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Fault-Tolerance-Oriented Topology, Routing and Wavelength Assignment Optimization for WDM All-Optical Networks

Permanent Link: http://ufdc.ufl.edu/UFE0042739/00001

Material Information

Title: Fault-Tolerance-Oriented Topology, Routing and Wavelength Assignment Optimization for WDM All-Optical Networks
Physical Description: 1 online resource (160 p.)
Language: english
Creator: Wang, Dexiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: all-optical -- candidate-routing -- circulant-graph -- fault-tolerance -- ordered-path-enumeration -- routing -- topology-optimization -- torus -- wavelength-assignment -- wdm-networks
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Wavelength-routed all-optical communication technologies have immense potential to become a qualified solution to next-generation communication networks satisfying both long-haul networking and special local communication requirements, as in avionic communication systems, due to its efficient one-shot data delivery, wide bandwidth provision, magneto-electrical interference resistance, light-weight signal carrying medium (fibers), etc. However, fiber optic components are susceptible to a range of operating faults, such as stability issues in both mechanical placements and electro-optic operations, especially under hazardous operating conditions. Therefore, it becomes more than desirable to propose efficient fault-tolerant network architectures and protocols to meet varied fault-tolerance requirements under certain resource provision limits. This dissertation is dedicated to studying optimal resource (in form of wavelengths and optical links) allocation problems in designing different types of fault-tolerant Wavelength Division Multiplexing (WDM) network architectures and then searching for best solutions. A range of classic topologies, such as torus and circulant graphs, are studied on which optimal fault-tolerant routing algorithms are developed. The Wavelength Assignment (WA) problem is investigated in depth and a Wavelength Allocation and Reuse (WAR) algorithm for the two-dimensional N×N torus of arbitrary sizes is developed which performs close to the best possible solution (lower bound). Spare sharing technology, in favor of reducing redundant resource utilization, is also studied in fault-tolerant architecture design and different levels of spare sharing are proposed on the torus topology to evaluate the tradeoff between network connection reliability and resource utilization. Circulant graph, featuring scalable network sizes and flexible connectivity, is exploited and a node-disjoint routing algorithm for arbitrary sizes and connectivity degrees of the circulant graph is proposed to facilitate the multi-level fault-tolerant implementation of all-optical Local Area Networks (LANs). From another perspective of fault-tolerant WDM architecture design, topological optimization under certain resource provision constraints is studied, in which a number of Integer Linear Programs (ILPs) are developed to model the problem in varied granularities. Based on the drawbacks analysis of the greedy approach, a two-phase heuristic algorithm is proposed that jointly considers the routing and wavelength assignment problems. Numerical simulations show that the proposed heuristic algorithm performs much better than the traditional method for the Routing and Wavelength Assignment (RWA) problems in which the routing and wavelength assignment are treated consecutively in a separate fashion. This dissertation also touches upon a fundamental problem: ordered path enumeration (or k-shortest path enumeration). Based on a series of graph-theoretical derivation, a new ordered path enumeration algorithm is proposed to help form a pool of possible paths for the flow requests. Then a problem-aware candidate routing scheme is developed to select candidate routes from the pool of enumerated paths. This ordered-path-enumeration-based candidate routing method is examined on two shared-path-protection RWA problems and the numerical results indicate its great performance advantage over the traditional k-shortest disjoint routing based method.
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.
Statement of Responsibility: by Dexiang Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Mcnair, Janise Y.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042739:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042739/00001

Material Information

Title: Fault-Tolerance-Oriented Topology, Routing and Wavelength Assignment Optimization for WDM All-Optical Networks
Physical Description: 1 online resource (160 p.)
Language: english
Creator: Wang, Dexiang
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: all-optical -- candidate-routing -- circulant-graph -- fault-tolerance -- ordered-path-enumeration -- routing -- topology-optimization -- torus -- wavelength-assignment -- wdm-networks
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Wavelength-routed all-optical communication technologies have immense potential to become a qualified solution to next-generation communication networks satisfying both long-haul networking and special local communication requirements, as in avionic communication systems, due to its efficient one-shot data delivery, wide bandwidth provision, magneto-electrical interference resistance, light-weight signal carrying medium (fibers), etc. However, fiber optic components are susceptible to a range of operating faults, such as stability issues in both mechanical placements and electro-optic operations, especially under hazardous operating conditions. Therefore, it becomes more than desirable to propose efficient fault-tolerant network architectures and protocols to meet varied fault-tolerance requirements under certain resource provision limits. This dissertation is dedicated to studying optimal resource (in form of wavelengths and optical links) allocation problems in designing different types of fault-tolerant Wavelength Division Multiplexing (WDM) network architectures and then searching for best solutions. A range of classic topologies, such as torus and circulant graphs, are studied on which optimal fault-tolerant routing algorithms are developed. The Wavelength Assignment (WA) problem is investigated in depth and a Wavelength Allocation and Reuse (WAR) algorithm for the two-dimensional N×N torus of arbitrary sizes is developed which performs close to the best possible solution (lower bound). Spare sharing technology, in favor of reducing redundant resource utilization, is also studied in fault-tolerant architecture design and different levels of spare sharing are proposed on the torus topology to evaluate the tradeoff between network connection reliability and resource utilization. Circulant graph, featuring scalable network sizes and flexible connectivity, is exploited and a node-disjoint routing algorithm for arbitrary sizes and connectivity degrees of the circulant graph is proposed to facilitate the multi-level fault-tolerant implementation of all-optical Local Area Networks (LANs). From another perspective of fault-tolerant WDM architecture design, topological optimization under certain resource provision constraints is studied, in which a number of Integer Linear Programs (ILPs) are developed to model the problem in varied granularities. Based on the drawbacks analysis of the greedy approach, a two-phase heuristic algorithm is proposed that jointly considers the routing and wavelength assignment problems. Numerical simulations show that the proposed heuristic algorithm performs much better than the traditional method for the Routing and Wavelength Assignment (RWA) problems in which the routing and wavelength assignment are treated consecutively in a separate fashion. This dissertation also touches upon a fundamental problem: ordered path enumeration (or k-shortest path enumeration). Based on a series of graph-theoretical derivation, a new ordered path enumeration algorithm is proposed to help form a pool of possible paths for the flow requests. Then a problem-aware candidate routing scheme is developed to select candidate routes from the pool of enumerated paths. This ordered-path-enumeration-based candidate routing method is examined on two shared-path-protection RWA problems and the numerical results indicate its great performance advantage over the traditional k-shortest disjoint routing based method.
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.
Statement of Responsibility: by Dexiang Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Mcnair, Janise Y.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042739:00001


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FAULT-TOLERANCE-ORIENTEDTOPOLOGY,ROUTINGANDWAVELENGTHASSIGNMENTOPTIMIZATIONFORWDMALL-OPTICALNETWORKSByDEXIANGWANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011DexiangWang 2

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Tomyfamily 3

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ACKNOWLEDGMENTS AlltheworkspresentedinthisdissertationareundertheguidanceofmyadvisorDr.JaniseMcNair.Herwisdom,generosity,andencouragingsmilehavebeensupportingmethroughone-after-anothertoughtimesthatIencounteredduringthislongprocessasaPh.D.pursuer.Hereby,althoughwayfarfromsufcient,Iwanttoexpressmysinceregratefulnessforeveryencouragementthatshegavetome,everypieceofguidancethatsheofferedme,andeverystepofprogressthatshehelpedmeachieve.ThespiritsthatIlearnedfromherinsomanyaspectsofscholarshipwillbecarriedonandplayapricelessrolethroughoutmyfuturecareer.IalsowanttothankallothermembersofmyPh.D.supervisorycommittee:Dr.AlanGeorge,Dr.HuikaiXie,andDr.MyThai,fortheiracademicadvicesandsupportonmyPh.D.proposalanddissertation.IstartedmyresearchunderDr.AlanGeorgeonagreen-internetprojectwhereIidentiedmyinterestsofresearchincomputernetworks.Dr.HuikaiXiesharedhisknowledgewithmeinfundamentalprinciplesofber-opticcommunications,whichformedmyessentialunderstandingintheareaofopticalcommunicationnetworks.Dr.MyThaibroughtmeintotheareaofapproximationalgorithmsandoptimizationtheory.TheknowledgethatIlearnedfromherfacilitatedsolvingmanyproblemsinthisdissertation.Alongtheentirewayofproducingthisdissertation,Ireceivednumeroushelpsfromsomanypeopleatdifferenttimesindifferentwaysthatthereisnowaytoenumerateallmythanks.Lastbutnotleast,myspecialthanksgotoallthemembersoftheWirelessAndMobileSystems(WAM)Laboratory:Dr.DawoodAl-Ari,ArvindhanKumar,MadhanSivakumar,GustavoVejarano,XiaoyuanLi,ObulapathiChalla,SeshupriyaAlluru,GunjanGupta,JingQin,XiangMao,JoseAlmodovar-Faria,JinJingPan,PaulMuri,RitwikDubey,GokulBhat,andJoeyMakar,forsomanybenecialdiscussionsandadvicesthattheyofferedmeonadailybasis. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 14 1.1Motivation .................................... 14 1.2DissertationOrganization ........................... 15 2TORUS-BASEDFOUR-WAYDISJOINT-LIGHTPATHSCOMMUNICATIONFORAVIONICWDMLANS ............................. 17 2.1RelatedWorks ................................. 18 2.1.1TopologicalOptions ........................... 18 2.1.2FaultToleranceinWDMOpticalNetworks .............. 18 2.1.3RoutingandWavelengthAssignment(RWA) ............. 19 2.2ContributionsandChapterOrganization ................... 20 2.3NetworkArchitecture .............................. 20 2.3.1SAERequirementsandEvaluationMetrics ............. 20 2.3.2Torus-BasedArchitecture ....................... 21 2.3.3Single-WavelengthLightpaths ..................... 22 2.4Non-OverlappingLightpathSetupAlgorithm:Four-wayOptimaLDisjointrouting(FOLD) ................................. 23 2.4.1Scenario1:X-YRouting ........................ 25 2.4.2Scenario2:XRouting ......................... 27 2.4.3Scenario3:YRouting ......................... 29 2.4.4DestinationGroup ........................... 29 2.5WavelengthAllocationandReuse(WAR) .................. 30 2.5.1ALowerBound(IdealWavelengthUtilization) ............ 30 2.5.2WavelengthAllocationandReuse(WAR)Algorithm ......... 32 2.6ControllerImplementation ........................... 45 2.7PerformanceAnalysis ............................. 46 2.7.1ProbabilisticAnalysis .......................... 46 2.7.2NetworkCapacityAnalysis ....................... 51 3TRADEOFFSTUDYONFAULTTOLERANCECAPACITYANDRESOURCEUTILIZATIONFORTHETORUS-BASEDALL-OPTICALWDMLANS ..... 54 3.1WavelengthAssignmentSchemes ...................... 55 5

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3.2FailureRecovery ................................ 58 3.3ReliabilityAnalysis ............................... 59 3.4SimulationandNumericalResults ...................... 61 4CIRCULANT-GRAPH-BASEDFAULT-TOLERANTROUTINGFORALL-OPTICALWDMLANS ...................................... 66 4.1RelatedWork .................................. 66 4.2Fault-TolerantRoutingAlgorithm ....................... 67 4.2.1CirculantNetworkArchitecture .................... 68 4.2.2Node-DisjointLightpathsSetup .................... 69 4.3NetworkResourceUtilization ......................... 74 4.4NetworkReliabilityAnalysis .......................... 77 5TOPOLOGICALOPTIMIZATIONFORSPARE-SHARING-BASEDWAVELENG-TH-ROUTEDALL-OPTICALNETWORKS ..................... 80 5.1Spare-Sharing-BasedTopologicalOptimization ............... 81 5.2RelatedWork .................................. 83 5.3ContributionsandChapterOrganization ................... 84 5.4ProblemFormulation .............................. 84 5.4.1Matrix-BasedRepresentation ..................... 85 5.4.2IntegerLinearProgramFormulation ................. 88 5.4.3K-ShortestDisjointRoutingBasedFormulation ........... 91 5.4.4ProblemSizeExemplication ..................... 93 5.5AGreedyApproach .............................. 94 5.5.1TheUnderlyingIdea .......................... 94 5.5.2DataStructures ............................. 95 5.5.3TheAlgorithm .............................. 97 5.5.4PerformanceComparison ....................... 97 5.5.5ApproximationRatioAnalysisforWorkingPathsAllocationunderAdequateWavelengthProvision .................... 100 5.5.6ComplexityandMemoryRequirementAnalysis ........... 101 5.6EnhancedHeuristics .............................. 102 5.6.1DrawbacksoftheGreedyApproach ................. 102 5.6.2TwoInitialSolutions .......................... 103 5.6.3SolutionPerfection(PER) ....................... 106 5.7Results ..................................... 107 5.7.1PerformanceComparison ....................... 107 5.7.2PerformanceIndicator ......................... 112 6ORDERED-PATH-ENUMERATION-BASEDCANDIDATEROUTING:AFACILI-TATINGAPPROACHTOSOLVINGRWAPROBLEMSFOROPTICALNET-WORKS ........................................ 116 6.1RelatedWork .................................. 117 6.2ContributionsandChapterOrganization ................... 118 6

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6.3OrderedPathEnumeration .......................... 118 6.3.1DenitionofTerminologies ....................... 118 6.3.2TheoremsregardingOrderedPathEnumeration ........... 119 6.3.3TheOrderedPathEnumerationAlgorithm .............. 123 6.3.4ContainerCoverMinimalityDetection ................. 123 6.3.5PotentialAlgorithmicAdvantages ................... 125 6.4ApplicationI:WavelengthUtilizationMinimizationforRWAwithShared--PathProtection ................................ 125 6.4.1ProblemDescription .......................... 126 6.4.2CandidateRouting ........................... 126 6.4.3ProblemFormulations ......................... 128 6.4.3.1NotationsUsedinThreeFormulations ........... 129 6.4.3.2TheOriginalFormulation .................. 130 6.4.3.3k-ShortestDisjointRoutingBasedFormulation ...... 132 6.4.3.4CandidateRoutingBasedFormulation ........... 133 6.4.3.5FormulationComparison .................. 133 6.4.4NumericalResults ........................... 135 6.5ApplicationII:TopologicalOptimizationforShared-PathProtectionRWA 136 6.5.1ProblemDescription .......................... 136 6.5.2CandidateRouting ........................... 137 6.5.3ProblemFormulations ......................... 138 6.5.3.1Notations ........................... 138 6.5.3.2TheOriginalFormulation .................. 138 6.5.3.3k-ShortestDisjointRoutingBasedFormulation ...... 139 6.5.3.4CandidateRoutingBasedFormulation ........... 139 6.5.3.5FormulationComparison .................. 140 6.5.4NumericalResults ........................... 140 7CONCLUSIONSANDFUTUREWORK ...................... 142 7.1Conclusions ................................... 142 7.2FutureWork ................................... 143 APPENDIX AOPTIMALITYPROOFOFTHEPROPOSEDNON-OVERLAPPINGLIGHTPATHSSETUPALGORITHM(FOLD) ............................ 146 BDERIVATIONOFLSEXPRESSIONS ....................... 151 REFERENCES ....................................... 154 BIOGRAPHICALSKETCH ................................ 160 7

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LISTOFTABLES Table page 2-1SummaryofLS,Dexpressionsfordifferentcases ................. 31 2-2SummaryofLSexpressionsfordifferentN ..................... 32 2-3SummaryofWWARexpressionsfordifferenttorussizes ............. 44 2-4Wavelengthrequirementforvariedtorussizes .................. 45 3-1Sparewavelengthrequirements .......................... 57 5-1Basicnotations .................................... 85 5-2Problemsizeexemplication:numberofvariables ................ 93 5-3Problemsizeexemplication:numberofconstraints ............... 94 5-4Topologicalcostcomparisonamongk-shortestpathbasedILPandthegreedyapproachforarandomlygeneratednetworkwith6nodesand6wavelengthsoneachlink ...................................... 97 6-1Problemsizecomparisonamongformulations:numberofvariables ...... 134 6-2Problemsizecomparisonamongformulations:numberofconstraints ..... 134 6-3Routeprocessingtimecomparison(insecond,runningonaWindowsmachinewitha3GHzprocessor) ............................... 134 6-4Averagecandidateroutedisjointednesscomparison(averagedoverows) ... 135 8

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LISTOFFIGURES Figure page 2-1A44torusbackboneconnectedviaopticalbers ............... 22 2-2Generalnon-overlappinglightpathsetupalgorithm ................ 24 2-3Source-destinationpositionalrelationship ..................... 24 2-4LightpathssetupforX-Yrouting ........................... 25 2-5CaseI'lightpathssetup ............................... 27 2-6CaseII'lightpathssetup ............................... 28 2-7Summaryoflightpathssetupcasesinthedestinationgroup(Nodd) ...... 29 2-8Summaryoflightpathssetupcasesinthedestinationgroup(Neven) ..... 30 2-9WARdemonstrationforthe33torus ....................... 33 2-10WARdemonstrationforthe44torus ....................... 34 2-11Groupinglightpaths ................................. 36 2-12Groupmirroringlightpaths .............................. 38 2-13NewlightpathsetupcaseswithconsiderationofWAR(Nisodd) ........ 38 2-14NewlightpathsetupcaseswithconsiderationofWAR(Niseven) ........ 39 2-15WARalgorithmperformance ............................ 44 2-16Receptionstructureofthecontroller ........................ 46 2-17Networkunreliabilityanalysisfora44torus ................... 49 2-18TTURdistributionfora44torus(f=0.1) .................... 49 2-19Conditionalprobabilitiesofconnectionfailuresfora44torus ......... 50 2-20Effectsofnetworkfailuresonnetworkcapacity .................. 52 2-21Averagecapacitydegradationcomparisonbetweentheproposed4-lightpathscommunicationandsingle-lightpathcommunication ............... 53 3-1Examplesof4disjointlightpathssetupbetweendifferentS-Dpairsina44torus(thelightpathindarkredistheworkingpathandthethreelightpathsinolivegreenaresparepaths) ............................. 56 3-2Wavelengthassignmentfortwosparesharingschemes ............. 56 9

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3-3TotalnumbersofwavelengthsrequiredforfourWAschemes .......... 57 3-4Lightpathstatetransitiondiagramforresource-sharedWAschemes ...... 58 3-5Anexampleofsparelightpathre-enablingina44torus ............ 59 3-6Connectionunreliabilitiesinthe44torus .................... 62 3-7Conditionalnetworkunreliabilitiesinthe44torus ............... 63 3-8Conditionalnetworkcapacityinthe44torus .................. 64 3-9Conditionalblocking/successratesinthe44torus ............... 65 4-1Circulant-graph-basednetworkarchitectureandexamplesoffault-tolerantroutingviaestablishingnode-disjointlightpaths .................. 68 4-2Fault-tolerantroutingfordestinationnodeswithmoduloindexdifferencefromthesourcenodebyW,greaterthanW,andsmallerthanW .......... 70 4-3LastDnode-disjointlightpathssetupforScenarioIandII(thelast-stopnodegroupandassociatedroutinglinksarecoloredgreen) .............. 72 4-4LinkutilizationfordifferentdestinationswithrespecttovariedW(N=16) ... 76 4-5WavelengthrequirementwithrespecttovariedWforall-nodesimultaneouscommunication(N=16) ............................... 76 4-6DisconnectionprobabilitychangewithfLandfN(N=16,W=2,S=0,D=8) 78 4-7DisconnectionprobabilitychangewithfLforvariedW(N=16,S=0,D=8) .. 78 4-8Disconnectionprobabilitydistributionacrossthenetwork(N=16,W=2,fL=0.1) .......................................... 79 5-1Topologicalsolutionswithoutandwithsparesharing ............... 81 5-2Validitydemonstrationofsparesharing ...................... 82 5-3ElementvaluetransitiondiagramforWAMW. ................... 96 5-4ElementvaluetransitiondiagramforWAMB. ................... 96 5-5Pseudocodeofthegreedyapproach ....................... 98 5-6SolvingprocessforthethreeILPinstanceswithoutreachingoptimalityafterrunningMOSEKfor8hours ............................. 99 5-7Originalgreedyapproachsolution. ......................... 103 5-8Linkpotentialbasedgreedysearchsolution .................... 104 10

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5-9Largestratiorstbasedsearchsolution ...................... 107 5-10Pseudocodeoftheperfectionalgorithm(PER) .................. 108 5-11Locationsof16USmajorcities ........................... 109 5-12Performanceimprovementsfromgreedysolutionsduetoheuristicalgorithmsforvariedwavelengthprovisions .......................... 110 5-13Convergenceprocessoftheperfectionalgorithmtakingthreedifferentinitialsolutions ....................................... 110 5-14Weightedwavelength/linkutilizationforworkingandbackuplightpathsintheLRF+PERinducedtopologicalsolutions ...................... 111 5-15Solutionperformanceindicationforthenetworkwith5wavelengthsprovision 113 5-16Weightedwavelengthutilization/averagebendingfactordistributionforvariedtopologicalsolutionsinthenetworkswith5,10,15,20,and25wavelengthsprovision ....................................... 114 6-1Pseudocodeoftheorderedpathenumerationalgorithm ............. 124 6-2NSFnetwork ..................................... 126 6-3Pseudocodeofthecandidateroutingscheme .................. 128 6-4Solutionoptimalitycomparisonbetweenk-shortestdisjointroutingand3-can-didateroutingafterrunningMOSEKfor8hours .................. 136 6-5Solutionoptimalitycomparisonbetweenk-shortestdisjointroutingand4-can-didateroutingafterrunningMOSEKfor8hours .................. 136 6-6Solutionperformancecomparisonbetweenk-shortestdisjointroutingand4-can-didateroutingafterrunningMOSEKfor8hours .................. 141 6-7Solutionperformancecomparisonbetweenk-shortestdisjointroutingand5-can-didateroutingafterrunningMOSEKfor8hours .................. 141 A-1Non-optimalitydemonstrationforagreedydisjointroutingsolution ....... 146 A-2Pathaugmentationbased2-shortestdisjointrouting ............... 147 A-3Progressiveandregressivelinks ........................ 148 11

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyFAULT-TOLERANCE-ORIENTEDTOPOLOGY,ROUTINGANDWAVELENGTHASSIGNMENTOPTIMIZATIONFORWDMALL-OPTICALNETWORKSByDexiangWangDecember2011Chair:JaniseY.McNairMajor:ElectricalandComputerEngineering Wavelength-routedall-opticalcommunicationtechnologieshaveimmensepotentialtobecomeaqualiedsolutiontonext-generationcommunicationnetworkssatisfyingbothlong-haulnetworkingandspeciallocalcommunicationrequirements,asinavioniccommunicationsystems,duetoitsefcientone-shotdatadelivery,widebandwidthprovision,magneto-electricalinterferenceresistance,light-weightsignalcarryingmedium(bers),etc.However,beropticcomponentsaresusceptibletoarangeofoperatingfaults,suchasstabilityissuesinbothmechanicalplacementsandelectro-opticoperations,especiallyunderhazardousoperatingconditions.Therefore,itbecomesmorethandesirabletoproposeefcientfault-tolerantnetworkarchitecturesandprotocolstomeetvariedfault-tolerancerequirementsundercertainresourceprovisionlimits. Thisdissertationisdedicatedtostudyingoptimalresource(informofwavelengthsandopticallinks)allocationproblemsindesigningdifferenttypesoffault-tolerantWavelengthDivisionMultiplexing(WDM)networkarchitecturesandthensearchingforbestsolutions.Arangeofclassictopologies,suchastorusandcirculantgraphs,arestudiedonwhichoptimalfault-tolerantroutingalgorithmsaredeveloped.TheWavelengthAssignment(WA)problemisinvestigatedindepthandaWavelengthAllocationandReuse(WAR)algorithmforthetwo-dimensionalNNtorusofarbitrarysizesisdevelopedwhichperformsclosetothebestpossiblesolution(lowerbound). 12

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Sparesharingtechnology,infavorofreducingredundantresourceutilization,isalsostudiedinfault-tolerantarchitecturedesignanddifferentlevelsofsparesharingareproposedonthetorustopologytoevaluatethetradeoffbetweennetworkconnectionreliabilityandresourceutilization.Circulantgraph,featuringscalablenetworksizesandexibleconnectivity,isexploitedandanode-disjointroutingalgorithmforarbitrarysizesandconnectivitydegreesofthecirculantgraphisproposedtofacilitatethemulti-levelfault-tolerantimplementationofall-opticalLocalAreaNetworks(LANs). Fromanotherperspectiveoffault-tolerantWDMarchitecturedesign,topologicaloptimizationundercertainresourceprovisionconstraintsisstudied,inwhichanumberofIntegerLinearPrograms(ILPs)aredevelopedtomodeltheprobleminvariedgranularities.Basedonthedrawbacksanalysisofthegreedyapproach,atwo-phaseheuristicalgorithmisproposedthatjointlyconsiderstheroutingandwavelengthassignmentproblems.NumericalsimulationsshowthattheproposedheuristicalgorithmperformsmuchbetterthanthetraditionalmethodfortheRoutingandWavelengthAssignment(RWA)problemsinwhichtheroutingandwavelengthassignmentaretreatedconsecutivelyinaseparatefashion. Thisdissertationalsotouchesuponafundamentalproblem:orderedpathenumeration(ork-shortestpathenumeration).Basedonaseriesofgraph-theoreticalderivation,aneworderedpathenumerationalgorithmisproposedtohelpformapoolofpossiblepathsfortheowrequests.Thenaproblem-awarecandidateroutingschemeisdevelopedtoselectcandidateroutesfromthepoolofenumeratedpaths.Thisordered-path-enumeration-basedcandidateroutingmethodisexaminedontwoshared-path-protectionRWAproblemsandthenumericalresultsindicateitsgreatperformanceadvantageoverthetraditionalk-shortestdisjointroutingbasedmethod. 13

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CHAPTER1INTRODUCTION Inthischapter,themotivationofinvestigatingthefault-tolerance-orientedresourceallocationoptimizationproblemsinthecontextofWDMall-opticalnetworksispresented.Thentheorganizationofthisdissertationfollows. 1.1Motivation ThetechnologicaladvanceintheareaofberopticcommunicationespeciallyonopticalswitchingtechniquesmakesitpossibletodesignWDMall-opticalnetworksthateliminateallintermediateOptical-Electrical-Optical(OEO)conversionsandqueuingprocesssuchthatthedatacanbedeliveredfromitssourcetoitsdestinationinaone-shotfashion[ 6 13 19 48 ].Togetherwiththetraditionalbandwidthadvantageoftheopticalnetworks,all-opticalnetworksenabledesignofanext-generationcommunicationarchitecturethatistargetedtosatisfymanytime-criticalandbandwidth-demandingapplications. DuetomanyadvantagesthatWDMopticalnetworkscanprovide,besidestheiruseintraditionallong-haulcommunicationnetworks,theyareexpandingtheirapplicationsintomanyothereldswhereothertypesofcommunicationtechnologieswerebeingused.Forexample,USNAVYistryingtoestablishnewSocietyofAutomotiveEngineering(SAE)standardsforber-optic-networks-basedavioniconboardcommunicationsystemsthatwereoperatingviatraditionalcopper-basedelectricalcommunication[ 29 39 44 45 63 ].ThenewberopticcommunicationbasednetworkdesigncangreatlyhelplowerequipmentSize,WeightandPower(SWaP)[ 11 ],improvemagneto-electricalinterferenceresistanceandprovideamuchhighercommunicationbandwidth[ 18 43 ]. Althoughthewavelength-routedall-opticalnetworksopenuparangeofopportunitiesforapplyingall-opticaltechnologiestonext-generationnetworkdesign,itisstillnoteasytoreachanoptimaldesignsolutionwithoutdeepunderstandingon 14

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resourceallocationproblemsduetothechallengefromthelimitedwavelengthresourceandwavelength-dependentimplementationcostintheswitchingfabrics. Inaddition,thesolutionstomoderncommunicationnetworksarefacingstrongerandstrongerfaulttolerancerequirementsespeciallyforthoseapplicationswithstringenttimelimitofdatadeliveryunderharshenvironmentalconditionsortheriskofdisasters[ 3 18 ]. Therefore,resource-utilization-efcientdesignsolutionsthatalsohavetosatisfythefaulttolerancerequirementsareneededindevelopingqualiedall-opticalarchitectures,routingandfailurerecoveryprotocols,andwavelengthassignmentalgorithms. Thisdissertationisdedicatedtoexploitingefcientwaystoaddressthosedesignchallengesviaacomprehensivestudyonrouting,wavelengthallocationandtopologicaloptimizationunderavarietyoffaulttolerancedemands. 1.2DissertationOrganization Thisdissertationisorganizedinto7chapters.Themotivationofinvestigatingrouting,wavelengthassignment,andtopologyoptimizationproblemsinall-opticalWDMnetworksispresentedinthischapter.ThroughChapters 2 to 6 ,thefocusesofdiscussionaremovedtosolvingaboveresource-allocation-relatedproblems,forvariousapplications,ingreatdetail.Chapter 7 concludesthisdissertationbyhighlightingndings,contributions,andfutureresearchgoalsonthisverytopicofthedissertation.Themainbodyofthisdissertationisasfollows. Chapter 2 iscenteredaroundthetorustopologyfocusingondevelopingtheFour-wayOptimalLink-Disjointroutingalgorithm(FOLD)andtheWavelengthAllocationandReuse(WAR)algorithminordertoenableafault-tolerantall-terminalcommunicationarchitecturewiththeminimumwavelengthrequirementforthenext-generationavioniconboardcommunicationsystems. Chapter 3 proposesfourwavelengthassignmentschemesforthe3redundantlightpaths(sparelightpaths)outofthe4link-disjointpathsdevelopedinChapter 15

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2 .Aenhancedfailurerecoveryalgorithmisproposedtofacilitatecommunicationswitchesuponfailureoccurrence.Atradeoffbetweenspareresourceallocationandfaulttoleranceperformanceisdiscernedviaanexhaustivesimulationovera44torus. Chapter 4 studiesthefault-tolerancepotentialofthecirculantgraphsviaexploitingnode-disjointroutinginacirculantgraphofanarbitrarysizeandconnectivitydegree.Anode-disjointroutingalgorithmthatfullyleveragesthecirculantgraphconnectivityisproposedforallpossiblesourceanddestinationpositions. InChapter 5 ,aspare-sharing-basedtopologicaloptimizationproblemisidentiedandaddressed,whichtargetstondalow-costtopologicalsolutiontoadaptthenetworktopologyoverthedisastrousnetworkattacks(earthquakes,hurricanes,oods,etc.)TheproblemisformulatedindifferentformsofIntegerLinearPrograms(ILPs)anditisshownthatthetraditionalroutingandwavelengthassignmentdecompositionbasedmethoddoesnotperformwellforthisproblem.Atwo-phaseheuristicalgorithm,basedondrawbackanalysisofthegreedyapproach,isproposedandsimulationresultsdemonstratesitsperformanceandcomputationaladvantagesoverthetraditionalmethodsthatareusedforsolvingtheRWA-relatedproblems. InChapter 6 ,aneworderedpathenumerationalgorithmisformallyproposedalongthewayofaseriesoftheoreticalderivations.Basedonthepoolofenumeratedpaths,acandidateroutingschemeisdevelopedtoidentifyasetofcandidateroutesforeachowrequestthattspecicproblems'nature.Finally,twoshared-path-protection-basedRWAproblemsaretestedandnumericalresultsshowgreatperformanceadvantagesoftheordered-path-enumeration-basedcandidateroutingoverthetraditionalk-shortestdisjointroutingbasedmethod. 16

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CHAPTER2TORUS-BASEDFOUR-WAYDISJOINT-LIGHTPATHSCOMMUNICATIONFORAVIONICWDMLANS OpticalNetworkingwithWavelengthDivisionMultiplexing(WDM)hasimmensepotentialtosatisfythefutureneedsofbothmilitaryandcommercialcommunicationsystems,duetoitshighbandwidthprovision,lowelectromagneticinterference,andlightweight[ 24 ].Inrecentyears,therehasbeenaninterestinreplacingcopperwithopticalberinavionicsystems.However,beropticcomponentsaresusceptibletofaultsduetotheiroperationaluncertainty.Inaddition,hazardousworkingconditionsmaketime-criticalcommunicationevenvulnerable[ 18 ].TheSocietyofAutomobileEngineers(SAE)hasspeciedvariousdesignRequirementsforOpticalNetworksinAvionic(RONIA)onboardcommunication,whicharebrieylistedinSection 2.3 .Therefore,thereisaneedtodesignappropriatecommunicationnetworkarchitecturesthatareabletoofferbothfaulttoleranceandefcientdatadeliverytoleveragetheadvantageousfeaturesofWDMtechnologies. Inthischapter,wefocusontherequirementsofcommunicationlatencyandfaulttolerance.Weproposesettingupmultiple(4)non-overlapping1lightpathsonthetorustoplogytoenablebothone-shotdatatransimissionandlightpath-switching-basedfailurerecoverycontrolledpurelyonthereceiverside.Werstdevelopanefcientnon-overlappinglightpathssetupalgorithm(calledFOLD(Four-wayOptimaLDisjointrouting))andproveitsoptimalityintermsofopticallinkresourceutilization.Then,basedonFOLD,awavelengthallocationandreuse(WAR)schemeenforcingwavelengthcontinuityisproposedtominimizethewavelengthutilizationforall-to-allcommunication. 1Inthischapter,thelightpathsarenon-overlappingaslongastheyarelink-disjoint. 17

