterthequeueiffcurrentresourcefullltheuser'srequest.[ 62 ]alsogroupsresourcesofsametypeinasinglelayersuchthatrequestinlayericanonlyaccesslayeri+1orlayeri)]TJ /F6 11.955 Tf 12.87 0 Td[(1afteritsworkinlayerinished.Actually,thislayeredmodelisnotonlywidelyusedinmosthardwarearchitectures,butisalsoappliedtomanysoftwares,suchasTCP/IPprotocolstack.Inthispaper,wealsousesuchlayeredmodeltopresentourlocalcomputationalresources.However,[ 62 ]doesnothandlethedependencyandcompatibilityconstrainsatthesamelevelofdetailordelitythatourMRRMformalismdoes.In[ 7 ],theresourcesaregroupedaccordingtotheirtyperatherthantheirphysical 100

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Figure5-1. GeneralModelofMRRM residency,whichbroughtconvenienceforcentralizedscheduling,alsotheresourceaccessibilityproblemisproposed,butonlythefullyaccessiblemodelisused. 5.2ResourceModelandDataStructure 5.2.1ResourceModel:MRRM Whendevelopingaresourcereservationsystem,onemustrstdesignarepresentationalformalism.Forexample,acomputationalsystemcanbeviewedasanetworkwithcomputationorstoragecentersattachedtosomeofitsedgenodes.Datasetsaregeneratedorprocessedusingtheseattachedresources,andaretransmittedthroughthenetworkfromoneedgeroutertoanother. Forthevastmajorityofe-Scienceapplications,resourcescanbeclassiedintonetworkresourcesandlocalcomputationalresources.Networkresourcestransferuserdatafromonesitetoanother,andincludebutarenotlimitedtoopticallinks,routersandswitches.LocalcomputationalresourcesincludeCPU,memory,harddiskandotherresourcesusedinprocessinguserrequests.Figure 5-1 providesageneralviewofMRRM'sbasicstructure. AsmentionedinSection 5.1 ,oneofthemajorgoalofourresourcemodelistoprovideauniedviewformultipletypesofresources.OurMRRMrepresentsanetworkintermsofagraphG=(V,E).TheorganizationofGmirrorstheconnectivityofthecomputationalnetworkbeingmodeled.EachswitchorrouterisrepresentedasanodeinV,whileeachnetworklinkismappedtoanedgeinE.Behindtheedgerouters,eachcomputationorstoragecentergroupsitsownlocalcomputationalresourcesinrelatedclusters.Eachclusterisrepresentedasasub-graphattachedtooneoftheedge 101

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routers.Asingleresourceunitismodeledasaresourcelinkandisassociatedwithanedge.Wechoosethisrepresentationforthefollowingreasons: 1. Itisnaturaltopresentnetworksasgraphsandassociatelinkbandwidthwithedgesinthegraph,asinmajorityofnetworkmodels. 2. Inthereservationaspect,mappingaCPUnode,amemoryblockoraharddiskintoaedgeinthegraphisfeasible,sinceourmainmodelingconcernpertainstorepresentingtheamountofavailableresources.WecanthuspresentaCPU'scapacityinthesamemannerasnetworklinkbandwidth. 3. Thecompatibilityanddependencyconstrainsamongdifferenttypesofresourcescanberepresentedasconnectivityconstraintsamongresourcelinksinthegraph. 4. Auniedviewofdifferentresourcetypesfacilitatesalgorithmdesignformultipleresourcesscheduling.Wecanextendtheknowngraph-based,singleresourceschedulingalgorithms[ 25 48 52 ]tothecurrentscenario,whileinheritingtheirattributes. Acomputationalsystemisthereforemodeledasagraphwiththecommunicationnetworkinthemiddle,andcomputationalresourcesattachedtonetworkedgenodes,asshowninFigure 5-1 .However,incontrasttotraditionalgraphrepresentationsofanetwork,twonewresearchchallengesemergedfromtheMRRMrepresentation: 1. Howdowemodelheterogeneousresourceswithauniedrepresentationthatdoesnotreduceoreliminatetheirdiversity? 2. Howdowemodelthecompatibilityandassignabilityconstraintsamongdifferentresources? Tosolvetheheterogeneityproblem,werstassigneachresourcelinkatypeID(T)]TJ /F4 11.955 Tf 12.1 0 Td[(ID),tospecifyitstype-forexample,CPU,Memory,ormoregenerally,Resource1,Resource2,etc.WiththetypeID,alllocalresourcescanbegroupedintooneofseveralmulti-partitionedresourceconstraintgraphs(MPRCGs),whichenablesresourceswiththesametypetobemanagedasonegroup.Thisgroupingstrategyfulllstherequirementsofauserrequestwithrespecttoresourcegranularity,asopposedtoasingleresourceunit.Typically,auseronlycaresaboutthequantityofagivenresourcetypethatwillbeavailable,butnotwhetherhisjobwillberunonagivenCPUnode 102

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orstoredinaparticularmemoryblock.Thisisreasonableinpractice,becausetheorganizationsofsupercomputersvaryfromonearchitecturetoanother,butshareasimilarlayeredstructure.Forexample,aCPUcanaccessmemorydirectly,butnotaharddisk.Ifapagefaultoccurs,thenthediskwillwritetothememory,butnotdirectlythroughthememorytotheCPU. CompatibilityisanotherimportantissueinMRRM.Usually,wehavemultiplechoiceswithinagivenresourcetype.However,notallresourceunitsofagiventypeareappropriatetoperformauser-orsystem-denedtask.Forexample,ifauserprogramiswritteninnativeIntelx86code,thenitwilllikelynotrunonaMIPSCPU.Inthiscase,computationalnodesthatdonotsupportIntelx86codeshouldbeexcludedforreservationbytheco-scheduler.Similarly,ifaprogramcanbeparallelized,thentheCPUsinvolvedinparallelcomputationmustbecompatibleintermsofcodetypeandmethodofdatasharingacrossmultipleresourceunitsinMRRM.Thus,wesupplyacompatibilityID(C-ID)toeachresource,whichfacilitatesgroupingofresourceswiththesameT-IDintodifferentcompatibilityclasses.Inourcurrentapproach,auserspeciesajobintermsofitscompatibilityID,andallresourceswiththesameC-IDwillbeconsideredforreservation.IfnoC-IDisspecied,thenalltheresourcesclassesareavailableforthisrequest. Afurtherconsiderationinvolvesaccessibility.Forexample,somecomputersuseDistributedSharedMemorytoprovideallCPUnodesfullaccesstothememorymodel.However,othersystemsonlyallowcertainCPUstoaccessspecicmemorypartitions,duetosecurityconcernsorphysicalconnectivity.Asimilarsituationalsocanexistwithmemoryandharddiskconnections.OurMRRMiscapableofsimulatingeachofthesescenarios.IftworesourcelinkswithindifferentMPRCGsareaccessibletoeachother,thenaspecicauxiliarylinkwithunlimitedcapacityconnectsthetwolinks.IfallresourcesinoneMPRCGareaccessibletoallresourcesinanotherMPRCG,thenaconnectionismadeattheMPRCGlevel,sothateveryresourcelinkineitherMPRCG 103

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Figure5-2. DetailedModelofMRRM isconnectedwitheveryresourcelinkintheotherMPRCG.Becausetheseauxiliarypathshaveunlimitedcapacity,theywillonlyidentifyresourceaccessibility,andwillnotinuencethereservationprocess. Figure 5-2 providesadetailedexampleofMRRMonalocalsite,whereallresourceswiththesametypeIDaregroupedwithinanMPRCG.ResourcesinoneMPRCGneedonlyaccesstheresourcesinthatMPRCG'sneighborhood.WithineachreservationMPRCG,resourceswiththesamecompatibilityIDarere-groupedtogethersuchthatparallelprogramscanrunusingallresourcesinanyoneoftheinnercycles.IftwoMPRCGsarecompletelyaccessibletoeachother,thenwe(a)addadummynodebetweentheMPRCGs,and(b)addanauxiliarylinkwithunlimitedcapacitybetweeneachresourcelink'sendnodeandthedummynode.InFigure2,thisprocessisshownbetweenresourcetypesAandB.Addingthisdummynode,asopposedtoconstructingafullyconnectedbipartitegraphdirectlybetweentwoMPRCGs,reducesthealgorithmscomplexitybyreducingthenumberofedgesinthegraph.Forexample,supposewehaveNresourcelinks.Then,afullyconnectedbipartitegraphrequiresN2auxiliarylinks,butonly2Nauxiliarylinkswhenanextradummynodeisadded.IncasetheresourcesinoneMPRCGcanonlyaccesspartoftheresourcesinitsneighborhood,thenadirectauxiliarylinkisrequiredtorepresentthisaccessibilitybetweentwodifferentresources.InFigure 5-2 ,thisisrepresentedbetweenresourcetypesBandC. 104

