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Optimization Techniques in Communication Networks

Permanent Link: http://ufdc.ufl.edu/UFE0022444/00001

Material Information

Title: Optimization Techniques in Communication Networks
Physical Description: 1 online resource (142 p.)
Language: english
Creator: Zheng, Xiaoying
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: control, multicast, network, optimization, p2p, scheduling, wireless
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: 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: Convex optimization techniques have found important applications in communications and signal processing. Recently, there has been a surge in research activities that apply the latest development in convex optimization to the design and analysis of communication systems. This research focuses on how to apply optimization techniques on both massive content distribution in high-speed internet core, and cross-layer design in wireless ad hod networks. One of the important trends is that the Internet will be used to transfer content on more and more massive scale. Collaborative distribution techniques such as swarming and parallel download have been invented and effectively applied to end-user file-sharing or media-streaming applications, but mostly for improving end-user performance objectives. We consider the issues that arise from applying these techniques to content distribution networks for improving network objectives, such as reducing network congestion. In particular, we formulate the problem of how to make many-to-many assignment from the sending nodes to the receivers and allocate bandwidth for every connection, subject to the node capacity and receiving rate constraints. The objective is to minimize the worst link congestion over the network, which is equivalent to maximizing the distribution throughput, or minimizing the distribution time. The optimization framework allows us to jointly consider server load balancing, network congestion control, as well as the requirement of the receivers. We develop a special, diagonally-scaled gradient projection algorithm, which has a faster convergence speed, and hence, better scalability with respect to the network size than a standard subgradient algorithm. We provide both a synchronous algorithm and a more practical asynchronous algorithm. However, on the negative side, swarming traffic has made capacity shortage in the backbone networks a genuine possibility, which will be more serious with fiber-based access. The second problem addressed is how to conduct massive content distribution efficiently in the future network environment where the capacity limitation can equally be at the core or the edge. We propose a novel peer-to-peer technique as a main content transport mechanism to achieve efficient network resource utilization. The technique uses multiple trees for distributing different file pieces, which at the heart is a version of swarming. In this chapter, we formulate an optimization problem for determining an optimal set of distribution trees as well as the rate of distribution on each tree under bandwidth limitation at arbitrary places in the network. The optimal solution can be found by a distributed algorithm. The results of the chapter not only provide stand-alone solutions to the massive content distribution problem, but should also help the understanding of existing distribution techniques such as BitTorrent or FastReplica. The joint optimal design of congestion control and wireless MAC-layer scheduling problem in multi-hop wireless networks has become a very active research area in the last few years. We solve this problem using a column generation approach with imperfect scheduling. We point out that the general subgradient algorithm has difficulty in recovering the time-share variables and experiences slower convergence. We first propose a two-timescale algorithm that can recover the optimal time-share values. Most existing algorithms have a component, called global scheduling, which is usually NP-hard. We apply imperfect scheduling and prove that if the imperfect scheduling achieves an approximation ratio rho, then our algorithm converges to a sub-optimum of the overall problem with the same approximation ratio. By combining the idea of column generation and the two-timescale algorithm, we derive a family of algorithms that allow us to reduce the number of times the global scheduling is needed.
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 Xiaoying Zheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Xia, Ye.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022444:00001

Permanent Link: http://ufdc.ufl.edu/UFE0022444/00001

Material Information

Title: Optimization Techniques in Communication Networks
Physical Description: 1 online resource (142 p.)
Language: english
Creator: Zheng, Xiaoying
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2008

Subjects

Subjects / Keywords: control, multicast, network, optimization, p2p, scheduling, wireless
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: 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: Convex optimization techniques have found important applications in communications and signal processing. Recently, there has been a surge in research activities that apply the latest development in convex optimization to the design and analysis of communication systems. This research focuses on how to apply optimization techniques on both massive content distribution in high-speed internet core, and cross-layer design in wireless ad hod networks. One of the important trends is that the Internet will be used to transfer content on more and more massive scale. Collaborative distribution techniques such as swarming and parallel download have been invented and effectively applied to end-user file-sharing or media-streaming applications, but mostly for improving end-user performance objectives. We consider the issues that arise from applying these techniques to content distribution networks for improving network objectives, such as reducing network congestion. In particular, we formulate the problem of how to make many-to-many assignment from the sending nodes to the receivers and allocate bandwidth for every connection, subject to the node capacity and receiving rate constraints. The objective is to minimize the worst link congestion over the network, which is equivalent to maximizing the distribution throughput, or minimizing the distribution time. The optimization framework allows us to jointly consider server load balancing, network congestion control, as well as the requirement of the receivers. We develop a special, diagonally-scaled gradient projection algorithm, which has a faster convergence speed, and hence, better scalability with respect to the network size than a standard subgradient algorithm. We provide both a synchronous algorithm and a more practical asynchronous algorithm. However, on the negative side, swarming traffic has made capacity shortage in the backbone networks a genuine possibility, which will be more serious with fiber-based access. The second problem addressed is how to conduct massive content distribution efficiently in the future network environment where the capacity limitation can equally be at the core or the edge. We propose a novel peer-to-peer technique as a main content transport mechanism to achieve efficient network resource utilization. The technique uses multiple trees for distributing different file pieces, which at the heart is a version of swarming. In this chapter, we formulate an optimization problem for determining an optimal set of distribution trees as well as the rate of distribution on each tree under bandwidth limitation at arbitrary places in the network. The optimal solution can be found by a distributed algorithm. The results of the chapter not only provide stand-alone solutions to the massive content distribution problem, but should also help the understanding of existing distribution techniques such as BitTorrent or FastReplica. The joint optimal design of congestion control and wireless MAC-layer scheduling problem in multi-hop wireless networks has become a very active research area in the last few years. We solve this problem using a column generation approach with imperfect scheduling. We point out that the general subgradient algorithm has difficulty in recovering the time-share variables and experiences slower convergence. We first propose a two-timescale algorithm that can recover the optimal time-share values. Most existing algorithms have a component, called global scheduling, which is usually NP-hard. We apply imperfect scheduling and prove that if the imperfect scheduling achieves an approximation ratio rho, then our algorithm converges to a sub-optimum of the overall problem with the same approximation ratio. By combining the idea of column generation and the two-timescale algorithm, we derive a family of algorithms that allow us to reduce the number of times the global scheduling is needed.
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 Xiaoying Zheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2008.
Local: Adviser: Xia, Ye.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2008
System ID: UFE0022444:00001


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andtomyparents. 3

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Iwouldliketothankmyadvisor,ProfessorYeXia,forhisconstantguidance,support,patience,andencouragement.Iamprivilegedtohavesuchawonderfuladvisor,whoisatalltimesenthusiastic,optimistic,patient,helpful,andencouraging.Hegavemecountlessadviceandinsightduringthecourseofmywork,withoutwhichcompletingthisdissertationwouldhavebeenmuchmoredifcult.Iwouldliketoexpressmyappreciationtomycommittee,ProfessorArunavaBanerjee,P.OscarBoykin,ShigangChenandRandyChow.ThankstoProfessorYuguangFang,whohasbeenworkingwithmeonthecross-layerdesignproject.Iamthankfultotheofce-matesandfriendsthatImetattheUniversityofFlorida,forsupportingmeovertheyearsandformakingmylifeatUFenjoyable.SpecialthankstoChunglaeChoandFengChen,whohavebeenworkingwithmeonthecontentdistributionandcross-layerdesignprojects,respectively.Iwouldliketothankmyparentswhohavealwayslovedmeandencouragedmeinmylife.Thanksalsogotomyparents-in-law,mybrotherandhiswifefortheirsupport.Finally,Iamparticularlygratefultomydarlinghusband,JunqiZhao.WehavebeengeographicallyseparatedbythePacicOceanthroughmyentirePh.D.life.Inthepastveyears,hehasbeenspendingalmosteverymorningandnightaccompanyingmebytheWebcam,whichmakesmeemotionallystrongandgivesmecouragetomakemyPh.D.dreamreal.Withouthisendlessloveandsupport,thisdissertationwouldnotbepossible. 4