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2.1RelatedWorks 2.1.1TopologicalOptions Regardingthetopologicalchoiceinavionicnetworkarchitecturedesign,variousopticalarchitecturesthatencompassawiderangeoftopologiesandroutingprotocolshavebeenproposedin[ 43 ].However,mostofthemdonotprovidethehigh-levelconnectedness,whichisrequiredtoachievehigh-levelfaulttolerance[ 61 ].Physicallybasedonatorustopology,[ 66 ]developsdifferenttypesoflogicaltopologiesusingak-hoproutingmodel.[ 64 ]and[ 51 ]discusstheroutingandwavelengthallocation(RWA)problemsundertheringtopology.However,thesepapersprovideverylimitedornosupportagainstnetworkfailures.In[ 60 ],weproposeapreliminarynon-overlappingfour-lightpathssetupalgorithmonthetorusstructure.However,thatworkdoesnotdetailroutingandwavelengthallocationfortoriofarbitrarysizes,whichwillbefullyaddressedinthischapter. 2.1.2FaultToleranceinWDMOpticalNetworks Withrespecttofault-tolerance-orientedstudiesforWDMopticalnetworks,[ 41 ]providesacomprehensiveclassicationofgeneralmesh-network-basedfault-toleranttechnologies.Itconcludesthatpath-basedprotectionoutperformslink-basedprotectionintermsofresourceutilization,andthatdedicated-pathprotectionoutperformsshared-pathprotectionintermsofconnectionreliabilityhoweverwiththecostofhigherresourceutilization.[ 53 ]discussesthecapacityprovisioningboundsforonefailurerecoveryintorus-basednetworksanddevelopsbothlink-basedandpath-basedrestorationstrategies.Actually,allrestoration-basedstrategiesrequirenon-negligibleprocessingtimeonfaultdetectionandresourcereallocation.Thereforetheymaynotmeettherequirementsoftime-criticalcommunication,asofavioniccommunication.[ 62 ]raisestheideaoflightpathdiversitytoenableamuchfasterfailureresponse,inwhichthesourcedeliversmultiplecopiesofdatatothedestinationbysplittingthelightontomultipleindependentlightpaths.However,itdoesnotdiscussanylightpath 18

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setupalgorithmsbasedonconcretetopologies.Moreover,sincethereplacementofthefailedlinksisalmostimpossibleduringmissionsofight,andmorethanonefailurecanhappenduringashortperiodespeciallyunderhazardousoperatingconditions,all+1-basedor:1-based2protectionsthatarewell-studiedintheliteratureasin[ 41 ],[ 21 ]and[ 37 ]maynottthefault-tolerantneedsofavioniccommunication.Therefore,adedicatedmulti-pathprotectiondesign,asproposedinthischapter,becomesdesirable. 2.1.3RoutingandWavelengthAssignment(RWA) Concerningroutingandwavelengthassignment(RWA)algorithms,[ 67 ]providesageneralintegerlinearprogram(ILP)formulationfortheRWAproblemsandofferssolutionsbydecouplingtheproblemintotherouting(R)partandwavelengthassignment(WA)part.[ 26 ]and[ 9 ]proposeanRWAalgorithmforsingle-lightpathall-to-allcommunicationunderthetorustopology.Itachievesoptimalwavelengthutilization(demandingN3=8wavelengths)fora2-dimensionaltoruswhentheone-dimensionalsizeofthetorus,N,iseven.Itisalsoshownin[ 47 ]that,foranoddNinthe2-dimensionaltorus,thereexistsanoptimalRWAschemerequiringN(N2)]TJ /F6 11.955 Tf 12.19 0 Td[(1)=8wavelengths.Duetotheroutingcomplexityof4-lightpathsetupforanycommunicationpairunderthetorustopology,theRWAproblembecomemuchharderanditisthefocusofthischapter.[ 31 ]discernsthetradeoffofdatadeliveryefciencyandwavelengthutilizationbetweentheone-shotall-opticalarchitectureandthemulti-hopoptical/electricalarchitecture.Itoffersageneralmulti-hoproutingalgorithmpursuingbalancebetweenfastdeliveryandwavelengthutilization.[ 46 ]developson-lineRWAalgorithmsforbidirectionalringandtorusarchitectures,whichattemptstominimizeaverageblockingprobabilityforanewtrafcsessiongivenaxednumberofwavelengths.However,thealgorithmis 2Intheliterature,+1referstotheprotectionschemeconsistingoftwodedicatedlightpathsforeachprotectedow,whereas:1correspondstotheprotectionschemeinwhichthesecondary(backup)light-pathcanbeusedforlow-prioritytrafctransmissionuntilafailurealongtheprimary(working)lightpathoccurs[ 35 ][ 14 ]. 19

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centralizedinnatureandrequirescorrectknowledgeoftheinstantaneousRWAovertheentirenetwork.Henceitisnotsuitablefordistributednetworkimplementation. 2.2ContributionsandChapterOrganization Inthischapter,weapplytheideaofredundantlightpaths,asin[ 62 ],toprotectthenetworkagainstfailuresandtoachievefastfailurerecoveryandzerodataloss.Becauseofitsconnectivityrichnessandsymmetry,thetorustopologyisexplored.TheformeroneisusedtodevelopdisjointlightpathsandthelatteroneisexploredwhendevelopingtheWARalgorithm. Themajorcontributionsofthischapterinclude:1.Atorus-based3-critical-fault-freeWDMbackbonearchitecturethatcansatisfyrequirementsofbothdatadeliveryeffectivenessandhigh-levelfaulttolerance.2.Anoptimal4-lightpathssetupalgorithm(calledFOLD)withefcientwavelengthallocationandreuseforall-to-allcommunicationinanarbitraryNNtorus.3.Acomprehensiveprobabilisticandnetworkcapacityanalysisforfaulttoleranceperformancedemonstration. Therestofthischapterisorganizedasfollows:Section 2.3 denesthenetworkarchitecture.Section 2.4 describesthenon-overlappinglightpathssetupalgorithm.Section 2.5 discussesWARforall-to-allcommunication.ThecontrollerimplementationisdescribedinSection 2.6 .Section 2.7 providesthefaulttoleranceperformanceanalysisoftheproposedarchitectureandsummarizesthischapter. 2.3NetworkArchitecture 2.3.1SAERequirementsandEvaluationMetrics TheSAEdesignrequirementsforopticalavioniccommunicationsystemsarespeciedbrieyasfollows[ 23 24 ]: TransparencyandHighBandwidth-theWDMLANsareexpectedtosupportsvarietyofsignalprotocolsforbothlegacyandnewapplicationswithoutanycompatibilityissue ScalableandSecure-scalableandrecongurablesystemswithpotentialMulti-LevelSecurity(MLS)support 20

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FlexibleNetworking-WDMLANscanbeoperatedwithsimplecontrol&managementtoenableeaseofuseandeaseoffuturenetworkcapacityupgrade FaultTolerant-reliabilityprovisionbyredundancyanddiversity ReduceSize,WeightandPower(SWaP)-compact,lowpowerandlowcostWDMsystems ThefocusofthisworkisprimarilyonthefaulttolerancerequirementsofSAEspecication,forwhichweproposeatorus-basedmulti-lightpatharchitecturewithabilitytotolerateupto3criticalberlinkfailures. AlongwiththeSAErequirementsforavionicWDMLANs,thereareasetofmetricsforpracticalorperformanceevaluationoftheproposedarchitecturedesign. RecoverySpeed.Ourworkappliesdedicatedredundantlightpathsprotectionandhencefailurerecoveryisbasedonswitchingreceptionamongdisjointlightpaths,whichleadstoveryfastrecovery. Reconguration(afterfailure).Inourwork,norecongurationisneededduringfailurerecoverybecausealldedicatedlightpathsaresetupinadvanceinthedesignphaseandnoswitchinglogicneedstoberecongured. Latency.Latencyisnegligibleinourcase,becauselightpathcommunicationeliminatesOptical-Electrical-Optical(OEO)conversionandqueuingprocessalongthedatadeliverypath. CapacityofFaultTolerance.Upto3criticallinkfailuresaresupportedduetothe4non-overlappinglightpathssetupproposedinthiswork. Size,WeightandPower(SWaP).Ourworkisexpectedtobemuchmorecapacity/size-,capacity/weight-,andcapacity/power-efcientthanthetraditionalcopperwiringbasedavionicsystemsduetotheweightadvantageofthebersandrecenttechnologicaladvanceinopticalswitchingdevice[ 11 ]. 2.3.2Torus-BasedArchitecture Duetotheadvantagesmentionedabove,thetorusisexploredasthebasicbackbonetopology,onwhichallfollow-onarchitecturalandprotocoldesignsarebased. 21

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Withoutlossofgenerality,takethe44torusasanexample.16controllers3,asshowninFigure 2-1 ,areconnectedviaopticalberscarryingWDMsignalsinacircularfashioninbothrow(orX)andcolumn(orY)directions.Neighboringcontrollersarebridgedviatwounidirectionalbers(Figure 2-1 onlyshowstheconnectioninsteadoftwoseparatelinks)inordertoallowforbidirectionalcommunications.Eachcontrollerhas4input/outputportsconnecting4neighborsfromtheeast,south,westandnorthdirectionsrespectively. Figure2-1. A44torusbackboneconnectedviaopticalbers 2.3.3Single-WavelengthLightpaths Time-criticalcommunicationhasrequirementsofminimumqueuingandtransmissiondelay,aswellasreliableprotectionagainstnetworkfaults,allofwhichrequiremultiplelightpathstobesetupbetweeneachsource-destination(S-D)pair.Dependingonwhetherwavelengthconvertersareusedinthecontrollers,alightpathcaneithertakeonasinglewavelengthormultiplewavelengthsondifferentlinksalongthepath.Theuseofwavelengthconverterscanleadtobetterroutingexibilityandeventuallybetterwavelengthutilization,butitalsoresultsinextra 3Hereweusecontrollerinsteadofnodebecausethearchitecturediscussedinthischapterisde-signedforbackboneuse.Differenttypesofsecond-tiernetworksmaybeconnectedtothebackboneviathecontroller. 22

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considerablephotoelectricdevicecost.Currentopticaltechnologieseitherrelyonoptical-electrical-optical(OEO)conversionorsemiconductoropticalamplier(SOA)toimplementwavelengthconversion[ 40 ].TheformertechnologyintroducesOEOdelaywhilethelatteroneisstillfacingstabilityissuesandsomeoperationalconstraints.Inthiswork,weenforcesinglewavelengthallocationonalightpath,whichleadstoalow-costcontrollerdesign.However,asaresult,thedesigndifcultyismovedfromthehardwareleveltotheroutingandwavelengthallocationlevel.Next,wedescribethelightpathssetupalgorithmtoestablishfournon-overlappinglightpathsforallS-DpairsinanNNtorus,whichenablesthenetworktotolerateatleastthreecriticallinkfailures. 2.4Non-OverlappingLightpathSetupAlgorithm:Four-wayOptimaLDisjointrouting(FOLD) Assumetheopticallinksfailinanindependentfashionwithafailureprobabilityf.Thenormaloperationprobabilitypisthereby1)]TJ /F3 11.955 Tf 12.22 0 Td[(f.Then,giventhatthefourlightpathsarelink-disjoint,theprobabilityofanS-DpairbeingdisconnectediscalculatedbyPdisconnection=[1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(f)l1][1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)l2][1)]TJ /F6 11.955 Tf 11.96 0 Td[((1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)l2][1)]TJ /F6 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(f)l2]=(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pl1)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pl2)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pl3)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pl4), (2) wherel1,l2,l3andl4denotethelengthsofthefournon-overlappinglightpathsinnumberofhops. Fromequation( 2 )weobservethatshorter-lengthlightpathscanleadtoalowerdisconnectionprobability.Besides,fromtheperspectiveofresourceutilization,alightpathssetupthatrequiresthelowernumberofopticallinksleadstothelighterlinkutilization.Sincebothoftheabovedesignconcernsagreeondevelopingshortlightpaths,weproposeagreedyapproachtosetupthefournon-overlappinglightpaths.ThegeneraldescriptionofthealgorithmislistedinFigure 2-2 Sincethenetworkarchitectureincludestwooppositeunidirectionalopticallinksforeachone-hopconnection,thereverselightpathssetupfromthedestinationtothesource 23

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Algorithm 1. Initialization:keepalltheopticallinksinthetorusstructure,S ,i 1(Sisthelightpathsetandiistheiterationindicatorofthealgorithm)2. Whilei4,do3:3. Findtheshortestpath,Path(i),fromthesourcetothedestinationinthecurrenttorusstructure,S S[Path(i),removeallthelinksinPath(i)fromthetorusstructure,i i+1,goto24. OutputS Figure2-2. Generalnon-overlappinglightpathsetupalgorithm canbeobtainedbyreversingthelightpathssetupfromthesourcetothedestination.Assuch,thelightpathssetupalgorithmworksforbi-directionalcommunication. Figure2-3. Source-destinationpositionalrelationship Inordertospecifyindetailhowthealgorithmworksforthetorustopology,werstintroduceseveralnotations.AsshowninFigure 2-3 ,dXanddYarethehorizontalandverticaldistancesfromasourcetoadestination,andNistheone-dimensionalsizeofthetorustopology.Ifthesourceandthedestinationarelocatedinthedifferentrowsanddifferentcolumns,wecalltheroutingX-Yrouting(Scenario1).Iftheyareinthesamerow,wecalltheroutingXrouting(Scenario2),andiftheyareinthesamecolumn,wecalltheroutingYrouting(Scenario3). 24

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2.4.1Scenario1:X-YRouting UsethetraditionalshortestX-YroutingtondtherstlightpathofthelengthdX+dY,thenusethetraditionalshortestY-XroutingtondthesecondlightpathofthelengthdY+dX.TherulesfortherstandsecondlightpathssetuparethesameregardlessofrelativeS-Dpositionsandthetorussize.However,dependingonthemagnituderelationamongdX,dYandN,theruleofdevelopingthethirdandfourthlightpathsvaries.WeillustrateallfourcasesinFigure 2-4 anddescribethembelowintermsofsettingupthethirdandfourthlightpaths. ACaseI BCaseII CCaseIII DCaseIV Figure2-4. LightpathssetupforX-Yrouting 25

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CaseI(Figure 2-4 (A)):1dXb(N)]TJ /F6 11.955 Tf 11.08 0 Td[(4)=2cand1dYb(N)]TJ /F6 11.955 Tf 11.09 0 Td[(4)=2c.Movefromthesourceverticallyandhorizontallybyonehopalongtheoppositedirectionsofthesecondandrstlighpaths,respectively,totheneighborcontrollersS1andS2,thenmovefromthedestinationhorizontallyandverticallybyonehopalongthesamedirectionsofthesecondandrstlightpathstotheneighborcontrollersD1andD2,thenrouteS1toD1usingshortestX-YroutingandS2toD2usingshortestY-XroutingtoobtainthethirdandfourthlightpathsofthesamelengthdX+dY+4. CaseII(Figure 2-4 (B)):dX>b(N)]TJ /F6 11.955 Tf 12.32 0 Td[(4)=2canddY>b(N)]TJ /F6 11.955 Tf 12.32 0 Td[(4)=2c.Movefromthesourcehorizontallyalongtheoppositedirectionoftherstlightpathuntilthecontrollerinthecolumnnexttothatofthedestinationisreached.ThenmoveverticallyusingshortestYroutingtothecontrollerrightnexttothedestination.ClosetheroutebythelasthoptoformalightpathofthelengthN)]TJ /F3 11.955 Tf 12.35 0 Td[(dX+dY.Similarly,movefromthesourceverticallyalongtheoppositedirectionofthesecondlightpathuntilthecontrollerintherownexttothatofthedestinationisreached,thenmovehorizontallyusingshortestXroutingtothecontrollerrightnexttothedestination,andclosetheroutebythelasthoptoformanotherlightpathofthelengthN)]TJ /F3 11.955 Tf 12.57 0 Td[(dY+dX.Thethirdlightpathistheshorteronewhilethefourthlightpathisthelongeroneofthesetwolightpaths,dependingonmagnitudeofdXanddY. CaseIII(Figure 2-4 (C)):dX>b(N)]TJ /F6 11.955 Tf 12.82 0 Td[(4)=2cand1dYb(N)]TJ /F6 11.955 Tf 12.82 0 Td[(4)=2c.UsethetechniqueforroutingtherstlightpathincaseIItoformthethirdlightpathoflengthN)]TJ /F3 11.955 Tf 12.16 0 Td[(dX+dY.Forthefourthlightpath,movefromthesourceverticallybyonehopalongtheoppositedirectionofthesecondlightpathtoitsneighborS1,thengohorizontallybyonehopalongtheshortestXroutingdirectiontoS2.MakeaverticalturnandroutealongtheshortestYroutingdirectionovertherowofthedestinationbyonehoptoD2.SwitchtherouteintohorizontaldirectionalongtheshortestXroutingdirectiontotheneighborofthedestinationD1,andnallyclosetheroutebythelastonehoptothedestinationwiththelightpathlengthdX+dY+4. 26

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CaseIV(Figure 2-4 (D)):1dXb(N)]TJ /F6 11.955 Tf 12.43 0 Td[(4)=2canddY>b(N)]TJ /F6 11.955 Tf 12.43 0 Td[(4)=2c.UsethetechniqueforroutingthesecondlightpathincaseIItoformthethirdlightpathoflengthN)]TJ /F3 11.955 Tf 12.64 0 Td[(dY+dX.Forthefourthlightpath,movefromthesourcehorizontallybyonehopalongtheoppositedirectionoftherstlightpathtoitsneighborS1.GoverticallybyonehopalongtheshortestYroutingdirectiontoS2.MakeaverticalturnandroutealongtheshortestXroutingdirectionoverthecolumnofthedestinationbyonehoptoD2.SwitchtherouteintotheverticaldirectionalongtheshortestYroutingdirectiontotheneighborthedestinationD1.Finally,closetheroutebythelastonehoptothedestinationwiththelightpathlengthdX+dY+4. 2.4.2Scenario2:XRouting UsethetraditionalshortestXroutingtondtherstlightpathwithlengthdX.TheruleofsettinguptherstlightpathisthesameforallcasesinScenario2regardlessofrelationshipbetweendXandN.However,thesetupisdifferentforthesecond,thirdandfourthlightpathsasdescribedbelow. Figure2-5. CaseI'lightpathssetup 27

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CaseI'(Figure 2-5 ):1dXb(N)]TJ /F6 11.955 Tf 12.85 0 Td[(8)=2c.MovefromthesourceverticallybyonehoptowardsthenorthtoS1,routealongtheshortestXroutingdirectiontothenorthneighborofthedestinationD1,andclosebythelasthoptheroutewithlengthdX+2toformthesecondlightpath.RoutethepathsymmetricallytowardsthesouthtoformthethirdlightpathwiththelengthdX+2.Wecallthesetwopathsmirroringpaths.Finally,movefromthesourcealongtheoppositedirectionoftherstlightpathbyonehoptotheneighborS3,turntothenorthbytwohopstoS4,switchintothehorizontaldirectionalongtheshortestXroutingdirectionoverthecolumnofthedestinationbyonehoptoD4,andthenturntothesouthandmovebytwohopstotheeastneighborofthedestinationD3.ClosetheroutebythelasthoptoformthefourthlightpathwiththelengthdX+8. Figure2-6. CaseII'lightpathssetup CaseII'(Figure 2-6 ):dX>b(N)]TJ /F6 11.955 Tf 12.26 0 Td[(8)=2c.UsethetechniqueabovetoformthetwosymmetricallightpathsofthelengthdX+2.ThenmovefromthesourcehorizontallyalongtheoppositedirectionoftherstlightpathtothedestinationtoformanotherlightpathofthelengthN)]TJ /F3 11.955 Tf 12.53 0 Td[(dX.Wecallthispaththecomplementarypath.IfdX+2is 28

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smallerthanN)]TJ /F3 11.955 Tf 11.98 0 Td[(dX,thersttwolightpathsofthelengthdX+2arethesecondandthirdlightpathsandtheoneofthelengthN)]TJ /F3 11.955 Tf 12.24 0 Td[(dXisthefourthlightpath.Otherwise,theorderisreversedaccordingly. 2.4.3Scenario3:YRouting ThisisexactlythemirroredscenarioofScenario2exceptthemainroutingdirectionistheYdirectionanddXisreplacedwithdY.CorrespondingtoCasesI'andII',weintroduceCasesIandIIforthisscenario(routingisnotshown). Theorem2.1.Theabove4-waynon-overlappinglightpathssetupreachesoptimallink-disjointroutingintermsofthetotalnumberoflinksused. Proof:SeeAppendix A forproof. Figure2-7. Summaryoflightpathssetupcasesinthedestinationgroup(Nodd) 2.4.4DestinationGroup Wedeneadestinationgroupwithrespecttoasourceasagroupofdestinations,asshowninFigures 2-7 and 2-8 ,whichuniquelycontributetothecompletesetofS-Dcombinations,ofcardinalityN2(N2)]TJ /F6 11.955 Tf 10.39 0 Td[(1)=2,inanNNtorus.Allthelightpathsetupcasescanbesummarizedintothedestinationgroup.AccordingtotherelationshipamongdX,dYandN,thedestinationgroupcanbepartitionedintosubgroupscorrespondingtoCasesI,II,III,IV,I',II',IandIIasshowninthedashedlinesurroundedareasinFigure 2-7 andFigure 2-8 29

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Figure2-8. Summaryoflightpathssetupcasesinthedestinationgroup(Neven) Inthenextsection,weproposeawavelengthallocationandreuseschemefortorusall-to-all4-waycommunicationswithatargettominimizewavelengthutilization. 2.5WavelengthAllocationandReuse(WAR) AnaivewaytoallocatewavelengthsistoassignaseparatewavelengthtoeachS-Dpair,suchthatthefournon-overlappinglightpathsforoneS-Dpairtakeonthesamewavelength.Asaresult,thenumberofwavelengthsrequiredforallS-DcommunicationsisgivenbythenumberofS-DpairsofatorusofsizeNN:Wnaive=N2(N2)]TJ /F6 11.955 Tf 11.96 0 Td[(1) 2. (2) However,thefournon-overlappinglightpathsarenotnecessarilyassociatedwiththesamewavelength.AnefcientwavelengthallocationalgorithmcanallowreuseofthesamewavelengthonmultiplelightpathsofdifferentS-Dpairs,withthegoalofachievingbetterwavelengthutilization. Wedenelinkwavelength(LW)asthewavelengthassociatedwithaspecicopticallink.AnLWisidentiedbyboththelinkindexandtheassociatedwavelength. 2.5.1ALowerBound(IdealWavelengthUtilization) Werstcalculatealowerbound,i.e.thelowestpossiblenumberofwavelengthsrequiredwhenalllightpathsforallS-Dpairsaresetupandnolinkwavelengthisleftidle. 30

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ThislowerboundcanbeexpressedasWLowerbound=Lall)]TJ /F4 7.97 Tf 6.58 0 Td[(to)]TJ /F4 7.97 Tf 6.58 0 Td[(all 2N2, (2) whereLall)]TJ /F4 7.97 Tf 6.58 0 Td[(to)]TJ /F4 7.97 Tf 6.58 0 Td[(allisthetotalnumberofLWsrequiredforall-to-allcommunicationusingtheoptimallightpathssetupalgorithmproposedinSection 2.4 ,and2N2isthenumberofLWsthatonewavelengthcanprovidethroughoutanNNtorus. Table2-1. SummaryofLS,Dexpressionsfordifferentcases LS,D Expressions LS,D(I) 4(dX+dY+2) LS,D(II)2(N+dX+dY) LS,D(III)N+2(2dY+dX+2) LS,D(IV)N+2(2dX+dY+2) LS,D(I0)4(dX+3) LS,D(II0)N+2(dX+2) LS,D(I00)4(dY+3) LS,D(II00)N+2(dY+2) WedenethetotalnumberofLWsrequiredforanS-DpairasLS,D,whichincludesthetotalnumberofhopsonthefourlightpathsoftheS-Dpair.Table 2-1 providesasummaryofLS,Dexpressionsderivedbasedontheproposednon-overlappinglightpathssetupalgorithm(FOLD),withrespecttodifferentcasesofthesourceandthedestinationlocations.ThenLall)]TJ /F4 7.97 Tf 6.59 0 Td[(to)]TJ /F4 7.97 Tf 6.59 0 Td[(allcanbecalculatedasLall)]TJ /F4 7.97 Tf 6.59 0 Td[(to)]TJ /F4 7.97 Tf 6.59 0 Td[(all=XS,DLS,D=N2LS, (2) whereLSisthenumberofLWsrequiredforaspecicsourcetosetupthe4-lightpathconnectionstoallthedestinationsinitsdestinationgroup(seeFigures 2-7 and 2-8 ).N2LSmakestheequalityholdduetothesymmetriccharacteristicofthetorusstructure,whichputsanevenpositiontoallthesourcesregardlessoftheirpositionsinthetorusstructure.ThecalculationofLSis:LS=XDLS,D, (2) 31

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wheretheexpressionsofLSwithrespecttovariousrangesandparitiesofNarelistedinTable 2-2 .ThedetailedderivationofthoseexpressionsisprovidedinAppendix B Table2-2. SummaryofLSexpressionsfordifferentN LS Expressions LS(2N5,Nisodd) (3N3)]TJ /F6 11.955 Tf 11.96 0 Td[(2N2+7N)]TJ /F6 11.955 Tf 11.95 0 Td[(8)=2 LS(2N5,Niseven)(3N3)]TJ /F6 11.955 Tf 11.96 0 Td[(2N2+8N)]TJ /F6 11.955 Tf 11.95 0 Td[(8)=2 LS(6N9,Nisodd)(2N3+9N2)]TJ /F6 11.955 Tf 11.95 0 Td[(28N+17)=2 LS(6N9,Niseven)(2N3+9N2)]TJ /F6 11.955 Tf 11.95 0 Td[(26N+16)=2 LS(N10,Nisodd)(N3+4N2)]TJ /F6 11.955 Tf 11.96 0 Td[(5N)]TJ /F6 11.955 Tf 11.96 0 Td[(32)=2 LS(N10,Niseven)(N3+4N2)]TJ /F6 11.955 Tf 11.96 0 Td[(4N)]TJ /F6 11.955 Tf 11.96 0 Td[(32)=2 AftercalculatingLS,wecanuseequations( 2 )and( 2 )tocalculateWLowerboundforanNNtorus. WenotethatduetothestructuralsymmetryoftheNNtorus,allthelinkshavethesamesetofpositionalrelationstoeachcontrollerinthenetwork.Inaddition,allthecontrollershavethesamelightpathsetupscheme.Therefore,thetotalnumberoflightpathsarrangedoneachlinkisthesamethroughoutthewholenetwork,whichmeansthatthenumberofwavelengthsrequiredforthewavelength-convertibletorusnetworkisalsoWLowerbound. Next,weshowhowthewavelengths,fordifferenttorussizes,areassignedtoallthelightpathstowhichwavelengthcontinuityisenforced. 2.5.2WavelengthAllocationandReuse(WAR)Algorithm Inordertominimizewavelengthutilizationandhowevertomaintainwavelengthcontinuityonallthelightpaths,weneedtondthebestlightpathsarrangementoneachwavelength,whichleadstothebestresourceutilization.Thefollowingdiscussionisorganizedintheorderofincreasingtorussizes. N=2:Wavelengthallocationforthe22torusistrivial,i.e.,thenaivewaytoallocatethewavelengthsdescribedatthebeginningofthissectioncanachieveoptimality(6wavelengths).SowestartdescriptionoftheproposedWARalgorithmfromthe33torusasshowninFigure 2-9 32

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A B C D E F Figure2-9. WARdemonstrationforthe33torus N=3:Intotalthereare36S-Dpairsforthe33torus,halfofwhichrequireX-YroutingandanotherhalfofwhichrequireXorYrouting.FortheS-DpairsrequiringX-Yrouting,Figure 2-9 (A)showsthearrangementoftwooffourlightpaths(inred)foreachof3S-Dpairs(horizontallycirculated)ononewavelength,andFigure 2-9 (B)showsthearrangementoftheothertwolightpaths(inred)foreachof3S-Dpairs(verticallycirculated)onanotherwavelength.AsobservedinFigure 2-9 (A)andure 2-9 (B),anunallocatedring(inblue)isleftunallocatedoneachwavelength.IfwerotatethearrangementinFigure 2-9 (A)verticallytwiceandrotatethearrangementinFigure 2-9 (B)horizontallytwice,followedbyareversionofallabovearrangements,wecanaccommodatethelightpathssetupsforallX-YroutingS-Dpairs(18pairs)on12wavelengthsandtherewillbeintotal12unallocatedringsondifferentwavelengthsleftevenlydistributedinthetorusstructure,whichcanbeusedtoarrangetwocomplementarylightpathsfor6X-routingand6Y-routingS-DpairsasshowninblueinFigures 2-9 (A)and 2-9 (B).FortheX-routingS-Dpairs,theothertwolightpathsareshowninFigures 2-9 (C)and 2-9 (D)respectively(inred)for3S-Dpairs.Thereisstillroomforarranging4mirroringlightpathsofY-routingS-DpairsshowninblueinFigures 2-9 (C)and 2-9 (D).AswerotatethearrangementsshowninFigures 2-9 (C)and 2-9 (D)horizontallytwice,allrequirementsforarrangingthemirroringlightpathsofall9X-routingS-Dpairsand6Y-routingS-Dpairsaresatised.Arrangementofmirroringlightpathsfortherest3Y-routingS-Dpairscanbemadeon2extrawavelengthsasshowninFigures 2-9 (E)and 2-9 (F).Thereby,23+2additionalwavelengthsare 33

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usedtothe12wavelengthsdiscussedpreviously.Finallyonelastwavelengthisneededtoprovide6ringstoaccommodate6pairsofcomplementarylightpathsthatthe12wavelengthscannot.Followingthisapproach,intotal,21wavelengths(WWAR(33)=21)arerequiredtoenableall-to-allcommunicationsforthe33torus,comparedwithWLowerbound(33)=19andWNaive(33)=36ascalculatedbyequations( 2 )and( 2 ). A B C D E F G H I J Figure2-10. WARdemonstrationforthe44torus N=4:Forthe44torus,thereareintotal120S-Dpairs,72ofwhichrequireX-Yrouting,24ofwhichrequireXroutingandtherest24ofwhichrequireYrouting.Figures 2-10 (A)and 2-10 (B)showhowtoarrangefourlightpaths(inred)ontwowavelengthsforthesourceandthedestinationonehopawayfromeachotherinbothXandYdirections.Bycircularlyrotatingandreversingthearrangements,wecanenable4-lightpathsconnectionsforallsuchS-Dpairs(32pairsintotal)using242=16wavelengthsandthereare32unallocatedringsevenlydistributedthroughoutthetorusstructureonthe16wavelengths.Figures 2-10 (C)and 2-10 (D)showhowtoarrangefourlightpathsontwowavelengthsforthesourceandthedestinationonehopawayfromeachotherinonedirectionandtwohopsawayfromeachotherintheotherdirection.Similarly,byrotatinghorizontallyandmirroringalongthediagonalthearrangements, 34