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AlsonoticethatwehavealltheresourcelinksinoneendofanMPRCG(e.g.,left-mostorright-mostMPRCGinFigure 5-2 )connectedtotheedgerouterwiththesameauxiliarylinks,soastohaveallcomputationalresourcesattachedtothenetwork.Wealsoaddadummynodeasasinknodeinthelocalgraph,suchthatresourcelinksattheoppositeendoftheMPRCGwillconnecttothisdummynode.Thus,allresources,whethernetworkorlocalcomputationalresources,areconnectedtoprovideasingleMRRMgraph.Tofulllauserjobschedulingrequest,theschedulingalgorithmthenattemptstondasingleormultiplepathfromthesourcesinknodetothedestinationsinknode.Thiscanprovideasmanyresourcesfromallresourcetypesastheuserspeciesforjobcompletion. Finally,withoutlossofgenerality,weassumethatinonesite,itslocalcomputationalresourcescaneitherbeallfullyaccessibleorallpartlyaccessible. 5.2.2DataStructures Anotherimportantaspectofourresourcemodelisitstemporalrepresentation,whichdescribestime-varyingresourcequantities.Theoptionsaretoeitherconsidertimeasdividedintoequalsizeslotsasisdonein[ 12 19 23 58 ]ortoconsidertimeasbeingcontinuousasin[ 32 46 48 52 ].Theslottedtimemodelusesanarrayforeachedgeinthegraphtorecorditsresourcestatusforeachtimeslot.Forexample,wemayuseatwodimensionalarrayRsuchthatR[d,t]givesthediskspaceavailableonharddiskdinslott. Inthecontinuoustimemodel,thestatusofeachresourceunitismaintainedusingatime-resourcelist(TRlist)thatsimulatetheTBListinthepreviouschapters.wechosecontinuoustimemodelwithTRListtopresentthetimedomainandthecorrespondingresourcequantity.Theadvantageofcontinuoustimedomainare:(i)continuoustimemodelisamorenaturalrepresentationforthetime.Thechangeoflinkbandwidthmayapplytoanypointinthetimedomain,whichisunabletobedescribedintheslottedtimemodel.(ii)thereisnoneedtopickatimegranularityortoplaceaboundonthe 105

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lengthofthelookaheadperiod.(iii)Theamountofmemoryrequiredtorepresentalinkstate(i.e.,theTRlist)isonlyafunctionofthetimevariationinresourceavailabilityratherthanthelengthschedulinghorizonTorthelengthoftheunittime.Moreover,therun-timeofreservationalgorithmsisafunctionofthesizeoftheTRlists.Thissizedependsonthenumberoftasksthathavebeenscheduled.Thelimitationsofthecontinuoustimemodelincludeitsrelativecomplexity.Ifarrayisused,thecomplexityofdeterminingthestatusofalinkatanygiventimewillbeO(logjTR[l]j)usingbinarysearch.Meanwhile,thesameoperationinslottedtimemodelonlytakesO(1)time.Becauseofthecorrespondencebetweenaslotandtime,weoftenusethetwotermsinterchangeably. Wealsoemploytwootherdatastructures:SteadyStageandST(StartTime)List.SteadyStagefromthepreviouschapters.ThewholeMRRM'schangingstatusovertimecanbeshownasaseriesofsteadystages.RecallthatthecapacityofalinkispresentedbyaTRlist,whichisaarrayoftime-resourcetuples.IfonlyconsiderthetimepartoftheTRlist,t0,t1,t2...tq,anytimeinterval[ti,ti+1]formsasteadystageofthatlink.Hence,ifweunionthetimepartsofeachresourcelink'sTRlisttogether,wewillobtainaglobaltimelist.LetT0
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ForaMRFSrequest,theuserwantstohavetheresourcereadyasearlyaspossiblewithinaReservationTimeWindow.Thiswindowspecifytheearlieststarttimeandlatestnishtimeoftheuser'sjob.AMRFSrequestisa6-tuple(s,d,dur,ResWin,RV,shareable).sanddarethesourceanddestinationnodeofthedatatransfer,thecomputationalresourcesattachedtosanddarethecomputationalresourcesthataregoingtobereserved.duristhedurationthatthoseresourcesneedstobereserved.ResWinisthereservationwindow,userjobmuststartandendwithinthiswindow,whichmeansthejob'sstarttimemustbeinthetimeinterval[st,et)]TJ /F4 11.955 Tf 13.04 0 Td[(dur].RVisavectorthatcontainstheallresourcerequirements.EachelementinRVspeciestheamountofonecertaintypeofresourcesthatneedtobereserved.Conventionally,weusetherstelementRV[0]topresenttherequiredbandwidthfromstod.ShareableisaBooleantagthatindicatewhetherthejob'sworkloadcanbesplitamongdifferentresourceunits.Ifshareableistrue,thentheuserrequestcanbefullledwithaggregatedcomputationalresourcesandmultiplenetworkpaths.Otherwise,wecanreservemultipleresourcesunitsineachlocalresourcestageandmulti-pathsinthenetwork. ForaMRFSrequesttobeaccepted,theschedulerneedstondouttheminimumt2[st,et)]TJ /F4 11.955 Tf 12.49 0 Td[(dur],suchthatwithin[t,t+dur],asinglepath/multipaththatconnectssanddcanprovideabandwidth/aggregatedbandwidthofRV[0].Also,withinthesametimeinterval,ineachresourcestagethatattachedtosandd,atleastoneresourceunit/compatiblesetcanprovideenoughresource/aggregatedbandwidthtofulllthecorrespondingrequirement. 5.4MultipleResourceSchedulingAlgorithm Basedonwhetherjobcanbesplitandwhetherthelocalresourcesarefullyaccessible,wedividedtheMRFSprobleminto4sub-problems: 107

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WS-RC:Theworkloadcanbesplit,andlocalcomputationalresourcesareconstrainedonaccessibility,whichmeansresourcesinonelayercanonlyaccesspartoftheresourcesinitsneighborhoodlayer. WN-RC:Theworkloadcannotbesplit,butlocalcomputationalresourcesareconstrainedonaccessibility. WS-RN:Theworkloadcanbesplit,andlocalcomputationalresourcesarenotconstrainedonaccessibility. WN-RN:Theworkloadcannotbesplit,butlocalcomputationalresourcesarenotconstrainedonaccessibility. FortheMRFSschedulingproblem,everytimearequestarrivesatthecentralizedcontroller,thecentralizedcontrollerwillselectthecorrespondingschedulingalgorithmaccordingtotheshareableagintherequestandthelocalresources'accessibility.Thisalgorithmwillcomputetheearliestnishtimethatfulllthejob'sresourceanddurationrequirement.Inthissection,wewillgivethedetailofall4algorithmsthatcorrespondtothe4sub-problemsproposedabove. 5.4.1WS)]TJ /F4 11.955 Tf 11.95 0 Td[(RCSchedulingAlgorithm Iftheworkloadcanbesplit,thengivenrequestriandthemulti-resourcegraphG,thealgorithmdiscoversthemaximumowfromsource'ssinktodestination'ssinkforeachbasicintervalwithinthereservationwindow.Theschedulerthenattemptstoidentify(a)ifthereareenoughresourceswithinthecurrentbasicintervalBIi[iftrue,BIiismarkedasfeasible];and(b)whetherornotthereexistsoneconsecutive(i.e.,temporallyconnected)sequencesofbasicintervals[BIi,BIi+1,...,BIj]withtotallengthlongerthantherequireddurationdur.Ifsuchasequenceisfound,thentheearliestpossiblestarttimeofthesequencebecomestherstpossiblestarttimeoftheuser'srequest. Whenrunningthemaximumowalgorithminourmultipleresourcegraph,theoriginalalgorithmcannotbeapplieddirectly.Imagingthemultipathfroms'ssinktod's 108

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sink,itmaygothroughresourcelinkswithvarioustypes.Theowalonesuchmultipathwillnotmakeanysenseontheresourcecapacityfromstod.Tosolvethisproblem,wescalethelinkcapacityofcertaintypes'resourcelinks.Foreachsinglerequest,wewillproduceacopyofcurrentmodel,let'ssayG0.Then,wewillchosetherequirementofoneresourceasbasevalue,andalltheotherresourcelinkswithdifferenttypewillscaletheircapacitynumericallybytheratiooftheircorrespondingrequirementstothebasevalue.Forexample,wechosenetworkbandwidthasourbaseresource.Letbandwidth'srequestbe10(MB=s)andCPU'srequestbe5(GHz).Inthiscase,wewillscalealltheCPUresourcelinks'capacitybyafactorof2andwealsoneedtosettheCPUtorequirementfrom5GHzto10GHz.Inthisway,weachievethenumericunicationamongdifferenttypeofresources,whichenablesthetraditionalmaximumowtorundirectlyonourscaledgraph.Aftertheowcomputation,theowsizeiscomparedwiththebasevalue,ifowsizeislargerthanbasevalue,currentsteadystageisconsideredtobefeasiblefortherequest.ThedetailofWS)]TJ /F4 11.955 Tf 11.47 0 Td[(RCSchedulingAlgorithmisshownbelow. Hereweusethemin-cutalgorithm[ 6 ]tosolvethemax-owproblem.ThecomplexityoftheaboveWS)]TJ /F4 11.955 Tf 12.94 0 Td[(RCschedulingalgorithmisO(jSSRWjN3),wherejSSRWjisthesizeofsteadystagelistwhatiswithinthereservationwindowandN=NN+Ns+Nd.NNisthenumberofnodesinthenetwork.NsandNdthenumberoflocalcomputationalresourcesattachedtosandd.WealsonotethatanypathintheMRRMmodelcanbedividedinto3separateparts:thepaththroughthelocalresourcesattachedtos,thenetworkpathandthepaththroughthelocalresourcesattachedtod.These3partsareindependentwitheachother,sotheowonthese3partscanbecalculatedseparately.Theglobalmaximumowequalstotheminimumofthethree.Withthisheuristic,thecomplexityofWS)]TJ /F4 11.955 Tf 12.26 0 Td[(RCschedulingalgorithmcanbereducedtoO(jSSRWj(Ns3+NN3+Nd3). 109