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page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 8 LISTOFFIGURES ....................................... 9 ABSTRACT ........................................... 11 CHAPTER 1INTRODUCTION .................................... 13 1.1ResearchMotivation ................................ 13 1.2WhyOptimizationFramework? ........................... 14 1.3ResearchOverview ................................. 17 2OPTIMIZINGNETWORKOBJECTIVESINCOLLABORATIVECONTENTDIS-TRIBUTION ....................................... 19 2.1Introduction ..................................... 19 2.2ProblemFormulationandMotivation ........................ 23 2.2.1OptimalFlowAssignmentProblem ..................... 23 2.2.2Motivation:PerformanceGain ....................... 26 2.3SubgradientMethod ................................. 27 2.4BarrierMethodwithGradientProjection ...................... 31 2.4.1NonlinearApproximationoftheMin-CongestionProblemwithBarrier .. 31 2.4.2GradientProjectionAlgorithm ....................... 34 2.4.3AnalysisofConvergence .......................... 37 2.5DiagonallyScaledAlgorithm ............................ 40 2.5.1Ill-ConditionedProblem ........................... 41 2.5.2DiagonallyScaledGradientProjectionAlgorithm ............. 41 2.5.3AsynchronousAlgorithm .......................... 42 2.6PerformanceEvaluation ............................... 46 2.6.1GradientProjectionvs.SubgradientAlgorithm .............. 46 2.6.2Scaledvs.UnscaledAlgorithm ....................... 47 2.7AdditionalRelatedWork .............................. 48 2.8Conclusion ..................................... 50 2.9ProofofConvergenceResultsfortheSynchronousAlgorithm(Algorithm1) ... 52 3OPTIMALPEER-TO-PEERTECHNIQUEFORMASSIVECONTENTDISTRI-BUTION ......................................... 61 3.1Introduction ..................................... 61 3.2ProblemDescription ................................ 64 3.2.1OptimalMulticastTreePacking ...................... 64 5

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....................... 67 3.2.3OptimallyAllocatedOverlayBandwidth .................. 68 3.3DistributedAlgorithm:DiagonallyScaledGradientProjection .......... 69 3.3.1FixedOverlayLinkBandwidth ....................... 69 3.3.2OptimallyAllocatedOverlayBandwidth .................. 77 3.3.3ConvergenceResults ............................ 78 3.4ColumnGenerationMethod ............................ 81 3.4.1IntroductionofColumnGenerationMethod ................ 81 3.4.2ApplytheGradientProjectionAlgorithmtotheRestrictedProblem .... 82 3.4.3GapbetweentheMasterProblemandtheRestrictedProblem ....... 82 3.4.4IntroduceOneMoreColumn(Tree) .................... 84 3.4.5SummaryoftheAlgorithm ......................... 84 3.4.6ConvergenceResult ............................. 85 3.5PracticalConsiderations .............................. 86 3.5.1OverlappingContent ............................ 86 3.5.2MixedArchitectureofFixedandAllocatedBandwidth .......... 86 3.5.3NetworkDynamicsandChurn ....................... 87 3.5.4ScalabilityandHierarchicalPartitionofSessions ............. 88 3.5.5AsynchronousAlgorithm .......................... 88 3.6PerformanceEvaluation ............................... 88 3.6.1PerformanceEvaluationMetrics ...................... 89 3.6.2BottleneckattheAccessLinks(Proles1to4) .............. 90 3.6.3BottleneckattheInternalofISPBackbone(Prole5) ........... 92 3.6.4BottleneckattheCross-ISPLinks(Prole6-7) ............... 94 3.6.5IntroduceTreesatVaryingDegreeofFrequency(Prole5) ........ 96 3.6.6ArrivalandDepartureDynamics(Prole8) ................ 97 3.7AdditionalRelatedWork .............................. 97 3.8Conclusion ..................................... 98 4ACLASSOFCROSS-LAYEROPTIMIZATIONALGORITHMSFORPERFOR-MANCEANDCOMPLEXITYTRADE-OFFSINWIRELESSNETWORKS ..... 105 4.1Introduction ..................................... 105 4.2ProblemDescription ................................ 107 4.2.1NetworkModel ............................... 108 4.2.2DualofMasterProblem ........................... 110 4.3Two-TimescaleAlgorithm ............................. 110 4.3.1SolveProblemMP-AwiththeSubgradientMethod ............ 111 4.3.2UpdateTimeFractiononaSlowerTimescale ............... 112 4.3.3SummaryoftheTwo-TimescaleAlgorithm ................. 117 4.4ColumnGenerationMethodWithImperfectGlobalScheduling .......... 118 4.4.1ColumnGenerationMethod ......................... 118 4.4.2ApplytheTwo-TimescaleAlgorithmtotheRMP ............. 119 4.4.3BoundingtheGapbetweentheMPandtheqth-RMP ........... 120 4.4.4IntroduceOneMoreExtremePoint(ColumnorSchedule) ......... 121 6

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............ 122 4.5PerformanceEvaluation ............................... 126 4.6Conclusions ..................................... 130 REFERENCES ......................................... 134 BIOGRAPHICALSKETCH .................................. 142 7

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Table page 3-1Distributionofbandwidthallocatedfordifferenttrees .................. 99 3-2BitTorrentsimulationparameters ............................. 99 3-3Comparisonofdownloadingtime(minutes)andnumberofactivetrees ......... 99 3-4Downloadingtime(minutes)comparisonofProle6and7 100 3-5Performancecomparisonofthefamilyofalgorithms(Prole5) ............. 100 4-1Performancecomparisonofthefamilyofalgorithms(largenetwork) .......... 131 8

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Figure page 2-1Server-clientdependency ................................. 57 2-2Ratioofworst-caselinkutilization:randomvs.optimalowassignment ........ 57 2-3ParalleldownloadunderTCP-Renocongestioncontrolvs.optimalowassignment .. 58 2-4Networkthatleadstoanill-conditionedproblem ..................... 58 2-5Convergenceofthesubgradientalgorithm ........................ 58 2-6Flowrateconvergenceofthesubgradientalgorithm ................... 59 2-7Convergenceofgradientprojectionalgorithm ...................... 59 2-8Convergenceofdiagonallyscaledgradientprojectionalgorithm ............. 59 2-9Poorperformanceofsubgradientandunscaledgradientprojectionalgorithms ..... 60 2-10Diagonallyscaledgradientprojectionmethod ...................... 60 3-1Node1sendstheletonode2and3 ........................... 101 3-2AllpossibledistributiontreesfortheexampleinFig. 3-1 ................ 101 3-3Performancecomparison(Prole1) ........................... 101 3-4Performancecomparison(Prole4) ........................... 102 3-5Structuresoftreesontheoverlaynetwork ........................ 102 3-6Performancecomparison(Prole5) ........................... 102 3-7Convergenceofthroughput(Prole5) .......................... 103 3-8Performancecomparison(Prole6) ........................... 103 3-9Performancecomparison(Prole7) ........................... 103 3-10Convergenceofthefamilyofalgorithms(Prole5) ................... 104 3-11Dynamicdepartureandarrivalofreceivers ........................ 104 4-1Smallnetworktopology .................................. 131 4-2Convergenceinconnectionrates(smallnetwork) .................... 132 4-3Convergenceinlinkowrates(smallnetwork) ..................... 132 4-4Convergenceinconnectionrates(largenetwork) ..................... 133 9

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................ 133 4-6BoundsfortheoptimalobjectivevalueoftheMP .................... 133 10

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Convexoptimizationtechniqueshavefoundimportantapplicationsincommunicationsandsignalprocessing.Recently,therehasbeenasurgeinresearchactivitiesthatapplythelatestdevelopmentinconvexoptimizationtothedesignandanalysisofcommunicationsystems.Thisresearchfocusesonhowtoapplyoptimizationtechniquesonbothmassivecontentdistributioninhigh-speedinternetcore,andcross-layerdesigninwirelessadhodnetworks. OneoftheimportanttrendsisthattheInternetwillbeusedtotransfercontentonmoreandmoremassivescale.Collaborativedistributiontechniquessuchasswarmingandparalleldownloadhavebeeninventedandeffectivelyappliedtoend-userle-sharingormedia-streamingapplications,butmostlyforimprovingend-userperformanceobjectives.Weconsidertheissuesthatarisefromapplyingthesetechniquestocontentdistributionnetworksforimprovingnetworkobjectives,suchasreducingnetworkcongestion.Inparticular,weformulatetheproblemofhowtomakemany-to-manyassignmentfromthesendingnodestothereceiversandallocatebandwidthforeveryconnection,subjecttothenodecapacityandreceivingrateconstraints.Theobjectiveistominimizetheworstlinkcongestionoverthenetwork,whichisequivalenttomaximizingthedistributionthroughput,orminimizingthedistributiontime.Theoptimizationframeworkallowsustojointlyconsiderserverloadbalancing,networkcongestioncontrol,aswellastherequirementofthereceivers.Wedevelopaspecial,diagonally-scaledgradientprojectionalgorithm,whichhasafasterconvergencespeed,andhence,betterscalability 11