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wecanenablethe4-lightpathssetupforallthoseS-Dpairs(32pairsintotal)using242=16wavelengthswith16unallocatedringsevenlydistributedacrossthetorus.The8S-DpairswiththesourceandthedestinationtwohopsawayfromeachotherinbothXandYdirectionscanhavethe4-lightpathssetupsasshowninFigures 2-10 (E)and 2-10 (F).Byrotatinghorizontallybyonehop,thelightpathssetupforthose8S-Dpairsismadeusing4fullyutilizedwavelengths.The32+16=48evenlydistributedunallocatedringscanbeusedforarrangingthecomplementarylightpathsofthe48S-DpairswhichrequireXorYroutings.Thetwomirroringlightpathsofthose48S-DpairscanbearrangedasshowninFigures 2-10 (G), 2-10 (H), 2-10 (I)and 2-10 (J).Figures 2-10 (G)and 2-10 (H)showhowthetwomirroringlightpathsarearrangedforthesourceanddestinationtwohopsawayfromeachotherinXdirectionviaroutingthesecondlightpathwiththesamelengthintheoppositeXdirection.Byrotatinghorizontallybyonehopandmirroringalongthediagonalthearrangements,allthemirroringlightpathsfor2-hop-away(bothhorizontallyandvertically)S-Dpairs(16pairsintotal)canbearrangedon8wavelengths,onwhichonemirroringlightpathfor32one-hop-awayS-Dpairscanalsobearrangedasshown.TheothermirroringlightpathsetupcanbenalizedasshowninFigures 2-10 (I)and 2-10 (J),togetherwiththeirmirroredsetupsalongthediagonal,whichintotalrequires4wavelengths.Therefore,the4-lightpathssetupfor44torusall-to-allcommunicationscanbeenabledby16+16+4+8+4=48wavelengths(WWAR(44)=48),comparedwithWLowerbound(44)=46andWNaive(44)=120. N5:WhenNgrowslarge,allroutingcases(I,II,III,IV,I',II',IandII)willappear.InsteadofillustratingtheWARalgorithmforeachtorussize,startingfromthe55torus,wedevelopageneralmethodtodealwithlightpathsarrangementinlightofthefollowingobservations. AsshowninFigure 2-11 (A),thethirdlightpaths(inred)ofNCase-I-routingS-Dpairscanbepiledupalongthediagonaldirectiontogetherwiththesecondlightpaths(in 35

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A B C D E F Figure2-11. Groupinglightpaths 36

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blue)ofotherNS-Dpairsonthesamewavelength,whichareformedbyinterchangingdXanddYoftherstNS-Dpairs.SimilarlythefourthlightpathsandrstlightpathscanbearrangedtogetherononewavelengthforaboveS-DpairsofCaseI. AsshowninFigure 2-11 (B),thethirdlightpaths(inred)ofNS-Dpairs(dX>dY)incasesIIorIIIcanbepiledupalongthediagonaldirectiontogetherwiththethirdlightpaths(inblue)ofotherNS-Dpairs(dX
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Figure2-12. Groupmirroringlightpaths Finally,forthesourceanddestinationlocatedinthesameroworcolumn,afullringcanaccommodatethetwocomplementarylightpathsandNmirroringlightpathscanbepiledupinawayshowninFigure 2-12 .SinceefcientarrangementofmultiplefourthlightpathsfortheS-Dpairs(refertoFigure 2-5 )ofcasesI'andIisalmostimpossible,weapplythecomplementaryroutingforCasesII'andIItotheS-DpairsofcasesI'andI. Figure2-13. NewlightpathsetupcaseswithconsiderationofWAR(Nisodd) Hence,newroutingdiagramsconsideringwavelengthallocationandreuseonthedestinationgroupforbothparitiesofNbecomeFigures 2-13 and 2-14 .Differentdashed 38

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Figure2-14. NewlightpathsetupcaseswithconsiderationofWAR(Niseven) linecoveredareasrepresentthedifferentroutingcasesdescribedinSection 2.4 .Next,weshowthegeneralWARalgorithmforN5infollowingprocedures. Firstly,foreachdestinationcontrollerinthepartitionofCaseI,twowavelengthsareneededtoarrangefourlightpathsofNsuchS-Dpairsofthesamepositionalrelationshipalongthediagonaldirection(asshowninFigure 2-11 (A)),andafterrotatingthearrangementalongeitherXorYdirectionNtimesallN2S-Dpairsofsuchpositionalrelationshiparearrangedwith4lightpathssetupon2Nwavelengths. ForeachdestinationcontrollerinthepartitionofcaseIIonthediagonal,Figure 2-11 (B)canbeappliedtoarrangeNthirdandNfourthshortestlightpathsononewavelength,andtheNrstandNsecondshortestlightpathscanbearrangedononewavelengthinthewayshowninFigure 2-11 (E). FortheedgecontrollerswhenNiseven,twowavelengthsareneededtopilefourlightpathsupforNsuchS-Dpairsofthesamepositionalrelationship,wherethesecondandfourthshortestlightpathsarearrangedusingFigure 2-11 (F),twothirdshortestlightpathscomingfromtwoS-DpairsmirroringeachotheralongthediagonalcanbearrangedononewavelengthusingFigure 2-11 (B)andtworstshortestlightpathscomingfromtwoS-Dpairsmirroringeachotheralongthediagonalcanbearrangedon 39

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onewavelengthusingFigure 2-11 (E).Hence,ingeneral,weneed2NwavelengthstocompletelightpathssetupforN2S-Dpairsofsuchpositionalrelations. ForthedestinationcontrollersintheareaofcasesIIIandIV,wenoticethatthefourthshortestlightpathingeneralcannotbepiledupwithotherlightpathssoonlyNofthemoccupyonewavelengthasshowninFigure 2-11 (C),althoughtherstandsecondshortestlightpathscanbepileduptogetherontoonewavelengthasshowninFigure 2-11 (D)andthethirdshortestlightpathforthedestinationcontrollerincaseIIIcanbepiledupwiththethirdshortestlightpathforthedestinationcontrollerincaseIVasshowninFigure 2-11 (B).Therefore,ingeneral2.5NwavelengthsareneededtosetupfourlightpathsforallsuchN2S-Dpairs. Secondly,weobservethattheunusedlinksforthedestinationcontrollersincaseIcanaccommodatethesameamountofarrangementforotherdestinationcontrollersincaseI.Forexample,aftertwolightpathsarrangementforthedestinationcontrollerwithdX=1anddY=1wecanhaveN)]TJ /F3 11.955 Tf 12.56 0 Td[(dX)]TJ /F3 11.955 Tf 12.57 0 Td[(dY)]TJ /F6 11.955 Tf 12.57 0 Td[(2=N)]TJ /F6 11.955 Tf 12.57 0 Td[(4connectedunusedlinksinbothXandYdirections,whichcanaccommodateanothertwolightpatharrangementforthedestinationcontrollerwithdX=(N)]TJ /F6 11.955 Tf 12.26 0 Td[(5)=2anddY=(N)]TJ /F6 11.955 Tf 12.26 0 Td[(7)=2(whenNisodd)orwithdX=(N=2))]TJ /F6 11.955 Tf 12.47 0 Td[(3anddY=(N=2))]TJ /F6 11.955 Tf 12.47 0 Td[(3(whenNiseven).Asaresult,infactweonlyneedonewavelengthtoarrangefourlightpathsofNS-Dpairswiththedestinationcontrollerscoloredinolivegreen.Besides,whenNisodd,inCaseIIIorIV,forthetwodestinationcontrollersbytwohopsawayfromthediagonal,thefourthshortestlightpathscanbepiledupwiththerstorthesecondshortestlightpaths(sincedX=dY+2ordY=dX+2),sothewavelengthrequirementforthesedestinationcontrollersdecreasesform2.5Nto2NforN2suchS-Dpairs.TheareasindifferentcolorsinFigures 2-13 and 2-14 indicatedifferentnumberofwavelengthsrequiredintheWARalgorithm. Finally,weneedtodealwiththelightpathsarrangementforthedestinationcontrollersinthesameroworcolumnofthesourcecontroller.SincetherearemanyunallocatedlinksleftduringlightpathssetupforX-YroutingS-Dpairs,manylightpaths 40

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forthoseXorYroutingdestinationcontrollerscanbearrangedonthoseunallocatedlinks. WhileNisodd: N11:Theunallocatedlinks(asshowninFigure 2-11 (E))fortherstshortestlightpathforthedestinationcontrollerofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.66 0 Td[(3)=2,dY=1)andthesecondshortestlightpathforthedestinationcontrollerofcoordinates(dX=1,dY=(N)]TJ /F6 11.955 Tf 12.28 0 Td[(3)=2)canaccommodateNmirroringlightpathsinthediagonaldirection(asshowninFigure 2-12 )forthedestinationcontrollerofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.7 0 Td[(1)=2,dY=0),theotherhalfofthemirroringlightpathscanbearrangedtogethersimilarlywithothertwolightpathsforabovetwodestinationcontrollers.Inaddition,theunallocatedlinks(asshowninFigure 2-11 (A))ontwoallocatedwavelengthsforthedestinationcontrollersofcoordinatesfrom(dX=(N)]TJ /F6 11.955 Tf 13.18 0 Td[(5)=2,dY=1)to(dX=(N)]TJ /F6 11.955 Tf 13.18 0 Td[(5)=2,dY=(N)]TJ /F6 11.955 Tf 11.98 0 Td[(5)=2)canaccommodatetwomirroringlightpathsofthedestinationcontrollersofcoordinatesfrom(dX=(N)]TJ /F6 11.955 Tf 12.93 0 Td[(3)=2,dY=0)to(dX=2,dY=0).Thelasttwomirroringlightpathsforthedestinationcontroller(dX=1,dY=0)canbearrangedwiththerstandsecondshortestlightpathsfortwodestinationcontrollersofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.34 0 Td[(1)=2,dY=(N)]TJ /F6 11.955 Tf 12.35 0 Td[(3)=2)and(dX=(N)]TJ /F6 11.955 Tf 12.35 0 Td[(3)=2,dY=(N)]TJ /F6 11.955 Tf 12.35 0 Td[(1)=2)separately.Followingthesameidea,wecanusetheunallocatedlinksforthecorrespondingdestinationcontrollersinthelefthalftoaccommodatetwomirroringlightpathsforthedestinationcontrollersofcoordinatesfrom(dX=0,dY=1)to(dX=0,dY=(N)]TJ /F6 11.955 Tf 12.03 0 Td[(1)=2).Forthetwocomplementarylightpathsofthelengthfrom1toN)]TJ /F6 11.955 Tf 12.8 0 Td[(1,weexploretheunallocatedlinksfortherstandsecondshortestlightpathsarrangedinthewayshowninFigure 2-11 (D)forthedestinationcontrollersofcaseIIIandIV.Weobservethat,exceptforthelightpathoflengthN)]TJ /F6 11.955 Tf 12.42 0 Td[(1,allthelightpathscanbearrangedusingthoseunallocatedlinks.Theunarrangedlightpaths(oflengthN)]TJ /F6 11.955 Tf 10.92 0 Td[(1)needextraNwavelengths. 41

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So,thetotalnumberofwavelengthsrequiredtoarrangealllightpathsisWWAR=N)]TJ /F6 11.955 Tf 11.95 0 Td[(7 2N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 221+N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 222+"N)]TJ /F6 11.955 Tf 11.96 0 Td[(1 22)]TJ /F19 11.955 Tf 11.95 16.86 Td[(N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 22#22.5)]TJ /F6 11.955 Tf 11.95 0 Td[(120.5)N+N=1 2N(N2+12N)]TJ /F6 11.955 Tf 11.95 0 Td[(55). (2) N=9:Nowtherstandsecondshortestlightpathstothedestinationcontrollersofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.27 0 Td[(3)=2,dY=(N)]TJ /F6 11.955 Tf 12.27 0 Td[(7)=2)and(dX=(N)]TJ /F6 11.955 Tf 12.28 0 Td[(7)=2,dY=(N)]TJ /F6 11.955 Tf 12.27 0 Td[(3)=2)needtobearrangedinthewayshowninFigure 2-11 (E)inordertoaccommodateXorYroutinglightpaths,sowavelengthrequirementbecomes2.5forthesedestinationcontrollersandWWARbecomesWWAR=N)]TJ /F6 11.955 Tf 11.95 0 Td[(7 2N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 221+N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 222+"N)]TJ /F6 11.955 Tf 11.96 0 Td[(1 22)]TJ /F19 11.955 Tf 11.95 16.86 Td[(N)]TJ /F6 11.955 Tf 11.95 0 Td[(5 22#22.5)]TJ /F6 11.955 Tf 11.95 0 Td[(80.5)N+N=1 2N(N2+12N)]TJ /F6 11.955 Tf 11.95 0 Td[(51). (2) N=7:Similarlysincetherstandsecondshortestlightpathstothedestinationcontrollersofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.63 0 Td[(5)=2,dX=(N)]TJ /F6 11.955 Tf 12.63 0 Td[(1)=2)and(dX=(N)]TJ /F6 11.955 Tf 12.63 0 Td[(1)=2,dY=(N)]TJ /F6 11.955 Tf 10.4 0 Td[(5)=2)needtobearrangedinthewayshowninFigure 2-11 (D),thewavelengthrequirementforthesedestinationcontrollersbecomes2.5andhenceWWARbecomesWWAR=N)]TJ /F6 11.955 Tf 11.95 0 Td[(7 2N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 221+N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 222+"N)]TJ /F6 11.955 Tf 11.96 0 Td[(1 22)]TJ /F19 11.955 Tf 11.95 16.86 Td[(N)]TJ /F6 11.955 Tf 11.95 0 Td[(5 22#22.5)]TJ /F6 11.955 Tf 11.95 0 Td[(40.5)N+N=1 2N(N2+12N)]TJ /F6 11.955 Tf 11.95 0 Td[(47). (2) N=5:Sincedestinationcontrollersofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.29 0 Td[(3)=2=1,dY=1)and(dX=1,dY=(N)]TJ /F6 11.955 Tf 12.31 0 Td[(3)=2=1)becomethesamecontrollers,theunallocatedlinksofthearrangementoftherstandsecondshortestlightpathsforthiscontrollercanonly 42

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accommodateNmirroringlightpathsinthediagonaldirection(asshowninFigure 2-12 )forthedestinationcontrollerofcoordinates(dX=(N)]TJ /F6 11.955 Tf 12.59 0 Td[(1)=2=2,dY=0),extra2Nwavelengthsareneededtoaccommodatetheotherhalfofabovemirroringlightpaths.HenceWWARbecomesWWAR=N)]TJ /F6 11.955 Tf 11.95 0 Td[(7 2N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 221+N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 222+"N)]TJ /F6 11.955 Tf 11.95 0 Td[(1 22)]TJ /F19 11.955 Tf 11.96 16.85 Td[(N)]TJ /F6 11.955 Tf 11.96 0 Td[(5 22#22.5)]TJ /F6 11.955 Tf 11.96 0 Td[(80.5)N+N+2N=1 2N(N2+12N)]TJ /F6 11.955 Tf 11.95 0 Td[(43). (2) WhileNisevenandN6: SimilartothemethodforNbeingodd,weusetheunallocatedlinksforthedestinationcontrollersofcoordinates(dX=(N=2))]TJ /F6 11.955 Tf 11.68 0 Td[(1,dY=1),(dX=(N=2))]TJ /F6 11.955 Tf 11.68 0 Td[(2,dY=1)through(dX=(N=2))]TJ /F6 11.955 Tf 12.1 0 Td[(2,dY=(N=2))]TJ /F6 11.955 Tf 12.11 0 Td[(2)toaccommodatethetwomirroringlightpathsforthedestinationcontrollersofcoordinatesfrom(dX=(N)]TJ /F6 11.955 Tf 12.22 0 Td[(2)=2,dY=0)to(dX=1,dY=0).However,thelightpathsforthedestinationcontroller(dX=N=2,dY=0)areleftwithoutbeingarranged,sotogetherwiththelightpathsforthedestinationcontroller(dX=0,dY=N=2),extra2Nwavelengthsareneededtoarrangethoselightpaths.Inaddition,3Nwavelengthsareneededtoarrange3complementarylightpathswithlengthN)]TJ /F6 11.955 Tf 12.22 0 Td[(1,N)]TJ /F6 11.955 Tf 12.22 0 Td[(2andN)]TJ /F6 11.955 Tf 12.22 0 Td[(3forthedestinationcontrollersinthesameroworcolumnofthesourceby1,2and3hopsaway.ThenWWARbecomesWWAR=(2N)]TJ /F6 11.955 Tf 11.95 0 Td[(6 221+2"N)]TJ /F6 11.955 Tf 11.96 0 Td[(1 22)]TJ /F19 11.955 Tf 11.95 16.86 Td[(N)]TJ /F6 11.955 Tf 11.95 0 Td[(6 22#2+N)]TJ /F6 11.955 Tf 11.96 0 Td[(4 240.5N+2N+3N=1 2N(N2+10N)]TJ /F6 11.955 Tf 11.96 0 Td[(32). (2) TheexpressionsofWWARwithrespecttodifferenttorussizesarelistedinTable 2-3 .BasedontheWWARexpressions,wecanplotthenumberofwavelengthsrequiredforall-to-allcommunicationswithcomparisonamongWNaive,WWARandWLowerbound, 43

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Table2-3. SummaryofWWARexpressionsfordifferenttorussizes WWAR Expressions WWAR(N=2) 6 WWAR(N=3)21 WWAR(N=4)48 WWAR(N=5)N(N2+12N)]TJ /F6 11.955 Tf 11.96 0 Td[(43)=2 WWAR(N=7)N(N2+12N)]TJ /F6 11.955 Tf 11.96 0 Td[(47)=2 WWAR(N=9)N(N2+12N)]TJ /F6 11.955 Tf 11.96 0 Td[(51)=2 WWAR(N11,Nisodd)N(N2+12N)]TJ /F6 11.955 Tf 11.96 0 Td[(55)=2 WWAR(N6,Niseven)N(N2+10N)]TJ /F6 11.955 Tf 11.96 0 Td[(32)=2 asshowninFigure 2-15 .Thenumericaldetailsfortoriwithone-dimensionalsizerangingfrom2to10areshowninTable 2-4 .Fromtheresults,wecanobservethegreatwavelengthsavingviausingtheproposedWARalgorithmandthatWWARisactuallyveryclosetothelowerboundWLowerbound.ItisalsonoticedthatWARperformsbetterwhenNiseventhanwhenNisodd,mainlyduetothelightpatharrangementeasinessforanevenNasshowninFigure 2-14 .Forthetoriinsmallsizes,suchas22,33,44,theWAR-derivedresultsareequalorveryclosetothebestpossibleresults(lowerbounds).Thishencecanleadtoeliminationofwavelengthconvertersfromthecontrollerdesignwithoutnoticeablewavelengthutilizationdegradation. Figure2-15. WARalgorithmperformance 44

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Table2-4. Wavelengthrequirementforvariedtorussizes N WLowerboundWWARWNaive 2 666 3192136 44648120 588105300 6154192630 72373011176 83524482016 94886213240 106648404950 Wenotethatthenumberofwavelengthsapproachesanunusuallylargenumber(tenthousandwhenthedimensionsizeNisabove25,)whichmayseemunrealistic.Theseresultsareacademicallyinformative.However,theresearchprojectthatthischapterisbaseduponislimitedtoamorerealisticsized44torus,whichsupports16backbonecontrollersandrequires48wavelengthsoneachone-hopconnection,orN=4inTable 2-4 [ 45 ]. 2.6ControllerImplementation Theactualcontrollerimplementationneedstointegratefunctionalitiesofthecontrollerasthesource,destinationandintermediateroutertogether.Asadatasource,datatargetingtoaspecicdestinationneedtobemodulatedontowavelengthsallocatedbytheWARalgorithmandthenmultiplexedwithallpass-throughwavelengthsbeforebeingsent.Asanintermediaterouter,all-optical13switchescanbeusedtoswitchthesignalfromaspecicinputporttoanyoftheremaining3outputportsbasedonthelightpathssetup.Theuseofall-opticalswitcheseliminatesOEOconversionalongalllightpathsandhenceenablesefcientone-shottransmission.Finally,asadatasink,thespecicwavelengthsaredroppedbasedontheWARalgorithmagainfromall4receivingdirections.ThecontrollerreceptionstructureisshowninFigure 2-16 Inthefault-freecase,4copiesofdemodulateddataarebufferedandtheonereceivedfromtheshortestlightpathispassedontotheapplicationlayerforprocessing. 45

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Figure2-16. Receptionstructureofthecontroller Upondatareceptionbytheapplicationlayer,thedatareceivedfromtheremaining3directionsaredeletedfromthecorrespondingbuffers. Uponoccurrenceofalinkfailure,aspecialtypeofmessagecalledLinkStatusNotication(LSN)willreacheachcontrollerthroughbroadcastfromthedetectingcontroller.Thecontrollerthencanmakedecisionofwhetherornottoswitchitsreceivingdirectionbasedonknowledgeofthelightpathssetup. 2.7PerformanceAnalysis Inthissection,werstintroduceprobabilisticmodelstoanalyzenetworkconnectionreliabilitiesfortheproposedarchitecture.Thenweshowtheimpactonnetworkthroughputcausedbyaseriesofnetworkfaultstoprovideadditionalinsightonthenetworkfaulttoleranceperformance. 2.7.1ProbabilisticAnalysis Thefaulttolerancemetricsusedaretwo-terminalreliability(TTR),one-to-all-othersreliability(OAR)andall-terminalreliability(ATR)forcomprehensiveanalysis.TTRindicatescommunicationreliabilityforasingleS-Dpair.OARsigniesthebroadcastreachabilityfromaspecicsource,whileATRreectsthefunctioningpossibilityofthewholenetwork. Tocarryouttheprobabilisticanalysis,weassociatetheprobabilityoffailure(f)toeachone-hopconnectionbetweenneighboringcontrollers.Undertheassumptionof 46

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independentone-hopconnectionfailurewithidenticalprobability,generalapproachestocalculatingprobabilitiesofTTR,OARandATRareasfollows. Theprobabilityoftwo-terminalreliabilityisPTTR=1)]TJ /F4 7.97 Tf 21.5 14.94 Td[(eXi=SDCSDifi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i, (2) wherefistheuniformprobabilityoffailureforallone-hopconnections,eisthenumberofone-hopconnectionsinthenetwork,SDistheminimumnumberofone-hopconnectionfailuresrequiredtodisconnecttheS-DpairandCSDiisthenumberoffailuresetswithrespecttothesource(S)anddestination(D)ofcardinalityi. Theprobabilityofone-to-all-othersreliabilityisPOAR=1)]TJ /F4 7.97 Tf 19.17 14.95 Td[(eXi=SCSifi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i, (2) whereSistheminimumnumberofone-hopconnectionfailuresrequiredtodisconnectthesourcefromatleastonecontrollerintherestofthenetworkandCSiisthenumberoffailuresetsofcardinalityiwithrespecttothesource(S). Theprobabilityofall-terminalreliabilityis:PATR=1)]TJ /F4 7.97 Tf 18.35 14.94 Td[(eXi=Cifi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i, (2) whereistheminimumnumberofone-hopconnectionfailuresrequiredtodisconnectanyS-DpairsandCiisthenumberoffailuresetsofcardinalityiwithrespecttoanyS-Dpairs. InsteadofshowingTTR,OARandATR,thischapterpresentsanalysisoftwo-terminalunreliability(TTUR),one-to-all-othersunreliability(OAUR)andall-terminalunreliability(ATUR),whichtakeoncomplementaryprobabilitiesofTTR,OARandATRrespectively.Thelowertheaboveprobabilities,thebetterthenetworkreliabilitycanbeachieved. 47

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Thegeneralcalculationapproachestocalculatingprobabilitiesofnetworkconnectedness,asshowninequations( 2 ),( 2 )and( 2 ),havebeenprovedtobelongtoNP-hardproblems[ 61 ].ResearchersusuallyapplyboundingtechniquesoruseMonteCarlosamplingmethodstoobtainreliabilitymeasures.However,withrespecttotheproposednon-overlapping4-lightpathssetupalgorithm(FOLD)andfaulttoleranceprotocol,theTTURprobabilitycanbecalculatedbyasimpleclosed-formexpression. Forthetwo-terminalconnectedness,itcanbeshownthatonlyafterallfourlightpathsareblockedcantheS-Dpairloseconnectivity.Hence,theprobabilityofTTURisgivenbythesameexpressionasinequation 2 ,rewrittenasfollowPTTUR=(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pl1)(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pl2)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pl3)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pl4). (2) Duetothesymmetricnatureofthetorusstructure,thereisnodifferenceforselectionofthesourcecontroller,sowexthesourceatthecontroller11andtakea44torusasanexampleofanalysis(refertoFigure 2-1 forcontrollerindices).Figure 2-17 showstheprobabilitiesoftwo-terminaldisconnection(PTTUR)of11from12andfrom33,whicharetwoextremecasesintermsofthedistancebetweenthesourceandthedestinationina44torus.Itcanbeobservedthattheconnectionreliabilitygetsgreatlyimprovedbycomparingwiththeprobabilityofonepathfailure(suchasthepathbetween11and33).ThedistributionofPTTURfromthecontroller11acrossthenetworktoallothercontrollersisgiveninFigure 2-18 ,whichprovidesinsightonrelativepositionalimpactofthesourceandthedestinationinatorusonTTUR. Withrespecttotheone-to-all-othersandtheall-terminalreliabilityanalysis,followingequations( 2 )and( 2 ),forallpossiblecombinationsofone-hopconnectionfailures,theone-to-all-othersconnectivityandtheall-terminalconnectivityareexaminedundertheproposedlightpathssetup.AnexhaustivecalculationofPOAURandPATURfora44torusiscarriedoutandtheresultsareshowninFigure 2-17 .Theresultsshowthattheproposedfault-tolerantarchitectureoffersgoodreliabilityevenfor 48

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Figure2-17. Networkunreliabilityanalysisfora44torus Figure2-18. TTURdistributionfora44torus(f=0.1) all-terminalconnectivityunderregularfailurestress(f0.1).Itisexpectedthattheone-to-all-othersconnectivityislessreliablethanthetwo-terminalconnectivityandtheall-terminalconnectivityisleastreliable. Sometimepeoplemaybeconcernedaboutthechanceofconnectionsustenanceuponoccurrencesofanumberofone-hopconnectionfailures.TheconditionalPTTUR,POAURandPATURonaxednumberofone-hopconnectionfailuresaregivenby 49

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equations( 2 ),( 2 )and( 2 ):PTTURjnone)]TJ /F4 7.97 Tf 6.59 0 Td[(hopconnectionfailures=CSDn )]TJ /F4 7.97 Tf 5.53 -4.38 Td[(en (2)POAURjnone)]TJ /F4 7.97 Tf 6.59 0 Td[(hopconnectionfailures=CSn )]TJ /F4 7.97 Tf 5.53 -4.38 Td[(en (2)PATURjnone)]TJ /F4 7.97 Tf 6.59 0 Td[(hopconnectionfailures=Cn )]TJ /F4 7.97 Tf 5.53 -4.38 Td[(en (2) where)]TJ /F4 7.97 Tf 5.52 -4.38 Td[(enisthenumberofcombinationsofnone-hopconnectionsoutoftheone-hopconnectionsetofcardinalitye.Themeaningsofrestitemsinaboveformulaearethesameastheyareinequations( 2 ),( 2 )and( 2 ).Figure 2-19 showstheresultsforthoseconditionalevaluationsofconnectionreliability.Theresultstestifyourarchitecturedesignagainstany3criticalcutswithoutlossofanynetworkconnectivity.Theresultsalsoshowthetrendsofconditionalreliabilityvariationwiththenumberoffailedone-hopconnections.Itcanbeobservedthatthereisabigprobability(>0.8)foratwo-terminalconnectiontosurviveovereven8one-hopconnectionfailures. Figure2-19. Conditionalprobabilitiesofconnectionfailuresfora44torus 50

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2.7.2NetworkCapacityAnalysis Notonlyistheconnectionreliabilityaffectedbythenetworkfaults,butalsothenetworkcapacity,becausetheblockedtransmissionswilldegradethenetworkcapacity.WedenethenetworkcapacityasthesumofcommunicationcapacitiesofallS-Dpairs.Assumethecommunicationisoperatedviaopticallaserswithatransmissionrateof1Gbps.Thenthenetworkcapacitysimplyequalstheproductofthenumberofcommunicationpairsandthebidirectionaltransmissionrate(2Gbps)inthefailure-freecase.Hereweconsiderall-to-allcommunicationandhencethefailure-freenetworkcapacityisgivenbyTfault)]TJ /F4 7.97 Tf 6.59 0 Td[(free=N2(N2)]TJ /F6 11.955 Tf 11.95 0 Td[(1) 2C, (2) whereCisthebidirectionaltransmissionrate. Thenetworkcapacityuponanumberofone-hopconnectionfailurescanbemeasuredintermsofaveragedegradednetworkcapacity,best-casedegradednetworkcapacityandworst-casedegradedcapacity.Theaveragedegradednetworkcapacityistheexpectednetworkcapacityunderacertainnumberofone-hopconnectionfailures(n)thatcanevenlytakeplaceacrossthenetwork.ItisdenedbyTavgjnfailures=Tfault)]TJ /F4 7.97 Tf 6.58 0 Td[(free)]TJ /F3 11.955 Tf 11.95 0 Td[(C(en)Xi=1Di )]TJ /F4 7.97 Tf 5.52 -4.38 Td[(en, (2) whereDiisthenumberofdisconnectedS-Dpairscausedbytheithnone-hopconnectionfailurecombination. Theworst-casedegradedcapacityundernone-hopconnectionfailuresisdenedbyTworstjnfailures=Tfault)]TJ /F4 7.97 Tf 6.58 0 Td[(free)]TJ /F3 11.955 Tf 11.95 0 Td[(CmaxiDi, (2) wheretheindexivariesfrom1to)]TJ /F4 7.97 Tf 5.53 -4.38 Td[(en. 51

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Thebest-casedegradedcapacityundernone-hopconnectionfailuresisdenedbyTbestjnfailures=Tfault)]TJ /F4 7.97 Tf 6.59 0 Td[(free)]TJ /F3 11.955 Tf 11.96 0 Td[(CminiDi. (2) Westillusethe44torustoexemplifythefaultimpactonnetworkcapacityanddemonstratefaulttoleranceperformanceoftheproposedarchitecture.Thecalculationofalldegradednetworkcapacitiesinequations( 2 ),( 2 )and( 2 )isviaexhaustivefailureenumerationandconnectioncheckduetotheacceptablesizeoftheselectedtorus. Figure2-20. Effectsofnetworkfailuresonnetworkcapacity Figure 2-20 showshowthenetworkcapacitydegradeswiththenumberofone-hopconnectionfailuresundertheproposed4-lightpathsprotectivearchitecture.First,itisprovedagainthatthenetworkcansurviveover3arbitraryfaultswithoutlossofnetworkcapacity.Second,thehugegapbetweenthebest-casecapacityandworst-casecapacityindicatesthegreatimpactofpositionaldifferenceoffailuresonthenetworkcapacity.Besides,thebest-caseandworst-casecapacitiesalsoserveasboundswhenpredictingthenetworkcapacityunderagivennumberofconnectionfailures.Last,onaverage,thenetworkcanstillmaintain92%ofitsoriginalcapacityafter8one-hopconnectionsfail,whichcountfor25%ofone-hopconnectionsinthenetwork. 52

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Figure2-21. Averagecapacitydegradationcomparisonbetweentheproposed4-lightpathscommunicationandsingle-lightpathcommunication Figure 2-21 showshowdifferentthenetworkcapacitydegradesfortheproposed4-lightpathsarchitectureandthe1-lightpatharchitecturewiththeroutingandwavelengthassignmentproposedby[ 9 ]respectively.Theadvantageof4-wayprotectionagainstnetworkcapacitydegradationisevidentasshowninthegureespeciallyforasmallnumberofone-hopconnectionfailures,whichisthenormalcasewhenanetworkoperates. Inthischapter,weproposeatorus-basedfault-tolerantall-opticalarchitecturethatappliesfouroptimalnon-overlappinglightpathstoachieveafast3-failures-freeprotection.Inordertominimizewavelengthutilization,wealsoproposeawavelengthallocationandreuseschemeviaacomprehensivediscussionovervariedsizesoftorus.Besidesefcientdatadeliveryvialightpathcommunicationandfastfailurerecoveryviaredundantnon-overlappinglightpathsprotection,theproposednetworkarchitectureshowsahugefaulttoleranceperformanceimprovementthatisdemonstratedviacomprehensiveconnectionreliabilityandnetworkcapacityanalysis.Allinall,thearchitectureproposedinthischapterhasagreatpotentialtobecomeasatisfyingsolutiontomodernavioniccommunicationsystems. 53