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WS-RCScheduling(reqi,G) { Randomlychooseresourcerkasbaseresource; G0=Scale(G,rk); BuildtheglobaltimelistLfromG',removealltheTioutsidetheschedulingwindowreqi.ResWin; IdentifyalltheSteadyStages; ForeachSteadyStageSSi { MFi=MaxFlow(G0,SSi); if(MFibasevalue) MarkSSiasafeasibleSteadyStage; } TraversetheSteadyStagelistagain.ndouttherstconsecutivefeasiblesteadystageslistwhichislongerthanreqi.dur; if(suchlistexist); accepttherequest; setlist'sthestarttimeasrequest'sstarttime; else rejecttherequest; } Figure5-3. WS)]TJ /F4 11.955 Tf 11.95 0 Td[(RCSchedulingAlgorithm 5.4.2WN)]TJ /F4 11.955 Tf 11.95 0 Td[(RCSchedulingAlgorithm IntheWN)]TJ /F4 11.955 Tf 12.23 0 Td[(RCcase,theworkloadcanonlybetransferredonasinglepathinthenetworkandcanonlybeprocessedusingasingleunitofeachtypelocalcomputationalresource.InourMRRMmodel,schedulingaWN)]TJ /F4 11.955 Tf 12.76 0 Td[(RCistondasinglepathfromsource'ssinktodestination'ssink.Here,wewillusetheExtendedBellman-Fordalgorithmproposedin[ 52 ]tosolvethisproblem. First,wewillextendtheconceptofanSTlistforanedgeapath.Letst(k,u)betheunionoftheSTlistsforallpathsfromvertexstovertexuthathaveatmostkedgesonthem.Clearly,st(0,u)=;foru6=sandweassumest(0,s)=[0,1].Also,st(1,u)=ST(s,u)foru6=sandst(1,s)=st(0,s).Fork>1(actuallyalsofork=1), 110

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weobtainthefollowingrecurrence st(k,u)=st(k)]TJ /F6 11.955 Tf 11.96 0 Td[(1,u)[f[v:(v,u)isanedgefst(k)]TJ /F6 11.955 Tf 11.96 0 Td[(1,v)\ST(v,u)gg (5) where[and\arelistunionandintersectionoperations.Forann-vertexgraph,st(n)]TJ /F6 11.955 Tf -448.71 -23.91 Td[(1,d)givesthestarttimesofallfeasiblepathsfromstod.TheBellman-Fordalgorithm[ 51 ]maybeextendedtocomputest(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1,d). Itiseasytoseethatthecomputationofthest(,)smaybedoneinplace(i.e.,st(k,u)overwritingst(k)]TJ /F6 11.955 Tf 13.14 0 Td[(1,u))andthecomputationoftheststerminatedwhenst(k)]TJ /F6 11.955 Tf 12.41 0 Td[(1,u)=st(k,u)forallu.Withtheaboveobservation,herewegivethedetailofExtendedBellman-Fordalgorithm. EachiterationoftheforlooptakesO(L)time,whereListhelengthofthelongeststlist.SincethisforloopisiteratedatotalofO(NE)times,thecomplexityoftheextendedBellman-FordalgorithmisO(NEL),whereNandEisthenumberofnodesandlinksinthewholemultipleresourcegraph. WhenusingtheextendedBellman-Fordalgorithmtosolvetherstslotproblem,werstndtheearlieststarttimetforafeasiblepathusingExtendedBellmanFord.Then,theactualpathmaybecomputedusingBFSwherethefeasibilityofeachlinkiscomputedbyxedthejob'ststart=tandtend=t+dur.AlsoBFSguaranteedtondtheshortestfeasiblepathinthegraph. 5.4.3WS)]TJ /F4 11.955 Tf 11.95 0 Td[(RNSchedulingAlgorithm IntheWS)]TJ /F4 11.955 Tf 12.97 0 Td[(RNscenario,theWS)]TJ /F4 11.955 Tf 12.97 0 Td[(RCschedulingalgorithmcandirectlybeapplied.However,thisdoesnotexploitthefullaccessibilityoflocalresourcesandtheMPRCGgraphalthoughthiswouldgreatlysimplifyourschedulingprocessforthelocalcomputationalresources.TheWS)]TJ /F4 11.955 Tf 11.95 0 Td[(RNalgorithmproceedsasfollows. Firstly,thenetworkpathcomputationproceedsseparatelyfromlocalresourcepathcomputation.Whencomputingthefeasibilityofagivenbasicinterval,themax-ow 111

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ExtendedBellman-Ford(s,d) { initializest()=st(0,); //computest()=st(n)]TJ /F6 11.955 Tf 11.95 0 Td[(1,) putthesourcevertexintolist1; for(intk=1;k
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Inthiscase,wedonotneedtoscaleourmultipleresourcegraphandwedonotneedtorunthemax-owalgorithmonthelocalcomputationalresources.Thus,weonlycheckeachlocalresourcestageforeachcorrespondingrequestbyvisitingeachresourcelinkforO(1)times.Sincethelistunionandintersectioncanalsobenishedinlineartime,wecanreducethealgorithmruntimetoO(jSSRWj(Ns+NN3+Nd). 5.4.4WN)]TJ /F4 11.955 Tf 11.95 0 Td[(RNSchedulingAlgorithm TheWN)]TJ /F4 11.955 Tf 12.36 0 Td[(RNproblemissimilartothepreviousWS)]TJ /F4 11.955 Tf 12.36 0 Td[(RNproblem.Inparticular,anetworkpathcanbecomputedviatheExtendedBellman-Fordalgorithmtoyieldtherststarttime.Thiscomputationisfollowedbybreadth-rstsearchtoidentifythepath.Forlocalcomputationalresources,wecanapplythesameapproachasinWS)]TJ /F4 11.955 Tf 12.5 0 Td[(RN.However,WN)]TJ /F4 11.955 Tf 12.02 0 Td[(RNneglectsthegroupingofresourcesaccordingtocompatibility.Sincetherequestedjobcannotbesplit,onlyoneresourceunitineachresourcepartitionisrequired.Inthatcase,thealgorithm'scomplexityisboundedbyO(NNENL+Ns+Nd) 5.5Evaluation 5.5.1EvaluationEnvironment WetestedourworkonaUSNETsimulatoratOakRidgeNationalLaboratory(ORNL).OuralgorithmswereintegratedwiththemiddlewarethatiscurrentlyusedatORNLandUniversityofMemphis,inordertoco-schedulenetwork,processorandstorageresourcessimultaneously.However,thecurrenttestbedatORNLhaslimitedstorageandcomputeresources.Thus,themaintestobjectivewastoensurethatthenetworkbandwidthschedulingcapabilitycorrectlyreservedbandwidthonthesoftwaretestbed.Basedonextensivetesting,ouralgorithmswereabletoeffectivelyreservebandwidth,aswellasgeneratereservationinstructionsforthevirtualprocessorandstoragesystemsthatweassumedtobepresentontheUSNETendnodes. Wemeasuredtheperformanceofthe4multipleresourceschedulingalgorithmsonrandomgeneratednetwork.Forourexperiments,therandomnetworkswetriedhad100,200,300,400or500nodesandtheout-degreeofeachnodewasrandomly 113

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selectedtobebetween3and7.Toensurenetworkconnectivity,therandomnetworkhasbidirectionallinksbetweennodesiandi+1forevery1i
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AAcceptanceratiovs.requestdensity BAcceptanceratiovs.networksize Figure5-5. Ourco-schedulingalgorithms'performanceonacceptanceratio. Acomputetimevs.requestdensity Bcomputetimevs.networksize Figure5-6. Ourco-schedulingalgorithms'performanceonconvergespeed. theaverageacceptanceratioandalgorithmruntimeasafunctionofnetworksize.Theresultisacquiredunderarequestdensityof3requestspersecond. Ourexperimentresultsshowthefollowing: 1. Theincreaseofrequestdensitywilldegradeeveryalgorithms'performance.Asmorerequestcomeintothesystemwithinthesametimeinterval,thenetworkbecomescongested,hencemorerequestswererejectedsincenotenoughresourcesareavailable.Inthemeantime,asmorerequestsarerunninginthesystemsimultaneously,thelengthofGlobaltimelistandSTlistincreases,whichleadstolongeralgorithms'runtime. 2. Asthenetworksizeincreases,thesystemgainslargercapacitytoaffordmorejobsrunningwithincertaintimeperiod.Sotherequests'acceptanceratioactuallyincreasedtogetherwiththenetworksize.However,theincreaseofnetworksizemakesthemax)]TJ /F4 11.955 Tf 12.11 0 Td[(owalgorithmandEBFalgorithmtakemoretimetoconverge.So,thealgorithmruntimestillincreased. 3. Whenmultiplepathsareallowedandresourcesarefullyshared,theschedulercanbetterutilizesystemresources,soastoacceptmorerequests.However,the 115