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However,onthenegativeside,swarmingtrafchasmadecapacityshortageintheback-bonenetworksagenuinepossibility,whichwillbemoreseriouswithber-basedaccess.Thesecondproblemaddressedishowtoconductmassivecontentdistributionefcientlyinthefuturenetworkenvironmentwherethecapacitylimitationcanequallybeatthecoreortheedge.Weproposeanovelpeer-to-peertechniqueasamaincontenttransportmechanismtoachieveef-cientnetworkresourceutilization.Thetechniqueusesmultipletreesfordistributingdifferentlepieces,whichattheheartisaversionofswarming.Inthischapter,weformulateanoptimizationproblemfordetermininganoptimalsetofdistributiontreesaswellastherateofdistributiononeachtreeunderbandwidthlimitationatarbitraryplacesinthenetwork.Theoptimalsolutioncanbefoundbyadistributedalgorithm.Theresultsofthechapternotonlyprovidestand-alonesolutionstothemassivecontentdistributionproblem,butshouldalsohelptheunderstandingofexistingdistributiontechniquessuchasBitTorrentorFastReplica. ThejointoptimaldesignofcongestioncontrolandwirelessMAC-layerschedulingprobleminmulti-hopwirelessnetworkshasbecomeaveryactiveresearchareainthelastfewyears.Wesolvethisproblemusingacolumngenerationapproachwithimperfectscheduling.Wepointoutthatthegeneralsubgradientalgorithmhasdifcultyinrecoveringthetime-sharevariablesandexperiencesslowerconvergence.Werstproposeatwo-timescalealgorithmthatcanrecovertheoptimaltime-sharevalues.Mostexistingalgorithmshaveacomponent,calledglobalscheduling,whichisusuallyNP-hard.Weapplyimperfectschedulingandprovethatiftheimperfectschedulingachievesanapproximationratio,thenouralgorithmconvergestoasub-optimumoftheoverallproblemwiththesameapproximationratio.Bycombiningtheideaofcolumngenerationandthetwo-timescalealgorithm,wederiveafamilyofalgorithmsthatallowustoreducethenumberoftimestheglobalschedulingisneeded. 12

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1 ],Kazaa[ 2 ]andGnutella[ 3 ],andvariouscontentdistributionnetworks.Anoverlay(includingP2P)networkislayeredontopofaphysicalnetwork,suchastheInternet,andreliesonthelatterforreachability.Thenodesofsuchanetworkareusuallyendsystemsrunninguser-levelapplications.Thisgivestheoverlaynetworktheexibilitytoformdifferenttopologiesandtheabilitytopoolresources,suchasstoragecapacity,computationalpowerandcommunicationbandwidth.Inanefforttoachievenewcapabilities,improvedperformance,andhigherreliability,researchersareoftenwillingtochallengethetraditionalnetworkingprinciplesandpractices. Onenotableattemptisthatmulti-pathorparalleltransmissionhasbeenproposedforvir-tuallyallthesenetworks,transcendingtheconventionofsingle-pathroutingintheInternet.Forinstance,inthecontent-distributionapplication,eachleorpiecesofthelecanbereplicatedatmanynodesinthenetwork.Auserwhowishestoretrievethelecanestablishconnectionswithmultipleserversandschedulethedownloadinparallel(See[ 4 5 ]forexamples.).Thissetupcanoftendramaticallyspeedupthedownloadingprocess,enhancetheperformance,andimprovethereliability. Anevenmoreaggressivecontentdistributionapproachistodoitbypackingmultiplemulticasttrees.Bymulticast,thecontentissimultaneouslydeliveredtoalltheusersovereach 13

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6 ].Thisfruitfulwayofviewingthemany-to-manyorcollaborativeapproachnotonlyachievesefcientnetworkresourceutilizationbutalsohelpstheunderstandingtheexistingdistributiontechniquessuchasBitTorrentorFastReplica. Thestudyofthecontentdistributionproblemhasledourresearchtothefarmorecompli-catedwirelessenvironment.Unlikethewirednetworkcase,thecapacitiesofwirelesslinksarenotconstant,butdependonmanyfactors,suchaspower,transmissioninterferenceandambientnoise,mediumcontentionandresolution(byscheduling),thestateofthereceivers(receivernum-bers,locations,andmobilityetc.),andrandomenvironmentalfactors.Conceptually,thestudyofthecontrolalgorithmsshouldbethejointdesignovertheowrates,andthepowerallocationandschedulingstrategies.Suchnetworkproblemiscurrentlyaverypopularthemeinnetworkresearch[ 7 19 ]. Themethodologicalframeworkforallthethreeproposedproblemsisthetheoryandalgorithmsofnetworkoptimization.Theunifyingtreatmentofthethreeproblemswithinthecommontheoreticalandalgorithmicframeworkisadistinguishingfeatureofthisresearch.Inaddition,thenetworkoptimizationtheoryandalgorithmsdevelopedintheresearchisapplicabletoalargenumberofsimilarproblemswithinorbeyondthisdissertation. 20 21 ]. Theoptimizationtechniqueshavebeenappliedtosolvenetworkowproblemasearlyasin[ 22 27 ].Inthesework,thetargetproblemisalinearlyconstrainedoptimizationproblemofthe 14

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23 ].TheFrank-Wolfetype[ 28 ]gradientprojectionmethodsonlyutilizetherstderivativeoftheobjectivefunction,whichhavebeenextensivelyappliedtomulticommoditynetworkowproblems[ 29 ],buttheconvergencerateofthesemethodsissublinearandthereforetooslowforapplications[ 23 ].Bertsekasetal.proposetousethesecondderivativeoftheobjectivefunctioninthegradientprojectionmethods,whichapproximatethewell-knowNewton'smethodandoftenresultinimprovedspeedofconvergenceinthenetworkowproblems[ 23 24 30 ]. ThevalidityofapplyingthetheoryandalgorithmsofoptimizationtonetworkproblemhasbeenmostconvincinglyestablishedbythesuccessfullineofresearchinoptimalowcontrolstartedbyKelly[ 31 ]andLow[ 32 ]andfollowedbymanyotherworks[ 33 40 ].Therstpapersrecognizethat,ratherthanvaguelyunderstandingow/congestioncontrolasmechanismstopreventreceiverornetworkcongestion,onecanclearlydescribethecontrolobjectiveandndthebestsolutionbyoptimizationorcontrolalgorithms,insteadofnon-optimal,heuristicsolutions.Furthermore,onecanbroadenthegoalasmaximizingallusers'utilitiessubjecttothelinkcapacityconstraints(hence,thereisnonetworkcongestion.).LetSdenotethesetofusers,xsbeuserss'sowrate,aconcavefunctionUsbeitsutilityfunction,andletS(e)bethesetofuser 15

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Thelinkalgorithmhasthestraightforwardinterpretationthatthelinkpriceincreaseswhentheaggregateratecrossingthelinkexceedsitscapacity,anddecreasesotherwise.Itisalocalalgorithm,requiringonlylocalinformation.Thesourcealgorithmisdistributedbutrequirestheend-to-endpathprice.Thisisnotadifcultysincenetworkowcontrolprotocolroutinelypassescontrolmessagesonthepath,whicharereturnedbacktothesourceafteraroundtrip.WeseethatastandarddistributedoptimizationalgorithmbecomesthemainingredientsofaTCP-stylenetworkcongestioncontrolalgorithm.Fordifferentutilityfunctions,therewillbedifferentcongestioncontrolalgorithms.Withtheoptimizationtheory,onecanalsoprovethealgorithminfactworks(converges),andshowhowfastthealgorithmconvergestotheoptimalsolution.Therangeofvalidparameterscanalsobeidentiedandlaterturnedforperformanceimprovementwhenthealgorithmsisinoperation.Manyalternativedistributedalgorithmshavealsobeenexploredforreasonssuchasimprovedconvergencespeed,lowerimplementationcomplexity,easeofdeployment,andsecondaryobjectivessuchasreducingthenetworkqueuesizes. 16