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CHAPTER3TRADEOFFSTUDYONFAULTTOLERANCECAPACITYANDRESOURCEUTILIZATIONFORTHETORUS-BASEDALL-OPTICALWDMLANS Inthelastchapter,wediscussthetorus-basedfaulttoleranceschemeforall-terminalcommunications,inwhichanoptimal4-wayfault-tolerantroutingalgorithm(FOLD)isproposedandawavelengthallocationandreuse(WAR)schemeisdevelopedtoallocatewavelengthresourcestodisjointlightpathsinadedicatedfashion.Actually,thewavelengthsallocatedtotheprotective(spare)lightpathshavepotentialtobesharedbythoselightpathsandthatresultsinalowerlevelofsparewavelengthutilization.Inthischapter,weexaminethewavelengthefciencyinfaulttolerancebycomparingthededicatedwavelengthassignmentschemeswithsharedwavelengthassignmentschemes. Themaincontributionsofthischapterareasfollows:1.atradeoffstudyisconductedtoexaminetheprotectionefciencyofsparewavelengthsforfourWavelengthAssignment(WA)schemesspanningfromnoprotection,throughsharedprotections,todedicatedprotectioninthesparewavelengthutilizationspectrum;2.twosparesharingschemesaredevelopedtoallocatewavelengthresourcestosparelightpathsinordertolowertheoverallsparewavelengthsdemand;3.acomprehensivereliabilityevaluationframeworkisexhibitedandextensivesimulationresultsprovideinsightintotheessentialperformance-costtradeoffoffaulttoleranceandprotectiveresourceallocation. Therestofthischapterisorganizedasfollows.Section 3.1 describesfourWAschemesforall-to-allcommunicationsspanningfromthelowestendtothehighestendofthesparewavelengthutilizationspectrum.ThefailurerecoveryschemeisillustratedinSection 3.2 .Section 3.3 providesmathematicalformulationsofnetworkreliabilities.Section 3.4 describessimulationtechniques,showssimulationresults,andconcludesthischapter. 54

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3.1WavelengthAssignmentSchemes Inthissection,wedevelopfourWAschemesalongthesparewavelengthutilizationspectrum,eachofwhichprotectsthenetworkatadifferentlevel.TherstWAschemedoesnotassignanyprotectivewavelengthstosparelightpathsandhenceprovidesnoprotectiontofailedconnections.Hence,thisWAschemeonlyneedstoconsiderWAforworkingpaths.Asindicatedinthelastchapter,thereexistoptimalsingle-lightpathRWAsolutionsforbothparitiesofNinaNNtorus(demandingN3=8andN(N2)]TJ /F6 11.955 Tf 11.96 0 Td[(1)=8wavelengthsforevenandoddNrespectively).IntheoptimalRWAsolution,eachlightpathisroutedviaitsshortestpathandallthewavelengthsallocatedarefullyoccupied.SincenosparewavelengthisallocatedinthisWAscheme,welabelthisWAschemeNoSpare(NS). ThesecondWAscheme,inadditiontoallocatingwavelengthstoworkingpathsinthewayasdescribedintherstWAscheme(NS),groupsallthesparepathsofallS-Dpairs,asdepictedinFigure 3-1 ingreen,ontoonesinglesparewavelength.SincethisverysparewavelengthisintensivelysharedbyallsparepathsofallS-Dpairs,high-levelresourcecontentionisexpectedduringfailure-recovery-inducedlightpathswitches.However,sincetherearethreealternativedisjointpathsthatarepossibletotakeoverthecommunication,theprobabilityofasuccessfulswitchishigherthanthetraditional:1-or+1-basedprotection,whichisanadvantageoffour-waydisjointroutinginfaulttolerance.WelabelthisWAschemeSingleSpare(SS).ThisWAschemeisillustratedinFigure 3-2 (A). ThethirdWAscheme,insteadofallocatingonlyonesparewavelengthtotheentiregroupofsparepaths,assignsonesparewavelengthtoasubgroupofthesparepathsthatbelongtotheS-Dpairswhoseworkingpathsareallocatedonthesamewavelength(workingwavelength).Duetoloweredcontentionlevelonthesparewavelengths,thesuccessfulswitchingprobabilityisexpectedtobehigherthanthesecondWAscheme(SS).Besides,thisWAschemecantolerateonearbitrarylinkfailurebecausethe 55

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ASourceanddestinationindif-ferentdimensions BSourceanddestinationinthesamedimension Figure3-1. Examplesof4disjointlightpathssetupbetweendifferentS-Dpairsina44torus(thelightpathindarkredistheworkingpathandthethreelightpathsinolivegreenaresparepaths) ASingleSpare(SS) BSpareWorkingInterleaving(SWI) Figure3-2. Wavelengthassignmentfortwosparesharingschemes communicationscarriedontheworkinglightpathsaffectedbythelinkfailurecanbeindividuallyswitchedtooneoftheirlink-disjointsparepathswithoutresourceconicts.WelabelthisWAschemeSpareWorkingInterleaving(SWI)andthisschemeisillustratedinFigure 3-2 (B). ThefourthWAscheme,byallocatingdedicatedwavelengthstoeachworkingandsparelightpaths,avoidsresourceconictsresultingfromthefailure-recovery-inducedswitches.Thereby,thenetworkwillbeabletotolerateatleastthreearbitrarylinkfailureswithoutlossofanyconnections.TheRWAsolutiontothisdedicatedfour-wayall-terminalcommunicationisdescribedinthelastchapterforallpossiblesizesofan 56

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NNtorus.Althoughthesolutionachievesgoodoptimality,thetotalwavelengthnumberrequiredismorethan4timesofthatforpure-working-pathcommunication(intheNoSparescheme),whichonlyrequires8wavelengthsforthe44torus.Thegaininfaulttolerancecapacitymaynotbalancethesparewavelengthexpenditure.WelabelthisWAschemeDEDICATED.Figure 3-3 showsthetotalnumbersofwavelengthsrequiredforthefourWAschemeswhenNvaries.Table 3-1 summarizesthenumberofprotective(spare)wavelengthsrequiredbythefourWAschemesdescribedabovefortheNNtorus. Figure3-3. TotalnumbersofwavelengthsrequiredforfourWAschemes Table3-1. Sparewavelengthrequirements WAschemeNSSSSWIDEDICATED NumberorequivalentnumberofprotectivewavelengthsforaNNtorus 0 1 Niseven:N3=8Nisodd:N(N2)]TJ /F15 10.909 Tf 10.91 0 Td[(1)=8 Niseven:WWAR)]TJ /F13 10.909 Tf 10.91 0 Td[(N3=8Nisodd:WWAR)]TJ /F13 10.909 Tf 10.91 0 Td[(N(N2)]TJ /F15 10.909 Tf 10.91 0 Td[(1)=8 WWARisthenumberofwavelengthsallocatedfordedicated4-wayprotection.TheexpressionsofWWARfordifferentNcanbefoundinTable 2-3 .Ingeneral,WWARismorethan4timesofthewavelengthnumberrequiredbypure-working-pathcommunicationespeciallywhenNgoeslarge. 57

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3.2FailureRecovery TherstWAschemedoesnotinvolveanyfaulttolerancesupportwhilethefourthWAschemesimplymakesthedestinationindependentlyswitchreceptionfromamongthefourdedicatedlightpaths.ThecomplicacyoffailurerecoveryonlycomesfromthesecondandthirdWAschemesbecauseoftheneedtohandletheresourceconictsamongsparelightpaths. Figure3-4. Lightpathstatetransitiondiagramforresource-sharedWAschemes Figure 3-4 showspossiblestatetransitionsoflightpaths,whichareessentiallytriggeredbyfailurerecovery.ThetransitionbetweenSPAREandUNAVAILmayresultfromresourceconicts,forwhichanexampleisdemonstratedinFigure 3-5 ,wherethesparelightpathfrom31to33experiencestwostatetransitions,SPARE!UNAVAILandUNAVAIL!SPARE,successively. Onceafailureoccurstoalink,morethanoneworkinglightpathcanbeaffected.Theorderofselectingfailedworkingpathsforrecoverymakesdifferencetothefaulttoleranceresultsbecauseoftheresourceconictsamongtheirsparepaths.Weexaminetwoselectionstrategiesasfollows. RANDOM:randomlyselectaworkinglightpathforrecoveryfromthepoolofaffectedworkinglightpaths SHORTEST:selecttheworkinglightpathwiththeshortestsparelightpath 58

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Ablockedsparelightpath(31!33) BRe-enabledsparelightpath C Figure3-5. Anexampleofsparelightpathre-enablingina44torus 3.3ReliabilityAnalysis WeevaluatethefaulttoleranceperformanceofallfourWAschemesviameasuringtheconnectionreliabilitiesandtheunder-failurenetworkcapacity. Theconnectionreliabilitiesareequivalentlycapturedbytheprobabilityoftheircomplementaryevents,connectionunreliabilities,asdenedasfollows. Two-TerminalUnReliability(TTUR),One-to-All-othersUnReliability(OAUR),andAll-TerminalUnReliability(ATUR): PTTUR=eXi=SD(ei)Xk=1CSDk Piifi(1)]TJ /F3 11.955 Tf 11.95 0 Td[(f)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i, (3)POAUR=eXi=S(ei)Xk=1CSk Piifi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)e)]TJ /F4 7.97 Tf 6.58 0 Td[(i, (3) andPATUR=eXi=(ei)Xk=1Ck Piifi(1)]TJ /F3 11.955 Tf 11.96 0 Td[(f)e)]TJ /F4 7.97 Tf 6.58 0 Td[(i, (3) ConditionalTTUR,OAUR,andATURonagivennumber(n)ofbidirectionalberlink(bilink)failures: 59

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PTTURjnbilinkfailures=(en)Xk=1CSDk Pnn, (3)POAURjnbilinkfailures=(en)Xk=1CSk Pnn, (3) andPATURjnbilinkfailures=(en)Xk=1Ck Pnn. (3) Thenotationsusedintheabovesixequationsareexplainedasfollows.fistheuniformbilinkfailureprobability.eisthetotalnumberofbilinksinthetorus.SD,S,andaretheminimumnumbersofbilinkfailuresrequiredtodisconnectaspecicS-Dpair,thesource(S)fromanyofothercontrollers,andanyS-Dpair,respectively.)]TJ /F4 7.97 Tf 5.48 -4.38 Td[(eiisthenumberofcombinationsofibilinkfailurescomingfromepossiblebilinks.Piiisthenumberofpermutationsfortheselectedibilinkfailures.CSDk,CSk,andCkarethenumbersofbilinkfailurepermutationsdisconnectingaspecicS-Dpair,thesource(S)fromanyofothernodes,andanyS-Dpair,respective,forthekthfailurecombination. Inordertoevaluatetheimpactofbilinkfailuresonthenetworkcapacity,wecalculatetheaveragenetworkcapacityconditionedonanumberofbilinkfailuresasfollows. Tavgjnbilinkfailures=Tfault)]TJ /F4 7.97 Tf 6.59 0 Td[(free)]TJ /F3 11.955 Tf 11.96 0 Td[(C1 PenPenXi=1Di (3) Tfault)]TJ /F4 7.97 Tf 6.59 0 Td[(freeisthefault-freenetworkcapacity,whichisactuallythesumofcapacitiesofallsource-destinationpairs.CisthefulltransmissionrateforaS-Dpair(1Gbpsassumedinthesimulationthatfollows).DiisthenumberofdisconnectedS-Dpairsfortheithpermutationofnbilinkfailures. 60

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3.4SimulationandNumericalResults Allofabovereliabilityperformancemetricsinvolvecountingthenumberofbilinkfailurepermutationsthatresultindisconnections.WeagainresorttotheMonte-Carlosamplingtechniquetouniformlygenerate100,000randombilinkfailurepermutationswhenthecardinalityofthebilinkfailuresetexceeds5.WethenapproximatethecalculationsoftheaboveequationsusingMonte-Carlosamplestriggeredsimulation.A44torusisselectedastheexamplenetworkandnumericalresultsarereportedasfollows. Figure 3-6 showstheconnectionunreliabilitydifferenceamongdifferentWAschemesandpathselectionorders.Thedifferenceisevidentbecauseofthequantitydifferenceinspareresourceprovision.AlsoobservedisthatthepathselectionordermakesdifferenceinthesecondWAscheme(SS)butmakesnegligibledifferenceinthethirdWAscheme(SWI).ThisisbecauseresourcecontentioninSSismuchsevererthaninSWIsuchthatselectionofshortersparepathsmakesthespareresourcesmoreefcientlyutilized.However,SHORTESTselectionfavorstheS-DpairwithashortersparelightpathwhiledisfavorstheS-Dpairwithalongersparelightpath.ThisisobservedinFigures 3-6 (A)and 3-6 (B)forwhichtheshortestsparelightpathsoftheS-Dpairs11!22(refertoFigure 3-1 fornodeindexing)and11!33areoflength2and4.Theconditionalconnectionunreliabilities,onanumberofbilinkfailures,areshowninFigure 3-7 .Besidesthesimilartrendobservedfromunconditionalunreliabilities,itisconrmedthattheSWIandDEDICATEDWAschemesmakethenetworkabletotolerateoneandthreearbitrarybilinkfailures,respectively. Figure 3-8 (A)showsthedifferenceofcapabilityinmaintainingnetworkcapacityuponacertainnumberofbilinkfailuresamongdifferentWAschemes.FurtherinsightintothewavelengthefciencycanbeobtainedinFigures 3-8 (B)and 3-8 (C).Theformeroneshowsthecomparisonofper-wavelengthcapacityamongdifferentWAschemes.Thelatteroneshowsthecomparisonofper-spare-wavelengthcapacitygain,whichis 61

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APTTUR(11!22) BPTTUR(11!33) CPOAUR(from11) Figure3-6. Connectionunreliabilitiesinthe44torus denedastheratiooftheoverallcapacitygain(fromthatofWAschemeNS)tothenumberofsparewavelengths.Thenumberofsparewavelengthsfora44torusis0,1,8,and48forthefourWAschemesrespectively.FromFigure 3-8 (B),weobservethat,althoughtheDEDICATEDschemeachievesthebestoverallcapacityprotection(asshowninFigure 3-8 (B)),itistheleastcost-efcientWAoption.FromFigure 3-8 (C),weobservethatSSisthemostspare-resource-efcientWAschemesincetheonlysparewavelengthismaximallyutilizedduetothehighest-levelfailure-recovery-inducedswitchingdemandonit. 62

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AConditionalPTTUR(11!22) BConditionalPOAUR(from11) CConditionalPATUR Figure3-7. Conditionalnetworkunreliabilitiesinthe44torus TheunderlyingreasonfortheperformancedifferenceamongdifferentWAschemesisrevealedbyFigure 3-9 .Figure 3-9 (A)showstheaverageswitchblockingrateconditionedonanumberofbilinkfailures,whichisdenedastheratioofthenumberofunsuccessfulfailure-recovery-inducedswitches(noavailablelightpathforswitch)tothetotalnumberofswitchingtrials.Reversely,Figure 3-9 (B)showsthesuccessfulswitchingratecontributed,onaverage,byasparewavelength.This,fromanotherperspective,indicatesthewavelengthefciencyintoleratingfailures.Thecomparisonshownin 63

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AAveragenetworkcapacity(Gbps) BAverageper-wavelengthcapacity(Gbps/wl) CAverageper-spare-wavelengthcapacitygain(Gbps/wl) Figure3-8. Conditionalnetworkcapacityinthe44torus Figure 3-9 (B)amongdifferentWAschemeshasanindicationsimilartothecomparisonshowninFigure 3-8 (C)inwavelengthefciency. ThischapterproposesandexaminesfourWAschemesonthetorustopology,threeofwhichprovidefaulttolerancesupportindifferentdegrees.Thepureworking-lightpath-orientedWAschemedemandsnosparewavelengthprovisionandhoweversupportsnofailureprotection.Thededicated4-wayprotectionWAschemeprovidesthehighestleveloffaulttolerancebutdoesnotefcientlyleveragethesparewavelengths.Inthemiddleofthesparewavelengthutilizationspectrum,the 64

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ABlockingrate(%) BPer-spare-wavelengthsuccessrate(%/wl) Figure3-9. Conditionalblocking/successratesinthe44torus twospare-resource-sharingWAschemesprovidereasonablebalancebetweenspareresourceutilizationandfaulttolerancecapacityinthesenseofwavelength-protectionefciencyinsupportingfailure-recovery-inducedswitchesandmaintainingnetworkcapacity. 65

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CHAPTER4CIRCULANT-GRAPH-BASEDFAULT-TOLERANTROUTINGFORALL-OPTICALWDMLANS Alluringfeaturesoftheall-opticalWDMnetwork,suchashugebandwidthprovision,negligibletransmissionlatency,andresistancetoelectromagneticinterference,identifytheall-opticalWDMnetworkitselfapromisingdesignoptionforthenext-generationmission-cruciallocalareanetworks(LANs),suchasavioniconboardcommunicationsystems.However,thecombinationofthehazardousworkingconditionandoperationaluncertaintyofber-opticdevicesmakesthecommunicationsubjecttofaults.Hence,itbecomescriticaltoequipthecommunicationsystemwithqualiedfaulttolerancecapabilityinthenetworkdesignphase. 4.1RelatedWork Intherecentliterature,redundantlightpathsprotectionbecomesafault-tolerantsolutionthatcanmeetthefastrecoveryrequirement.In[ 62 ],theauthorsfocusonphysicallayerissuessuchasthemodelofcombiningsignalsfromdifferentlightpaths,optimalityofthedecisionruleanderrorprobabilitybound.Atthenetworklayer,ourpreviousworkstudiestheroutingandwavelengthallocationissuesbasedona2-Dtorustopology[ 60 ].However,the2-Dtorustopologyputscertainrestrictionsbothonthenumberofsupportednodesandonthenetworkconnectivity,becausethenumberofsupportednodeshastobeaproductoftwointegers(numbersofnodesinarowandacolumnrespectively)andeachnodeisofaconnectiondegree4.Thereforenomorethan4disjointlightpathscanbeestablishedsimultaneously.In[ 61 ],areliabilityanalysismodelisconstructedfortheall-opticalnetworkovervariousnetworktopologies:circulant,Harary,CagesandMooregraphs.Itconcludesthatcirculantgraphsareprincipalcandidatetopologies.However,itsreliabilityanalysisispurelybasedonassociatingafailureprobabilityontoeachlinkanddoesnotconsideranyroutingissues. Inthischapter,wefocusoncirculantgraphsandexploretheirfault-tolerantpotentials.In[ 32 ],adirectedcirculantgraphisstudiedandanalgorithmisprovided 66

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forestablishingnnode-disjointpathsbetweenanodepair.However,thealgorithmputsalimitationonthenumberofsupportednodesandonthenetworkconnectivity.Inthiswork,insteadofthedirectedgraph,weconsiderundirectedgraphandhencedoublethenetworkconnectivityandfault-tolerancecapacity.Inaddition,werelaxthelimitationsonthenumberofsupportednodesandonthenetworkconnectivity.Inotherwords,anynumberofsupportednodesandnodedegreecanbeaccommodatedinourarchitecture.In[ 16 ],basedonacirculantgraphcontainingkredundantnodes,ak-fault-tolerantsolutionisproposedtomaintainisomorphismoftheoriginalgraphunderanyknodefailures.Thisdesignappliesonlyforparallelcomputingprocessorsbecausethecomputationcanbeswitchedbetweenanytwoprocessorsafteroneofthemfails.However,inourarchitectureweassumeeachnodeispositionallytiedtoadatasourceorsinkandhencearbitraryfunctionalswitchbetweennodesisalmostimpossible. Themajorcontributionsofthischapterarefollows:1.weproposeacirculant-graph-basedall-opticalWDMLANarchitecture;2.wedevelopafault-tolerantroutingalgorithmthatfullyexploresthecirculantgraphconnectivityviasettingupamaximumnumberofnode-disjointlightpaths;3.weanalyticallycalculatenetworkresourceutilizationmeasuredbythenumbersofrequiredlinksandwavelengths;and4.wederiveareliabilitymodelcombiningeffectsofbothnodeandlinkfailures. Therestofthischapterisorganizedasfollows:Section 4.2 denesthenetworkarchitectureanddescribesthenode-disjointlightpathssetupalgorithm.Basedonthelightpathssetup,networkresourceutilizationisanalyzedinSection 4.3 .Section 4.4 providesaprobabilisticmodelandshowsthenetwork-reliabilityrelatednumericalresults,followedbyachaptersummaryintheend. 4.2Fault-TolerantRoutingAlgorithm Inthissection,wedescribethecirculant-graph-basedall-opticalnetworkarchitecturethatoffersexibilityinthenumberofsupportednodesandnetwork 67

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connectivity.Thenbasedonthearchitecturedenition,wedevelopanode-disjointlightpathsetupalgorithmtofullyexplorefault-tolerancecapability. 4.2.1CirculantNetworkArchitecture AC12(f1,2,3g) B6node-disjointlightpathsbetweennodes0and3 C6node-disjointlightpathsbetweennodes0and6 D6node-disjointlightpathsbetweennodes0and1 Figure4-1. Circulant-graph-basednetworkarchitectureandexamplesoffault-tolerantroutingviaestablishingnode-disjointlightpaths AcirculantgraphCN(A),whereNisapositiveintegerandAfaj1abN=2cg,isagraphofNvertices1inwhichtheithvertexisadjacenttothe(i+j)thand(i)]TJ /F3 11.955 Tf 12.18 0 Td[(j)thvertices2foreachjinsetA.Forinstance,CN(f1,...,bN=2cg)correspondstoacompletegraphandCN(f1g)representsaring.Therebywedenethecirculant-graph-based 1WeindextheNverticesfrom0toN)]TJ /F23 9.963 Tf 9.96 0 Td[(1clockwise.2If(i+j)or(i)]TJ /F25 9.963 Tf 9.97 0 Td[(j)isbeyondtherange(0,N)]TJ /F23 9.963 Tf 9.96 0 Td[(1),theirmodulovalues(byN)shouldbetaken. 68

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all-opticalarchitectureasfollows:1.replaceallverticesbydatadeliveryandreceptionnodes;2.replacealledgesbytwounidirectionalopticalbersrunninginoppositedirections.Eachnodehasdirectopticalconnectionsto2jAjothernodes3.Sincetransmissionlatencyisnegligibleinall-opticalnetworkswhichdeliverdatathroughall-optically-switchedlightpaths,thegraphdiameterisnotagreatconcernandhencewextheoffsetsetA=f1,...,Wg,whereW2f1,...,bN=2cg.ThevalueofWdependsonfault-tolerancerequirement(thenumberofdisjointlightpaths).Acirculant-graph-basedarchitectureexampleisshowninFigure 4-1 (A),inwhich12nodesareconnectedinacirculantfashionandeverynodehas6directopticalconnectionstoitsindex-closestneighbors. 4.2.2Node-DisjointLightpathsSetup Sinceinthenetworkarchitecturedescribedaboveeachnodecansimultaneouslysendandreceivedatathroughits2Wneighboringnodes4,theremayexist2Wdisjointpathsforanycommunicationpairs.Weassumethatthedestinationnodeusesthesamelightpathinreversetoreachthesourcesodirectedlightpathcanbesimplyreplacedbylightpath.Inthissectionwedevelopa2W-node-disjoint-lightpathssetupalgorithmforanysourceanddestination.Duetothesymmetricnatureofcirculantgraphs,inourdiscussionwexthesourceatnode0andvarythedestinationnodeindexfrom1tobN=2c.Accordingtothepositionalrelationshipbetweenthedestinationnode(indexedbyD)andthenodeindexedbyW,thediscussiononfault-tolerantroutingfallsintofollowingthreecasesrespectively. 3IfNisevenandA=f1,...,N=2g(acompletegraph),thenumberofdirectconnectionsfromnodeibecomes2jAj)]TJ /F23 9.963 Tf 14.94 0 Td[(1becausenode(i+N=2)andnode(i)]TJ /F25 9.963 Tf 9.96 0 Td[(N=2)areactuallythesamenode.4ForacompletegraphcontainingaevennumberofnodesinwhichW=N=2,thenumberofneigh-boringnodesofeachnodebecomes2W)]TJ /F23 9.963 Tf 19.92 0 Td[(1.Thefault-tolerantroutinginthiscaseistrivialtodiscussbecauseanysourcecantakeone-hopdirectconnectionandother2W)]TJ /F23 9.963 Tf 20.14 0 Td[(2two-hopnode-disjointcon-nectionstoreachanydestinationregardlessofthesourceanddestinationpositions.Hencethefollowingdiscussiononlyfocusesoncirculantgraphsinwhicheachnodehasexactly2Wneighbors. 69

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A B C Figure4-2. Fault-tolerantroutingfordestinationnodeswithmoduloindexdifferencefromthesourcenodebyW,greaterthanW,andsmallerthanW Case1:D=W .AsshowninFigure 4-2 (A),therstWnode-disjointlightpathscanbesetupclockwiseasfollows: 0!1!D0!2!D...0!W)]TJ /F6 11.955 Tf 11.95 0 Td[(1!D0!D OtherWnode-disjointlightpathsaresetupcounter-clockwiseasfollows: 0!N)]TJ /F6 11.955 Tf 11.96 0 Td[(1!N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)]TJ /F3 11.955 Tf 11.96 0 Td[(W!...!N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)]TJ /F6 11.955 Tf 11.96 0 Td[((dN)]TJ /F5 7.97 Tf 6.59 0 Td[(1)]TJ /F4 7.97 Tf 6.59 0 Td[(W We)]TJ /F6 11.955 Tf 19.92 0 Td[(1)W!W 0!N)]TJ /F6 11.955 Tf 11.96 0 Td[(2!N)]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F3 11.955 Tf 11.96 0 Td[(W!...!N)]TJ /F6 11.955 Tf 11.96 0 Td[(2)]TJ /F6 11.955 Tf 11.96 0 Td[((dN)]TJ /F5 7.97 Tf 6.59 0 Td[(2)]TJ /F4 7.97 Tf 6.59 0 Td[(W We)]TJ /F6 11.955 Tf 19.92 0 Td[(1)W!W ... 0!N)]TJ /F3 11.955 Tf 11.96 0 Td[(W!N)]TJ /F3 11.955 Tf 11.96 0 Td[(W)]TJ /F3 11.955 Tf 11.95 0 Td[(W!...!N)]TJ /F3 11.955 Tf 11.96 0 Td[(W)]TJ /F6 11.955 Tf 11.96 0 Td[((dN)]TJ /F4 7.97 Tf 6.59 0 Td[(W)]TJ /F4 7.97 Tf 6.58 0 Td[(W We)]TJ /F6 11.955 Tf 19.93 0 Td[(1)W!W TheideaofroutingaboveWlightpathsistomakeeachlightpathstrideoverthemaximumnumberofnodesateachhopwithouthittingthesamenodethatotherlightpathsmaychoosetopassthrough.Themaximumstridingcanalsominimizethenumberofhopsneeded,whichcorrespondstoaresourcesavingandconnectionreliabilityimprovementundercertainlinkornodereliability.Obviouslyabove2Wlightpathsarenode-disjointandvalidpathsbecausethereisnonodeoverlapbetween 70

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differentpathsexceptatthesourceandthedestination.Figure 4-1 (B)showsanexampleoflightpathssetupforCase1. Case2:W
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TherstDpathsarepurelyclockwiserouted,followedbyW)]TJ /F3 11.955 Tf 13.21 0 Td[(Dclockwise/counterclockwiseroutedpathsandthenbyW)]TJ /F3 11.955 Tf 12.27 0 Td[(Dcounterclockwise/clockwiseroutedpaths. Finally,theremainingDnode-disjointlightpathscanbedevelopedinawaysuchthateachlightpathtriesthebiggeststridecounter-clockwiseuntilhitsthegroupofnodesindexedfromW+1toW+D.ThenthedirectconnectionsfromthatgroupofnodestodestinationnodeDnalizethelightpathssetup.TherearetwoscenariosinwhichtheDlightpathshitthegroupofnodesindexedfromW+1toW+Dindifferentways,asshowninFigures 4-3 (A)and 4-3 (B)respectively.Thelightpathssetupisdetailedasfollows. A B Figure4-3. LastDnode-disjointlightpathssetupforScenarioIandII(thelast-stopnodegroupandassociatedroutinglinksarecoloredgreen) ScenarioI:[N)]TJ /F6 11.955 Tf 11.96 0 Td[((W+D+1)]%W[(N)]TJ /F3 11.955 Tf 11.95 0 Td[(W+D)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[((W+D+1)]%W %isthemodulooperator.Theleft-handsiderepresentstheindexdistancethatthelightpathtakingthebiggeststride(W)initsrsthop,namedasheadinglightpath,hasfromthenodeW+D+1beforehittingthenodegroup(indexedfromW+1toW+D).Theright-handsiderepresentstheindexdistancethatthelightpathtakingthesmalleststride(W)]TJ /F3 11.955 Tf 12.6 0 Td[(D)initsrsthop,namedastailinglightpath,hasfromthenode 72

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W+D+1beforehittingthenodegroup(indexedfromW+1toW+D).TheequalityholdswhenD=1,i.e.,theheadinglightpathbecomesthesameasthetailinglightpath.WetermtherstdistanceasHandtheseconddistanceasT.ThisscenariostatesthatthelastclosestcounterclockwisetouchbeforehittingthenodegroupisfromtheheadinglightpathandastrideofH+DwillmakeallDlightpathsuniquelyhitonenodeinthenodegroup.TheDlightpathsarethendevelopedasfollows. 0!N)]TJ /F3 11.955 Tf 9.69 0 Td[(W+D)]TJ /F6 11.955 Tf 9.69 0 Td[(1!...!N)]TJ /F3 11.955 Tf 9.69 0 Td[(W+D)]TJ /F6 11.955 Tf 9.69 0 Td[(1)]TJ /F6 11.955 Tf 9.69 0 Td[((d(N)]TJ /F4 7.97 Tf 6.58 0 Td[(W+D)]TJ /F5 7.97 Tf 6.58 0 Td[(1))]TJ /F5 7.97 Tf 6.58 0 Td[((W+D) We)]TJ /F6 11.955 Tf 15.39 0 Td[(1)W!W+D!D 0!N)]TJ /F3 11.955 Tf 12.21 0 Td[(W+D)]TJ /F6 11.955 Tf 12.21 0 Td[(2!...!N)]TJ /F3 11.955 Tf 12.21 0 Td[(W+D)]TJ /F6 11.955 Tf 12.21 0 Td[(2)]TJ /F6 11.955 Tf 12.22 0 Td[((d(N)]TJ /F4 7.97 Tf 6.59 0 Td[(W+D)]TJ /F5 7.97 Tf 6.59 0 Td[(2))]TJ /F5 7.97 Tf 6.58 0 Td[((W+D)]TJ /F5 7.97 Tf 6.58 0 Td[(1) We)]TJ /F6 11.955 Tf 20.44 0 Td[(1)W!W+D)]TJ /F6 11.955 Tf 11.95 0 Td[(1!D ... 0!N)]TJ /F3 11.955 Tf 11.96 0 Td[(W!...!N)]TJ /F3 11.955 Tf 11.95 0 Td[(W)]TJ /F6 11.955 Tf 11.95 0 Td[((d(N)]TJ /F4 7.97 Tf 6.59 0 Td[(W))]TJ /F5 7.97 Tf 6.59 0 Td[((W+1) We)]TJ /F6 11.955 Tf 19.93 0 Td[(1)W!W+1!D ScenarioII:[N)]TJ /F6 11.955 Tf 11.96 0 Td[((W+D+1)]%W[(N)]TJ /F3 11.955 Tf 11.95 0 Td[(W+D)]TJ /F6 11.955 Tf 11.96 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[((W+D+1)]%W ThisscenariostatesthattheheadinglightpathwillrunintothenodegrouptogetherwithotherD)]TJ /F3 11.955 Tf 11.61 0 Td[(T)]TJ /F6 11.955 Tf 11.61 0 Td[(1successivelightpaths.TheremainingTlightpathscantakeastrideofindexdistanceDtouniquelyhittherestTnodesinthenodegroup.ThedetailedDlightpathssetupisasfollows. 0!N)]TJ /F3 11.955 Tf 9.67 0 Td[(W+D)]TJ /F6 11.955 Tf 9.66 0 Td[(1!...!N)]TJ /F3 11.955 Tf 9.67 0 Td[(W+D)]TJ /F6 11.955 Tf 9.67 0 Td[(1)]TJ /F6 11.955 Tf 9.66 0 Td[((d(N)]TJ /F4 7.97 Tf 6.59 0 Td[(W+D)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F5 7.97 Tf 6.59 0 Td[((W+D) We)]TJ /F6 11.955 Tf 15.35 0 Td[(1)W!W+T!D ... 0!N)]TJ /F3 11.955 Tf 12.22 0 Td[(W+D)]TJ /F3 11.955 Tf 12.22 0 Td[(T!...!N)]TJ /F3 11.955 Tf 12.22 0 Td[(W+D)]TJ /F3 11.955 Tf 12.22 0 Td[(T)]TJ /F6 11.955 Tf 12.22 0 Td[((d(N)]TJ /F4 7.97 Tf 6.58 0 Td[(W+D)]TJ /F4 7.97 Tf 6.58 0 Td[(T))]TJ /F5 7.97 Tf 6.59 0 Td[((W+D) We)]TJ /F6 11.955 Tf 20.45 0 Td[(1)W!W+1!D 0!N)]TJ /F3 11.955 Tf 10.22 0 Td[(W+D)]TJ /F3 11.955 Tf 10.22 0 Td[(T)]TJ /F6 11.955 Tf 10.22 0 Td[(1!...!N)]TJ /F3 11.955 Tf 10.22 0 Td[(W+D)]TJ /F3 11.955 Tf 10.22 0 Td[(T)]TJ /F6 11.955 Tf 10.22 0 Td[(1)]TJ /F6 11.955 Tf 10.22 0 Td[((d(N)]TJ /F4 7.97 Tf 6.59 0 Td[(W+D)]TJ /F4 7.97 Tf 6.59 0 Td[(T)]TJ /F5 7.97 Tf 6.59 0 Td[(1))]TJ /F5 7.97 Tf 6.59 0 Td[((W+D) We)]TJ /F6 11.955 Tf 16.46 0 Td[(1)W!W+D!D ... 0!N)]TJ /F3 11.955 Tf 11.96 0 Td[(W!...!N)]TJ /F3 11.955 Tf 11.95 0 Td[(W)]TJ /F6 11.955 Tf 11.95 0 Td[((d(N)]TJ /F4 7.97 Tf 6.59 0 Td[(W))]TJ /F5 7.97 Tf 6.59 0 Td[((W+D) We)]TJ /F6 11.955 Tf 19.93 0 Td[(1)W!W+T+1!D Fromthediscussionof2Wlightpathsdevelopmentforcase3,itcanbeconcludedthattheyarealsonode-disjointandvalidpaths.Figure 4-1 (D)showsanexampleoflightpathsetupforCase3. 73