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resultingmulti-pathalgorithmsrequiremorecomputationtimetoobtainafeasibleresult. 4. Generallyspeaking,all4algorithmsscalesverywellwitheithersystemsizeorrequestdensity.Evenwhentheworkloadishigh,onerequestaveragelytakeslessthanaminutetondoutschedulingresult. 116

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CHAPTER6SCHEDULINGINTIME-DOMAINWAVELENGTHINTERLEAEDNETWORKS 6.1ProblemDenition Time-domainWavelengthInterleavedNetworking(TWIN)isanoptics-basedtransportnetworkarchitecturethataimstoprovidecosteffectiveopticalgrooming[ 43 50 65 ].Traditionalopticalnetworksworkinoneofthefollowingtwomodes:opticalcircuitswitching(OCS)oropticalpacketswitching(OPS).InOCSnetworks,thenestbandwidthgranularityofferedbyanopticalswitchisatawavelengthlevel,i.e.onesinglewavelengthonabercanbeusedbyonlyoneend-to-endtrafcandcannotbesharedwithothertrafc.Thisisnoteffectivewhenthetrafcdemandismuchlowerthanthewavelengthcapacity.Attheotherextreme,OPSnetworkspermitsharingofopticallinksbytrafcwithdifferentsourcesanddestinations.Thesenetworks,whichareenabledbyoptical-electronic-optical(OEO)conversionateachnodeinthenetwork,tendtoincurarelativelyhighsystemcostandtransmissiondelay,asOEOconvertersaregenerallyexpensiveandtheconversionprocessistime-consumingcomparingtodirectcircuitswitching.SometechniqueshavebeenintroducedtoimprovetheutilizationofopticallinksbysimulatingOPSoverOCS,suchasopticalburstswitching(OBS)[ 60 ].However,OBSstillneedshigh-speedopticalswitchesandacontentionalgorithmateachswitch. Widjjaetal.etc.proposedTWINtoovercomelinkutilizationproblemsinOCSbutavoidthehighcostanddelaysresultingfromOEOconvertersbeingdeployedatalltheopticalswitches[ 65 ].TWINperformsopticalgroomingonlyatitsedgeswitchesandthenetworkcoreispurelybasedonpassivewavelength-selectiveswitches(WSS)thatroutethewavelengthsfromtheiringressportstotheappropriateegressports[ 55 ].InaTWINnetwork,theedgenodescanbeeithersourcesordestinations.Atransmitterwithamulti-frequencylaserislocatedateachsourcenode.Withthistransmitter,sourcenodescanchangethewavelengthoftheiropticalsignalinsub-nanoseconds[ 34 ].Sourcenodescollectdataunitsfromvariousclientsandassembledataunitsforthesame 117

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destinationintooneburst.Whensendingtheburst,thesourcechangesitsfast-tunablelasertothewavelengthuniquelyassignedtothatdestination.Theintermediatenodesrouteopticalburstsbasedpurelyonthewavelengthoftheburst.Whentheburstisreceivedatitsdestination,itisdisassembledandforwardedtothecorrespondingclients.Ascurrentopticalswitchescannotseparatetheburststhatsharethesamewavelength,onlytrafcwiththesamedestinationmayshareawavelengthinthetime-domain.Thisconstraintleadstotree-likeroutesinthenetworkforeverydestination,wherethedestinationistherootandthesourcesaretheleaves. Inthischapter,wediscussthewavelengthassignmentproblemforTWINnetworks(TWIN-WA).Wesolvethisproblemusingatwo-phaseprocess:TreeConstructionandTree-WavelengthAssignment.Tree-Constructiongroupsthetrafcdemandswiththesamedestinationtogetherandconstructsthecorrespondingdestinationtrees.Tree-WavelengthAssignmentprocessassignswavelengthstothedestinationtreesconstructedinthepreviousstep.ThegoaloftheTree-Wavelengthassignmentphaseistominimizethetotalnumberofwavelengthsneededtoaccommodatethetrafcdemands. WeshowthattheminimumnumberofdestinationtreescanbeconstructedusingagreedyapproachintheTreeConstructionphase.FortheTree-WavelengthAssign-mentproblem,weproveitsNP-CompletenessbyreducingtheGraph-Coloringproblemtoit.Weproposeagreedystrategythatmatchesdestinationtreesandwavelengthsonebyone.Wealsoproposedtwotreesortingmethodsandtwowavelengthsortingmethodstoregulatetheorderoftree-wavelengthmatching.Whendifferenttreesortingandwavelengthsortingmethodsareappliedtothetree-wavelengthassignmentscheme,fourheuristicsarepresented:MC-BF,MC-MF,MP-BFandMP-MF.Extensivesimulationsareconductedtoevaluatetheperformancesoftheseheuristics.Theresultsshowthatperformingsortingondestinationtreesandwavelengthsimprovestheassignmentresults,especiallyunderlowtrafcloads.However,performingsorting 118

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bringssomeextraoverheadtothesortheuristics'runningtime,butoverallcomputationcostsremainacceptable.Inlargetopologieswithheavyworkload,theheuristicwithoutanysortingbecomescompetitiveasitcanprovidesimilarschedulingperformancewithmuchlesscomputationalcost. Therestofthischapterisorganizedasfollows.InSection 6.2 ,wediscussrelatedwork.InSection 6.3 ,weexplaintheTWINarchitectureindetailanddenetheTWIN-WAproblemformally.InSection 6.4 ,agreedyalgorithmforTree-Constructionispresented.InSection 6.5 ,weprovetheNP-CompletenessofTree-Wavelengthassign-mentproblemandfourheuristicsarediscussed.Section 6.6 presentsanexperimentalevaluationofthefourheuristicsforTree-Wavelengthassignment.Section??givestheconclusions. 6.2RelatedWork OpticalCircuitSwitching(OCS)withwavelength-dimensionmultiplexing(WDM)[ 30 ]providesthemosteconomicalsolutionforhighspeedopticalnetworks.However,theinexibleroutingschemeandcoarsemultiplexinggranularitymakeitonlysuitableforthelong-livedlargebulkdatatransfers.Ontheotherhand,OpticalPacketSwitching(OPS)[ 59 ]andOpticalBurstSwitching(OBS)[ 13 ]havebeenproposedtoprovidesub-wavelengthschedulinggranularityandthecapabilityofdynamicrouting.However,theultra-highspeedoptical-electronic-opticalswitchesthatarerequiredintheOPS/OBSnetworksarenormallyexpensiveanddifculttomaintain.ThehighcostindeploymentandmaintenanceinhibittheuseofOPSandOBSinmodernnetworks. Time-domainWavelengthInterleavedNetworking(TWIN)hasbeenproposedtollthegapsbetweenOCSandOPS/OBS.ThearchitectureofTWINisintroducedin[ 65 ].ThegoalofTWINistoprovidesub-wavelengthgranularityfortrafcschedulingwithoutusingexpensivehighspeedopticalswitchesinthenetworks.TWINachievesthisbyonlyallowinglightpathswiththesamedestinationtoshareawavelength.Asthetrafconthesamewavelengthwillnotbesplitagain,theeconomicalswitches 119

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usedinOCSnetworksareabletoroutetheburstsinTWINnetworks.TWINbringsnewchallengestotraditionalopticalschedulingapproaches.[ 53 ]presentssomebasicideasintheroutingtheburstschedulinginTWINnetworksandproposedaperformancemeasurementframework.[ 50 ]investigatedtheopticalburstschedulingprobleminTWINnetworks.Theyshowthatachievingthemaximumthroughputwithzeropropagationdelayisequivalenttotheoptimalmatchingprobleminbipartitegraphs.Theyalsodemonstratethatevenwhenpropagationdelayisnon-negligible,afactor-2approximateschedulingalgorithmexiststomaximizethethroughput.Meanwhile,[ 66 ]focusedontheprovidingbetterQoSinTWINnetworks.TheyintroducedanIntegerLinearProgrammingformulationthatminimizethequeueingdelayoftheopticalbursts.TheyalsoproposedtheDestinationSlotSet(DSS)algorithmtoapproximatelysolvetheproblemwithinreasonabletime. Inthischapter,wefocusonthewavelengthassignmentproblemforTWINnetworks.Traditionalwavelengthassignmentstrategiestaketheavailablewavelengthnumberasthemainconstraint.However,asfractionalwavelengthisallowed,andthegeneraltrafcowisassumedinsub-wavelengthlevel,TWINwavelengthassignment(TWIN-WA)isrelaxedfromtheintegercapacityconstraintintraditionalnetworks.ThemainconcerninTWIN-WAistheconictontopologiesamongmultipledestinationtreeswhentheyshareonewavelength.Moreover,insteadofassigningwavelengthtoeachlightpath,TWINassignswavelengthstoadestinationtree.TraditionalwavelengthassignmentproblemarenormallyequivalenttotheBin-Packingproblem[ 1 ].However,TWINnetworks'wavelengthassignmentproblemisavariationoftheGraph-Coloringproblem[ 3 ],asshowninSection 6.5.1 [ 23 33 40 ]togetherprovideasummaryontheexistingwavelengthassignmentstrategies.ThemostpopularwavelengthassignmentstrategiesareFirst-FitandBest-Fit,wherethewavelengthsarematchedwiththerequestaccordingtoarandomorderortotheirremainingcapacities.[ 33 ]proposedadeferredwavelengthassignmentstrategy 120