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34 35 41 45 ]towirelessadhocnetworks[ 7 19 ]. 2 andchapter 3 Inchapter 2 ,thecontentdistributioniscarriedbyend-to-endunicasttrafc.Themainproblemishowtoselectasubsetoftheserversforeachclientanddecidethetransmissionratefromeachselectedservertotheclient,sothattheclientsgettheirrequiredbandwidth,theserversarenotoverloadedandtheworst-caselinkcongestioninthenetworkisminimized.Werstdevelopasubgradientalgorithm,whichworksonthedualproblemandisthemostfrequentlyusedalgorithminthenetworkingliterature.However,sincethisproblemisoftenill-conditioned,ourexperiencehasshownthatthesubgradientalgorithmconvergesslowly.Wethendevelopaspecialdiagonallyscaledgradientprojectionalgorithmoperatingontheprimalproblem,whichtriestoemulatethefasterNewton'smethodandovercomesthisdifculty. Inchapter 3 ,thecontentdistributionismoreaggressiveandiscarriedbyallpossiblemulti-castdistributiontrees.Thedifcultyisthatthenumberofallpossibledistributiontreesincreasesexponentiallyasthenetworkincreases.Fortunately,bythegradientprojectionalgorithm,weavoidenumeratingallpossibletreesandareabletoidentifyandusetheoptimaldistributiontrees,andatthesametime,allocatecorrectbandwidthontheselectedtrees.Furthermore,weintroducethecolumngenerationmethod,whichintroducesonemoretreeatatimeandgraduallyexpandsthetreeset.Bycombiningtheideaofcolumngenerationandthegradientprojectionalgorithm, 17

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Motivatedbythecolumngenerationapproach,inchapter 4 ,wesolvetheproblemofajointoptimaldesignofcongestioncontrolandwirelessMAC-layerschedulingusingacolumngener-ationapproachwithimperfectscheduling.Wepointoutthatthegeneralsubgradientalgorithmhasdifcultyinrecoveringthetime-sharevariablesandexperiencesslowerconvergence.Werstproposeatwo-timescalealgorithmthatcanrecovertheoptimaltime-sharevalues.Mostexistingalgorithmshaveacomponent,calledglobalscheduling,whichisusuallyNP-hard.Weapplyimperfectschedulingandprovethatiftheimperfectschedulingachievesanapproximationratio,thenouralgorithmconvergestoasub-optimumoftheoverallproblemwiththesameapproximationratio. 18

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46 ]. Animportantnetworkingproblemaddressedbythischapterishowtoconductmassivecontentdistributionefcientlyinthefuturenetworkenvironmentwherethecapacitylimitationcanequallybeatthecoreortheedge.Ourworkhasapplicationsinthefollowingtwo-stepcontentdistributionprocess,whichisprevalenttodayandisexpectedtobecomemoreimportantinthefutureforreducingwide-areanetworktrafc.Intherststep,thecontentisdistributedoverinfrastructurenetworks,suchascontentdistributionnetworks,ISPnetworksorIPTVnetworks.Inthesecondstep,theendusersretrievethecontentfromoneormorenearbycontentservers.Ineitherstepbutparticularlytherst,collaborativedistributiontechniquesarebecomingveryattractive.Bycollaborativedistribution,wemeandifferentnodesinadistributionsessionhelpeachothertospeedupthedistributionorimproveotherperformance 19

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1 ].Althoughswarmingwasoriginallyinventedinend-systemle-sharingapplications,itisreallyafundamentaldistributiontechniquethatcanbeemployedbytheoperatorsofcontentdistributionnetworks. Thischapterdescribeshowtoimprovecollaborativedistributiontechniquesforachievingnetworkobjectivesincontentdistributionnetworks.Sincemostofthesetechniquesweredesignedfortheend-userenvironment,targetingend-userperformanceobjectives,theyneedconsiderablemodicationandimprovementbeforetheycanbeappliedtocontentdistributionnetworksandachieveimportantnetworkperformanceobjectives,suchaslownetworkcongestionorhighthroughput.Inparticular,onecommonassumptionofthecurrentend-usersystemsisthatthenetworkisaccess-limited.Asaresult,theydonothavebuilt-incongestioncontrolorbandwidthallocationmechanismsthatcancoordinatetheentiredistributionsessionandefcientlycopewithinternalnetworkcongestion.Instead,theyeitherrelyonthedefaultTCPcongestioncontrol,workingindependentlyoneachindividualconnection,ordonothaveanycongestioncontrolatallifUDPisused.Aswillbedemonstratedinthechapter,theyeithercauseunnecessarilyheavynetworkcongestionatpartsofthenetwork(duetopoorlybalancednetworkload),ormisstheopportunitytoachieveashorterdistributiontime(orequivalently,ahigherthroughput)giventhesamenetworkcongestionlevel.Theperformancegapbetweenwhatthesesystemscanaccomplishandthebestpossiblecanbeverywide. Thischapterproposesaschemethatmakescoordinatedbandwidthassignmentamongdifferentconnectionsinthesamedistributionsessionsoastominimizetheworst-casenetworkcongestion,orequivalently,maximizethedistributionthroughput.Thecoordinationisachievedthroughfullydistributedalgorithms.Foreaseofdiscussion,wecallthereceivingnodesthe 20

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2-1 asanexample.)Ourproblemistoselectasubsetoftheserversforeachclientanddecidethetransmissionratefromeachselectedservertotheclient,sothattheclientsgettheirrequiredbandwidth(e.g.,forstreamingrequirement),theserversarenotoverloadedandtheworst-caselinkcongestioninthenetworkisminimized. OurproblemformulationandsolutionfollowthenetworkoptimizationapproachintroducedbyKellyetal.[ 31 ]andLowetal.[ 32 ].Theproblemcontainsafractionalserver-selectionproblem.Forinstance,aclientcanget1=3ofitsdownloadfromoneserverand2=3fromanotherserver.Ifaconnectionisassignedazeroornearzerobandwidth,theclientessentiallyhasnotselectedthecorrespondingserver.Oursolutiontotheoptimizationproblemleadstodistributedalgorithmsthatcombineserverassignmentwithcongestioncontrol(orequivalently,bandwidthallocation). Ourcontributionsareasfollows.Theresultsfromthischapterwillbeusefulforcontentdistributionnetworks,ISPsandIPTVdistributionnetworks.Uptoordersofmagnitudeimprove-mentinthroughput(orreductionincongestion)ispossiblewithourscheme.Withrespecttonetworkoptimization,oursolutionisaspecialgradientprojectionalgorithmoperatingontheprimalproblem,insteadofthesubgradientalgorithm,whichworksonthedualproblem.For 4 ].Withsourcecoding,thelechunksarecodedandareceivercanreconstructtheentireleaslongasitreceivesasufcientnumberofchunks,irrespectiveoftheidentityofthechunks.Eachservermaycontainanarbitrarycollectionofcodedchunks.Thenodesexchangecodedchunkswithouttheneedofknowingwhattheyare. 21

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47 ].Withthegradientpro-jectionalgorithm,weareabletoovercomethisdifcultywithdiagonalscaling,whichtriestoemulatethefasterNewton'salgorithm.Forimprovedpracticality,wehavealsodevelopedanasynchronousversionofthealgorithm.Thecorrectness(i.e.,convergence)ofallversionsofthealgorithmshasbeenproven.Wealsogiveimportantresultsontheconvergencespeed. Thechapterisorganizedasfollows.Intheremainingpartoftheintroduction,wesummarizethecommonnotationusedthroughoutthechapter.InSection 2.2 ,weintroducealinearopti-mizationmodelanddiscussitsperformanceadvantagecomparedwithrandomserverassignmenttogetherwithUDPorTCP.Thisservestofurthermotivateouroptimization-basedproblemformulation.InSection 2.3 ,wedescribeafullydistributedsubgradientalgorithmforthelinearproblem.InSection 2.4 ,weapproximatethelinearproblemusingabarrier-functionapproach.Wethendevelopagradientprojectionalgorithm.Next,weextendthegradientprojectional-gorithmtoadiagonally-scaledversion.Finally,weprovideanasynchronousversionofthealgorithm.InSection 2.6 ,wepresentexperimentalresultstocomparetheperformanceofthesubgradientalgorithmandthegradientprojectionalgorithmwithoutandwithscaling.InSection 2.7 ,additionalrelatedworkisreviewed.TheconclusionisinSection 2.8 LetSVbethesetofserversandCVbethesetofclients.SandCmayoverlap.Letpijbetheallowedpath,forinstance,theshortestpath,fromserveritoclientj.Weregard 22