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Theorem4.1.ThecirculantgraphCN(f1,...,Wg),whereW2f1,...,bN=2cg,isbotha2W(or2W)]TJ /F6 11.955 Tf 12.39 0 Td[(1foracompletegraphconsistingofanevennumberofvertices)-edge-connectedanda2W(or2W)]TJ /F6 11.955 Tf 11.96 0 Td[(1)-vertex-connectedgraph. Proof:Theaboveredundantlightpathsetupalgorithmshowsthatthereexist2W(or2W)]TJ /F6 11.955 Tf 12.9 0 Td[(1)edge-disjointpathsbetweenanypairsofnodes,sotheremovalofarbitrary2W)]TJ /F6 11.955 Tf 12.62 0 Td[(1(or2W)]TJ /F6 11.955 Tf 12.62 0 Td[(2)edgescannotdisconnectanynodepairsandthereforeCN(f1,...,Wg)is2W(or2W)]TJ /F6 11.955 Tf 11.93 0 Td[(1)-edge-connected.Inaddition,sincethe2W(or2W)]TJ /F6 11.955 Tf 11.93 0 Td[(1)pathsarealsovertex-disjoint,whichmeansnotwopathsshareanynodesexceptthesourceanddestination,theremovalofarbitrary2W)]TJ /F6 11.955 Tf 12.9 0 Td[(1(or2W)]TJ /F6 11.955 Tf 12.89 0 Td[(2)nodescannotdisconnectanynodepairsintheremaininggraphandhenceCN(f1,...,Wg)isalso2W(or2W)]TJ /F6 11.955 Tf 11.96 0 Td[(1)-vertex-connected.2 Therefore,basedonabovenode-disjointlightpathssetupalgorithm,acirculant-graph-basedall-opticalarchitecturecanbeestablishedtosatisfycommunicationlatencyandfaulttolerancerequirementsfordesignatednumbersofnodesandtolerablenetworkfaults(linkornodefaults). 4.3NetworkResourceUtilization Inthissection,wecalculatelinkresourceutilizationforanysource-destinationpairswithvariedfaulttolerancesupportindicatedbyW.Basedontheresults,wederivewavelengthrequirementforallnodepairs'simultaneouscommunicationsgivenwavelengthconversionisprovided,whichisalsothelowerboundofachievablewavelengthnumberunderthewavelength-continuityconstraint. Thenumberoflinksthroughwhich2WdisjointlightpathsconnectingSandDpassisthesumofthelengthsofindividuallightpathsasfollowsLSD=2WXi=1LiSD. (4) BasedonthedisjointlightpathssetupalgorithmdescribedinSection 4.2 ,LSDcanbefurtheranalyticallyderivedbycasesandscenariosdenedinSection 4.2 asfollows. 74

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Fornon-completeCN(f1,...,Wg):LSD=WN)]TJ /F3 11.955 Tf 11.96 0 Td[(W W+(N)]TJ /F3 11.955 Tf 11.96 0 Td[(W)%W+3W)]TJ /F6 11.955 Tf 11.96 0 Td[(2,forD=W (4)LSD=WN)]TJ /F3 11.955 Tf 11.96 0 Td[(D W+D W+((N)]TJ /F3 11.955 Tf 11.95 0 Td[(W)%W+D%W)+2W)]TJ /F6 11.955 Tf 11.96 0 Td[(2,forD>W (4)LSD=N)]TJ /F6 11.955 Tf 11.96 0 Td[((W+D) WD+4W)]TJ /F3 11.955 Tf 11.95 0 Td[(D)]TJ /F6 11.955 Tf 11.96 0 Td[(1,forD
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Figure4-4. LinkutilizationfordifferentdestinationswithrespecttovariedW(N=16) Figure4-5. WavelengthrequirementwithrespecttovariedWforall-nodesimultaneouscommunication(N=16) decreaseforall-nodesimultaneouscommunicationasshowninFigure 4-5 Sincethenumberofavailablewavelengthstieswiththecomplexityofthenodestructure,suchasthesizeofMUX/DEMUXandswitchingmatrix,densely-connectedCN(f1,...,Wg)offersextrabenetinwavelengthrequirementbesidesnetworkreliabilitygain.Furtherwavelengthreductionmightresorttotrafcgroomingtechniques,whichisasubjectoffuturework. 76

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4.4NetworkReliabilityAnalysis Bothnodesandlinksmaybesubjecttofaults.WemodelthenetworkfaultsviaassociatingafailureprobabilityfNtoeachnodeandanotherfailureprobabilityfLtoeachlinkunderanassumptionthatallnodesandlinksfailinanindependentfashion.Inaddition,itisalsoassumedthatanodefailurewillblockallincomingandoutgoingcommunications.Therebytheprobabilityofdisconnectionforaspecicsource-destinationpaircanbederivedasfollowing:PS=Ddisconnection=P(S=DdisconnectionjnofaultonSandD)P(nofaultonSandD)+P(S=DdisconnectionjfaultySorD)P(faultySorD) (4) Thefourtermsintheaboveequationareasfollows:P(S=DdisconnectionjnofaultonSandD)=2WYi=1h1.0)]TJ /F6 11.955 Tf 11.96 0 Td[((1.0)]TJ /F3 11.955 Tf 11.95 0 Td[(fL)LiSD(1.0)]TJ /F3 11.955 Tf 11.95 0 Td[(fN)(LiSD)]TJ /F5 7.97 Tf 6.58 0 Td[(1)i (4)P(nofaultonSandD)=(1.0)]TJ /F3 11.955 Tf 11.95 0 Td[(fN)2 (4)P(S=DdisconnectionjfaultySorD)=1.0 (4)P(faultySorD)=1.0)]TJ /F6 11.955 Tf 11.96 0 Td[((1.0)]TJ /F3 11.955 Tf 11.95 0 Td[(fN)2 (4) Figure 4-6 showshowthedisconnectionprobabilityforthesource-destinationpair(0!8)inC16(f1,2g)varieswithdifferentfNandfL.ItcanbeobservedthatfNplaysabiggerrolethanfLinthedisconnectionprobabilitybecausebothsourceanddestinationnodesaresubjecttonodefailureswhilethepressureoflinkfailuresismitigatedbymultipledisjointlightpaths.TheeffectsofdifferentWvaluesontheconnectionreliabilitycanbeobservedinFigure 4-7 inwhichallnodesareassumedtobefault-free(fN=0). 77

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Figure4-6. DisconnectionprobabilitychangewithfLandfN(N=16,W=2,S=0,D=8) Figure4-7. DisconnectionprobabilitychangewithfLforvariedW(N=16,S=0,D=8) ThedisconnectionprobabilitydecreasesalmostproportionallytotheincreaseofWinthelogarithmicscale,whichdemonstratesvastfault-toleranceperformanceimprovementbyincreasingnetworkconnectivity.ThedisconnectionprobabilitydistributionforasourcewithrespecttoalldestinationsacrossthenetworkisshowninFigure 4-8 .Itcanbeobservedthatthedestinationswithcloserindicestothesourceareofhigherconnectionreliabilitybecausetheycanberoutedtothroughmore2-hoplightpathsand 78

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hencerequirealowernumberoflinksintheirdisjointlightpathsetups,asshowninFigure 4-4 Figure4-8. Disconnectionprobabilitydistributionacrossthenetwork(N=16,W=2,fL=0.1) Inthischapter,weproposeacirculant-graph-basedall-opticalWDMnetworkarchitectureanddevelopanode-disjointfault-tolerantroutingalgorithmthatoffersexiblefault-toleranceoptions.Bothnodefailureandlinkfailurearemodeledandprobabilisticanalysisresultsshowevidentreliabilityimprovementwithmoderatelinkresourceincrease.Infuture,weplantodevelopawavelengthassignmentmethodforallnodepairs'communicationwiththeproposedfault-tolerantroutingalgorithmunderthewavelength-continuityconstraint. 79

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CHAPTER5TOPOLOGICALOPTIMIZATIONFORSPARE-SHARING-BASEDWAVELENGTH-ROUTEDALL-OPTICALNETWORKS WavelengthDivisionMultiplexing(WDM)provideshugebandwidthpotentialanddesignexibilitybyarrangingwavelengthstoservedifferenttrafcowsonthesamecommunicationlink.Wavelength-routedall-opticalnetworksofferinstantaneouscommunicationbyavoidingtheslowOptical-Electrical-Optical(OEO)conversionandintermediatequeuinginallswitchingnodesalongthetransmissionpath(lightpath).Therestrictionofwavelengthcontinuitythroughoutthepathcanbeimposedtoeliminatewavelengthconversioncostintheswitchingnode[ 40 ].However,therestrictionalsoincreasesRoutingandWavelengthAssignment(RWA)difculty[ 40 ].Inaddition,inordertoprotectthenetworkagainstfailures,itisdesirabletodesignafault-tolerantnetworkinwhichanarbitrarylinkfailurecanbetoleratedwithoutthelossofanycommunicationsessions. Inthischapter,weattempttodevelopalow-costone-link-failure-freenetworktopologythatobeyswavelengthcontinuityandisabletoaccommodatearequirednumberoftrafcowswithauniformnumberofwavelengthsavailableoneachlink.Althoughtheresultingtopologicalsolutioncanbethechoiceofphysicaldeployment,itwouldalsobeappliedtoredenethelogicalber-communicationtopologyontheexistingphysicalinfrastructureoncesingleormultipledisastrousattacks(earthquakes,hurricanes,oods,aswellaselectromagneticpulses)happentocertainpartsofthenetwork[ 3 ].Wecallthistopologicaladaptationtohazardousphysicalattacks,whichpotentiallyaffectalargegeographicalarea.Thetraditionalsmall-scaleprotections,suchasSharedRiskLinkGroup(SRLG)basedschemes[ 20 ][ 49 ],maynotbefullycapableoftidingthecommunicationoverthoseunpredictablelarge-scaledisasters.Thetopologicaladaptationcanbeachievedbysolvingatopologicaloptimizationproblem,asstudiedinthischapter,accordingtothechangednetworkresourceprovision.Weproposetheuseofthetopologicaladaptationtorespondtolarge-scaledisaster-induced 80

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resourceoutageandleveragethetraditionalshared-path-basedprotectiontotoleratenormalsingleberlinkfailures.Thereby,twoprotectionlevelsaddressingdifferentfailurescalescanberealized. 5.1Spare-Sharing-BasedTopologicalOptimization Sparesharingprovidesopportunitiestosavewavelengthandlinkresources[ 34 ][ 52 ],whichresultsinalower-costtopologicalsolution.AsshowninFigure 5-1 ,therearetwobidirectionalowsA$DandB$D.Givenonlyonewavelength,ifnosparesharingisapplied,thetopologicalsolutiontoaccommodatetwoworkingandtwobackuppathsmustincludelinksB)]TJ /F3 11.955 Tf 11.77 0 Td[(EandD)]TJ /F3 11.955 Tf 11.76 0 Td[(E(asshowninFigure 5-1 (A)).However,theuseofthesetwolinkscanbeavoidedbyallowingthetwobackuppathstosharethelinksB)]TJ /F3 11.955 Tf 11.96 0 Td[(CandC)]TJ /F3 11.955 Tf 11.96 0 Td[(D(asshowninFigure 5-1 (B)). AWithoutsparesharing BWithsparesharing Figure5-1. Topologicalsolutionswithoutandwithsparesharing Asdescribedinmanyshared-pathprotectionbasedworks[ 41 ][ 21 ][ 37 ],theroutingandwavelengthassignmentfortheworkingandbackuppathshastoobeythefollowingrules: 1.Theworkingpathanditsbackuppathforanyowrequestmustbelink-disjoint 2.Nowavelengthonthesamelinkcanbesharedneitherbetweentwoworkingpathsnorbetweenaworkingpathandabackuppath 3.Notwobackuppathscansharethesamewavelengthonthesamelinkiftheirworkingpathsjoineachotheranywhereinthenetwork Therstruleensuresthatatleastonepath(workingorbackup),foranyow,cansurviveoveranyonelinkfailure.Thesecondruleavoidscollisionbetweentwo 81

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workingpathsaswellasenablesavalidswitchfromaworkingpathtoitsbackuppathbyguaranteeingavailabilityofthebackuppath.Thethirdrulepreventsanytwoswitchesfromconictingontheirbackuppathswhenalinkfailuredisablestwoworkingpaths.TheaboverulesonsparesharingcanbedemonstratedinFigure 5-2 .Figure 5-2 (A)isavalidsparesharingexample(onlinkB!E)becausethetwoworkingpathsofowB!CandowA!Earelink-disjointandhenceanylinkfailureoverthenetworkcannottriggerbothowstoswitch.Figure 5-2 (B)isaninvalidsparesharingexamplebecausethefailureonlinkA!DwouldforcebothowstoswitchandcollisionwillhappenontheirbackuppathsifthetwobackuppathssharethesamewavelengthonlinksA!BandB!C.Therefore,twoseparatewavelengthshavetobeusedonthetwobackuppaths. AValidsparesharing BInvalidsparesharing Figure5-2. Validitydemonstrationofsparesharing Ingeneral,theroutingandwavelengthassignment(RWA)problemwithoutinvolvingfaultprotectionisNP-complete[ 42 ].Whenfaultprotectionisconsidered,theproblembecomesevenharderbecausethefeasiblesolutionshavetoaccountforbackupresourceallocationandaresubjecttomorecomplicatedconstraints,asshowninIntegerLinearProgram(ILP)formulationsinSection 5.4 .WeanticipatethatthetopologicaloptimizationproblemstudiedinthischapterisNP-hardbecausemanyofitssubproblemshavebeenshowntobeNP-complete.Forexample,thewavelengthassignmentinaStaticLightpathEstablishment(SLE)problemisshowntobeequivalenttographcoloringandhenceisNP-complete[ 13 ].Besides,theroutingandwavelengthassignmentforasingleworkingandbackuppathpairsatisfying 82

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spare-sharingconstraintsisalsoNP-complete[ 38 ][ 57 ].Thestudiedtopologicaloptimizationproblemrequiresallowrequeststobeequippedwithvalidworkingandbackuppathpairsatthelowesttopologicalcost,andithenceiscomputationallymorecomplicated.Thetopologicalcostisdenedasthetotalcostofthegroupofbidirectionallinksincludedinthenaltopology. 5.2RelatedWork Pastresearchonfault-tolerance-orientedtopologicaloptimizationproblemsmainlyfocusedonminimizationoftotallinkcostonageneralgraphconstrainedonsatisfactionofeitherk-connectivityoraconnectionreliabilitythreshold.Theappliedmethodsarebranch&boundbaseddecompositionasin[ 27 ],approximationalgorithmsasin[ 68 ],orevolutionaryalgorithmsasin[ 55 ].However,withrespecttothetopologicaloptimizationforwavelength-routedall-opticalnetworks,previouslynospecicworkwasinvolvedtothebestofauthors'knowledge. Inthecontextofwavelength-routedWDMnetworks,theRWAproblemhasbeenextensivelystudied[ 67 ][ 7 ].Sincethen,moreandmorefocusesaregiventofaulttoleranceenhancedRWAproblems.[ 41 ]providesacomprehensiveclassicationinfaulttoleranceschemesandshowsthetradeoffamongresourceutilizationandconnectionreliability.[ 33 ]focusesondevelopingasetoftrafc-pattern-awarelink-disjointlightpathstoachievelowerconnectionblockingprobability.[ 36 ]exploitsprimary-backupmultiplexingtechniquetoallowaprimarylightpathsharewavelengthswithoneormorebackuplightpaths,inordertoincreasethenumberofestablishedlightpathsatthecostofreducedrecoveryguarantee.[ 56 ]appliesowaggregationtodevelopingcomputationallylesscomplicatedILPformulations.However,theproblemsaddressedbythosepapersarebasedonapredeterminednetworktopologyandnotopologicalchangecanbemade.Besides,in[ 41 ],thegoalistominimizetheresourcecapacityutilizationdenedasthetotalnumberofwavelengthlinks.Theoptimumsolutiontotheproblemmayleadtoverydifferentwavelengthrequirementsondifferentlinks.Inour 83

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work,weconsideruniformwavelengthprovisiononeachlink,whichwillrenderdesignuniformitytotheMUX/DEMUXlogicsforeachnode.Moreover,mostofpreviousworksuseasetofalternativeroutesthatarepredeterminedinordertocontroltheproblemsize[ 41 ][ 21 ][ 37 ][ 36 ].However,weshow,inSection 5.5 ,thisdecompositiondoesnottthenatureofthetopologicaloptimizationwellandhencecannotleadtosatisfyingperformance.Wejointlyconsiderroutingandwavelengthassignmentinoneproblemwhoseoptimalsolutionwillperformbetterthantheonewithoutroutingbeingconsideredsimultaneously. 5.3ContributionsandChapterOrganization Themajorcontributionsofthischapterarelistedasfollows.First,wedevelopanILPmodelconsideringbothroutingandwavelengthassignment,whichfullyexploitsthebestpossiblesolutiontothestudiedspare-sharing-basedtopologicaloptimizationproblem.Second,wediscernthatapplyingtraditionalformulations,whichbasethesolutionsonasetofpredeterminedalternativeroutes,cannotresultinconvincingsolutionsbycomparingtothegreedyapproach.Thirdly,weproposeatwo-phaseheuristicalgorithmbasedonobservationofthedrawbacksofthegreedyapproachandexperimentallyexhibittheefciencyofthealgorithm. Therestofthischapterisorganizedasfollows:Section 5.4 providesanoriginalILPformulationandadecomposedILPformulationwhichisbasedonasetofk-shortestdisjointroutes.TheperformanceofthedecomposedILPformulationiscomparedwithagreedyapproachdevelopedinSection 5.5 .InSection 5.6 ,weproposeatwo-phaseheuristicalgorithmtoimprovetheperformanceofthegreedyapproach.ThenumericalresultsandalgorithmperformanceanalysisarediscussedinSection 5.7 ,whereachaptersummaryisalsoprovided. 5.4ProblemFormulation Inthissection,toprovidehigh-levelinsightindesignconstraintsandobjective,werstintroduceamatrix-basedrepresentationforthetopologicaloptimization 84

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problem,consideringbothroutingandwavelengthassignment,forspare-sharing-basedwavelength-routedall-opticalnetworks.Then,anequivalentILPisdevelopedtomodeltheprobleminaformthatcanbeprocessedbyprofessionalsolverssuchasMOSEK1.Finally,ak-shortestdisjointroutingbasedILPformulationthatonlyconsidersrouteselectionandwavelengthassignmentisdeveloped.ThedecompositionoftheproblemintoroutingandwavelengthassignmentisalsothetraditionalwaytosolvetheRWA-relatedproblems[ 41 ]. 5.4.1Matrix-BasedRepresentation Table5-1. Basicnotations NotationDenition NNumberofnodesLNumberofdirectedlinksinthecompletegraphofNnodes,i.e.N(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)FNumberofowrequestsWNumberofprovidedwavelengths ThebasicnetworkparametersarelistedinTable 5-1 ,basedonwhichthefollowingparameterizedmatricesaredened.P=[pij]FLandQ=[qij]FLPandQareow-linkincidencematricesfortheworkingandbackuppathsrespectively,inwhichpijandqijindicatewhetherowipassesthroughlinkjornotfortheworkingpathandbackuppathrespectivelybytakingon1or0.A=[aij]WFandB=[bij]WFAandBarewavelength-owincidencematricesfortheworkingandbackuppathsrespectively,inwhichaijandbijindicatewhetherwavelengthiisassignedtoowjornot 1MOSEKisalarge-scalemixed-integerlinearprogramsolverusingacombinationoftheinteriorpoint,branchandcuttechnologies[ 1 ][ 2 ]. 85

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fortheworkingpathandbackuppathbytakingon1or0.D=[dij]FNDistheow-nodeincidencematrixinwhichdij=1ifnodejisthesourceofowianddij=)]TJ /F6 11.955 Tf 9.3 0 Td[(1ifnodejisthedestinationofowi.Otherwisedij=0.Foraspecicproblem,Disknownandindicatedbytheowsetup.G=[gij]NLGisthenode-linkincidencematrixforagraphofNnodesandLdirectedlinks(completegraph),inwhichgij=1ifnodeiisthetailoflinkj,gij=)]TJ /F6 11.955 Tf 9.3 0 Td[(1ifnodeiistheheadoflinkjandgij=0ifnodeiisnotincidenttolinkj. Inlightoftheabovematrixdenitions,thespare-sharing-basedroutingandwavelengthassignmentrulestogetherwithpathvaliditycanbeparaphrasedequivalentlyusingthefollowingmatrix-basedconstraints: 1.Pathvalidityforworkingandbackuppaths.PGT=D (5)QGT=D (5) isaregularmatrixmultiplicationoperator. 2.Disjointednessofworkingandbackuppaths.P+Q1FL (5) 1isaFLmatrixinwhichallelementsare1. 3.Conictavoidancebetweenpaths. 86

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a.Working/workingandworking/backupconictavoidance.AP+BQ1WL (5) whereworksasaregularmatrixmultiplicationoperatorexceptthatitusesBooleanadditioninwhich1+1=1whencalculatingelementsintheresultingmatrix. b.Conictavoidancebetweenbackuppaths. Hereweintroducetwomatricesasbelow:C=[cij]FF=PPT Ciscalledtheworkingpathconictmatrixinwhichcij=1whentheworkingpathofowijoinstheworkingpathofowjatleastononelink.Cij=[cijkm]FL CijisinducedfromCandcalledtheconictpossessionmatrixwithrespecttoowiandowj,wherecijkm=1ifk=iorjandcij=1.Otherwisecijkm=0.Therefore,thebackup/backupconictavoidanceisgovernedbythefollowinginequality:B(CijQ)1,foralli6=j (5) whereisanelement-to-elementmultiplicationoperatorandCijQrepresentstheinducedow-linkincidencematrixinwhichonlythepathinformationofthetwoconictedowsiskept. Finally,theobjectivefunctioncanbeformulatedthroughthelasttwostepsasfollows:T=maxjGj (5)minST (5) 87

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wheremaxjjoperationselectsallthelinksthatareallocatedintopologyformationandformsalinkexistencevectorinwhichtheexistenceofalinkisdenotedby1or0.Sisthelinkcostvectorinwhicheachelementrepresentsthecostofselectingthecorrespondinglink.TheoptimizationgoalisthereforetominimizetheinnerproductST,whichcorrespondstothetopologicalcost. 5.4.2IntegerLinearProgramFormulation Thematrix-basedrepresentationdiscussedinSection 5.4.1 fullyconsidersroutingandwavelengthassignmentissues,andhenceitsoptimalsolutionleadstothelowestpossibletopologicalcost.Equivalenttothematrix-basedrepresentation,thetopologicaloptimizationproblemcanalsobeformulatedintoanintegerlinearprogramasfollows. Constants: N:Numberofnodes W:Numberofprovidedwavelengthsoneachlink Indices: i,j,k:Nodeindextakingintegersfrom0toN)]TJ /F6 11.955 Tf 11.96 0 Td[(1 s:Sourcenodeindextakingintegersfrom0toN)]TJ /F6 11.955 Tf 11.96 0 Td[(1 d:Destinationnodeindextakingintegersfrom0toN)]TJ /F6 11.955 Tf 11.96 0 Td[(1 w:Wavelengthindextakingintegersfrom0toW)]TJ /F6 11.955 Tf 11.95 0 Td[(1 b:Bidirectionallinkindextakingintegersfrom0toN(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)=2)]TJ /F6 11.955 Tf 11.96 0 Td[(1 Data: cb:Costofthebidirectionallinkb Decisionvariables(integer): xsdwij:Representwhetherowrequests!droutesitsworkingpaththroughwavelengthwonunidirectionallinki!jbytakingon1or0 ysdwij:Representwhetherowrequests!droutesitsbackuppaththroughwavelengthwonunidirectionallinki!jbytakingon1or0 88

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Auxiliaryvariables(integer): sdw1,sdw2:Representwhethertheworkingandbackuppathsofows!d,respectively,takeonwavelengthwbytakingon1or0 zb:Representwhethertheworkingorbackuppathofanynodepairpassesthroughbidirectionallinkbbytakingon1or0 Model: .minXbcbzb (5) subjecttofollowingconstraints: 1.Pathvalidityandwavelengthcontinuityforworkingandbackuppaths. Xixsdwij)]TJ /F19 11.955 Tf 11.96 11.36 Td[(Xkxsdwjk=8>>>><>>>>:)]TJ /F10 11.955 Tf 9.3 0 Td[(sdw1,ifs=j+sdw1,ifd=j0,O.W. (5)Xiysdwij)]TJ /F19 11.955 Tf 11.96 11.36 Td[(Xkysdwjk=8>>>><>>>>:)]TJ /F10 11.955 Tf 9.3 0 Td[(sdw2,ifs=j+sdw2,ifd=j0,O.W. (5)Xwsdw1=1 (5)Xwsdw2=1 (5) wheresdw1andsdw2representwhethertheworkingandbackuppathsofows!dtakesonwavelengthwbytakingon1or0. 2.Linkdisjointednessbetweenworkingandbackuppathsforthesameowrequest. 89

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Xwxsdwij+Xwysdwij1 (5) 3.Conictavoidancebetweenpaths. a.Working/workingandworking/backupconictavoidance.Xsdxsdwij+ysdwij1 (5) b.Backup/backupconictavoidance.Xwxs1d1wi1j1+Xwxs2d2wi1j1+ys1d1wi2j2+ys2d2wi2j23s1d16=s2d2andi1j16=i2j2 (5) Equation( 5 )indicatesthat,onanywavelengthofanylink,eitheronlyoneworkingpathorseveralbackuppathsareallowedtobeallocated,whichenforcesrules2and3mentionedinSection 5.1 .Equation( 5 )makestherule4holdinwhichtwobackuppathscannottakethesamewavelengthonthesamelink(thenys1d1wi2j2+ys2d2wi2j2=2)ifthecorrespondingtwoworkingpathsjoineachotheranywhere(Pwxs1d1wi1j1+Pwxs2d2wi1j1=2). 4.Inclusionofallbidirectionallinks,throughwhicheitherworkingorbackuppathsarearranged,intothenallinksetonwhichthetopologicalcostiscalculated. xsdwijzb,xsdwjizb(i
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5.4.3K-ShortestDisjointRoutingBasedFormulation SincethetraditionalwayintheliteraturetosolvetheRWA-relatedproblems(usuallybasedonaxedtopology)istodecomposetheproblemintotheroutingpartandthewavelengthassignmentpart.Intheroutingpart,k-shortestdisjointroutingbasedalternativeroutesareoftenapplied.Althoughthisproblemdecompositionshrinksthefeasiblesolutionspace(becauseoflimitedroutingchoices),thesizeoftheproblemcanbereducedtocertaindegreesuchthatasmall-scaleproblemmaybeoptimallysolved.Inordertoevaluatethefeasibilityofdecompositiontothestudiedtopologicaloptimizationproblem,wedevelopak-shortestdisjointroutingbasedILPasfollows. Constants: W:Numberofprovidedwavelengthsoneachlink F:Numberofowrequests K:Numberoflink-disjointroutesdevelopedbythek-shortestpathalgorithmforeachnodepairinthecompletegraphofNnodes Sets: Rl:Setofallpossibleroutespassingthroughtheunidirectionallinklgivenbythek-shortestpathalgorithmforallnodepairs R0b:Setofallpossibleroutespassingthroughthebidirectionallinkbgivenbythek-shortestpathalgorithmforallnodepairs Indices: i:Nodepairindextakingintegersfrom0toF)]TJ /F6 11.955 Tf 11.96 0 Td[(1 w:Wavelengthindextakingintegersfrom0toW)]TJ /F6 11.955 Tf 11.95 0 Td[(1 r:Routeindexforaspecicnodepairtakingintegersfrom0toK)]TJ /F6 11.955 Tf 11.96 0 Td[(1 l:Unidirectionallinkindextakingintegersfrom0toN(N)]TJ /F6 11.955 Tf 11.95 0 Td[(1))]TJ /F6 11.955 Tf 11.95 0 Td[(1 b:Bidirectionallinkindextakingintegersfrom0toN(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)=2)]TJ /F6 11.955 Tf 11.96 0 Td[(1 Data: cb:Costofthebidirectionallinkb 91

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Decisionvariables(integer): uwir:Representwhethertheworkingpathofnodepairitakesonrouteronwavelengthwbytakingon1or0 vwir:Representwhetherthebackuppathofnodepairitakesonrouteronwavelengthwbytakingon1or0 Auxiliaryvariables(integer): zb:Representwhethertheworkingorbackuppathofanynodepairpassesthroughbidirectionallinkbbytakingon1or0 Model: .minXbcbzb (5) subjecttofollowingconstraints: 1.Pathvalidityandwavelengthcontinuity(automaticallysatised). 2.Linkdisjointednessbetweenworkingandbackuppathsforthesameowrequest. Xwuwir+Xwvwir1 (5)XrXwuwir=1 (5)XrXwvwir=1 (5) 3.Conictavoidancebetweenpaths. a.Working/workingandworking/backupconictavoidance(oneachunidirectionallinkl,nowavelengthwcanbesharedamongmorethanoneworkingpathsorbetweenworkingandbackuppaths). 92

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X(i,r)2Rluwir+vwir1,for(i,r)2Rl (5) b.Backup/backupconictavoidance(foranunidirectionallinkl1,throughwhichtheworkingpathsoftwonodepairsi1andi2pass,thebackuppathsofthesetwonodepairscannotshareanywavelengthwonanyunidirectionallinkl2). Xwuwi1r1+Xwuwi2r2+vwi1r3+vwi2r43i16=i2,r16=r3andr26=r4(i1,r1)2Rl1,(i2,r2)2Rl1,(i1,r3)2Rl2,(i2,r4)2Rl2 (5) 4.Inclusionofallbidirectionallinks,throughwhicheitherworkingorbackuppathsarearranged,intothenallinksetonwhichthetopologicalcostiscalculated.uwirzb,if(i,r)2R0b (5) andvwirzb,if(i,r)2R0b (5) Table5-2. Problemsizeexemplication:numberofvariables Numberofows51015202530 k=2135255375495615735 k=31953755557359151095 k=425549573597512151455 k=5315615915121515151815 OriginalILP1875373555957455931511175 5.4.4ProblemSizeExemplication InordertoprovideinsightinthesizeoftheoriginalproblemformulatedinSection 5.4.2 andthedecomposedproblemformulatedinSection 5.4.3 ,weuseanexemplarnetworkinwhichthereare6nodeswith6wavelengthsoneachunidirectionallink,and 93

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Table5-3. Problemsizeexemplication:numberofconstraints Numberofows51015202530 k=2218550864119015641872 k=337910041779258434734458 k=4600183035945352765610284 k=594731486783106881564921630 OriginalILP1076204762401105860199648031481004560720 thebidirectionallinkcostsarerandomlygeneratedbetweenanypairsofnodes.Foreachnodepair,thereareatmost5link-disjointroutesandatleast2link-disjointroutesareneededforaowrequesttorouteitsworkingandbackuppaths.ThenumbersofdecisionandauxiliaryvariablesareshowninTable 5-2 whereasthenumbersofconstraintsarelistedinTable 5-3 fordifferentformulationsandownumbers.FromthetablesweobservethattheoriginalILPisofahugesizeevenforasmallnetwork,whichmakessolvingitalmostunaffordableforanymodernILPsolver[ 41 ][ 7 ].Comparablythek-shortestdisjointroutingbasedILPformulationsholdingafewthousandvariablesandconstraintsforasmallnetworkhavebetterpotentialtobesolvedbyanILPsolver. 5.5AGreedyApproach Inthissection,wedevelopagreedyapproachtosolvethetopologicaloptimizationproblemwhichcanbeimplementedbyalow-orderpolynomial-timealgorithm. 5.5.1TheUnderlyingIdea Wetreatthepotentialnetworkresource(linksandwavelengthsinthecompletely-connectednetwork)asanumber(W)ofwavelength-associatedgraphsonwhichwavelength-routedlightpathscanbeestablished.Forthestudiedtopologicaloptimizationproblem,thegoalistondagroupofbidirectionallinksthatcanaccommodateallowrequestsatacostaslowaspossiblewhilesatisfyingalldesignconstraintsspeciedinSection 5.1 .Theideaofthegreedyapproachisto,ateachiteration,searchforandarrangeaowrequestwhichleadstominimumtopologicalcostincreaseuntilallowrequestsareaccommodated.Thegreedyapproachmakesthe 94