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foropticalnetworkswithwavelengthconverterprovides.Thisstrategyimprovestherequestacceptingratebydeferringthewavelengthassignmentfromschedulingtimetoactualjobstarttime.[ 38 ]proposedtheleast-conversionassignmentschemethatattemptstoreducethewavelengthconversionoverheadinsparsewavelengthconverternetworks. InTWINnetworks,thesimplestwavelengthassignmentstrategyistoassigneachdestinationtreeanindividualwavelength.However,thisstrategyrequiresthatthenumberofavailablewavelengthsbeequaltothenumberofdestinationtrees.[ 43 ]investigatesascenariowherethenumberofavailablewavelengthsarelessthanthenumberofdestinationnodes.AwavelengthreuseschemeisproposedasanextensiontoTWIN.ItallowsmultipledestinationtreestoshareawavelengthusingTime-DomainMultiplexing.Toavoidthecollisionsamongtrafcows,thesourcesnodesthatbelongtodifferentdestinationtreesshouldworkinatime-sharingmannerandacomprehensiveburstschedulingalgorithmisneeded.Toprovidefairnessamongtreesthatshareawavelength,thedatabufferforeachsourcenodeismonitored.Adestinationtreewillcontendforaacertainwavelengthwhenthelengthofitssourcenodes'inputbuffersgrowsbeyondacertainthreshold.Astheinputdatarateisalwaysassumedtobelessthanthelinkcapacity,nodestinationtreewillneedtokeepoccupyingawavelengthandsendingdata.Ouralgorithmconsidersthewavelengthreuseprobleminadifferentdirection:multipledestinationtreesshareonewavelengthiftheirtopologiesarecompatible.Oncethewavelengthisassigned,thesourcenodescanworkinafullloadtotransmittheburstwithoutworryingabouttheowcollisions,whichgreatlysimpliestheburstscheduling. 6.3NetworkModelandProblemDenition Onthespectrumofopticalnetworks,TWINnetworksresidebetweentheOCSandOPSnetworks.Comparedtothesetraditionalopticalnetworks,theTWINposesfollowingnewfeatures: 121

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1. SimilartotraditionalOCSnetworks,TWIN'sdataburststravelalongthelightpathusingapre-assignedwavelength.However,awavelengthcanbesharedbymultipletrafcowsfromdifferentsources,onlyiftheirtotalowsizedoesnotexceedthewavelengthcapacity. 2. Ascurrentopticalswitchescannotseparatetheburststhatsharethesamewavelength,trafcononeopticallinkhastoberoutedtothesamedestinationiftheyareusingthesamewavelength.SoTWINlightpathswithsamedestinationaregroupedtogetherasatreestructurewherethedestinationistherootandthesourcesaretheleaves.Wavelengthsareassignedtoeachofthesetrees,ratherthantoasinglelightpath. Figure 6-1 showsasimpleTWINnetwork.Twosourcenodes,S1andS2aresendingtrafctotwodestinationnodes,D1andD2.A5-nodecommunicationnetworkconnectsthesourcesandthedestinations.Inthenetwork,thereare4differentlightpaths:(S1,D1),(S2,D1),(S1,D2)and(S2,D2).Beforethewavelengthsareassigned,theselightpathsaregroupedinto2treestructuresaccordingtotheirdestination,denotedasT1=<(S1,S2),D1>andT2=<(S1,S2),D2>,respectively.D1istherootofT1whileD2istherootofT2.Thenetworkcontainstwowavelengths:W1andW2.Eachdestinationtreehastobeassignedtoawavelengthbeforethetransmissioncanstart.Inthesimplestcase,T1isassignedwavelengthW1andT2isassignedwavelengthW2.Duringthetransmission,S1andS2interleavetheirtrafctoD1andD2bytuningthecoloroftheirlasertothecorrespondingwavelength.Foreachnodeinthecommunicationnetwork,aroutingtableismaintainedtoindicatetheoutgoingportfordifferentwavelengths.Whenthetrafcarriveattheinternalswitches,routingisperformedusingonlytherulesintheroutingtableandthecoloroftheincomingbursts.Thisguaranteesthatopticalburstsofagivenwavelengthwillberoutedtotheintendeddestination.Forexample,inFigure 6-1 ,nodeamustcombinethetrafcfromnodeS1anddonwavelengthW1andforwardistothelinkthatconnectstonodeb,accordingtotheroutingtable.Nodeb,afterreceivingtheburstsonwavelengthW1,willforwardthemtonodeD1,whichistheirdestination. 122

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Figure6-1. AnexampleofTWINnetwork. Fortrafcwhoserequiredbandwidthisfractiontothewavelengthcapacity,TWINnetworkswillgreatlyfacilitatetheirschedulingbyprovidingmoreexibleandner-grainedroutingandwavelengthassignmentscheme.Mostopticalnetworksstillusestaticroutinginthehighspeedmodeaschangingtherouteson-the-yincursveryhighoverheads.Therefore,inthispaper,wealsoassumethatthepathforeachsource/destinationpairaspre-computed,andfocusourresearchonthewavelengthassignmentproblemforTWINnetworks(TWIN-WA).GivenaTWINnetworkG=,atrafcdemandisdenedasr=(s,d,bw),wheres,d2Visthesourceanddestinationnodeofthetrafcow,andbw2(0,1]isthefractionofthewavelengthcapacityrequired.TWIN-WAtakesasetoftrafcdemandsRasinput.Thegoalistoaccommodatethealldemandsr2Rusingaminimumnumberofwavelengths. AsdescribedinSection 6.1 ,TWIN-WAissolvedusinga2-stepprocess.IntheTree-Constructionphase,weconstructthedestinationtreesandintheTree-Wavelengthassignmentphase,weperformthewavelengthassignment.DuringtheTree-Constructionphase,thetrafcdemandsinRaregroupedtogetherbytheirdestination.Ineachgroup,thecorrespondinglightpathsaremergedtogethertoformadestinationtree.Asfractionaljobassignmentisallowed,asimplegreedyalgorithmwillgeneratethedestinationtreesetwithminimumsize.Tree-Wavelengthassignmentalgorithmsassigneachdestinationtreeawavelength.Weshowthatndinganoptimal 123

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assignmentthatusesthefewestnumberofwavelengthisNP-Hard.Severalheuristicsarethenproposedforwavelengthassignment. 6.4TreeConstruction InaTWINnetworkG,adestinationtreefornodeDiisdenotedasT(Di)=(,Di),whereisthesetofallsourcenodesinT(Di).GiventhesetoftrafcdemandsR,weneedtorstconstructdestinationtreesfromthelightpathsbeforewecanactuallyassignthewavelength.ThisprocessiscalledTree-Construction.ThegoalofthisprocessistominimizethetotalnumberofthedestinationtreesintheresultsetT(D). Figure6-2. AnexampleofTWINTreeConstruction. Inthischapter,weallowthetrafcrequest(s,d,bw)tobepartitionedintomultiplesub-requeststhatcanbeassignedtodifferentdestinationtrees.Thisisreasonableasmostmodernopticalswitchesarecapableoftransmitting/receivingdataburstsondifferentwavelengthssimultaneously.Aslongasthedestinationnodesarecapableofpackageorderingandre-assembly,fulllingonerequestwithmultipledataowsistotallyfeasible.Ontheotherhand,ifwesimplymergingallthelightpathswiththesamedestination,theresultingdestinationtreemaynotbeadmissibletothenetwork,asthetotalowsizeforonedestinationmayexceedthewavelengthcapacity.Figure 6-2 showsanexampleoftreeconstruction.Threesourcenodes,S1,S2andS3,aretosenddata 124

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TreeConstruction(G,R) { results=;Groupthetrafcdemandsaccordingtotheirdestinations. for(eachdestinationgroupDG(Di)) { InitializeanewdestinationtreeTj(Di). Tj(Di).capcity=wavlengthcapacity. curTree=Tj(Di). for+(eachtrafcdemandsrinDG(Di)) { Mergethelightpathfromr.stor.dintoTj(Di). if(r.bwcurTree.capacity) Insertanewdemand(s,d,r.bw)]TJ /F4 11.955 Tf 11.95 0 Td[(capacity)intoDG(Di). AddcurTreeintoresults. InitializeanewdestinationtreeTj+1(Di). curTree=Tj+1(Di). } } } returnresults; } Figure6-3. ThegreedyalgorithmforTree-Construction tonodeDsimultaneously.Thedatarateateachsourcenodeis0.6.Ifwemergeall3lightpathsintoonedestinationtree,thetotaltrafconlink(a,D)wouldexceedthewavelengthcapacity.Sothedemandshavetobesplitintotwoseparatedestinationtrees,i.e.T0(D)=(,D)andT1(D)=(,D).Moreover,whencomposingthedestinationtrees,weshouldtrytouseupallthewavelengthcapacities,astheunutilizedcapacitycannotbesharedbyotherdestinationtrees.Basedontheaboveobservations,weproposedthefolloinggreedyalgorithmtocomputetheminimumdestinationtreeset,asshowninFigure 6-3 OurTree-Constructionalgorithmrstgroupsthetrafcdemandsaccordingtotheirdestination.Thiscanbedonebysimplyscanningthedemandsetonce.Foreachgroup, 125