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Theowrate(i.e.,trafcrate)fromserveritoclientjisdenotedbyxij,orxpforp=pij2P,interchangeably.Letye=Ppij3exijdenotetheowratethroughlinkeforanye2E.Theutilizationoflinke,ameasureoflinkcongestion,isdenotedbye,ande=ye=ce.Letzi=Pj2Cxijdenotethetotalsendingrateofserveriforanyi2S. Inournotation,allvectorsarecolumnvectors.LetjjxjjdenotetheusualEuclideannormofavectorx,andh;idenotetheusualvectorinnerproduct.Foranyvectorx,wedenoteby[x]+thepositivepartofx(i.e.,thevectorobtainedbyreplacingeachnegativecomponentofxwithzero).ForanymatrixE,wedenoteitsinducednormbyjjEjj,whichisequaltomaxjjxjj=1jjExjj,andwedenoteby[E]ijtheentryonrowiandcolumnj.Letdiag[1;2;;n]denotethenndiagonalmatrixwhosediagonalelementsarethei's.ForanynitesetS,wedenotebyjSjthecardinality(numberofelements)ofS. Inaddition,letussupposetheroutebetweeneachserver-clientpairisxed,forinstance,bytheshortestpathrouting.Letdenotetheworstlinkutilization(i.e.,thehighestlinkutilization 23

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Condition( 2 )saysthatthelinkutilization(i.e.,congestionlevel)ateachlinke,(Ppij3exij)=ce,mustbenogreaterthan.Here,Ppij3exijistheaggregatebandwidthofallconnections(paths)crossinglinke.Condition( 2 )istheservercapacityconstraint:Theaggregatesendingrateoutofservericannotexceeditstotalcapacity,Ki.Forsomeapplica-tionssuchasmediastreaming,aminimumreceivingrateisrequired.Condition( 2 )istheclientrequiredbandwidthconstraint:Theaggregatereceivingrateatclientjshouldsatisfytherequestedreceivingrate,Qj.ThereisnolossofgeneralitytowritePi2Sxij=QjinsteadofPi2SxijQjfortheminimumraterequirement,sinceifanoptimalsolutionexistsfortheMin-Congestionproblem,theremustexistanoptimalsolutionforwhichtheequalityholds.Condition( 2 )mayalsobeinterpretedasrequiringsaturationoftheclients'capacityorthedownlinkcapacityattheclients'accesslinks. 24

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2-1 asanexample.)Theformulationcanbemodiedaccordinglyandthestructureofthesolutionalgorithmsremainsthesame. 2-1 showsamoderatelycomplexcollaborativedistributionexample,wherenodesmayservebothasclientsandserverstorelaythecontent.Theserver-clientdependencygraphneedsnotbeacyclic.Forinstance,wecanadd(directed)edgesfromnode6tonodes1and4.AswarmingsessionsuchasthoseinBitTorrenttypicallycorrespondstocyclicgraphs.Ourformulationandsolutionsapplytothecycliccase.But,sidearrangementisneededtoensurethat,inanycycle,thedatafromnodeatobisnotthesameasthedatafrombbacktoa.Theyeachshouldhavedifferentinformationcontent(e.g.,differentpartsofthele). TheMin-Congestionproblemisrelatedto,butdifferentfrom,thewell-knownmaximumconcurrentow(MCF)problem[ 48 ].SimilartotheMCFproblem,theMin-Congestionproblemhasanequivalentthroughput-maximizationformulationasfollows.Let=1=,and~xij=xij=. 25

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2.7 .).Inthisscheme,eachclientselectsarandomnumberofservers.Tohavefaircomparison,wealsorequirethatconditions( 2 )and( 2 )aresatised.Thiscomplicatestheactualsever-selectionprocedurealittle.Butthedetailsarenotessential. 49 ]combinesourcecodingwithUDPasthedistributionmechanismformulticastcontentdistribution,toavoidmanytechnicaldifcultiesofusingTCPinthemulticastsituation.Ourexperimentsareconductedonrandomnetworks.TheprecisemodelfortherandomnetworksisdescribedinSection 2.6 .Weonlypointoutthattheusualtransit-stubnetworkmodelisinappropriateforourpurpose,sincewearedealingwithcontentdistributionnetworksorISPnetworksinsteadoftryingtocapturetheprovider-customernetworkrelationshipoftheInternet. IntherandomserverassignmentschemewithUDP,eachclientrequeststhesameowratefromeachrandomlyselectedserver.Therandombehaviorofthesystemissimulated.TheMin-CongestionproblemissolvedoptimallyusingtheGnuLinearProgrammingKit(glpk-4.9).Weconductexperimentsonanetworkwith1000nodes,16144directedlinks,and500clients.Wevarythenumberofserversfrom100to500.TheclientsidereceivingrateisnormalizedtobeQj=1:0forallj2C,andallservershavethesameKivalue. Fig. 2-2 showsthelinkutilizationatthemostcongestedlinkforeachscheme.Weseethattheoftherandomassignmentisthousandstimeslargerthantheoptimalone.Thus,itisverybenecialtoapplytheoptimizationsolution.Infact,onecancreatenetworkexampleswheretheworst-caselinkcongestionintherandomassignmentschemeisarbitrarilyworsethantheoptimalscheme.ThereasonisthattherandomassignmentschemewithUDPhasnomechanismtocope 26

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Fig. 2-3 (a)showsthat,whenTCPisapplied,theworst-caselinkutilizationoscillatesaround0.1,representingafactorof30increaseovertheoptimalresult,whichis0:0033.Undertheoptimalbandwidthallocation,themaximumthroughputis1=0:0033=300,implyingthatthenetworkcanhandlethetrafcratecorrespondingtoKi=1200andQj=300.ForthecasewithTCP,ourexperimentsshowthatTCPatmostcanhandlethetrafcratecorrespondingtoKi=400andQj=100. 31 32 34 35 41 43 ]fornetworkcontrolandresourceallocationproblems.Aconvexseparable 27

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47 ]. WewillusethesubgradientapproachontheMin-Congestionproblem,inawaysimilartoMadanandLall[ 50 ]'ssolutiontoasensornetworklifetimemaximizationproblem.Theideaisthattheproblemremainsthesameiftheobjectivefunctionisreplacedby2.Wethenhaveanonlinearconvexseparableproblem,andweexpecttheproblemcanbedecomposedintoanumberofsub-problems,whichcanbecarriedoutbyeachnodedistributively,hopefullyusingonlylocalinformationorinformationfromasmallnumberofnodes.Aftersomecomputation,itcanbeobservedthat,inthedistributedalgorithm,eachnodestillneedstocommunicatewithallothernodesforpartofthecomputation.Tomakethealgorithmcompletelydistributedanddecentralized,theproblemisreformulatedintothefollowingproblem. Foranye=(k;l)2E,letNebethesetofneighboringlinksofe,denedasfollows.Ne=fe0:eande0areconsecutivelinksonsomeroutingpathg[fe0=(k;l0)2E:ifk2Sg[fe0=(k0;l)2E:ifl2Cg: 28

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2 )andthedenitionofNeguaranteethatalleareidenticalforalle2E.TheparticulardenitionofNeistoremovemanyredundantconstraintsoftheform( 2 )andtomaketheresultingdistributedalgorithmsimpler.Thesecondtermintheoptimizationobjectiveisaregularizationterm.Thenewproblemisidenticaltomodel1when=0. Toderivethedistributedalgorithm,theideaistowritedowntheLagrangefunctionandthenndthedualfunctionbyminimizingtheLagrangefunctionwithrespecttotheprimalvariables.Inessence,thesubgradientalgorithmissimilartothegradient(steepestascent)algorithmforthedualproblem.Ingeneral,thegradientforthedualproblemdoesnotnecessarilyexist.Thesubgradientisageneralizationofthegradientandalwaysexists.PartofthedecompositioncomesfromthestepofminimizingtheLagrangefunction.Inourcase,letthevectorsw,,andvbethedualvariablesassociatedwiththeconstraints( 2 ),( 2 ),( 2 )and( 2 ),respectively.LetL(x;;;;w;v)betheLagrangefunctionandletg(;;w;v)bethedualfunction.Then,g(;;w;v)=infx0;0L(x;;;;w;v)=Xi2SKiiXj2CQjj+Xe2Einfe0(2e+(ceweXe02Nevee0+Xe02Neve0e)e)+Xi2SXj2Cinfxij0(x2ij+(i+j+Xe2pijwe)xij): Thedualproblemis, Dual:maxg(;;w;v)s:t:we0;8e2Ei0;8i2S:

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2 )resultingfromthedecomposition.Thesolutionstothedecomposedminimizationproblemswillbewrittenasfollows.xij=argminxij0fx2ij+(i+j+Xe2pijwe)xijg Thesubgradientalgorithmisthenappliedtothedualproblem.Byaverygeneraltheory(Proposition6.1.1in[ 29 ]),thedualproblemhereisdifferentiableandthesubgradientisequaltothegradient.Furthermore,thesubgradientisrelatedtotheconstraintsoftheprimalproblem.Thecomponentsofthesubgradientofgcorrespondingtowe,i,j,vee0fore2E,i2S,j2C,ande2E;e02Neare,@ge Finally,theactualsubgradientalgorithmatiterationsteptis,x(t)ij=xij((t);(t);w(t);v(t))(t)e=e((t);(t);w(t);v(t))w(t+1)e=[w(t)e+t(Xpij3ex(t)ijce(t)e)]+(t+1)i=[(t)i+t(Xj2Cx(t)ijKi)]+(t+1)i=(t)i+t(Xi2Sx(t)ijQj)v(t+1)ee0=v(t)ee0+t((t)e0(t)e);

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2 ),thepathowxijcanbecomputedusingtheaccumulatedlinkcostsalongthepathfromserveritoclientj,andi,jfromtheserverandclient.Thiscanbeaccomplishedbyanend-to-endfeedbackcontrolprotocol.From( 2 ),thecomputationofeneedsonlyPe02Nevee0andPe02Neve0e,wheree0isaneighboringlinkofe.Hence,thesubgradientalgorithmiscompletelydecentralized.Asanalcomment,accordingto( 2 ),theratexijisadjustedbasedonboththepathcongestioninformationandthecoststhatreecttherateviolationattheserverandtheclient.Hence,thecouplingofcongestioncontrolandserver-clientassignmentisembodiedin( 2 ).InSection 2.6 ,thesimulationresultswillshowthatthesubgradientalgorithmdoesnotachievegoodscalabilityandconvergesslowlycomparedwiththegradientprojectionalgorithmstatedinthenextsection. 50 ].Foreache2E,lete=Ppij3exij=ce,whichistheutilizationoflinke.Let~denotethevector(e),8e2E.Letjj~jj1=maxe2Eebethemax-norm,andjj~jjq=(Pe2Eqe)1=qbetheq-norm,forq>0.Theobjectiveofminimizingtheworstlinkcongestioncanbewrittenasminjj~jj1.Asq!1,jj~jjq!jj~jj1.Hence,theobjectiveofminjj~jj1canbeapproximatedbyminjj~jjq,forsomereasonablylargeq.Theoptimalsolutionunderthelatterobjectiveisthesameasthatunderminjj~jjqq.Wenowhave 31

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But,moreisneededtoapplytheaforementionedspecializedgradientprojectionalgorithm.Wewritetheserversidecapacityconstraintsin( 2 )asabarrierfunction.Then,theMin-Congestionproblemcanbeapproximatedbythefollowingproblem. s.t.Ppij3exij=cee;8e2E Condition( 2 )isnotreallyconstraintbutdenitionofwhateisintermsof(xij),whichcanbesubstitutedintotheobjectivefunction.Theaboveproblemnowhasonlyconstraintsofthetypein( 2 )andthenon-negativityconstraint.Constraintsofthistypearecalledintersectionsofsimplices.Aconvexproblemontheintersectionofsimplicesadmitsaspecialgradientprojectionalgorithm,whichwillbeexplained. Wewillnextsimplifythedescriptionoftheproblemforeasymanipulationandwillalsoremovecertaintechnicaldifculty.Letxdenotethevector(xij),8i2S,8j2C(orequivalently,(xp),foranyp2P).Foreachlinke,letyebethetotaltrafcowthroughe(i.e.,ye=Ppij3exij).Letydenotethevector(ye),8e2E.Inaddition,denezi=Pj2Cxij,whichisthetotalsendingratefromserveri,andletzdenotethevector(zi),8i2S.DenotebyX
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Foreachlinke2E,denethelinkcostfunctionsintermsofpathowvectororthelinkowrate,respectively,as and Denethecostfunctionsofthenetworkintermsofthepathowvectororthelinkowvector,respectively,asf1(x)=Pe2Ef1e(x)and^f1(y)=Pe2E^f1e(ye).Thetwocostfunctionsarerelatedbyf1(x)=^f1(y)=^f1(G1x). Next,considereachsummandofthebarrierfunctionln(Kizi),wherezi=Pj2Cxij.Thefunctionln(Kizi)isonlydenedontheinterval[0;Ki).Thiscreatessometechnicaldifcultyindevelopinganalgorithm,sinceweneedtoworryaboutthefeasibilityoftheowratevectorwithrespecttotheservercapacityconstraintduringeachstepofthealgorithmiteration.Wewillresolvethisdifcultybyextendingthedenitionofthefunctionln(Kizi)totheinterval[0;1),foralli2S.Forconvenience,letusreplacebyq.Foranyserveri2S,denethenewbarrierfunctionintermsofthepathowvectorortheserversendingrate,respectively,as 2)2(1 42+ln))ifPj2Cxij>Ki,(2) and 2)2(1 42+ln))ifzi>Ki,(2) where0<<1isasmallconstant.Thepointisthat^f2i(zi)thusdenediscontinuouslydifferentiableon[0;1).Theoverallbarrierfunctionsintermsofthepathowvectororthe 33

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Finally,wecanwritetheapproximationtotheMin-Congestionproblemasfollows,whichwewillsolvedistributedly. s.t.x2X: 2 ),theoptimalityconditionisespeciallysimple[ 47 ].Furthermore,thereexistsaspecialgradientprojectionalgorithm[ 51 ][ 52 ].Inthissubsection,wedescribetheoptimalityconditionandthealgorithm.Inthenextsubsection,wegivetheconvergenceresults. Throughout,wewillassumeq2unlessotherwisementioned.Wewillneedvariousrstandsecondderivativesrelatedtothefunctionf.Thederivativeoffwithrespecttoxij(thepathowrate)isgivenby andthesecondderivativeoffwithrespecttoxijandxklisgivenby Thefollowingfactsareimportantforthedevelopmentoffullydistributedalgorithms. 1. 34

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Asshownin( 2 ),@f(x) Eachlinkcostandservercostcanbecomputedlocallybyeachlinkandserver,usingonlylocalinformation. 4. Tocomputethecostofpathpij,clientjcancollectalllinkcostsonthepathandthecostofserveri,usinganend-to-endcontrolprotocol. 5. Inviewof( 2 ),similarstatementsas2)4)canbesaidaboutthesecondderivative,@2f(x) Thecomputationof( 2 )and( 2 )requirestherstandsecondderivativesof^f1eand^f2i.Therstderivativesof^f1eand^f2iaregivenby 2)ifzi>Ki.(2) Thesecondderivativesof^f1eand^f2iaregivenby (2) SincethefeasiblesetXisaconvexsetandtheobjectivefunctionisaconvexfunction,wecancharacterizeanoptimalsolutionxtotheproblem( 2 )-( 2 )bythefollowingoptimality 35

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Thisconditionisequivalentto,foranyi2Sandj2C(See[ 47 ].), Importantinsightiscontainedin( 2 ).Itsaysthat,inanoptimalsolution,aclientjonlyselectsthoseservers(i.e.,receivespositivetrafcratesfromthem)thathavetheminimumpathcost.Here,thepathforserverireferstotheonefromserveritoclientj(i.e.,pij).Thepathcostunderxrefersto@f(x) 2 ),whichcontainsbothlinkcostsforthelinksonthepathandtheservercost.Underx,foreachclientj,letusdenetheserverthathasthesmallestpathcosttoclientjbysj(x).Thatis, Iftherearemultiplesuchservers,anarbitraryoneamongthemischosen. Basedonasimilaralgorithmin[ 51 ],weapplythesynchronousiterativegradientprojectionalgorithmlistedinAlgorithm 1 tosolvetheproblem( 2 )-( 2 ). where,forallj2C,i6=sj(x(t)), and 36