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pathsearchtendtoreusethoselinksthathavebeenallocatedbutstillholdassignablewavelengths. Withrespecttotheorderofworkingpathandbackuppatharrangement,wechoosetoarrangealltheworkingpathsbeforethebackuppathsareestablished.Thisisbecause,underthespare-sharingscheme,theworkingpathsdominatetheresourceutilizationandtherefore,followingtheintuition,lowerworkingpathsresourceutilizationisdirectlyrelatedtoabetteroveralltopologicalcost.Hencetheresourceallocationpriorityshouldbegiventoworkingpaths.Anotheradvantageofprioritizingworkingpathsarrangementisthatthiscanhelpworkingpathstopickuprelativelyshortroutes,whichmakesworkingpathslessvulnerabletolinkfailuresalthoughthisisnotexplicitlyincludedintheproblemspecication. 5.5.2DataStructures Wavelength-associatedAdjacencyMatricesforWorkinglightpaths(WAMW):anadjacencymatrixtaggedbyaspecicwavelengththatcontainelementsindicatingassignmentavailabilityandcostofcorrespondinglinksonthatspecicwavelengthforworkinglightpathsearch.Therearethreepossiblevaluesonwhichanelementcantakeduringalgorithmoperation: Originalcost:therealcosttosetupabidirectionallinkbetweentwonodes,meaningthatthelinkcarryingthisverywavelengthhasnotbeenselectedforroutinganypath. Innity:thisvalueindicatesthatthewavelengthonthatspecicunidirectionallinkhasalreadybeenallocatedandhenceisnotbeavailableforallocation. Zero:thisvalueindicatesthatabidirectionallinkcarryingthisverywavelengthhasalreadybeenselectedforroutingsomepathsbutthisspecicwavelengthhasnotbeenallocatedandisstillavailableforallocationwithnotopologicalcostincrease. ThealgorithmstartswithallWAMWsequaltotheoriginaladjacencymatrixinwhicheachelementtakesonthecostoftheoriginalbidirectionallink.Asthealgorithmoperates,thevalueofamatrixelement,correspondingtoaspeciclink 95

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andawavelength,maytransitamongthreevaluesaccordingtotransitiondiagramshowninFigure 5-3 Figure5-3. ElementvaluetransitiondiagramforWAMW. Wavelength-associatedAdjacencyMatricesforBackuplightpaths(WAMB):denedinthesamewayasaboveforworkinglightpaths.Theonlydifferenceisinthevaluetransitionsinwhichalinkwavelengthallocationforabackuppathdoesnotswitchthevaluetoinnitybuttozerobecausealaterbackuppathhaspotentialtoreusethiswavelengthgiventhatthespare-sharingRWArulesarefollowed,asshowninFigure 5-4 Figure5-4. ElementvaluetransitiondiagramforWAMB. ConictTable(CT):asquarematrixofFFelements,whereFisthenumberofowsandeachelementindicateswhetherthetwoworkinglightpathsofthetwoowsarelink-disjointornotbytakingonvalue1or0.Thistablecanbeincrementallyestablishedbycheckingconictionbetweenthenewlyallocatedworkingpathandallocatedworkingpaths. 96

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5.5.3TheAlgorithm Foreachiterativeworkinglightpathsearch,theDijkstrashortestpathalgorithmisapplieddirectlyoneachofWAMWsandtheshortestpath(correspondingtotheoneleadingtolowestcostincrease)amongallwavelengthsispickedforcandidacyamongallunsatisedowrequestsinthatveryiteration. Foreachiterativebackuplightpathsearch,thesimilarprocedureasaboveforworkinglightpathsearchisusedexceptthattheadjacencymatricesonwhichtheDijkstraalgorithmworksarenottheWAMBsdirectlybutareconstructedfromWAMBsinthefollowingway:makeacopyfromtheWAMBonthecorrespondingwavelength,settoinnitiestheelementscorrespondingtotheunidirectionallinksthroughwhichtheunder-searchbackuplightpath'sworkinglightpathpasses,andalsosettoinnitiestheelementsofunidirectionallinksiftheyhavealreadybeenallocatedtootherbackuplightpathswhoseworkinglightpathsarenotlink-disjointwiththeunder-searchbackuplightpath'sworkinglightpath(usingConictTable).ThepseudocodeofthealgorithmisshowninFigure 5-5 5.5.4PerformanceComparison Table5-4. Topologicalcostcomparisonamongk-shortestpathbasedILPandthegreedyapproachforarandomlygeneratednetworkwith6nodesand6wavelengthsoneachlink Numberofows 51015202530 k=2 43.3951.6771.9679.3679.3687.08k=3 37.3242.6153.1653.1653.1653.16k=4 37.3240.1440.1445.7645.7645.76k=5 36.6640.1443.8945.7645.7645.76Greedy 25.1525.1532.5444.8844.8852.60Greedy+PER 20.9920.9932.5432.5435.8241.21 Optimalsolutionsfork-shortestdisjointroutingbasedILPformulations Thebestresults(allwithin3.08%relativegaptothebestpossiblesolutions)afterrunningMOSEKfor8hours.ThesolvingprocessisshowninFigure 5-6 thatdemonstratestheconvergingprocessofboththeobjectivefunctionvalueandtherelativegapforthethreeILPinstances 97

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Algorithm:GreedySearch 1. procedureGreedySearch(WAMWs,WAMBs)2. //Searchforworkinglightpaths3. fori=0toF)]TJ /F15 10.909 Tf 10.91 0 Td[(14. LowestCostW 15. forj=0toF)]TJ /F15 10.909 Tf 10.91 0 Td[(16. ifthejthowrequestisnotsatised7. fork=0toW)]TJ /F15 10.909 Tf 10.91 0 Td[(18. runDijkstra'salgorithmonWAMWs[k]9. ifresultingcost
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Figure5-6. SolvingprocessforthethreeILPinstanceswithoutreachingoptimalityafterrunningMOSEKfor8hours Table 5-4 showstheperformancecomparisonbetweenthegreedyapproachandthek-shortestdisjointroutingbasedILPformulations2forthenetworkexempliedinSection 5.4.4 .Thekdisjointroutesforeachowrequestaredevelopedusingthepath-augmentation-baseddisjointroutingalgorithmwhichgeneratesklink-disjointroutesbetweenanodepairwithminimumtotallinkcost[ 54 ][ 65 ][ 22 ].DuetotheoverwhelmingsizeoftheoriginalILPasshowninSection 5.4.4 ,thecorrespondingextremelylargememoryrequirementmakessolvingtheoriginalILPformulationalmostimpossible.Theresultslistedinthelastrowarefromanalgorithmiccombinationofthegreedyapproachandasolutionperfectionprocess(PER)whichwillbediscussedinthenextsection.First,weobservethat,ingeneral,thehighernumberofalternativeroutes(k)leadstolowertopologicalcostbecausetheowrequestscanhavemorechoicesinpicking 2Allthek-shortestdisjointroutingbasedILPinstancesaresolvedbyMOSEK5.0.0.90ontheNEOSserverwith8hoursrunningtimelimit. 99

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routesfortheirworkingandbackuppaths.Theonlyexceptionhappenstotheinstancewith15owrequestsinwhich5alternativeroutesresultinahighertopologicalcostthan4alternativeroutes.Thisisbecausethe4alternativeroutesarenotnecessarilyasubsetofthe5alternativeroutesbetweenanodepairandhencethesolutionspacefor4-shortestroutingbasedILPisnotnecessarilyasubsetofthatfor5-shortestroutingbasedILP.Actually,bythedenitionofk-shortestdisjointrouting,thetotalcostof4shortestdisjointroutesshouldbelowerthanoratmostequaltothecostofanyof4disjointroutesoutofthe5shortestdisjointroutes,whichalsomakesthisexceptionpossible.Second,wecanobservethatthesolutionsofthegreedyapproachonaveragearemuchbetterthanthoseofk-shortestdisjointroutingbasedILPformulations.Thereasonforthatisthekshortestroutesaredevelopedbasedonaconnectivity-richtopology(usuallyacompletegraphifanytwonodescanhaveadirectconnection)thanthetopologyassumedtobexedbymostofpreviousworks.Therebythedisjointroutesdevelopedfordifferentnodepairsmayhavelesschanceofbeingoverlappedandhenceahighernumberoflinksareexpectedtobeincludedintothetopologicalsolution.Thisshowsthatthetraditionalalternativeroutesbasedmethodsarenotefcientinsolvingthetopologicaloptimizationproblem.Lastly,bytakingthegreedysolutionasaninitialsolution,aperfectionprocesscanimprovetheresultstoagreatextent,showingthattherestillexistsconsiderableimprovementspace,whichwillbefurtherexploredbyheuristicsdiscussedinthenextsection. 5.5.5ApproximationRatioAnalysisforWorkingPathsAllocationunderAde-quateWavelengthProvision Sinceingeneraltheworkingpathsdominateresourceutilizationcomparedwiththebackuppaths,asshowninSection 5.7 ,wederiveanapproximationratioanalysisforresourceallocationofworkingpathstocapturetheoptimalityofthegreedyapproach.Weassumethatthereareadequatewavelengthsoneachlinks,i.e.,atleastFwavelengthsareprovided.TheoptimaltopologicalcostisdenotedbyOPT.The 100

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topologicalcostofthegreedysolutionisdenotedbyCostgreedy.LetP1,P2,...,PFbetheworkingpathsthatthegreedyapproachdevelopsfortheFowrequestsintheorderofbeinggenerated.ForpathPi,wedenePrice(Pi)asthetopologicalcostincreaseaftergeneratingworkingpathPi.WealsodenetheoptimaltopologicalcostincreasetogenerateworkingpathsfortherestowrequestsafterP1,...,Pi)]TJ /F5 7.97 Tf 6.59 0 Td[(1havebeendevelopedbythegreedyapproachasOPT0i.SinceCost(Pi)leadstotheoptimalcostincreasefortheworkingpathPi,wehavePrice(Pi)OPT0i. (5) Inaddition,sinceOPT0iistheoptimalcostincreaseforthesubsetofowrequestsi,...,Funderthecost-reducedgraph(thecostsofthelinksonwhichthepathsdevelopedbythegreedyapproacharereducedto0),wealsohaveOPT0iOPTiOPT, (5) whereOPTirepresentstheinducedtopologicalcostofowrequestsi,...,Fintheoptimalsolution. Hence,nallywehave Costgreedy=FXi=1Price(Pi)FXi=1OPT0iFXi=1OPTiFXi=1OPT=FOPT (5) andCostgreedy OPTF (5) 5.5.6ComplexityandMemoryRequirementAnalysis Byobservingthestructureofthealgorithm,wendthat,whenanO(N2)Dijkstra'salgorithmisapplied,thecomputationalcomplexityofthealgorithmisO(F2WN2),whereFisthenumberofowrequests,WisthenumberofavailablewavelengthsandNisthe 101

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numberofnodes.Withrespecttothememoryrequirement,aworkingspaceintheorderofO(WN2+F2)isneededforalgorithmoperationwhichcomesfromtwodominatingdatastructures:wavelength-associatedadjacencymatricesandtheconicttable. 5.6EnhancedHeuristics Inthesection,werstidentifytwodrawbacksofthegreedyapproachthatmayprohibititfromachievingbetterperformanceincertainscenarios.Basedonthedrawbackanalysis,weproposetwoapproachesfeaturingdifferentinitialsolutiongenerationtothetopologicaloptimizationproblem.Then,wedevelopaperfectionprocessworkingonthegeneratedinitialsolutionswiththegoalofloweringthetopologicalcostoftheinitialsolutions.Thecombinationoftheinitialsolutiongenerationandtheperfectionprocessformstheproposedheuristicalgorithmastherstandthesecondphasesrespectively. 5.6.1DrawbacksoftheGreedyApproach Sincethegreedyapproach,describedinSection 5.5 ,onlypursuestheiteration-wideoptimality,thereexistspotentialforittoloseglobaloptimality.Figure 5-7 (A)showsanetworkthatstillhastwoowrequests(A!CandB!D)towhichresourceneedstobeassigned.Weassumethatallthelinksmissinginthegureeitherrunoutofwavelengthresourceorcosttoohightobeconsidered.Thedashedlinksrefertothoselinksonwhichnowavelengthisassignedtoanyow.Thesolidlinksrepresentthoselinksthathavebeenassignedbutstillholdavailablewavelengths.Wealsoassumethateachlinkinthegurehasatleast2availablewavelengthsforallocation.Therearetwovaluesoneachlink.Therstindicatesthecurrentcostincreasebyroutingthroughthecorrespondinglinkandthesecond(intheparentheses)indicatestheoriginalcostofthebidirectionallink. ThegreedyapproachwillndrouteA!CforowrequestA!CandthenndrouteB!A!C!DforowrequestB!D,asshowninFigure 5-7 (B),whichresultsincostincrease17+8+8=33.However,ifwemaketheowA!Ctakeontheroute 102

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A B Figure5-7. Originalgreedyapproachsolution. A!E!F!CandmaketheowB!DtakeontherouteB!E!F!D,thecostwillbeonlyincreasedby18.Thisexampleillustratesmyopiaofthegreedyapproachundercertainscenario. Besidespotentialmissofglobaloptimality,thegreedyapproachtendstoroutepathsinazigzagwayinordertoreusethoseallocatedlinksasshowninFigure 5-7 (B)inroutingowrequestB!D.Sucharoutingfashionmakestheroutesmorelikelybetwistedtogether,whichnotonlypotentiallyleadstohigheroverallresourceconsumptionbutalsorendersdifcultytoimprovingthesolutionindissolvingthetwistedsituation.Forexample,thetworoutescoupledonlinkA!Carenoteasytobemovedtotheiroptimalpositionscomparedwiththescenarioinwhichtheyareseparateandroutedclosetotheirshortestpaths.Thewaytomove(orreroute)pathswillbediscussedinSection 5.6.3 5.6.2TwoInitialSolutions Inordertoassisttheroutingprocessoftheoriginalgreedyapproachtoidentifyrightpathsforroutingowrequests,weproposeusingthelinkpotentiall,anumericalfactorrangingfrom0to1,toevaluatethepotentialthatalinkshouldbeincludedintothetopologicalsolution.Theunderlyingideaisthatthelink,byroutingthroughwhichthepathsareclosertotheirshortestpathsinlength,shouldhavehigherpotentialtobe 103

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pickedforallocation.AswecanseeinFigure 5-7 (A),althoughowA!CfavorspickinglinkA!C,owB!DdoesnotbecauserouteB!A!C!Dismuchlongerthantheshortestroute(B!D)ofowB!D.However,linkE!FcanbalancetheclosenessrequirementofbothowA!CandowB!Dtotheirshortestpaths.IflinkE!FcanbepickedforroutingowA!C,theresultingroutesofthetwoowrequestswillbecomeoptimal,asshowninFigure 5-8 (B). A B Figure5-8. Linkpotentialbasedgreedysearchsolution Werstdeneow-associatedlinkpotential,fl,astheratiooftheoriginalpathcostpcfofroutingtheowintheoriginalnetworktotheoriginalpathcostpc0fofroutingtheowthroughtheunidirectionallinklundercurrentnetworkresourcecondition.pc0fcanbedecomposedintothreeparts:thecostofroutingtheowfromitssourcetothetailnodeoflinklpc0f1,thecostoflinklcl,andthecostofroutingtheowfromtheheadnodeoflinklpc0f2tothedestination.fl=pcf pc0f=pcf pc0f1+cl+pc0f2 (5) Wedenelinkpotential,l,astheaverageoffloverowsthatstillwaitforroutingandresourceallocation,asfollowingl=1 F0Xffl, (5) 104

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whereF0representsthenumberofunassignedowrequests.Sincepcfistheshortestpathcostintheoriginalnetwork,flisavaluesmallerthanoratmostequalto1,soisl.Intuitively,lwithahighervalueindicateslinkl,onaverage,isclosertotheshortestpathsofunassignedowrequestsandhenceshouldhavehigherpotentialtobeselected.Thisisbecausetheclose-to-shortestroutescanhelpsavenetworkresourceand,combinedwiththegreedysearch,eventuallymayleadtoalowertopologicalcost. Intheshortestpathsearchalgorithm,thelinkwithlowercosthashigherpotentialtobeselectedintotheshortestpath.Hence,wedenelinkcostcoefcient,l,asl=1)]TJ /F10 11.955 Tf 11.96 0 Td[(l (5) andusetheproductoflandthelinkcurrentcosttoreplacethelinkcostusedinthegreedysearchalgorithm.Inthisway,weintegratetheuseoflinkpotentialintothegreedysearchalgorithmandwecallthenewalgorithmlinkpotentialbasedgreedysearch.StilltakethenetworkshowninFigure 5-7 (A)forexample,underthecurrentnetworkresourcecondition,owA!Ccosts17andowB!Dcosts33toroutethroughlinkA!C.TheshortestpathcostsofbothowA!CandowB!Dare17intheoriginalnetwork.HenceA!C=(17=17+17=33)=2=25=33andA!C=1)]TJ /F6 11.955 Tf 12.32 0 Td[(25=33=8=33.ThecostoflinkA!Cisreplacedwith178=33=136=33.Followthesameprocess,thecostsoflinkB!DandlinkE!Farereplacedwith136=33and27=10respectively.TheupdatednetworkwithnewlinkcostsisshowninFigure 5-8 (A).Thecostsoftherestlinksarenotshownbecausetheydonotaffecttheroutingdecisions.ApplythegreedyalgorithmontothenewnetworkshowninFigure 5-8 (A).ThenrouteA!E!F!CbecomesthedecisionrouteinsteadoftheoriginalrouteA!C.AfterthatowB!DwouldpickuprouteB!E!F!D,insteadofrouteB!A!C!D,becauseofzerocostincrease.TheresultingroutesareshowninFigure 5-8 (B),whichisoptimalforthetwoowrequests. 105

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WithrespecttotheseconddrawbackidentiedinSection 5.6.1 ,weproposeanothermodicationontheoriginalgreedysearchalgorithmwiththeaimtoavoidgeneratingzigzagpaths.Insteadoftryingtochangethelinkcostsasinlinkpotentialbasedgreedysearch,ateveryiteration,thealgorithmselectstheowrequestthatleadstothelargestpathratio.Pathratioisdened,withrespecttoaspecicowrequest,astheratiooftheshortestpathcostintheoriginalnetworktothecostoftheshortestroutedevelopedinthegreedysearchalgorithmonthecurrentnetwork.Thisowselectionordertendstoselecttheroutethatisclosetoitsshortestpathinlengthandhencecanavoidgenerationofthosezigzagpathstoagreatextent.Wecallthismodiedsearchlargestratiorst(LRF)search. TaketheexampleasshowninFigure 5-9 (A),weassumewavelengthresourceonlinkA!CareallassignedandhencelinkA!Cisnotshowninthegure.TheoriginalgreedyapproachwillndrouteA!G!Cwithcostincrease9forowrequestA!CandthenrouteowrequestB!DthroughB!A!G!C!D,asshowninFigure 5-9 (B).Theresultingtworoutesarecloselycoupledtogetheronmanylinksawayfromtheirshortestpaths.However,LRFsearchwillselectrouteB!DforowrequestB!CandthenrouteowrequestA!CthroughA!G!C,asshowninFigure 5-9 (C).Thisisbecause,insteadofpursuingminimumcostincrease,LRFsearchrendersprioritytoselectingtherouteclosertoitsshortestpath.AlthoughLRFsearchinducedroutingleadsthecosttobeincreasedhigher(by17+9=26)thantheoriginalgreedysearch(by9+8+8=25),thepotentialtomovethetwolooselycoupledroutestotheiroptimalpositionswouldbehigher.Thedetailedroutesmovingprocedurewillbediscussedinthenextsubsection. 5.6.3SolutionPerfection(PER) Thesolutionsgeneratedbytheoriginalgreedyapproach,linkpotentialbasedgreedysearch,andLRFsearchcanbeusedasinitialsolutionsandweproposeasolutionperfectionprocessworkingonthoseinitialsolutions.Theinitialsolution 106

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A B C Figure5-9. Largestratiorstbasedsearchsolution generationandtheperfectionprocessaretwophasesoftheheuristicalgorithmproposedinthissection. Theideaoftheperfectionprocessistryingtoreroutealltheowspassingthroughaspecicbidirectionallinkbyleveragingtherestofunassignednetworkresource.Ifthereroutedsolutionleadstoalowertopologicalcost,thereroutedsolutionbecomesthenewsolution.Thisprocesswillcontinueonanotherbidirectionallinkuntilnoimprovementcanbeachievedbyreroutingtheowsonwhicheverbidirectionallink.ThepseudocodeoftheperfectionalgorithmislistedinFigure 5-10 5.7Results Inthissection,weevaluatetheperformanceoftheproposedheuristicmethodsproposedinthelastsection.Theadvantageofapplyingsparesharingtechniquesinresourcesavingisalsoshown.Finally,bydeningperformanceindicator,werevealtheunderlyingreasonforperformancedifferentresultingfromvariedalgorithmoptions. 5.7.1PerformanceComparison Weuseanexemplarnetworkinwhich16USmajorcities,asshowninFigure 5-11 ,aretreatedasnodesandthebidirectionallinkcostsareassignedtobeproportional 107

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Algorithm:Perfection 1. procedurePerfection(init solution,WAMWs,WAMBs,CT)2. current solution init solution3. do4. improvement 05. foreachassignedbidirectinallinkl6. TWAMWs WAMWs,TWAMBs WAMBs,TCT CT7. releasewavelengthresourcealongworkingandbackuppathspassing throughbidirectionallinklbyupdatingTWAMWsandTWAMBs8. reroutetheworkingpathsthroughrestofnetwork,updateTWAMWs, TWAMBs,andTCT9. reroutethebackuppathsfollowingallspare-sharingconstraintsthroughrest ofnetwork,updateTWAMWsandTWAMBs10. ifreroutingissuccessful11. calculateresultingtopologicalimprovement12. ifresultingimprovement>improvement13. improvement resultingimprovement14. updatecurrent solution15. endif16. endif17. endfor18. ifimprovement>019. WAMWs TWAMWs,WAMBs TWAMBs,CT TCT20. endif21. whileimprovement>022. outputcurrent solution23. endprocedure Figure5-10. Pseudocodeoftheperfectionalgorithm(PER) tothedistancebetweencities3.Eachnodeisrequiredtoestablishcommunicationtoallothernodesinthenetwork,whichindicatesthenumberofowsrequestsis16(16)]TJ /F6 11.955 Tf 11.95 0 Td[(1)=240. 3Incaseofdisaster-inducedlinkfailures,thecorrespondinglinkcostsaresettoinnityinordertore-ectthechangednetworkresourcecondition.Therebytheresultingtopologysolutionwillavoidutilizinganyofthelinksaffectedbythedisaster.Inthischapter,focusingpurelyonperformancecomparisonforthetopologicaloptimizationproblemamongdifferentalgorithms,wedonotmodelandconsideraspecicsetoflinkfailures.Thiswillbethefocusoffuturework. 108

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Figure5-11. Locationsof16USmajorcities Wetesttheaboveproblemcongurationviatheoriginalgreedyapproach(Greedy),linkpotentialbasedgreedysearch(Potential),largestratiorstsearch(LRF),andthecombinationsofthoseapproachesandtheperfectionprocess(Greedy+PER,Potential+PER,andLRF+PER).Thesolutionsoftheapproachesotherthantheoriginalgreedyapproacharecomparedwiththeinitialgreedysolutionsandtheirimprovements,denedasImprovementX=TopologicalCostgreedy)]TJ /F3 11.955 Tf 11.96 0 Td[(TopologicalCostX TopologicalCostX, (5) arerecorded,whereXrepresentstheapproachusedforcomparison.Theresults,withdifferentwavelengthprovision,areshowninFigure 5-12 .Aswecanobserve,exceptthesolutionsofLRF,allotherapproachesingeneralleadtobetterperformancethantheoriginalgreedyapproach.Intermsofinitialsolutions,linkpotentialbasedgreedysearchonaverageproducesthelowest-costtopologicalsolutions.Withrespecttothenalsolutions,allPER-inducedsolutionsaremuchbetterthantheircorrespondinginitialsolutions.TheLRF+PERinducedsolutions,withimprovementof20%30%fromthegreedysolutions,outperformallothersolutionsalthoughtheinitialLRF-inducedsolutionsareworsethanthecorrespondinggreedysolutions.Thisisbecausetheinitial 109

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LRF-inducedsolutionsarebettershapedintermsoftwistingcondition,whichmakesthePEReasiertoreroutetheowstotheiroptimalpositions. Figure5-12. Performanceimprovementsfromgreedysolutionsduetoheuristicalgorithmsforvariedwavelengthprovisions Figure5-13. Convergenceprocessoftheperfectionalgorithmtakingthreedifferentinitialsolutions Figure 5-13 demonstratestheconvergenceprocessoftheperfectionalgorithmtakingtheGreedy,Potential,andLRFsolutionsasinitialsolutionsfortheproblemcongurationwith5wavelengthsavailableoneachlink.ItshowsthatPotentialcanproducethelowestinitialsolutionwhereastheinitialsolutionofLRFisverycostlyand 110

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howeverithashigherpotentialtobeperfectedbyrunningtheperfectionalgorithmafteralargenumberofiterations. Figure5-14. Weightedwavelength/linkutilizationforworkingandbackuplightpathsintheLRF+PERinducedtopologicalsolutions Figure 5-14 illustratesthebenetofapplyingsparesharingtechniquesinresourcesaving.Theweightedwavelength/linkutilizationisdenedasU=PlPwOlwCl PlWCl, (5) whereClisthecostoftheunidirectionallinklandOlwindicatesifthewavelengthwonlinklisoccupiedornotbytakingon1or0.Thisutilizationtakesonvaluebetween0and1,andthehighervalueindicatesmoreefcientresourceutilization.Sinceonwavelengthonthesamelinkcanbesharedbetweenworkingandbackuppaths,thetotalweightedresourceutilizationcanbedecomposedintoworkingpathutilizationandbackuppathutilization,asindicatedbytheblueandredbarsinFigure 5-14 .Theadvantagethatthebackuppathscansharewavelengthsamongeachothermakesbackupresourceutilizationextremelylowcomparedwithresourceutilizedbyworkingpaths,especiallyconsideringthefactthattheworkingpathsareroutedbeforethebackuppathsandhencetheycantakecomparablyshorterroutes.Thetrendthatthebackupresource 111

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utilizationisincreasedwiththenumberofavailablewavelengthsisobserved.Thisisbecausethenetworktopologybecomesthinner(includinglessnumberoflinkswhenincreasingthenumberofwavelengths,asshowninFigure 5-14 )andthesparesharingconstraintsstartplayingaheavierroleinroutingthebackuppaths.Forthesamereason,thenarrowerroutingchoicesforbothworkingandbackuppathsmaketheoverallwavelength/linkutilizationdecreasewiththenumberofwavelengths. 5.7.2PerformanceIndicator Inordertobetterdiscernhowdifferentlyalgorithmsperform,weidentifytwoessentialfactorsthatcanindicatethesolutionperformanceforthetopologicaloptimizationproblem.OneistheweightedresourceutilizationUasdenedinthelastsubsectionandanotherisaveragebendingfactorB,denedasB=1 2F Xfpcf pwf+Xfpcf pbf!, (5) wherepcfistheshortestpathcostofowf,pwfisthesolutionworkingpathcostofowf,andpbfisthesolutionbackuppathcostofowf.Theaveragebendingfactorshowshowfaronaveragethesolutionpathsdeviate(orbend)fromtheirshortestpaths.Avaluecloseto1signiesthatthesolutionpathsareincostclosetotheirshortestpathsandhenceareindividuallycost-efcientinwavelengthutilization,whereasavaluetowards0indicatesthesolutionpathsarecostlyrouted. Intuitively,anidealtopologicalsolution(best-possibleorlower-boundsolution)wouldfullyutilizethewavelengthinthesolutiontopologywithallpathsroutedthroughtheirshortestpaths.Hence,atopologicalsolutionhavinghighwavelengthutilizationandhighaveragebendingfactorsimultaneouslywouldleadtogoodperformance(lowtopologicalcost.)TherebywedeneperformanceindicatorofthetopologicalsolutionsasPI=BU. (5) 112

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Figure5-15. Solutionperformanceindicationforthenetworkwith5wavelengthsprovision Thevalidityofthisdenition(logicalcorrespondencebetweenPIvaluesandtopologicalcosts)isveriedbyobservingFigure 5-15 inwhichtheperformanceindicatorvaluesandtopologicalcostsresultingfromdifferentalgorithmsareshown.Asobserved,thehighestPIappearswiththeLRF+PERinducedsolutionthatleadstothebestperformance,whereasthelowestPIhappensontheinitialsolutionofLRFthatisworst-performedamongthecomparingapproaches.PIcorrespondingtotheinitialgreedysolutionisthelowest,indicatingthehighesttopologicalcostoftheinitialgreedysolution.AlsoobservedisthatallPIscorrespondingtoPER-enhancedsolutionsarehigherthanPIsoftheirinitialsolutions. Figure 5-16 showsmoredetailsonhowthetopologicalsolutionsfromdifferentalgorithmsaredistributedwithrespecttoweightedwavelengthutilizationandaveragebendingfactor.Ingeneral,theLRFinducedsolutionshavehigheraveragebendingfactors.ThereasonforLRF+PERtoperformthebestisinthatitssolutionsscorehighinbothweightedwavelengthutilizationandaveragebendingfactorcomparedwithotheralgorithms.ThePERinducedimprovementforthegreedysearchandlinkpotential 113

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Figure5-16. Weightedwavelengthutilization/averagebendingfactordistributionforvariedtopologicalsolutionsinthenetworkswith5,10,15,20,and25wavelengthsprovision basedgreedysearchisbecausethePERprocessimprovesthewavelengthutilizationbyreroutingpathswithoutdegradingthebendingsituations. Inthischapter,wefocusonunderstandingthecomplexityofthetopologicaloptimizationproblemforspare-sharingbasedall-opticalnetworksbystudyingitsILPformulation.Next,duetocomputationaldifcultyofsolvingreasonablylargeproblemsthroughILPformulation,oureffortisdedicatedtosearchingalgorithmicsolutionstotheproblem.WerstdevelopagreedysearchalgorithmwhoseperformanceisveriedtobebetterthanthetraditionalRWAmethodsbasedonroutingandwavelengthassignmentdecomposition.Thenweproposeatwo-phaseheuristicalgorithminwhichtwodifferentinitialsolutionsearchapproachesaredeveloped.Throughexhibitingresultsonanexemplarnetwork,thegoodnessoftheproposedheuristicalgorithmisveriedbycomparingwiththegreedysolutions.Finally,ametric,performanceindicator,isdened 114

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andveriedtorevealtheunderlyingreasonofperformancedifferencefromvariedalgorithms. 115

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CHAPTER6ORDERED-PATH-ENUMERATION-BASEDCANDIDATEROUTING:AFACILITATINGAPPROACHTOSOLVINGRWAPROBLEMSFOROPTICALNETWORKS Asthestudyofroutingandwavelengthassignment(RWA)problemsmigratesfromsolvingsinglelightpathestablishmentproblemstodealingwithprotective(spare)lightpathsetupinordertotoleratenetworkfaults,thehardnessofsolvingthoseproblemsrisesdramatically.Thisisnotonlybecauseoftheextradecisionvariablescreatedfortheprotectivelightpaths,butalsoduetomuchmorecomplicatedconstraintsgeneratedtoavoidresourceconictandenforcepath-disjointednessbetweentheworkingandsparelightpaths.ThecombinedeffectisthecreationofhugeconstraintmatricesassociatedwiththeoptimizationmodelsforthoseRWAproblems,whichseverelychallengesthecapacityofstoragesystemsandthecomputationalpowerofcurrentstate-of-the-artmachines. Torelaxthehardnessoffullysolvingthoseproblemstotheirglobaloptimality,researcherstraditionallydecoupletheproblemsintotheroutingpartandthewavelengthassignmentpart[ 41 67 ].Bypotentiallysacricingtheglobaloptimality,thesizeoftwosubproblemscanbewellcontrolledfornetworksofamoderatesize.Actually,oftenforaspecicRWAproblem,thereexistsanintrinsiccouplingrelationbetweentheroutingpartandthewavelengthassignmentpartinorderfortheproblemtobesolvedclosetoitsglobaloptimality.Inthischapter,weareespeciallyinterestedinthoseRWAproblemsthataredesignedtoprotectthenetworkinashared-pathfashion[ 21 36 38 41 57 ].Theroutingpartsofthoseproblemsaretraditionallysolvedbysearchingforasetoflink-disjointpathsasroutingcandidates,becauseotherwisetheworkingandsparelightpathswouldbesubjecttosimultaneousfailuresinducedbythefaultonthelinksharedbetweenthem.Thenthewavelengthassignmentparttakesthoseroutingoptionsasworkingandsparelightpathcandidatesforbestwavelengthresourcearrangement.Theclassiclink-disjointpathsearchwidelyappliedinthe 116