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thecorrespondingdestinationtreesareconstructedgreedily.Ifaddingthecurrentlightpathtothecurrentdestinationtreewouldexceeditswavelengthcapacity,wesplitthecurrentrequestintotwosub-requests.Therstpartjoinsthecurrenttreeandusesallitsremainingcapacity.Thesecondpartstartsanewdestinationtreeintowhichweattempttomergetheremainingpathsinthecurrentgroup.Theoptimalityofthisgreedyalgorithmisobviousasthenumberofresulttreesisminimizedforeachdestinationnodes.ThetimecomplexityofthistreeconstructionalgorithmisO(jVjjRj),wherejRjisthesizeofthetrafcdemandsetandjVjisthenumberofnodesinthenetwork,whichboundslengthofallpossiblelightpaths. 6.5Tree-WavelengthAssignment Whenadataburstisreadytobesentout,thesourcenodeneedtoknowwhichwavelengthitwillusetotransmittheburstforitsintendeddestination.InTWINnetwork,thisisdecidedbythetree-wavelengthassignmentprocess.Intraditionalopticalnetworks,wavelengthsareassignedtospeciclightpaths.However,inTWINnetworkseachdestinationtreeisassignedawavelength.Inthissection,wediscussdifferentstrategiesofassigningwavelengthstodestinationtrees.Thedestinationtreesareconstructedintheprevioustreeconstructingphase.Ourgoalistominimizethenumberofwavelengthsthatweusetoaccommodateallthetrees.InSection 6.5.1 ,weintroducethegenericformofthetree-wavelengthassignmentproblemandprovethatcomputingtheoptimaltree-wavelengthassignmentisNP-Hard.InSection 6.5.2 ,fourgreedyheuristicsareproposedtoapproximatelysolvetheprobleminreasonabletime. 6.5.1GenericFormoftheTree-WavelengthAssignmentProblem WenotethatinTWINnetworks,twodestinationtreesthatsharesomelinkscannotbeassignedtothesamewavelength,astheTWINswitcheswillnotbeabletodistinguishtheirtrafc.So,treesthathavecommonlinksareconsideredinconictforwavelengthassignment.Ontheotherhand,treesthatdonotnotshareanylink 126

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canbeassignedthesamewavelengthwithoutinterference.Suchtreesaresaidtobecompatible. Anotherobservationfortree-wavelengthassignmentisthatadestinationtreemaybeassignedmorethanonewavelength.Thatis,somesource-destinationpathsmayuseonewavelengthwhiletheotherpathsuseadifferentwavelengths.Inparticular,wecandivideadestinationtreeintoacompatiblepartandaconictpartwithrespecttoacurrentwavelengthsthathasalreadybeenassignedtosomeothertrees,andassignthecompatibleparttothecurrentwavelength.Notethatthesplitalwaysstartsfromthesourcenodes(leafnodes),andendsatthedestination(root).Sincethedestinationnodesisabletoreceivedataowfrommultiplewavelengthssimultaneously,splittingdestinationtreeasdescribeddoesnotaffectthecorrectnessofthedatatransmission.However,itprovidesmoreexibilitywhenweresolvetheconictsamongdestinationtrees. Basedontheaboveobservations,thegenericformofthetree-wavelengthas-signmentproblemisasfollow:GivenasetofdestinationtreesDT=(t0,,ti)onaTWINnetworkG,minimizethetotalnumberofwavelengthsthatareneededtoaccommodateallthetreesinDT,withoutviolatingthefollowingconstraints:1).Destinationtreesthatshareawavelengthshouldbecompatiblewitheachother.2).Destinationtreeobtainedfromthetreeconstructionphaseiseitherassignedasinglewavelength,orsplitintoseveralpartswitheachpartbeingassignedtodifferentwavelengths. Theorem6.1. Theabovetree-wavelengthassignmentproblemisNP-Hard. Proof. WeprovethisbyreducingtheGraph-Coloringproblemtothetreeassignmentproblem.Graph-Coloringisawell-knownNP-Completeproblem.GivenagraphG,wewanttocoloralltheverticeswithaminimumnumberofcolorssuchthatnotwoadjacentverticeshavethesamecolor. 127

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WerstconstructacorrespondingTWIN-WAinstancebasedonaGraph-ColoringinstanceG.ForeachnodeviinG,weinitializeacorrespondingtreeti,whichonlycontainsitsrootnoderi.Foreachlink(vi,vj)inG,weinsertanewedge(nij 1,nij 2)tobothtreestiandtj.Weappendthisnewedgetothelastinsertednodeinthetree,sothetreehasachain-likestructure.Figure 6-4 givesasimpleexample.Nodev1andv2areadjacentinG.Sowehaveedge(n12 1,n12 2)appendedtonodesr1andr2fortreest1andt2respectively.Forthesamereason,edge(n13 1,n13 2)isappendedtonoden12 2int1andnoder3int3.AfterwenishtheabovestepsforalllinksinE,wehaveadestinationtreesetDT=(t1,t2,,tn).WeconstructaTWINnetworkGtfromDTbymergingthetopologyofallthetreesinDT.Intheexample,weobtaina7-nodegraphGtbymergingtreest1,t1andt3inDT. Figure6-4. ReductionfromGraph-Coloringproblemtotree-wavelengthassignmentproblem. FromtheconstructionofDTandGt,wecanseethatiftwoverticesviandvjareadjacentinG,treetiandtjmusthaveacommonlinknij 1andnij 2,whichmeanstiandtjareinconictinthetree-wavelengthassignmentprocessfornetworkGt.Ontheother 128

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hand,iftwotreestiandtjareinconictforwavelengthassignment,theymustsharetheedgefromnij 1tonij 2andthatedgeistheonlylinkthatiscommontobothtrees.Fromtheconstruction,theremustbealinkbetweenverticesviandvjinG.Meanwhile,ifthetreesareallintheshapeofachain,asinourconstruction,splittingatreebringsnobenettothewavelengthassignmentprocess.Therefore,intheoptimalassignmentforDTonGt,everytreeinDTisassignedtoasinglewavelength. Now,letkbetheminimumnumberofcolorsweneedtocolorGandmbetheminimumnumberofwavelengthsweneedtoaccommodateallthetreesinDT.Basedontheaboveobservation,weclaimthatk=m.First,weshowthatkwavelengthsissufcient,ifwecancolorGusingatmostkcolors.OurwavelengthassignmentschemeistoassigntreetithewavelengthWj,jkifthecorrespondingvertexviinGiscoloredusingcolorCj.Sincetiwillnotbesplit,andalltheverticesinGthatarecoloredwithCjcannotbeadjacenttoeachother,wecanguaranteethattiwillbecompatibletoanyothertreesthatareassignedthewavelengthWj.Now,weshowthatGalsocanbecoloredwithoutconictusingatmostmdifferentcolors,whenevermwavelengthsaresufcientfortheconstructedtreesetDT.ForeachnodeviinG,ifitscorrespondingtreetiisassignedtowavelengthWj,itwillbecoloredwithCj.SincethereisnoconictinWj,nodeswithcolorCjwillnotbeadjacenttoeachotherinG.Thereforethecolorofviisvalid. Fromtheabovestatements,Graph-Coloringcanbereducedtothetree-wavelengthassignmentprobleminpolynomialsteps.Sotree-wavelengthassignmentisaNP-Hardproblem. 6.5.2GreedyHeuristics Inthissection,weproposeasetofgreedyheuristicstocomputeanapproximatelyoptimalassignmentinreasonabletime.Theseheuristicshaveasimilarmainprocesswhencomputingthewavelengthassignment.However,theydifferfromeachotherintheordertheinputdestinationtreesandtheexistingwavelengthsareassigned. 129

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Themainideaofourgreedyheuristicsisasfollows.ThedestinationtreesinDTarecheckedonebyoneaccordingtothetreesortingorder.Adestinationtreeismatchedagainstalreadyassignedwavelengthsaccordingtothewavelengthsort-ingorder.FortreetiandwavelengthWj,ifpartofticantintowavelengthWj,tiisdividedandapartofitisassignedtheWj.TherestoftiisthenmatchedagainstthewavelengthsWj+1andsoon.Ifallthein-usewavelengthstogethercannotaccommodateti,anewwavelengthisopenedfortheunassignedpartofti. Wepropose2differentapproachestosortthedestinationtrees. 1. MostConictsTreeFirst(MC):ThetreesaresortedindecreasingorderoftothenumberothertreesinDTwithwhichtheyhaveaconict,denotedasCNi.Thisissortingcriterionisbasedontheideathatifweassigntreeswithmoreconictsrst,wemayreachtheminimumnumberofrequiredwavelengthsveryquickly.Then,forthosetreeswithlessconicts,thereisahigherchancethattheywilltintotheexistingwavelengths. 2. MostProcessedTreeFirst(MP):LetPibethenumberofconictedtreesoftithathavealreadybeenassignedwavelengths.InsteadofchoosingtreeswithlargerCNivalues,wepickuptreesthathashigherPivalues.Eachtimeafteratreeisassigned,thePivaluesofalltheunassignedtreesareupdatedandtheonewiththelargestPivalueischosenasthenexttreetobeassignedwavelengths.WhenmultipletreeshavethesamePi,thetiebreakerwillbethevalueoftheirCNivalue.ThethoughtbehindthisorderingissimilartotheMCordering.Moreover,MPorderishopedtoimprovetheMCorderbykeepingtheprioritiessynchronizedwiththeresultoftheexistingassignments. Wealsoproposetwosortingordersforwavelengths. 1. Best-FitWavelengthFirst(BF):Thein-usewavelengthsaresortedinthedecreasingorderofthenumberoflinksinthenetworkthatdonotusethiswavelength.Thisorderisupdatedeverytimeatree-wavelengthassignmentiscompleted. 2. Most-FitWavelengthFirst(MF):Everytimebeforeadestinationtreeisbeingassigned,theexistingwavelengthsaresortedbythesizeofthesubtreetheycanaccommodateforthecurrenttree.Wemeasurethesubtreesizebycountingthenumberofsourcenodesthatcanbecontainedinthecurrentwavelength.Ifonewavelengthcanholdalargernumberofthesourcenodesandtheircorrespondinglightpaths,itwillhavehigherpriorityduringthematching.Thewavelengthsre-orderingistriggeredatruntimewheneverthecurrenttreeischanged.Eitherasplitonthecurrenttree,oranewtreeistakenoutfromDTforassignment.We 130