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2 )isascalaron[a ;1],forsomea 2 )and( 2 )computetheendpoint,x(t),ofafeasibledirection,entrybyentry.Foreachclientj,therearetwocases. Notethatthedescriptionincase2ensuresthatx(t)isfeasible(i.e.,inX).Sincex(t)isalsofeasible,by( 2 ),thenewratevectorx(t+1),whichisonthelinesegmentbetweenx(t)andx(t),isfeasible.Hence,ifwestartwithafeasibleratevectorx(0)inX,thenx(t)isinXforallt. 2 )and( 2 ),respectively,allbasedonthelocalaggregateratepassingthroughtheserverorthelink.Thepathcost,@f @xij,canbecomputedbyclientjbasedonthelinkcostsandtheservercostalongthepathfromserveritoclientj,accordingto( 2 ).Hence,thegradientprojectionalgorithmiscompletelydecentralized. 53 ]tondanupperboundonthatguaranteestheglobalconvergenceofthesynchronousgradientprojectionalgorithmtoanoptimalsolutionoftheproblem( 2 )-( 2 ).Furthermore,whenthesequencefx(t)ggetsnearanoptimalsolutiontowhichitconverges,theconvergencespeedislinear(i.e.,geometric).Sometechnicalconditionorminorreformulationoftheproblemareneededforsomeoftheseresultstohold. 2 )canbereplacedwithamoregeneralupdatex(t+1)=A(t)x(t)+(IA(t))x(t),whereA(t)isajSjjCjjSjjCjdiagonalmatrixwithdiagonalentriesintheinterval[a ;1],forsomea 37

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2 )-( 2 ).Themainproofsareadaptedfromtheproofsin[ 53 ].MoredetailsabouttheproofsaregiveninAppendix 2.9 2 )-( 2 )withx(0)2X0isoptimal. 2 .TheproofisgiveninAppendix 2.9 Allourexperimentalresultsonrandomnetworkshaveshownthatthealgorithmactuallyconvergestoanoptimalpointanddoessoquitefast.Wenextshowthat,underadditionalconditions,thealgorithmindeedconvergestoanoptimalpoint.Furthermore,itconvergeslinearly(i.e.,geometrically)oncethesequencefx(t)ggetssufcientlyneartoanoptimum. TheconditionA1isnotastringentone.Itcanbenaturallysatisedorforcedtobesatised.Ifalinkisnotusedatallintheend,itprobablywouldnothavebeendeployedoractivatedintherstplace.Alternatively,itmaybeeasytospotthoselinksthatwillnotgetusedinthenalsolution,possiblybasedonpastknowledge.Then,theselinkscanbeexcludedfromthealgorithm. Animportantfactisthat,underA1,thesecondderivative(^f1e)00(ye)isstrictlypositiveforeverye2Eatanyoptimalxandy=G1x(See( 2 ).).Asaresult,thediagonalentriesofr2^f1areboundedawayfromzerobysomepositivescalaratanoptimalx.Bycontinuity,the 38

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2 ).).Wecanthenapplytheresultin[ 53 ]andconcludelocalgeometricconvergenceintheneighborhoodNofx,aswellasglobalconvergence.Theprecisestatementistechnical.Werstneedthefollowinglemma. Proof. 2.9 Dene 2 )-( 2 )convergestoanelementofXandtheconvergencerateislinear(i.e.,geometric)inthesensethatforalltt0, Theconstantsandparametersareasfollows.1=a =(LjSj),andL>0istheupperboundofthenormofr2foverX0;D5=a =(D4+1),D4=((5L+1)(D3)2+1+21+6L(1)2=a 1 .^2^L2areanytwopositivescalars

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Proof. 2.9 2 )-( 2 )withx(0)2X0convergestoanoptimalpoint. Proof. 1 ,thesequencefx(t)gwillgetnearanoptimalsolution.ByTheorem 2 ,onceitgetssufcientlynearthatoptimum,itconvergestotheoptimumatageometricspeed. Wesuspectthatthealgorithmalmostalwaysenjoysglobalgeometricconvergenceinrandomnetworksofreasonablesizeandtrafc.However,onemaynotclaimthistheoretically.But,theoriginalMin-Congestionproblemcanbeapproximatedslightlydifferently,anditcanbeshownthatthereformulatedproblemalwaysenjoysglobalgeometricconvergence.Notethatminjj~jj1hasthesamesolutionasminjj~+~jj1,wheree=foranyconstant>0,8e2E.Wethenconsidertheapproximationofminjj~+~jj1.Moreprecisely,wereplacethetermjj~jjqqbyminjj~+~jj1intheobjectivefunctionin( 2 )andkeepthesameconstraints.Withthismodication,^f1e(ye)=(ye 53 ]andconcludeglobalgeometricconvergence. Anotherpossiblereformulationistodene^f1e(ye)=exp(ye=ce+)forsome>0and>0.Theobjectiveofminimizingjj~jj1isapproximatedbyminimizingPe2Eexp(ye=ce+).Globalgeometricconvergencecanalsobeclaimed. 2.6 willshow,thegradientprojectionalgorithmin( 2 )-( 2 )doesnotworkwellenoughintermsofpracticalconvergencespeedwhenthenetworkbecomeslarge.Inthissection,wewilldiscussthecauseoftheproblemandgiveamorescalablevariantofthealgorithm. 40

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2 )-( 2 )isill-conditionedbecauseofthepoorrelativescalingoftheoptimizationvariables.Bythiswemeanthatsingleunitchangesofdifferentvariableshavedisproportionateeffectsonthecost.Forinstance,inthenetworkofFig. 2-4 (Thenumbersaroundthelinksarethelinkcapacities.),theeffectonthefunctionfduetooneunitincrementofx11canbeverydifferentfromtheeffectduetooneunitincrementofx21.Thiscanbeseenfromthepartialderivativesoffwithrespecttox11orx21inthefollowingexample.ThistypeofsituationcorrespondstoalargeconditionnumberoftheHessianoff.(Seepage71of[ 47 ]forarelevantdiscussion.) Letuslookattwoparticulariterationstepsofthealgorithm.AssumeKi=1:5foralli2SandQj=1:0forallj2C.OneoftheoptimalassignmentsoftheMin-Congestionproblemisx=(1;0:5;0:05;0;0:5;0:95)T.Here,theowratesarerstorderedaccordingtotheserversandthenaccordingtotheclients.With=1e04and=1e05in( 2 )and( 2 ),thestepsizeupperbound1isabout5:6e+06(SeeTheorem 2 .).Assumetheowassignmentinthecurrentiterationisx=(0;1;0:5;1;0;0:5)T.Themostcongestedpathisp21andweneedtodecreasetheowonthispathaccordingtothealgorithmin( 2 )-( 2 ).Therstderivativesoffwithrespecttox11andx21are3:31041and1:6109,respectively.Hence,by( 2 ),atthecurrentstep,thechangeoftheowonp21isabout0:009,whichisreasonableinsize.Nowassumethecurrentowassignmentisx=(1;0;0:5;0;1;0:5)T.Themostcongestedpathisp13.Therstderivativesoffwithrespecttox13andx23are6:81018and5:01031,respectively.Consideringthelinkcapacities,itiseasytoseethatpathp13ismuchmorecongestedthanp23andweneedtodecreasetheowonp13.Butwiththestepsize1,thechangeoftheowonp13isonly3:81011,whichistoosmalltodecreasex13from0:5toward0:05substantially.Hence,thestepsizecannotleadtofastconvergenceforallcasesofx(t). 41

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47 ].).Suchscalingofthevariablescanalsobeinterpretedasusingdifferentstepsizesfordifferent(unscaled)variables.Thescaledalgorithmisasfollows. whereforallj2C,i6=sj(x(t)), with Here,@f(x(t)) 2 ),( 2 ),and( 2 ),and(t)isasdescribedforthenon-scaledgradientprojectionalgorithminSection 2.4 .TheonlychangefromAlgorithm 1 isthescalingfactord1ij(t)=1 2 )and( 2 ),respectively,allbasedonthelocalaggregateratepassingthroughtheserverorthelink.dijcanbecomputedbyclientjbasedontheaccumulatedvaluesalongthepathsfromserveritoclientj,andfromserverSj(x)toclientjaccordingto( 2 )( 2 ).Hence,thescaledalgorithmiscompletelydecentralized. Thedevelopmentwillcapturetwoissuesthatmaypossiblymaketheasynchronousal-gorithmincorrect:theasynchronousoperationsofdistributednetworkelementsanddelayed 42