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literatureisk-shortestlink-disjointrouting[ 22 54 ],whichndsklink-disjointpaths(ifexist)inadirectedgraphwiththelowesttotallinkcost. However,wewillshowinthischapterthatthepathsdevelopedbyk-shortestlink-disjointroutingmaynotbesttthecouplingrelationbetweenthetwosubproblemsbecauseoftwomajorreasons:(1)theamountofdisjointpathsdevelopedmaynotbeenoughforless-denselyconnectednetworks,and(2)thesetsofdisjointpathsfordifferentowrequestsmaynotbethebestoptionsfortheproblemasawhole.Infact,inordertondthebestsetofcandidateroutes,thesetofpossiblepathsforroutingselectionhastobesufcientlylarge.Inthischapter,weproposeanewpathenumerationalgorithmwhichcanndapre-describednumberofpathsinanincreasingorderofcostforaspecicnodepair.Basedonthepoolofenumeratedpaths,certainselectionrulesareappliedtoformcandidateroutingsetsofapropersizeforwavelengthassignmentoptimization. 6.1RelatedWork Theshared-pathprotectionschemehasbeeninvestigatedinmanyworks,suchasin[ 21 36 38 41 57 ],withdifferentproblemsetups.Bothmathematicalprogrammingbasedformulations[ 37 41 ]andalgorithmicheuristics[ 21 36 38 57 ]havebeenappliedtosolvetheproblem.Ithasbeenprovedin[ 38 ]thatestablishingasingleworking/sparelightpathpairisanNP-completeproblem.FullysolvingtheRWAproblemforallowrequestsbecomesprohibitiveevenforasmall-sizenetwork.Hence,k-shortestdisjointroutingiswidelyappliedastheroutingsolution[ 41 ].However,in[ 33 ],theauthorsobservethatthelackofawarenessoftherealtrafcpatternbyk-shortestdisjointroutingleadstoapoorconnectionsuccessrate.Itisindicatedthatthecandidateroutingsetshouldbedevelopedinaproblem-dependentfashion. Thestudyonpathenumeration(commonlycalledk-shortestpathenumeration)datesbacktoYen'sandLawler'salgorithms[ 30 65 ].Thebasicideaofthosealgorithmsistopartitionthesearchingspace(allun-enumeratedpaths)intomutuallyexclusive 117

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subsetsbasedontheenumeratedpaths,andthenextenumeratedpathispickedastheshortestonefromamongallthosesubsets.[ 12 ]providesacomparativestudyoverseventyrelatedworksandconcludesLawler'smethodgivesthebestperformancewhenconsideringonlyacyclicpathsanddirectedgraphs.Cyclic-path-allowedpathenumerationisstudiedin[ 15 ].Amorerecentworkin[ 25 ]leveragesthereplacementpathsalgorithmandappliesittoworkwiththeshortestpathbranchingstructure,whichleadstoafactor-(n)improvementwhenthereplacementpathsrarelyfails.[ 50 58 ]furtherconsiderpathenumerationunderasetofconstraints. 6.2ContributionsandChapterOrganization Themajorcontributionsofthischapterareasfollows.First,weformallyproposeanewpathenumerationalgorithmbasedonaseriesoftheoreticalderivation.Second,wedevelopacandidateroutingschemetondcandidateroutingsetsthatbesttspecicproblems.Third,byformulatingandsolvingtwoconcreteRWAproblems,wenumericallyshowevidentpotentialofthepath-enumeration-basedcandidateroutinginimprovingresourceallocationperformance. Therestofthischapterisorganizedasfollows:Section 6.3 describestheproposedorderedpathenumerationalgorithm.ThecandidateroutingschemeisdiscussedinSections 6.4 and 6.5 ,wheretwospecicRWAproblemsarestudiedforperformanceillustration. 6.3OrderedPathEnumeration Thissectionformallydescribestheorderedpathenumerationalgorithmthatweproposetodevelopapoolofpossiblepathsforcandidateroutesselection. 6.3.1DenitionofTerminologies Network:denotedbyG(N,L),whereNisasetofnodesandLisasetofdirectedlinkscomposingthenetwork. OrderedPathContainer:denotedbyPk(s,d),asequenceofkshortestpathsconnectingthesourcesandthedestinationdinanincreasingorderofcost(orlength) 118

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inwhichtherstpathp1isindeedtheshortestpathamongallpossiblepaths,thesecondpathp2isthesecondshortestpath,...,andthekthpathpkisthekthshortestpath.Wecallitcontainerforshortintherestofthechapterifthecontextisclear.WealsoassumethatallthepathsinPk(s,d)areloop-freebecauseitisofnobenettoconsiderthepathswithloopsfortheresourceallocationproblemsstudiedinthischapter. OrderedPathContainerCover:denotedbyC(Pk(s,d)),agroupofdirectedlinkswhoseremovalfromthenetworkleadsallthepathsinthecontainerPk(s,d)tobebroken.Wecallitcontainercoverintherestchapterifthecontextisclear. MinimalOrderedPathContainerCover:denotedby~C(Pk(s,d)),agroupofdirectedlinksthatisanorderedpathcontainercoverbutisnotanorderedpathcontainercoveranymoreifanylinkinthegroupisremoved. CompleteMinimalContainerCoverSet:denotedbyS(Pk(s,d)),thecompletesetofallpossible~C(Pk(s,d))thatcontainminimalnumbersofdirectedlinkstocovertheorderedpathcontainerPk(s,d). CompleteCoveringLinkSet:denotedbyL(Pk(s,d)),thecompletesetofdirectedlinksthatcoveratleastonepathinPk(s,d)(i.e.,thesetofdirectedlinkstakenbythepathsinPk(s,d)). 6.3.2TheoremsregardingOrderedPathEnumeration Theorem6.1.(k-shortestPathEnumeration)GiventheorderedpathcontainerPk(s,d),Pk+1(s,d)canbeformedbyaddingthe(k+1)thshortestpaththatisob-tainedbyselectingtheshortestpathamongshortestpathsofthenetworksinducedbyremovingalllinksin~C(Pk(s,d))fromtheoriginalnetworkG(N,L). Proof:Assumethatpk+1isthe(k+1)thshortestpathfromstodintheoriginalnetwork.Accordingtotheuniquenessofpathpk+1andtheassumptionthatallthepathsconsideredareloop-free,alongeachpathinPk(s,d)theremustbeatleastonelinknotbelongingtopathpk+1.Thenwegroupthoselinksintoasetwhichbydenition 119

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becomesancontainercoverC(Pk(s,d)).Clearly,theremustexistaminimalcontainercover~C(Pk(s,d))asasubsetofC(Pk(s,d)).Sincepk+1isavalidpathinthenetworkinducedby~C(Pk(s,d)),inwhichhoweverallpathsinPk(s,d)arenotvalid,andpk+1isthe(k+1)thshortestpathintheoriginalnetwork,pk+1mustbetheshortestpathofthenetworkinducedby~C(Pk(s,d)).Inotherwords,the(k+1)thshortestpathpk+1canbefoundamongtheshortestpathsofthenetworksinducedbyall~C(Pk(s,d))ofS(Pk(s,d)).2 Theorem6.1indeedshowsawaytoenumeratethepathsforaspecicsource-destinationpair,startingfromtherstshortestpath,inanincreasing-costorder. Lemma6.1.Anyminimalcontainercover~C(Pk+1(s,d))inS(Pk+1(s,d))mustcontainasubsetthatisaminimalcontainercover~C(Pk(s,d))inS(Pk(s,d)). Proof(bycontradiction):If~C(Pk+1(s,d))doesnotcontainanysubsetthatisa~C(Pk(s,d)),therstkshortestpathsthencannotbecoveredby~C(Pk+1(s,d))andtherefore~C(Pk+1(s,d))isnotevenavalidcontainercover,whichisacontradictiontothedenitionof~C(Pk+1(s,d)).2 Thislemmaimpliesthatall~C(Pk+1(s,d))canbedevelopedbyexpanding~C(Pk(s,d))inS(Pk(s,d)). Lemma6.2.Thecardinalityofanyminimalcontainercover~C(Pk+1(s,d))isatmostgreaterthanthecardinalityofitscontainedminimalcontainercover~C(Pk(s,d))by1. Proof(bycontradiction):Assumethatthecardinalitydifferenceisgreaterthan1,whichmeansthattherearemorethan1linkin~C(Pk+1(s,d))thatarenotincludedin~C(Pk(s,d)).Sinceallrstkshortestpathsarealreadycoveredbythelinksin~C(Pk(s,d))andatmostoneextralinkisneededtocoverthe(k+1)thpath,~C(Pk+1(s,d))willnotbeaminimalcover,whichcontradictsitsdenition.2 Lemma6.3.Anyminimalcontainercover~C(Pk+1(s,d))canonlycontainone~C(Pk(s,d)). 120

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Proof:If~C(Pk+1(s,d))isthesameas~C(Pk(s,d)),theproofbecomestrivial. If~C(Pk+1(s,d))isnotthesameas~C(Pk(s,d)),accordingtolemma6.2,~C(Pk(s,d))canbedifferentfrom~C(Pk+1(s,d))thatcontainsitbyatmostonelink.Assumethattherearemorethanone~C(Pk(s,d))containedin~C(Pk+1(s,d))andwenametwoofthem~C1(Pk(s,d))and~C2(Pk(s,d)).Thenthecardinalityof~C1(Pk(s,d))and~C2(Pk(s,d))mustbej~C(Pk+1(s,d))j)]TJ /F6 11.955 Tf 20.97 0 Td[(1.Moreover,sinceneither~C1(Pk(s,d))nor~C2(Pk(s,d))cancontainalinkcoveringthe(k+1)thshortestpath(otherwise~C(Pk+1(s,d))isnotaminimalcontainercover),thereisnowaytodevelop~C(Pk+1(s,d))from~C1(Pk(s,d))or~C2(Pk(s,d))byaddingonlyonelinkunless~C1(Pk(s,d))isthesameas~C2(Pk(s,d)).2 Inlightoflemmas6.1,6.2,and6.3,wehaveaformalmechanismtoderivethecompleteminimalcontainercoversetviacontainercoverexpansion,asstatedintheorem2. Theorem6.2.(ContainerCoverExpansion)GivenS(Pk(s,d)),S(Pk+1(s,d))canbedevelopedfromall~C(Pk(s,d))ofS(Pk(s,d))asfollows:keepa~C(Pk(s,d))inS(Pk+1(s,d))ifitcancoverthe(k+1)thshortestpathorexpanda~C(Pk(s,d))thatcannotcoverthe(k+1)thshortestpathbytryingtoincludeonelinkonthe(k+1)thshortestpath.AccordingtowhetherthelinkbelongstothecompletecoveringlinksetL(Pk(s,d)),therearetwocases: 1.IfthelinkdoesnotbelongtoL(Pk(s,d)),includeitinto~C(Pk(s,d))forminga~C(Pk+1(s,d)). 2.IfthelinkdoesbelongtoL(Pk(s,d)),includeittoforma~C(Pk+1(s,d))iftheresultingcontainercoverisminimal. Proof:Accordingtolemmas6.1and6.2,all~C(Pk+1(s,d))canbedevelopedfrom~C(Pk(s,d))byaddingatmostonelink.Accordingtolemmas6.2and6.3,notwodistinct~C(Pk(s,d))canbeexpandedtothesame~C(Pk+1(s,d)),whichmeanstherewillbenoredundantexpansionindevelopingS(Pk+1(s,d)). 121

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Withrespecttotheexpansionfroma~C(Pk(s,d))toa~C(Pk+1(s,d)),sinceaddinganylinkthatisnotonthe(k+1)thpathwillnothelpforma~C(Pk+1(s,d)),checkingthroughthelinksonthe(k+1)thpathisenoughtondallpossible~C(Pk+1(s,d))expandedfromaspecic~C(Pk(s,d)). Iftheaddedlinkonthe(k+1)thshortestpathdoesnotbelongtoL(Pk(s,d)),thenitcannotcoveranypathinPk(s,d).Takingoutanylinkfrom~C(Pk(s,d))willleavePk(s,d)notfullycovered.Thereforetheexpandedlinksetbyincludingsuchalinkisavalid~C(Pk+1(s,d)). Iftheaddedlinkonthe(k+1)thshortestpathdoesbelongtoL(Pk(s,d)),itmaycoversomepath(s)inPk(s,d).Takingoutsomelinkfrom~C(Pk(s,d))mayleavePk(s,d)stillfullycovered.Ifnot,theexpandedlinksetisavalid~C(Pk+1(s,d)).2 Theorem6.2actuallydescribesaformalwaytobuildupS(Pk+1(s,d))fromS(Pk(s,d))throughindependentexpansionsoneach~C(Pk(s,d))inS(Pk(s,d)). Theorem6.3.Ifthe(k+1)thshortestpathhasauniquecostintheorderedpathspectrum(asequenceofallpossiblepaths),theminimalcontainercoverexpansionsfromS(Pk(s,d))toS(Pk+1(s,d))onlyhappenon~C(Pk(s,d))thatleadstogenerationofthe(k+1)thshortestpathbyremovingallthelinksin~C(Pk(s,d))fromtheoriginalnetworkG(N,L). Proof(bycontradiction):Assumeanexpansionhappensona~C(Pk(s,d))thatdoesnotleadtogenerationofthe(k+1)thshortestpathbutanotherpathp.Accordingtotheprocessoforderedpathenumerationandthecostuniquenessofthe(k+1)thshortestpath,pathpmusthaveahighercostthanthe(k+1)thshortestpath.Sincebyassumptionthe~C(Pk(s,d))doesnotcoverthe(k+1)thshortestpath(becauseitisexpandedtocoverthe(k+1)thshortestpath),pathpcannotbetheshortestpathinducedby~C(Pk(s,d)),whichcontradictspathp'sdenition.2 Theorem6.3providesinsightinthescopeofminimalcontainercoverexpansionsafteranewpathisenumerated. 122

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Theorem6.4.Allthe~C(Pk(s,d))thatleadtogenerationofashortestpathwithacostgreaterthanthecostofthe(k+1)thshortestpathwillnotbeexpandedwhendevelopingS(Pk+1(s,d)). Proof(bycontradiction):Assumesucha~C(Pk(s,d))getsexpandedwhendevelopingS(Pk+1(s,d)),whichmeansthe~C(Pk(s,d))cannotcoverthe(k+1)thshortestpath.Sincetheshortestpathgeneratedbyremovingallthelinksin~C(Pk(s,d))hasagreatercostthanthe(k+1)thshortestpathhas,thatpathisnotavalidshortestpathassociatedwith~C(Pk(s,d)).Thatisacontradiction.2 Theorem6.4statesthatexpansions,ateachroundofpathenumeration,arewellboundedandhencewastedexpansionsaretherebyavoided. 6.3.3TheOrderedPathEnumerationAlgorithm Basedonthederivationofabovetheoremsandlemmas,weformulatetheorderedpathenumerationalgorithminFigure 6-1 12,whichenumeratesKshortestpathsconnectingsanddinnetworkG(N,L).Theorem6.4canbeappliedinline9ofthealgorithmtoeasetheconditioncheckbecause~C(Pk(s,d))mustcoverpk+1iftheshortestpathcostofthenetworkG(N,Ln~C(Pk(s,d)))ishigherthanthecostofpk+1. 6.3.4ContainerCoverMinimalityDetection Asindicatedinline24inFigure 6-1 ,decisiononminimalityoftheexpandedcontainercover~C(Pk(s,d))[flgisanecessarysteptoguaranteetheminimalityofandtocontrolthegrowthofthecompletecontainercoversetS(Pk+1(s,d)). Anaivewaytoexaminetheminimalityoftheexpandedcontainercover~C(Pk(s,d))[flgisbydenitiontocheckifanyofitssubsetsformedbyremovingonelinkin~C(Pk(s,d))canstillcoverallthepathsinPk(s,d)(thereisnoneedtocheck 1Thesetoperatorninline11referstotherelativecomplement,i.e.,Ln~C(Pk(s,d)),fljl2L,l=2~C(Pk(s,d))g.2Thefunctioncallshortest path()inline11canbesavedbystoringtheshortestpathanditscostinnetworkG(N,Ln~C(Pk(s,d)))thersttimewhen~C(Pk(s,d))isformed. 123

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Algorithm:Ordered Path EnumerationInput:G(N,L),s,d,K//network,source,destination,requirednumberofpathsOutput:paths[K]//orderlyenumeratedpaths 1. procedureOrdered Path Enumeration2. P0(s,d) ,~C(P0(s,d)) 3. L(P0(s,d)) ,S(P0(s,d)) f~C(P0(s,d))g4. (paths[1],paths cost[1]) shortest path(G(N,L),s,d)5. k 06. while(kcost13. next cost cost,next path path14. endif15. else16. foreachlinklonpaths[k+1]17. ifl=2L(Pk(s,d))18. S(Pk+1(s,d)) S(Pk+1(s,d))[f~C(Pk(s,d))[flgg19. (path,cost) shortest path(G(N,Lnflgn~C(Pk(s,d))),s,d)20. ifnext cost>cost21. next cost cost,next path path22. endif23. else24. if~C(Pk(s,d))[flgisaminimalcontainercover25. S(Pk+1(s,d)) S(Pk+1(s,d))[f~C(Pk(s,d))[flgg26. (path,cost) shortest path(G(N,Lnflgn~C(Pk(s,d))),s,d)27. ifnext cost>cost28. next cost cost,next path path29. endif30. endif31. endif32. endfor33. endif34. endfor35. L(Pk+1(s,d)) L(Pk(s,d))[fljl2paths[k+1]g36. k k+137. paths[k+1] next path38. endwhile39. endprocedure Figure6-1. Pseudocodeoftheorderedpathenumerationalgorithm 124

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thecoverageofthe(k+1)thshortestpathbecauseitiscoveredbylinkl).WhenthereissuchasubsetfoundthatisabletocoverallthepathsinPk(s,d),wedecidethat~C(Pk(s,d))[flgisnotminimal.InsteadofcheckingcoverageofallthepathsinPk(s,d),asmarterwayistoonlycheckiflinklcancoverthepathsthatareuniquelycoveredbytheremovedlink(butnotbyanyotherlinkin~C(Pk(s,d))).Thisuniquecoveragerelationneedstobemaintainedthroughthecontainercoverexpansionprocess,whichisnotacomplexoperationandisnotdescribedindetailintheinterestofspace.Therealsoexistheuristicopportunitiesandthedetaileddiscussionisskippedaswell. 6.3.5PotentialAlgorithmicAdvantages Asweshowinalgorithmdescription,theminimalcontainercoverexpansionbasedpathenumerationfeaturesboundedexpansions.Besides,eachtimetheshortestpathcalloperatesonanetworkdifferentfromitslastcall(beforecontainercoverexpansion)onlybyonelinkabsence(suchdifferencehoweverishugeinLawler'smethod).Hence,incrementalimplementationoftheshortestpathcallispotentiallypossible. 6.4ApplicationI:WavelengthUtilizationMinimizationforRWAwithShared-PathProtection Asmentionedatthebeginningofthischapter,oneofthereasonsforwhichk-shortestdisjointroutingmaynotperformwellisthatthenetworkisless-denselyconnectedandhencenotmanydisjointpathscanbedeveloped.Forexample,intheNSFnetworkasshowninFigure 6-2 ,theaveragenumberoflink-disjointpathsdevelopedbythek-shortestdisjointroutingalgorithmoverallpossiblenodepairsis2.32.However,fortheRWAproblemwithshared-pathprotection,asstudiedin[ 21 36 38 41 57 ]anddescribedafterwards,thereshouldbetwolink-disjointpathsforeachowrequest(onefortheworkinglightpathandtheotherforthesparelightpath).Hence,thepoolofcandidateroutesgivenbyk-shortestlink-disjointroutingmaynotprovideenoughroutingchoicesforsolvingtheproblemtogoodoptimality. 125

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Figure6-2. NSFnetwork 6.4.1ProblemDescription GivenanetworkG(N,L)andagroupofowrequests,theproblemistondandassigncontinuouswavelengthstoapairofworkingandsparelightpathsforeachowrequestwithasetofdesignrules.Theobjectiveofthesolutionistominimizethetotalwavelengthsusedthroughoutthenetwork.Thedesignrulesaredescribedasfollows: 1.Theworkingpathanditssparepathforanyowrequestmustbelink-disjoint; 2.Nowavelengthonanylinkcanbesharedneitherbetweentwoworkingpathsnorbetweenaworkingpathandasparepath; 3.Notwosparepathscansharethesamewavelengthonanylinkiftheirworkingpathsjoineachotheranywhereinthenetwork. Shared-pathprotectioncaneffectivelyhelplowertheresourcerequirementfortheprotectivepathsanditguaranteesthatthenetworkcansurviveoveronearbitrarylinkfailure. 6.4.2CandidateRouting Sincethelackofroutingchoicescausedbyapplyingk-shortestdisjointroutingtoaless-denselyconnectednetworkpotentiallydegradestheoptimalityinsolvingtheproblem,moreroutingoptionsneedtobeexploredwiththefollowingthreeconsiderations:(1)thecandidateroutesareexpectedtohaveshortlengthforresourcesaving;(2)thedisjointednessrelationamongthecandidateroutesforaowrequestshouldbehighenoughforbroaderchoicesonselectingdisjointworking/sparepathpairs 126

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amongthem;(3)thenumberofcandidateroutesneedstobemoderatelycontrolledinorderfortheproblemcomplexitytostaycomparabletothek-shortestdisjointroutingbasedmethod. WetreatthepathsinPK(s,d)developedbytheproposedpathenumerationalgorithmasvertices,andthereisanundirectededgebetweentwoverticesifthetwopathsthatthetwoverticesrepresentarelink-disjoint.Wethencallsuchgraphpath)]TJ /F3 11.955 Tf -440.45 -23.91 Td[(disjointednessgraph,denotedbyG(PK(s,d)).Inaddition,wesortalltheverticesbasedontheircorrespondingpathcosts.Accordingtothethreeconsiderationsdescribedabove,ourgoalistondasubgraph(containingacontrollednumberofvertices)inG(PK(s,d))thatisasdenseaspossibleandmeanwhilecontainsverticeswithcostsaslowaspossible. Withrespecttothelateststudiesondensesubgraphs[ 4 5 17 28 ],althoughndingasubgraphofmaximumdensity3withoutsizeconstraintsonitisshowntobeapolynomial-timeproblem[ 28 ],solvingthesize-constraineddensestsubgraphproblemremainsNP-hard[ 5 17 ].Inaddition,thosestudiesdonotconsidervertexcostsinselectingsubgraphs.Weproposeasimpleheuristicschemetobalancethedensityandthecostconsiderationsatthesametime,whichsearchesforasubgraphwithapre-describednumberofverticesinG(PK(s,d)). Theideaistostartthesearchfromthelowest-costvertex.Fromthereeachtimewetrytondapaththatisofthelowestcostamongalltheunselectedpathsthatarelink-disjointwiththepathselectedatthelastround.Ifthereisnosuchpathavailable,thenthesearchbacktracksreturningtothemostrecentvertexalongthesearchtreeandstartsagain.Themeritofthismethodisthatthesearchisalwaystryingtondlow-costpathsandatthesametimekeepsacertainlevelofpathdisjointedness.Figure 6-3 3ThedensityofasubgraphonvertexsetSisdenedasd(S),jE(S)j=jSj,whereE(S)isthesetofedgesinthesubgraphinducedbyverticesinS. 127

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showsthepseudocodeofthecandidateroutingscheme,fromwhichwecanseethealgorithmisindeedadepth-rst-search(DFS)traversaloverapre-describednumberofverticesinavertex-cost-sortedgraph. Algorithm:Candidate Routes SearchInput:PK(s,d),G(PK(s,d)),M//p1,...,pKaresortedbycostOutput:RM(s,d)//candidateroutesetofcardinalityM 1. procedureCandidate Routes Search2. RM(s,d) fp1g3. setallelementsoflast visit[K]toUNVISITED4. i 2,last visit[1] DEADEND5. while(iM)//searchforithcandidatepath6. j indexofthelatestincludedpathinRM(s,d)7. found NO8. while(j6=DEADEND)9. for(kfrom1toK)10. if((pj,pkarelink-disjoint)and(last visit[k]==UNVISITED))11. RM(s,d) RM(s,d)[fpkg//foundanewpath12. found YES,last visit[k] j//recordthesearchtree13. break14. endif15. endfor16. if(found==YES)17. break18. else19. j last visit[j]//backtrack20. endif21. endwhile22. i i+123. endwhile24. endprocedure Figure6-3. Pseudocodeofthecandidateroutingscheme 6.4.3ProblemFormulations Weprovidethreeintegerlinearprogramming(ILP)formulationsfortheshared-path-protection-basedRWAproblem.Therstoneprovidesanoriginalformulationthatleavesthechoicesonroutingandwavelengthassignmentfullyopen.Thesecondonelimitstheroutingoptionstothepathsdevelopedbythek-shortest 128

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link-disjointroutingalgorithm.Thethirdoneappliescandidaterouting,asdescribedabove,tocreateroutingoptions. 6.4.3.1NotationsUsedinThreeFormulations Constants: N:NumberofnodesinnetworkG(N,L) L:NumberofdirectedlinksinnetworkG(N,L) F:Numberofowrequests W:Numberofprovidedwavelengthsoneachlink Rf:Numberofcandidateroutesprovidedtoowf,whichdependsontheroutingschemeused Indices: i:Nodeindextakingintegersfrom0toN)]TJ /F6 11.955 Tf 11.95 0 Td[(1 l:Unidirectionallinkindextakingintegersfrom0toL)]TJ /F6 11.955 Tf 11.96 0 Td[(1 f:Flowindextakingintegersfrom0toF)]TJ /F6 11.955 Tf 11.95 0 Td[(1 r:Routeindextakingintegerfrom0toRf)]TJ /F6 11.955 Tf 11.95 0 Td[(1 w:Wavelengthindextakingintegersfrom0toW)]TJ /F6 11.955 Tf 11.95 0 Td[(1 Sets: Rl:Setofallcandidateroutesofallowspassingthroughtheunidirectionallinkl Lini:Setofunidirectionallinksterminatingatnodei Louti:Setofunidirectionallinksstartingfromnodei Cfr:Setofconictingroutesinowf'scandidateroutingsetthatarenotlink-disjointwithrouterofowf Decisionvariables(integer): xfwl,yfwl:Representwhetherowfroutesitsworkingandsparepaths,respectively,throughwavelengthwonlinklbytakingon1or0 129

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ufwr,vfwr:Representwhethertheworkingandsparepaths,respectively,ofowftakerouterthroughwavelengthwbytakingon1or0 Auxiliaryvariables(integer): fw1,fw2:Representwhethertheworkingandsparepathsofowf,respectively,takeonwavelengthwbytakingon1or0 ewl:Representswhetherwavelengthwonlinklisassigned(toeitheraworkingorasparelightpath)bytakingon1or0 Ew:Representswhetherwavelengthwisassignedanywhereinthenetworkbytakingon1or0 6.4.3.2TheOriginalFormulation Model: .minXwEw (6) subjecttofollowingconstraints: 1.Working/sparepathvalidityandwavelengthcontinuity .Xl12Linixfwl1)]TJ /F19 11.955 Tf 16.4 11.35 Td[(Xl22Loutixfwl2=8>>>><>>>>:)]TJ /F10 11.955 Tf 9.3 0 Td[(fw1,ifs(f)=i+fw1,ifd(f)=i0,O.W. (6)Xl12Liniyfwl1)]TJ /F19 11.955 Tf 16.4 11.36 Td[(Xl22Loutiyfwl2=8>>>><>>>>:)]TJ /F10 11.955 Tf 9.3 0 Td[(fw1,ifs(f)=i+fw1,ifd(f)=i0,O.W. (6)Xwfw1=1 (6)Xwfw2=1 (6) 130

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wheres(f)andd(f)in( 6 )and( 6 )refertothesourceanddestinationnodesofowf. 2.Working/sparepathlinkdisjointedness .Xwxfwl+Xwyfwl1 (6) 3.Conictavoidancebetweenpaths a.Resourceconictavoidanceoneachwavelengthlink:Xfxfwl+yfwl1 (6) b.Spare/spareresourceconictavoidanceonthesamewavelengthlink:Xwxf1wl1+Xwxf2wl1+yf1wl2+yf2wl23f16=f2andl16=l2 (6) Equation( 6 )indicatesthat,onanywavelengthlink,eitheronlyoneworkingpathormultiplesparepathsareallowedtobeestablished,whichenforcesdesignrule2describedinSection 6.4.1 .Equation( 6 )makesdesignrule3holdinwhichtwosparepathscannottakethesamewavelengthlink(thenyf1wl2+yf2wl2=2)ifthecorrespondingtwoworkingpathsjoineachother(thenPwxf1wl1+Pwxf2wl1=2). 4.Wavelengthlinkoccupationaccounting .xfwlewl (6)yfwlewl (6)ewlEw (6) 131

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6.4.3.3k-ShortestDisjointRoutingBasedFormulation Model: .minXwEw (6) subjecttofollowingconstraints: 1.Pathvalidityandwavelengthcontinuity(automaticallysatisedsinceallpathsdevelopedbyk-shortestdisjointroutingarevalidpaths) 2.Working/sparepathlinkdisjointedness .Xwufwr+Xwvfwr1 (6)XrXwufwr=1 (6)XrXwvfwr=1 (6) 3.Conictavoidancebetweenpaths a.Resourceconictavoidanceoneachwavelengthlink:X(f,r)2Rlufwr+vfwr1,for(f,r)2Rl (6) b.Spare/spareresourceconictavoidanceonthesamewavelengthlink:Xwuf1wr1+Xwuf2wr2+vf1wr3+vf2wr43,f16=f2,r16=r3,andr26=r4,(f1,r1)2Rl1,(f2,r2)2Rl1,(f1,r3)2Rl2,and(f2,r4)2Rl2,l16=l2. (6) 132

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4.Wavelengthlinkoccupationaccounting .ufwrewl,for(f,r)2Rl (6)vfwrewl,for(f,r)2Rl (6)ewlEw (6) Itshouldbenoticedthat,inaboveformulation,routeindexrrangesoverallpossibleroutesdevelopedbyk-shortestdisjointrouting,anditcanvaryfromowtoowbecauseofdifferentlocalconnectivityamongnodepairsinthenetwork. 6.4.3.4CandidateRoutingBasedFormulation Thecandidateroutingbasedformulationisessentiallyverysimilartothek-shortestdisjointroutingbasedformulationexceptthattherangeoftherouteindexrnowdependsonthecandidateroutingset,andthedisjointednessconstraintformulatedin( 6 )isreplacedby Xwufwr+Xwvfwr+X(f,r)2CfrXwvfwr1. (6) TheadditionaltermP(f,r)2CfrPwvfwraddedtotheconstraintexpressionistoavoidselectingtheworkingandsparepathsthatarenotlink-disjoint,whichisnotneededinthek-shortestdisjointroutingbasedformulationbecauseallroutesdevelopedtherearelink-disjoint. 6.4.3.5FormulationComparison WecompareabovethreeformulationsontheNSFnetworkinthefollowingaspects:problemsize,routeprocessingtime,andaveragecandidateroutedisjointedness.Forthek-shortestdisjointroutingbasedformulation,allthelink-disjointroutesdevelopedfor 133