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alsonotethatthereisnoneedtocompletelysortallthewavelengthsduringtheupdates.Theonlywavelengthweareinterestedinistheonethatcanaccommodatethelargestsubtree.ThereforeweonlyneedtondtheMost-Fitwavelengths,ratherthansortallwavelengths. Combiningthedifferenttreesortingandwavelengthsortingmethodstogether,weobtain4differentheuristicsfortree-wavelengthassignment:MC-BF,MC-MF,MP-BFandMP-MF.Thecomplexityofeachofourheuristicsisasfollows: 1. MC-BF:LetjVjbethenumberofverticesintheTWINnetworkandjTjbethenumberofdestinationtreesinDT.Whenwedeterminetheconictsbetweeneachpairoftrees,ittakesO(jVj)timeaseachtreemaycontainatmostjVj)]TJ /F6 11.955 Tf 18.82 0 Td[(1edges.SinceeverypairoftreesinDTischecked,countingtheconictsforthewholeDTsettakesO(jVjjTj2)time.ThesortingtakesanotherO(jTjlog(jTj))time.ThereforecomputingtheMCordertakesO(jVjjTj2)time.Duringtheassignmentprocess,themaximumnumberofwavelengthneededisjTj.SothenumberofmatchesforeachdestinationtreeisO(jTj).Foreachtreewavelengthpair,ittakesO(jVj)timetomatchthem.SotheprocessingtimeforonesingledestinationtreeisboundedbyO(jVjjTj).TomaintaintheBForder,weneedtoupdatethewavelengthcapacitiesandsortthem.IttakesanotherO(jTjlog(jTj))time.SotheoverallprocessingtimeforonedestinationtreeisO(jVjjTj+jTjlog(jTj)).ThetotalcomplexityforMC-BFalgorithmisO(jVjjTj2+(jVjjTj+jTjlog(jTj)jTj)=O(jVjjTj2). 2. MC-MF:TondtheMost-Fitwavelengthstothecurrenttree,weneedtomatchthetreeagainstallthewavelength,ThistakesO(jVjjTj)time.AdestinationtreewillsplitatmostjVj)]TJ /F6 11.955 Tf 18.73 0 Td[(1timesduringtheassignment,soO(jVj2jTj)timeistakentoprocessonedestinationtree.TheoverallcomplexityforMC-MFisO(jVjjTj2+jVj2jTj2)=O(jVj2jTj2)),whereO(jVjjTj2)istheMCsortingtimeandO(jVj2jTj2)isthetree-wavelengthmatchingtime. 3. MP-BF:Ifthetreeorderisupdateddynamically,extraO(T)operationsareaddedtotheprocessingofeachdestinationtree.However,theseextraoperationsdonotchangedtheasymptoticcomplexityforthetree-wavelengthmatchingprocess.TheoverallMP-BFcomplexityisthesameasforMC-MF:O(jVjjTj2). 4. MP-MF:SimilartoMP-BF,theextraoperationsrequiredtomaintaintheMPorderisdominatedbytheothertree-wavelengthassignmentoperations.Thus,theseextraoperationsdonotaffecttheasymptoticcomplexityofMP-MF,whichisstillO(jVj2jTj2). 131

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6.6Evaluation 6.6.1ExperimentalFramework Inthissection,wemeasuretheperformanceofthewavelengthassignmentheuristicsdescribedinSection 6.5 andevaluatehowdifferentsortingschemesaffecttheperformancesinvariousscenarios.Besidescomparisonontheoptimalityofthetheirassignments,wealsomeasuretheexecutiontimeofeachheuristicandstudyhowexecutiontimevarieswithnetworksizeandworkloads.Weimplementedano-sortversionofthegreedyheuristicsthatdoesnotdothesortingstepsforeitherthedestinationtreesorthewavelengths.Bycomparingtheno-sortheuristicwiththeonesweproposedinSection 6.5.2 ,wecaninvestigatetheimpactofthesortingsteps.Foreverytestcase,wealsoprovidealower-boundfortheoptimalsolution(LB).Thelower-boundiscomputedbycountingtheoccurrencesofeachnetworklinksinalldestinationtrees.Themaximumcountamongallthelinksisthelowerboundfortheminimumnumberofwavelengthsweneed.Withthisbound,wecanestimatehowwellourheuristicscandointheexperiments. Tosimulateaopticalnetwork,weusea25-nodemesh-torustopology,arealworld19-nodeMCInetwork(Figure 6-5 )andseveralrandomlygeneratedtopologies.Forrandomlygeneratedtopologies,wesettheout-degreeofeachnodetobearandomintegersbetween5and7.Toensurenetworkconnectivity,therandomnetworkhasbidirectionallinksbetweennodesiandi+1forevery1i
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AMCI BMesh Figure6-5. NetworkTopologies istotallyrandom.Thesourcesanddestinationdarethenselectedusingauniformrandomnumbergeneratorfromtherespectivesetssothattheworkloadisdistributeduniformlyamongdifferentnodepairs.TherequiredowsizeBWisgeneratedusingachoppedNormalDistributionN(0.1,2.510)]TJ /F9 7.97 Tf 6.59 0 Td[(3).Usingthisdistribution,about96%oftheowssizesareintheinterval(0,0.2).Generatedowsizearediscardedifitsvalueisoutsidetherange(0,1).Astheexpectationoftrafcdemandsisonly0.1,mostadmissibledestinationtreesgeneratedwillcomprisemultiplelightpaths. Foreachtestcase,themaximumnumberoftrafcdemandsisboundedbythenumberofsource-destinationpairs.ThisnumberisdenotedasMaxLoad.Forexample,ina100-noderandomnetwork,ifwemark40%ofthenodesassourcenodesand20%nodesasdestinationnodes,wewillhaveatmost800differentsource-destinationpairs.Thatwouldbethemaximumnumberoflightpathsthatweneedtohandleinthetestcase.Duringtheexperiments,ourworkloadsarevariedfrom20%ofMaxLoadto100%ofMaxLoad. 6.6.2EvaluationResults Figures 6-6 and 6-7 presenttheevaluationresultsforourwavelengthassignmentheuristicsundervarioustrafcloadsinMCIandrandomnetworks.Intheexperiments,weproducetrafcloadsthatare20%,40%,60%,80%and100%oftheMaxLoad.Fromtheexperimentalresults,wemakethefollowingobservations: 133

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Figure6-6. TheperformancesofwavelengthassignmentheuristicsunderdifferentnumberofrequestsinMCInetwork. Figure6-7. Theperformancesofwavelengthassignmentheuristicsunderdifferentnumberofrequestsin100-noderandomtopologies. 1. All4heuristicsthatweproposeinSection 6.5.2 generatebetterassignmentsthantheno-sortheuristicinalltestscenarios.Theseheuristicsoutperformtheno-sortheuristicwithmoreobviousmarginsinthelighttrafcloads(lessthan60%)thaninheavytrafcloads.Thisshowsthatthesortingthetreesandwavelengthsprovidesmorehelptothewavelengthassignmentwhenthenetworkislessoccupied.Whenthenetworkslinksaresaturated,rearrangetheorderofmatchwillnotbeabletoimprovetheschedulingmuch. 134

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2. Amongthefourgreedheuristics,MP-MFheuristicgivesthebestperformanceinalltestcases.Regardingthesortingmethodsforthedestinationtrees,theMPheuristicsprovidesbetterassignmentsthantheMCheuristics.Thismeansadjustingthetreeorderdynamicallyprovidesmorereasonablematchingordersduringthetree-wavelengthassignment.Ontheotherhand,theMFheuristicsoutperformtheBFheuristicswhentheworkloadishigh(morethan80%).However,whentheworkloadislessthan40%,theperformancesofMFheuristicsandBFheuristicsarecomparable.Thisshowsthatwhenthetrafcloadishigh,amorecarefulchoiceonthewavelength,likeMF,isnecessarytoprovideabetterassignment.Whenthereareplentyofresourcesavailable,arelativelycrudesorting,likeBF,issufcient. 3. Whenthetrafcloadislight,thesortedheuristicsprovidearesultsclosetothelowerbound,i.e.averygoodapproximationontheoptimalsolution.Whentrafcloadishigh,theassignmentsfromtheheuristicsarerelativelyfarawayfromthelowerbound.However,thisdoesnotnecessarilymeanthattheheuristicscannotapproximatetheoptimalsolutionsunderhighworkloads,asthelowerboundsmaynottightlyboundtheoptimalsolutionswhentrafcloadishigh. 4. Thenumberofwavelengthsneededincreasedwiththetrafcload.InsmallnetworkslikeMCIandMesh,theneedforextrawavelengthsincreasesfasterthaninlargerandomnetworks.Thereasonisthatitislesslikelytonddisjointlightpathsfordifferentsource-destinationpairs.Whentrafcloadincreases,conictsaremorefrequentinsmallnetworksthaninlargenetworks. Figure6-8. Theperformancesofwavelengthassignmentheuristicsinrandomnetworkswithvarioussizes. Figure 6-8 presentstheperformanceoftheheuristicsonrandomtopologiesofvarioussizewhenthenumberoftrafcdemandsis800.Wecanseethatwiththe 135