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eachowareincludedforroutingoptions.Forthecandidateroutingbasedformulation,thecandidateroutingsetsofsizes3and4,respectively,areconsideredforeachow. Table6-1. Problemsizecomparisonamongformulations:numberofvariables Numberofowrequests1020304050 Originalformulation819015990237903159039390k-shortestdisjointroutingbased85013301790225027103-candidateroutingbased99015902190279033904-candidateroutingbased11901990279035904390 Table6-2. Problemsizecomparisonamongformulations:numberofconstraints Numberofowrequests1020304050 Originalformulation12803805372380122763802199238034520380k-shortestdisjointrouting300380171442023703349063-candidateroutingbased4690131802503041310584904-candidateroutingbased97203504070840123020185990 Table6-3. Routeprocessingtimecomparison(insecond,runningonaWindowsmachinewitha3GHzprocessor) Numberofowrequests1020304050 Originalformulation0(noroutesearchrequired)k-shortestdisjointroutingbased<0.001<0.001<0.001<0.001<0.0013-candidateroutingbased0.0160.0470.0780.0940.1254-candidateroutingbased0.0160.0470.0780.0940.125 TheproblemsizecomparisonischaracterizedinTables 6-1 and 6-2 ,whichrecordthenumbersofvariablesandconstraintsgeneratedfordifferentproblemformulationswhen10wavelengthsareavailableoneachlink.Aswecanobserve,theoriginalformulationleadstoahugeproblemsizeevenforasmallproblem,whichindeedprohibitsmodernILPsolvers4fromobtainingavalidsolution.However,thesizesofthosecandidatepathbasedformulationsaremoderatelycontrolled. TherouteprocessingtimeiscomparedinTable 6-3 ,whichisdenedasthetimespentongeneratingroutesfordifferentformulations.Forthecandidateroutingbased 4ModernILPsolverscan,dependingontheproblemstructure,solveaproblemwithuptoafewtensofthousandsvariables/constraintsingeneral. 134

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Table6-4. Averagecandidateroutedisjointednesscomparison(averagedoverows) Numberofowrequests1020304050 k-shortestdisjointroutingbased1.601.701.671.651.64 3-candidateroutingbased2.302.352.332.302.28 4-candidateroutingbased3.904.003.973.983.90 formulation,thistimeincludesenumeratingallpossiblepathsforeachowontheNSFnetworkandrunningthecandidateroutingalgorithm(listedinFigure 6-3 ).Itisobservedthat,althoughthecandidateroutingbasedformulationrequireshigherrouteprocessingtime,thetimecostisonaveragestillverylow. Table 6-4 showsthepotentialbenetofapplyingcandidateroutingbycomparingaveragecandidateroutedisjointedness,whichisdenedastheaveragenumberoflink-disjointpathpairsinacandidaterouting(ork-shortestdisjointrouting)setoverallowrequests.Thisquantityindicatestheexibilityinchoosingawork/sparepathpairwhensolvingtheRWAproblem.AsobservedfromTable 6-4 ,candidateroutingleadstomuchhigherexibilityinworking/sparepathsselection,whichcanbeleveragedforbettersolutionoptimality. 6.4.4NumericalResults WetesttheaboveILPformulationsandevaluatetheirperformancebysendingthemtoanILPsolverMOSEK,alarge-scalemixed-integerlinearprogramsolverapplyingacombinationoftheinteriorpoint,branchandcuttechnologies[ 1 ].Figures 6-4 and 6-5 showsthebestresultsafterrunningMOSEKondifferentformulationsfor8hours.Wecanclearlyobservetheperformanceadvantageofcandidateroutingoverk-shortestdisjointroutingwhenthenumberofowrequestsandthesizeofthecandidateroutingsetincrease.Onaverage,theperformanceimprovementreaches8.25%and14.68%,respectively,for3-candidateroutingand4-candidateroutingovertheprobleminstanceswithownumbersfrom20to50. 135

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Figure6-4. Solutionoptimalitycomparisonbetweenk-shortestdisjointroutingand3-candidateroutingafterrunningMOSEKfor8hours Figure6-5. Solutionoptimalitycomparisonbetweenk-shortestdisjointroutingand4-candidateroutingafterrunningMOSEKfor8hours 6.5ApplicationII:TopologicalOptimizationforShared-PathProtectionRWA Anotherreasonforpotentialunsatisfactoryperformanceofk-shortestdisjointroutingisthattheroutingsetdevelopedmaynotbestttheproblemnatureaswediscussinthissection. 6.5.1ProblemDescription Theproblemissimilartotheproblemdiscussedinthelastsectioninthesensethatalltheowrequestshavetobeaddressedbyallocatingworking/sparepathpairsandallthedesignrulesdescribedinSection 6.4.1 musthold.Thedifferenceintopologicaloptimizationisthatthereisnoxedtopologyandtheobjectiveistondtheleast-cost 136

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topologythatcanaccommodatealltheowrequests.Thetopologicalcostisdenedasthesumofthecostsofallbidirectionallinksthatareallocatedtoowrequests.Thesolutiontothisproblemcanbepotentiallyusedasatopologicalchoiceofinitialnetworkdeployment,oritcanalsobeusedforgeneratinganewlogictopologytoadaptthetrafcoveranewphysicalnetworkcondition,aswediscussinthelastchapter. 6.5.2CandidateRouting Thenatureoftheproblemindicates,inordertouselesslinkresources(correspondingtohavingalower-costtopology),thewavelengthsoneachassignedlinkshouldbeasfullyutilizedaspossibleandthereforethelinkresourcescanbeefcientlysharedamongtheows.Suchindicationimpliesthatthecandidateroutesofowsareexpectedtohavehighpotentialtooverlap.However,thek-shortestdisjointroutingalgorithmdevelopsroutingsetsforeachowindependentlyandnooverlappotentialamongtheroutingsetsisconsidered. Werstdenebidirectionallinkpotentialbasb,FXf=1KfXk=1Ifkb, (6) whereIfkbindicateswhetherthekthenumeratedpathofowfpassesthroughbidirectionallinkbbytakingon1or0,Fisthenumberofowrequests,andKfisthenumberofenumeratedpathsforowf.Hence,bbecomesafrequencycounterindicatingtheoverlappotentialofallpossiblepathsonlinkb. Wethendenepathpotentialfkforenumeratedpathsasfk,(Xb2pfkbcb)=(Xb2pfkcb), (6) wherecbisthecostofbidirectionallinkbandpfkisthekthenumeratedpath(asetofbidirectionallinks)ofowf.fkessentiallycanbetreatedasascore,averagedoverthelinksthatpathpfkpassesthrough,indicatingthepath'soverlappotentialwithotherpaths. 137

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Thecandidateroutingforthetopologicaloptimizationproblemtakessimilarconsiderationsasfortheproblemdiscussedinthelastsectionexceptthat,insteadofchoosingshort-lengthpaths,theselectedcandidateroutesareexpectedtohavehighpathpotentials.ThenthecandidateroutingselectionschemedescribedinFigure 6-3 canstillapplytothetopologicaloptimizationproblemwiththeonlychangethattheverticesinthepath)]TJ /F3 11.955 Tf 12.68 0 Td[(disjointednessgraphG(PK(s,d))aresortedbasedontheircorrespondingpathpotentials. 6.5.3ProblemFormulations Thethreetypesofformulationscorrespondingtotheonesinthelastsectionareshownafterintroducingseveralnewnotations. 6.5.3.1Notations Onlythenotationsthatuniquelyapplytothetopologicaloptimizationproblemarelisted.Therestofnotationscanbereferredtointhelastsection. Indices: l:Unidirectionallinkindexrangingfrom0toN(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1))]TJ /F6 11.955 Tf 11.96 0 Td[(1 b:Bidirectionallinkindexrangingfrom0toN(N)]TJ /F6 11.955 Tf 11.96 0 Td[(1)=2)]TJ /F6 11.955 Tf 11.96 0 Td[(1 Sets: Rb:Setofallcandidateroutesofallowspassingthroughbidirectionallinkb Data: cb:Costofbidirectionallinkb Auxiliaryvariables(integer): zb:Indicateswhetherbidirectionallinkbisallocatedbytakingon1or0 6.5.3.2TheOriginalFormulation Model: .minXbcbzb (6) 138

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subjecttothefollowingconstraints1-4: Constraints1,2,and3arethesameasthoselistedfortheoriginalformula-tioninthelastsection.Constraint4isasfollows 4.Bidirectionallinkutilizationaccounting .xfwlzb,b=bl=2c (6)yfwlzb,b=bl=2c (6) 6.5.3.3k-ShortestDisjointRoutingBasedFormulation Model: .minXbcbzb (6) subjecttothefollowingconstraints1-4: Constraints1,2,and3arethesameasthoselistedforthek-shortestdisjointroutingbasedformulationinthelastsection.Constraint4isasfollows 4.Bidirectionallinkutilizationaccounting .ufwrzb,for(f,r)2Rb (6)vfwrzb,for(f,r)2Rb (6) Itshouldbenoticedthatthereisnoharshlimitonthecardinalityofk-shortestdisjointroutingsetsbecausethetopologynowisopenfortheproblemdiscussedinthissection. 6.5.3.4CandidateRoutingBasedFormulation Thecandidateroutingbasedformulationforthetopologicaloptimizationproblemisverysimilartotheabovek-shortestdisjointroutingbasedformulationwiththesameexceptionasdescribedinthecorrespondingplaceofSection 6.4.3.4 139

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6.5.3.5FormulationComparison Sincethehardlimitonthecardinalityofk-shortestdisjointroutingsetsisremoved,wecanapplythesamenumberofcandidateroutestoboththek-shortestdisjointroutingbasedandthecandidateroutingbasedformulations.Therefore,theresultingproblemsizeisveryclosefortheabovetwoformulationswhiletheproblemsizeoftheoriginalformulationisstilloverwhelmingforanymodernILPsolver.TherouteprocessingtimeforallthecandidateroutingbasedILPinstancesthattaketherst100enumeratedpathsintocandidateroutingselectionisbelow5secondsonanetworkdescribedinthenextsubsection.Intheinterestofspace,thosecomparisondetailsarenotlisted. 6.5.4NumericalResults TheperformanceoftheILPformulationsisshowninFigures 6-6 and 6-7 ,wherethebesttopologicalcostsforthevarioustrafcloadsamong16USmajorcities(asshowninFigure 5-11 )areillustrated.Thereare10wavelengthsavailableoneachlinkandthelinkcostisassumedproportionaltothedistancebetweencities.Wecanobservethat(1)increaseofthenumberofcandidaterouteshelpsachievingbetteroptimality(aswecompareFigures 6-6 and 6-7 ),and(2)thecandidateroutingbasedformulationbringsdownthetopologicalcostinducedbythek-shortestdisjointroutingbasedformulation,onaverage,by14.14%and14.92%forthe4-candidateroutingand5-candidateroutingrespectively. Althoughtheapparentbenetofk-shortestdisjointrouting(algorithmiceasinessandlowlinkutilization)traditionallymakesitaneasyoptionfordevelopingcandidateroutes,thelackofawarenessofthespecicproblems'naturemayleadthesolutionstodeviatefromtheirglobaloptimality.Hence,in-depthexplorationofpossibleroutingchoicesforthebestttotheproblemisneeded.Inthischapter,werstformallyproposeanewcontainer-cover-expansion-basedpathenumerationalgorithmandthendevelopacandidateroutingschemewithspecialconsiderationontheproblem 140

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Figure6-6. Solutionperformancecomparisonbetweenk-shortestdisjointroutingand4-candidateroutingafterrunningMOSEKfor8hours Figure6-7. Solutionperformancecomparisonbetweenk-shortestdisjointroutingand5-candidateroutingafterrunningMOSEKfor8hours nature.Thenumericalresultsshowthatagreatperformancebenetcanbeobtainedbyapplyingsuchmethodstotwoshared-path-protection-basedRWAproblems. 141

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CHAPTER7CONCLUSIONSANDFUTUREWORK 7.1Conclusions Thisdissertationcarriesoutanin-depthstudyonfault-tolerantall-opticalcommunicationnetworksinvolvingmanyaspectsoftheresource-allocation-relatedproblems,whichincluderouting,wavelengthassignment,andtopologyoptimization.Westartwithafault-tolerantroutingandwavelengthassignmentproblemundertheNNtorusstructure,inwhichweproposeanoptimalnon-overlappinglightpathsetupalgorithm(FOLD)toestablish4link-disjointlightpathsforallsource-destinationpairs.Thedevelopmentofanefcientwavelengthassignmentandreuse(WAR)schemefollowswhichefcaciouslytslightpathstogetherintoalownumberofwavelengths.Theefciencyoftheschemeisveriedbyshowingaverysmallperformancegapbetweentheschemeandthelowerboundsolutioninwhichwavelengthconversionisapplied.Intheend,wevalidatetheproposedfault-tolerance-enhancedtorusarchitecturewithrespecttotheconnectionreliabilityandunder-failurethroughputdegradationviaextensivesimulationresults.Theobservationfromthenumericalcomparisonsstatesthat,viaapplyingtheproposedfault-tolerantarchitecture, thenetworkcantolerateupto3arbitrarycriticallinkfailureswithoutlossofanyconnection, theconnectionreliabilityisimprovedbyanorderofmorethan104undertheregularfailureprobability(102)inthe44torus, andtheaveragenetworkthroughputcansustainoveramuchlargernumberoffailureswithoutevidentdegradationthantheregularnetworkarchitecturewithoutfault-tolerancedesignenabled. Inordertoreducethewavelengthutilizationwhilekeepingthefault-tolerancefeature,variedsparesharingschemesaredevelopedandappliedtothetorus-basedarchitecture.ThetradeoffbetweenthecapacityoffaulttoleranceandresourceutilizationisdemonstratedviaaMonteCarlosamplingbasedsimulationovera44torus. 142

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Theotherexaminedclassictopologyforfault-tolerancestudyisthecirculantgraphs,inwhichanynumberofcommunicationnodesandarbitraryconnectivitycanbesupported.InlightofMenger'stheorem,wedevelopanode-disjointroutingalgorithmforallsource-destinationpairsinthecirculantgraphsoftheformCN(f1,...,Wg),inwhichalltheroutesareanalyticallycalculatedwithrespecttothenodeindices.Wealsoprovideathoroughresourceutilizationanalysisandderiveaprobabilisticmodeltocapturebothnodeandlinkfailures'effecttotheconnectionreliability.Anumericalanalysisbasedona16-nodecirculantgraphshows: thelinkresourceutilizationdoesnotvarymuchoverdifferentsource-destinationpairlocationsforthesamedegreeofnetworkconnectivity, andtheconnectionreliabilityincreasealmostlinearlyinthelogarithmicscalewiththeincreaseofthedegreeofnetworkconnectivity,showingthevastfaulttolerancepotentialofthecirculantgraphs. Besidesdevelopingfault-tolerantroutingandwavelengthassignmentalgorithmbasedontheclassictopologies,thestudyisalsogiventotheproblemthattargetstoconstructordene,upondisasteroccurringcertainpartofexistingnetworkinfrastructure,anewfault-toleranttopologywiththelowestcost.Wenamethistypeofproblemsasthetopologicaloptimizationoradaptationproblems,whichareanticipatedtobecomputationallyNP-hard.TheproblemisformulatedintotwoformsofILPswithdifferentcharacterizinggranularity.Weshowthelackofoptimalityofthetraditionalsolvingmethodinwhichroutingandwavelengthassignmentaretreatedastwoindependentsubproblemsbycomparingwithagreedyapproach.Finally,basedonthestudyoftheproblemnature,weproposeatwo-phaseheuristicsfortheproblemandthenumericalresultsshowavasttopologicalcostimprovementfromthegreedyapproach. 7.2FutureWork Alongtheprocessofthefault-tolerancestudywithndingsonmanyresource-allocation-relatedproblems,thisdissertationisawarethattherestillexist 143

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manyfacetsofresearchworthyoffurtherinvestigationandthefocusesofthenext-stageeffortmayowintofollowingdirections: Inthetorus-basednon-overlappinglightpathssetupalgorithm,thecontrollerdisjointednessisnotexplicitlyemphasizedalthoughonlyCasesIIIandIVcancauselightpathstojoinattwocontrollers.Inthefuture,eitheraclearanalyticalanalysisonthefault-toleranceperformancedifferencebetweentheproposedFOLDandacontroller-disjointedness-enforceddesignoracontroller-disjointedness-orientedRWAoptimizationcouldbestudied. Intheprobabilisticstudyofthetorus-basedfault-tolerantarchitecture,auniformlydistributedtrafcpatternandfailurepatternacrossthetorusareassumed.However,thetrafcpatternmaynotbedistributedinthiswayandmayevenvarywithtime.Thefailureprobabilitydistributionmaydependonthespecicstructuraloroperationalvulnerabilityoftheavionicsystemsontheaircrafts.Hence,atrafc-pattern-awareorfailure-distribution-awareroutingandresourceallocationadaptationstudymayneedfurtherstudyinorderforthetorusarchitecturetobeexaminedinmorepracticalaspects. Inthefault-tolerantroutingstudyonthecirculantgraphs,weassumethattheoffsetAisrestrictedtotakeasetofcontinuousintegersf1,...,Wg.However,thisrestrictionmaynotleadtheproposedroutingalgorithmtooptimalityinlinkresourceutilizationorconnectionreliability.FuturestudymaybededicatedtooptimizingtheintegeroffsetAinCN(A)withthegoalofmaximizingconnectionreliabilityorminimizingresourceutilizationbyrelaxingtheelementsofAtotakeonanyintegersin[1,bN=2c]. Inthetopologicaloptimizationproblem,weanticipateitscomplexitytobeNP-hardviabeingawarethatmanyofitssubproblemsareNP-complete.However,thatisnotarigorousstatementwithoutastrictproof.Properwell-knownNP-completeproblemsneedtobeidentiedforthetopologicaloptimizationproblemtobereducibletoandthentheNP-hardstatementaboutthetopologicaloptimizationproblemcanstrictlyhold. Weprovideapreliminaryanalysisontheapproximationofthegreedyapproachtothetopologicaloptimizationproblem.However,theanalysisworksonlyonworkingpathsallocationandtheresultingratioisnottightenoughtoshowtheperformanceofthegreedyapproach.Theremaystillexistspaceofanalysistofurthertightentheratioandweexpecttoincludebothworkingandbackuppathsallocationintotheanalysis. Weformallyproposeanewordered-path-enumerationalgorithm.However,acomprehensiveanalysisonitsalgorithmefciencyisnotfullyconducted.Inthefuture,weplan(1)toanalyzethealgorithmcomplexitybycomparingitindetailwiththewell-knownpathenumerationmethodssuchasYen'sandLawler's 144

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algorithms,and(2)toutilizethepathenumerationtoidentifypotentialmultipleoptimalk-shortestdisjointroutingsolutionsthatthecurrentk-shortestdisjointroutingalgorithmcannot. 145

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APPENDIXAOPTIMALITYPROOFOFTHEPROPOSEDNON-OVERLAPPINGLIGHTPATHSSETUPALGORITHM(FOLD) Webasetheoptimalityproofoftheproposednon-overlappinglightpathssetupalgorithm(FOLD)onthek-shortestlink-disjointpathsalgorithm[ 10 ][ 54 ][ 22 ].Herekisequalto4withrespecttoFOLD. Ingeneral,thegreedyalgorithmdescribedinFigure 2-2 doesnotnecessarilyleadtoanoptimalsolution,asshowninFigure A-1 .Apairoflink-disjointpaths(inorangeandblue)generatedviathegreedyalgorithmtakes3+6=9links(asshowninFigure A-1 (A)),whichisgreaterthanthenumberoflinksthattheoptimalsolution(asshowninFigure A-1 (B))takes. AGreedysolution BOptimalsolution FigureA-1. Non-optimalitydemonstrationforagreedydisjointroutingsolution Theoptimalk-shortestlink-disjointpathsalgorithmisbasedonpathaugmentationanditguaranteesreachingoptimalityviandingoneofoptimalsolutionsiftherearemany.Thealgorithmndsoptimalkshortestlink-disjointpathsbyaugmentingoptimalk)]TJ /F6 11.955 Tf 12.36 0 Td[(1shortestlink-disjointpaths.Take2-shortestlink-disjointpathsasanexample,asshowninFig.22,toillustratehowthealgorithmworks. Step1:FindtheshortestpathP1fromthesourcetothedestination,asshowninFigure A-2 (A); Step2:ReplaceP1with)]TJ /F3 11.955 Tf 9.3 0 Td[(P1(reversethelinkdirectionandchangethelinkweightfrom+1to)]TJ /F6 11.955 Tf 9.29 0 Td[(1),asshowninFigure A-2 (B); Step3:FindashortestpathP2fromthesourcetothedestinationinthemodiedgraph,asshowninFigure A-2 (C); 146

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A B C D FigureA-2. Pathaugmentationbased2-shortestdisjointrouting Step4:TaketheunionofP1andP2,removefromtheunionthelinksetthatconsistsoftheP1linkswhosereversedlinksappearinP2,thengrouptheremaininglinksintotwolink-disjointpathsP01andP02,whichareanoptimal2-shortestlink-disjointpaths,asshowninFigure A-2 (D). WecalltheshortestpathgeneratedinStep3theaugmentingpath.Basedontheoptimal2-shortestlink-disjointsolutionjustobtained,wecanextendaboveprocedurestoobtainanoptimal3-shortestlink-disjointsolutionbyreplacingP01andP02with)]TJ /F3 11.955 Tf 9.3 0 Td[(P01and)]TJ /F3 11.955 Tf 9.3 0 Td[(P02andfollowingsimilarstepstoSteps3and4.Then,anyoptimalk-shortestlink-disjointpathscanbeobtainedbasedonoptimal(k)]TJ /F6 11.955 Tf 11.95 0 Td[(1)-shortestlink-disjointpaths. Fromabovedescriptionoftheoptimalk-shortestlink-disjointrouting,weobservethatthegreedyalgorithmwillperformthesameastheoptimalalgorithmiftheshortestpathgeneratedinStep3doesnotoverlapanypathsintheoptimallink-disjointpathssetgeneratedafterthepreviousiteration. Hence,ifeachgreedypathdevelopedinSectionIIIcansatisfytheabovecondition,wecanconcludethattheproposedlightpathssetupalgorithm(FOLD)cangenerateoptimalsolutionsforanyS-Dpositionalrelation. 147

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FigureA-3. Progressiveandregressivelinks Tofacilitatetheproofoftheaboveclaim,were-examineallthelinksinthetoruswithrespecttoadestination(D),asshowninFigure A-3 ,intermsoftheirfacilitationinroutingtowardsthedestination.Thelinksinorangeindicatethoselinksbytakingwhichtheroutegetsclosetothedestination(wecallthemprogressivelinks).Thelinksinblueindicatethoselinksbytakingwhichtheroutegetsawayfromthedestination(wecallthemregressivelinks). RegardingX-Yrouting,sincetherstandsecondshortestpathsareboththeshortestpathsthroughoutthenetwork,theyareoptimal2-shortestdisjointpaths.Thenwendoptimal3-shortestdisjointpathsviaaugmentingthe2-shortestdisjointpathsbyreversingthe2pathsandassociatingeachreversedlinkwithweight)]TJ /F6 11.955 Tf 9.3 0 Td[(1.Itcanbeobservedthatallreversedlinksonthe2pathsareregressivelinksforallcases(I,II,IIIandIV)ofS-Dpositionalrelationship.Althoughbytakinganyofthoselinksaroutecangain)]TJ /F6 11.955 Tf 9.3 0 Td[(1weightbenet,atleastoneturn-backhophastobepaidfortheroutetoreachthedestination.Hence,takingthosereversedlinksleadstonobenetinroutingashortestpathasinStep3duringpathaugmentation.Therefore,theaugmentingpathdoesnotneedtotakeonanylinksonthe2-shortestpathsandtherst3greedypathsgeneratedbyFOLDareoptimal.Forthefourthgreedypaths,thediscussionhastobe 148

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basedoncases.ForCasesII,IIIandIV,itcanbeobservedthatallreversedlinksonthethirdshortestpathareregressivelinksbutpossiblythelinkincidenttothesourceisnot.Accordingtotheloop-freepropertyofthepathaugmentationalgorithm[15],theaugmentingpathwillnotretakeanylinksincludingthelinkincidenttothesource.Therefore,theaugmentingpathforgenerating4-shortestpathsdoesnotneedtotakeonanylinksonthe3-shortestpaths,andallthefourgreedypathsareoptimal.ForCaseI,itcanbeobservedthatallreversedlinksonthethirdshortestpathareeitherregressiveorincidenttothesourceexceptonehorizontallinkattheuprightcornerofthethirdgreedypath(asshowninFigure 2-4 (A)).WelabelthetailcontrollerasD3andtheheadcontrollerasS3incidenttothatlink.However,thebenetoftakingthislinkisoverwhelmedbythecostofroutingfromthesourcetoD3followedbyroutingfromS3tothedestinationafterremovaloflinksonthreeoptimalshortestpaths.Hence,theaugmentingpathcanalsoberoutedindependentlyofthethreeoptimalpaths.Accordingly,thefourgreedypathsproposedforallX-Yroutingcases(I,II,IIIandIV)areoptimal. RegardingXroutingandYrouting,sincethepathssetupofYroutingisexactlymirroredfromthatofXrouting,weonlyneedtoconsidertheoptimalityofXroutingpathssetup.Sinceallreversedlinksoftherstshortestpathareregressive,theaugmentingpathcanberoutedindependentlywithouttakinganylinksoftherstshortestpath.Ifthetwomirroringpathsarethesecondandthirdshortestpaths,sinceallthereversedlinksonthemareregressivebutthetwolinksincidenttothesource,forthesamereasonmentioned,theaugmentingpathscanberoutedindependentlywhendevelopingthe3-and4-shortestlink-disjointpaths.Ifthecomplementarypathisthesecondshortestpath,itcanbeobservedthatallreversedlinksonitareregressive.Then,theaugmentingpathcanalsoberoutedindependently.Therefore,thefourgreedypathsareoptimallink-disjointpathsforbothXandYrouting. 149

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CombiningthediscussiononX-Y,XandYrouting,itcanbeconcludedthattheproposednon-overlappinglightpathssetupreachesoptimality. 150

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APPENDIXBDERIVATIONOFLSEXPRESSIONS ThecalculationsofLSarederivedasfollows. While2N5,onlyroutingcasesII,II'andIIappearineachdestinationgroup NisoddLS=N)]TJ /F11 5.978 Tf 5.75 0 Td[(1 2XdX=1[N+2(dX+2)]+N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdY=1[N+2(dY+2)]+2N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdX=1N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdY=12(N+dX+dY)=1 2(3N3)]TJ /F6 11.955 Tf 11.96 0 Td[(2N2+7N)]TJ /F6 11.955 Tf 11.95 0 Td[(8) (B) NisevenLS=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdX=1[N+2(dX+2)]+[N+2(N=2+2)]=2+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdY=1[N+2(dY+2)]+[N+2(N=2+2)]=2+2N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdX=1N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdY=12(N+dX+dY)+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdX=12(N+dX+N=2)+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdY=12(N+dY+N=2)+(N+N=2+N=2)=1 2(3N3)]TJ /F6 11.955 Tf 11.95 0 Td[(2N2+8N)]TJ /F6 11.955 Tf 11.96 0 Td[(8) (B) While6N9,casesI,IIIandIVareincludedadditionallyinthedestinationgroup 151

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NisoddLS=N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdX=1[N+2(dX+2)]+N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdY=1[N+2(dY+2)]+28<:N)]TJ /F11 5.978 Tf 5.75 0 Td[(5 2XdX=1N)]TJ /F11 5.978 Tf 5.76 0 Td[(5 2XdY=14(dX+dY+2)+N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdX=N)]TJ /F11 5.978 Tf 5.75 0 Td[(3 2N)]TJ /F11 5.978 Tf 5.76 0 Td[(5 2XdY=1[N+2(2dY+dX+2)]+N)]TJ /F11 5.978 Tf 5.75 0 Td[(5 2XdX=1N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdY=N)]TJ /F11 5.978 Tf 5.75 0 Td[(3 2[N+2(2dX+dY+2)]+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdX=N)]TJ /F11 5.978 Tf 5.76 0 Td[(3 2N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdY=N)]TJ /F11 5.978 Tf 5.76 0 Td[(3 22(N+dX+dY)9=;=1 2(2N3+9N2)]TJ /F6 11.955 Tf 11.96 0 Td[(28N+17) (B) NisevenLS=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdX=1[N+2(dX+2)]+[N+2(N=2+2)]=2+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdY=1[N+2(dY+2)]+[N+2(N=2+2)]=2+28<:N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdX=1N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(2XdY=14(dX+dY+2)+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdX=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(2XdY=1[N+2(2dY+dX+2)]+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdX=1N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdY=N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1[N+2(2dX+dY+2)]+N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdX=N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdY=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(12(N+dX+dY)9=;+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdY=1[N+2(2dY+N=2+2)]+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdX=1[N+2(2dX+N=2+2)]+2[N+N=2+(N=2)]TJ /F6 11.955 Tf 11.96 0 Td[(1)]+2[N+(N=2)]TJ /F6 11.955 Tf 11.95 0 Td[(1)+N=2]+(N+N=2+N=2)=1 2(2N3+9N2)]TJ /F6 11.955 Tf 11.95 0 Td[(26N+16) (B) WhileN10,casesI'andIjoininthedestinationgroup 152

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NisoddLS=N)]TJ /F11 5.978 Tf 5.76 0 Td[(9 2XdX=14(dX+3)+N)]TJ /F11 5.978 Tf 5.75 0 Td[(1 2XdX=N)]TJ /F11 5.978 Tf 5.76 0 Td[(7 2[N+2(dX+2)]+N)]TJ /F11 5.978 Tf 5.76 0 Td[(9 2XdY=14(dY+3)+N)]TJ /F11 5.978 Tf 5.75 0 Td[(1 2XdY=N)]TJ /F11 5.978 Tf 5.76 0 Td[(7 2[N+2(dY+2)]+28<:N)]TJ /F11 5.978 Tf 5.75 0 Td[(5 2XdX=1N)]TJ /F11 5.978 Tf 5.76 0 Td[(5 2XdY=14(dX+dY+2)+N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdX=N)]TJ /F11 5.978 Tf 5.75 0 Td[(3 2N)]TJ /F11 5.978 Tf 5.76 0 Td[(5 2XdY=1[N+2(2dY+dX+2)]+N)]TJ /F11 5.978 Tf 5.75 0 Td[(5 2XdX=1N)]TJ /F11 5.978 Tf 5.76 0 Td[(1 2XdY=N)]TJ /F11 5.978 Tf 5.75 0 Td[(3 2[N+2(2dX+dY+2)]+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdX=N)]TJ /F11 5.978 Tf 5.76 0 Td[(3 2N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdY=N)]TJ /F11 5.978 Tf 5.76 0 Td[(3 22(N+dX+dY)9=;=N3+4N2)]TJ /F6 11.955 Tf 11.95 0 Td[(5N)]TJ /F6 11.955 Tf 11.95 0 Td[(32 (B) NisevenLS=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(4XdX=14(dX+3)+N 2XdX=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(3[N+2(dX+2)]+1 2[N+2(N=2+2)]+N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(4XdY=14(dY+3)+N 2XdY=N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(3[N+2(dY+2)]+1 2[N+2(N=2+2)]+28<:N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdX=1N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(2XdY=14(dX+dY+2)+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdX=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(2XdY=1[N+2(2dY+dX+2)]+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdX=1N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdY=N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1[N+2(2dX+dY+2)]+N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(1XdX=N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(1XdY=N 2)]TJ /F5 7.97 Tf 6.58 0 Td[(12(N+dX+dY)9=;+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdY=1[N+2(2dY+N=2+2)]+N 2)]TJ /F5 7.97 Tf 6.59 0 Td[(2XdX=1[N+2(2dX+N=2+2)]+2[N+N=2+(N=2)]TJ /F6 11.955 Tf 11.96 0 Td[(1)]+2[N+(N=2)]TJ /F6 11.955 Tf 11.95 0 Td[(1)+N=2]+(N+N=2+N=2)=N3+4N2)]TJ /F6 11.955 Tf 11.96 0 Td[(4N)]TJ /F6 11.955 Tf 11.96 0 Td[(32 (B) 153

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BIOGRAPHICALSKETCH DexiangWangreceivedhisPh.D.fromtheUniversityofFloridainthefallof2011.DuringhisPhDstudyintheDepartmentofElectricalandComputerEngineeringattheUniversityofFlorida,hewasamemberoftheWirelessandMobileSystemsLaboratorydirectedbyDr.JaniseMcNair.HereceivedhisB.E.degreein1999andM.E.degreein2002fromHuazhongUniversityofScience&Technology(Wuhan,China),bothinMaterialScience&Engineering.BeforestartinghisPhDresearchattheUniversityofFloridain2006,heworkedforthetelecommunicationR&DteamofHuaweiTechnologiesonSDHopticalnetworkswitchingequipments.Afterthat,heworkedasasoftwareengineerfortheR&DteamofUTStarcomTelecomonWCDMA-RNCproducts.HehasbeenconductinghisPhDresearchintheareasoffault-tolerantall-opticalcommunicationsystems,robustmultimediacommunicationoverheterogeneousnetworks,greeninternet,wirelessnetworkthroughputoptimizationviatransmissionpowercontrol,andenergy-efcientcognitiveradiosensornetworks. 160