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increaseofthenetworkcapacity,fewerwavelengthsarerequiredtoaccommodatetherequestset.However,whenthenetworksizeismorethan400nodes,theimprovementsarealmostnegligible.Recallthatduringthetreeconstructionphase,multipledestinationtreesarebuiltifthetotaltrafcsizeexceedsthewavelengthcapacity.Sowhenthenetworktopologyislargeenoughtoresolvemostconictsinthetreetopologies,theminimumnumberofwavelengthsneededinsuchnetworksisheavilyinuencedbythemaximumnumberofadmissibletreesthatsharethesamedestination,i.e.thecapacityofthewavelengthagainbecomesthemainconstraint. Figure6-9. Thealgorithmrunningtimeofwavelengthassignmentheuristicsunderdifferentnumberofrequestsin100nodenetworks. Figure 6-9 presentstherunningtimeofourwavelengthassignmentheuristicsunderdifferentworkloadsandFigure 6-10 givestherunningtimeasafunctionofthenetworksize.Weseethattheno-sortheuristicisalwaysthefastestalgorithms.Thedifferenceintherunningtimeincreaseasthenetworksizegrows,aswellasthetrafcloadsincrease.Heuristicsusingthesametreesortingalgorithmsgenerallyhavethesamerunningtime,whichmeanstheoverheadbroughtbythetwowavelengthsortingmethodsaresimilartoeachother.Forheuristicsusingdifferenttreesortingscheme,wenotethattheMCsortingisfasterthantheMPsorting.However,theirperformancegap 136

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Figure6-10. Thealgorithmrunningtimeofwavelengthassignmentheuristicsinrandomnetworkswithvarioussizes. ismuchsmallerthanthegapwiththeno-sortheuristic.Althoughthesortedheuristicsarerelativelyslowcomparedtotheno-sortheuristics,theiroverallcomputationalcostsarestillacceptable.Inbestcases,theaverageschedulingtimeforonerequestislessthan5seconds.Intheworstcase,theaverageschedulingtimeislessthan30secondsfortheslowestheuristic. Asasummary,thesortingschemesprovideconsiderablebenetstotheTWINwavelengthassignments.Theimprovementismorewithrelativelylowworkload.However,thesortheuristics'runningtimeisaffectedbytheextraoverheadbroughtbythesortingprocess.Nevertheless,therunningtimesarestillreasonableevenintheworstcase.Theno-sortheuristiciscompetitivewhenthenetworksarelargeandtrafcdemandsareheavy.Itprovidesmuchfasterschedulingspeedwhileyieldinglittleintheassignmentoptimality. 137

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CHAPTER7CONCLUSION Thisdissertationhasfocusedonsolvingvariousresourceschedulingproblemsinhighspeednetworks.Ourcontributionsareconcludebelow. Wedenedasetofdatastructurestorepresentthechangingstatusofavailableresources.Wediscussedindetailtheprosandconsofthecontinuoustimemodelanddiscretetimemodel.WeusedTime-BandwidthListtoassociatetemporalinfowiththeresourceavailability.WeusedStartTimeListtoindicatethefeasibilityofanetworkpathforcertainuserrequests.WealsousedSteadyStagestorepresenttheperiodsduringwhichresourceavailabilityisstaticforthewholenetwork.Foropticalnetworks,anextendednetworkmodelwaspresentedtodealwiththewavelengthconvertersinthenetwork.Formultipleresourcescheduling,theMRRMmodelwaspropose.Inthismodel,differenttypesofresourcesareuniformlyrepresentedusingagraphbasedmodel. Severalschedulingproblemsforsinglepathschedulingingeneralnetworks,includingxedslot,maximumbandwidthinslot,maximumduration,rstslot,allslotsandall-pairsall-slotswereconsidered.Foreachproblem,weproposedseveralalgorithms(DAFP,kDP,andkSPforxedslotproblem,LSWandEBFforrstslotproblemandsoon).Wealsoconductedextensiveevaluationsofeachalgorithminvarioustestenvironmentstoassessitsperformance. WedenedtheEarliestFinishTimeFileTransferProblem(EFTFTP)toexplorethebenetbroughtbymulti-pathroutingforlargeletransfersfrommultiplesourcestomultipledestinations.Wedevelopedseveralmulti-pathreservationalgorithmstosolvethisproblemfortheonlineandbatchschedulingcases.Anewmax-owbasedgreedyalgorithm(GOS)andseveralnovelvariantsofthek-shortestpathsalgorithmswereproposedforonlinescheduling.AnovelLPformulationwasusedtodevelopanoptimalalgorithmforbatchscheduling.Extensivesimulationsusingbothrealworldandrandom 138

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networksshowthatourGOSalgorithmprovidesagoodbalanceamongmaximumnishtime,averagenishtime,andcomputationalcomplexity.Thisalgorithmmaybeextendedtothecasewhenswitchingoverheadisnotnegligible. Wehaveextendedthealgorithmsoriginallyproposedforgeneralnetworkstoincorporatethewavelengthsharingandwavelengthcontinuityconstraintsofopticalnetworks.Wemodiedtwoexistingopticalnetworkschedulingalgorithms(MSPFandMSWF)toachievebetterperformance.Wealsoshowedthatadeferredwavelengthassignmentstrategycanbeeffectivelyusedinconjunctionwithmanyroutingalgorithms.Thiseffectivelyalleviatestheneedtokeeptrackofthebandwidthallocationstatusofeachwavelength.OurresultsshowthattheadaptedEBFalgorithmperformsbetterthanotheralgorithms.Forheterogeneousnetworks,LSWalsoprovidedcomparablesolutions;whileforhomogeneousnetworksMSPFandMSWFprovidecomparablesolutions. Besidesfull-wavelengthconversion,wealsoexploredtheimpactofsparsewavelengthconversiononrst-slotscheduling.Weproposedanewnetworkmodeltoemulatethefull-conversionalgorithmsinsparseconversionnetworks.Usingthismodel,weconductedextensiveexperimentstoassesstheimpactofwavelengthconvertersonFirst-SlotRWAalgorithms'performance.Ourexperimentsindicatedthatincreasingwavelengthconvertershaspositiveimpactonblockingperformance,butverylittleimpactontheavailabilityofearlierstarttimes.Wealsoshowedthatfornetworksnolargerthanseveralhundrednodes,deployingwavelengthconvertersonatmost60%ofnodeswouldbeenoughtoprovideasatisfyingperformance.Additionally,analgorithmswitchingstrategythatadaptstheschedulingalgorithmasthecurrentworkloadchangeswasproposed.Whenthenetwork'strafcpatterndidnotchangingdramatically,thisstrategyresultedinconsiderableperformanceimprovement. Weconsideredthemultipleresourceschedulingproblem,andpresentedseveralsolutionsintermsofamulti-resourcemodel.Weproposedaexibleandefcient 139

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multi-resourcereservationmodel(MRRM)andsolvedfourinstancesofthemultiplereservationrstslot(MFRS)problem.Basedonourmodel,fouralgorithmsweredevelopedforeachindividualinstanceofMRFS.Experimentsonaheterogeneouscomputernetworkshowedthatouralgorithmsscalelinearlyintermsofnetworksizeandrequestratio. Weproposeda2-stepprocesstosolvethewavelengthassignmentproblemforTWINnetworks.WeshowedthatdeterminingthewavelengthassignmentthatusetheminimumnumberofwavelengthsisaNP-Completeproblem.Fourgreedyheuristicsarepresentedtocomputetheapproximatedsolutionwithinreasonabletime.Theevaluationresultsshowthatperformingsortingondestinationtreesandwavelengthsimprovestheassignmentresults,especiallyunderlowtrafcloads.However,performingsortingbringssomeextraoverheadstothesortheuristics'runningtime,butoverallcomputationcostsarestillacceptable.Meanwhile,inlargetopologieswithheavyworkloads,theno-sortheuristicbecomescompetitiveasitcanprovidesimilarschedulingperformancewithmuchlesscomputationalcost. 140

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BIOGRAPHICALSKETCH YanLireceivedhisPh.D.oncomputerscienceinUniversityofFloridainDecember2010.HewasworkingunderthesupervisionofDr.SartajSahniandDr.SanjayRanka.Hisresearchinterestsarethealgorithmsanddatastructuresfortheresourceschedulinginhighspeednetwork.YanLireceivedhisB.S.inHuazhongUniversityofScienceandTechnologyin2003andhisM.S.inInstituteofSoftware,ChineseAcademyofSciencein2006.HeisnowpursuingPhDinUniversityofFlorida. 147