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54 ][ 53 ]. Intheasynchronousalgorithm,wereplace( 2 )with andsetxsj(x(t));j(t)accordingto( 2 ),whereforanyclientj,sj(x(t))satisessj(x(t));j(t)=mini2Sij(t).Here,ij(t)and^dij(t)aresomeestimatesof@f @xij(t)anddij(t),respectively,whichare,ingeneral,inexactduetoasynchronyanddelaysinobtainingmeasurements. Incorrectingtheassumptionofperfectsynchronizationofcomputation,weonlyassumethatthetimebetweenconsecutiveupdatesisbounded.Moreprecisely,foreachserveri,clientjandlinke,letthesubsetsofupdatingtimesbeTif0;1;2;g,Tjf0;1;2;gandTef0;1;2;g.Assumethesesubsetssatisfyft;t+1;;t+B11g\Ti6=;,ft;t+1;;t+B11g\Tj6=;andft;t+1;;t+B11g\Te6=;forallt.Thatis,foreachclient,serverorlink,atleastoneupdateoccursineveryB1consecutivetimeslots. Wenowdescribetheprocessbywhichij(t)and^dij(t)areformed,andtheupdateiscarriedout.Foreachlinkeandeacht2Te,wesete(t),theestimatorof(^f1e)0,tobe where,foreveryt,he(t;)arenonnegativecoefcientssummingtoone.Thatis,e(t)istheweightedsumoftheB2+1truemeasurementsof(^f1e)0(ye())ontheB2+1timeslotsatorpriortot.Hence,e(t)isaweightedmovingaverageof(^f1e)0(ye(t)).Similarly,^de(t),theestimatorof 43

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Foreacht62Te,wesete(t)=e(t1)and^de(t)=^de(t1).Thatis,noupdateoccursonthetimeslotsnotinTe. Similarly,foreachserveriandeacht2Ti,weseti(t),theestimatorof(^f2i)0,tobe and^di(t),theestimatorof(^f2i)00,tobe where,foreachxedt,hi(t;)arenonnegativecoefcientssummingtoone.Foreacht62Ti,weseti(t)=i(t1)and^di(t)=^di(t1). Intheabovecomputationsofthemovingaverages,(^f1e)0,(^f2i)0,(^f1e)00and(^f2i)00aregivenby( 2 )-( 2 ). Attimet2Tj,foreachclientj,ij(t)and^dij(t)arenaturallyestimatedby and where 2 )-( 2 ).Foreacht62Tj,wesetij(t)=ij(t1)and^dij(t)=^dij(t1). 44

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2 )and( 2 ).Givenxij(t)andxij(t)foralli2S,clientjderivesthenextowassignment,xij(t+1)foralli2S,accordingto( 2 ).Ateacht62Tj,wesetxij(t)=xij(t1).Inthiscase,x(t+1)=x(t). TheasynchronousalgorithmissummarizedinAlgorithm 2 .Weassumethatthereisanend-to-endcontrolprotocoloperatingoneachpathfromaservertoaclient,ifthepathiscarryingapositiveow.Thefunctionofthisprotocolistocarrythemeasurementinformationfromeachlinkandservertotherelevantclientsthatusethelinkandserver. 2 )and( 246 ),respectively. 2 )and( 248 ),respectively. 2 ),( 250 ),respectively.Then,itcomputesxij(t)foralli2Saccordingto( 2 )and( 2 ).Clientjchoosesnewratesxij(t+1)foralli2Saccordingto( 2 )andcommunicatesthechangedowratestotheservers. >0;andthestepsizeischosensmallenough.

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2 )-( 2 ),theconditionsofTheorem 4 areallmet.Hence,wehavethefollowingconclusion. 2 )-( 2 ),f(x(t))generatedbytheasynchronousdiagonallyscaledalgorithmconvergestofandanylimitpointoffx(t)gisaminimizingpoint.Moreover,xij(t)xij(t)convergestozeroforallserver-clientpairiandj. SincewearedealingwithcontentdistributionnetworksorISPnetworks,wewillnotusethetypicaltransit-stubnetworkmodel,whichismoresuitabletocapturetheprovider-customernetworkrelationshipintheInternet.Instead,allexperimentsinthechapterareconductedonthefollowingclassofuniformrandomnetworks.Theparametersarethenumberofnodesnandthenumberofdirectedlinksm.Letp=m=2 2;p)distributed,anditsexpectedvalueism=2.Hence,theexpectednumberofdirectedlinksism.Iftheresultingnetworkisnotconnected,werepeatthesameprocedure.Thecapacityofeachdirectedlinkisuniformlydistributedon[0:1;1000].Theserversandclientsarerandomlydistributedoverthenetwork. 46

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2-5 showsthatafterabout600iterations(not150iterationsasitmightappearinthegure,becausethealgorithmstillneedstosatisfytheuniformclientsidereceivingrateasFig. 2-6 (b)shows),convergestotheoptimalvalue.Fig. 2-6 (a)showstheaverageowratesfromtheserversortotheclients.Afterabout50iterations,theyeachbecomeclosetotheirnalvalues,4:0and1:0,respectively.Thismeansthattherateoftrafcenteringandexitingthenetworkateachnodebecomesstablequickly.Fromiterationstep50to150,thealgorithmisdevotedtobalancingthelinkutilization.Fig. 2-6 (b)showsthestandarddeviationoftheclientrates.Itislessthan6%oftheaveragerateafter50iterations.Thealgorithmspendsabout600iterationstomakeQj=1:0foreachclientj. Inthegradientprojectionalgorithm,wechooseconstantvaluesfor,and.InFig. 2-7 (a),asqgrows,theoptimalvaluefortheapproximateproblem( 2 )-( 2 )becomesincreasinglyclosetotheoptimumoftheMin-Congestionproblem.Consideringthescaleoftheverticalaxis,thetwooptimalvaluesarenevertoodifferentforallq2.Again,thenon-zeropreventsthemfrombecomingexactlythesame.Fig. 2-7 (b)showsthat,forq=10,ittakes600iterationsfortheobjectivevaluetoconverge,butonly100iterationsforittocomeveryclosetotheoptimum.Theconvergenceisfasterthanthesubgradientalgorithm.Notethatunlikethesubgradientalgorithm,thegradientprojectionalgorithmalwaysguaranteesthatQj=1:0foreachclientjateachiteration. 2-8 (a)showsthatthescaledalgorithmconvergestotheoptimalvalueafter350iterations,whichisslightlyfasterthantheunscaledalgorithm.However,thesubgradientalgorithmandtheunscaledgradientprojectionalgorithmdonotworkwellforlargernetworks.Wenextconductexperimentsonaslightlylargernetworkwith100nodes,1606links,20serversand60clients.WesetKi=4:5andQj=1:0.AsFig. 2-9 (a)shows,thesubgradientalgorithmstartstostablizeatiteration200,but,towardsavalue=0:00096,whichisabout2:0timesoftheoptimumoftheMin-Congestionproblemandcannotbeimprovedby 47

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2-9 (b),theunscaledgradientprojectionalgorithmproducesabettersolutionthanthesubgradientalgorithm,butcosts10000iterationstogetclosetotheoptimum.Inbothalgorithms,thepoorperformanceismostlikelyduetothefactthattheproblemisill-conditioned.Asanevidence,whenweapplythescaledprojectionalgorithmtothisnetworkof100nodes,Fig. 2-8 (b)showsthatitonlytakes250iterationstoreachtheoptimum,whichisveryclosetotheoptimumoftheMin-Congestionproblem. Thescaledalgorithmachievesgoodconvergenceresultsformuchlargernetworks.Weconductexperimentsonanetworkwith1000nodes,16036links,150servers,and750clients.Thisnetworkismuchbeyondwhatthesubgradientalgorithmortheunscaledgradientprojectionalgorithmcansolve.WesetKi=10:0foralli2S,Qj=1:0forallj2C,andq=10.Otherparametersarechosenmanually.Fig. 2-10 (a)showsthat,after2000iterations,thealgorithmconvergesnicely.Incontrast,fortheunscaledgradientalgorithmorthesubgradientalgorithm,extensivedegreeoftuningofthestepsizestillwillnotmakethealgorithmworkforsuchanetworksize.Itisalsoworthnotingthatthisnetworkisevenchallengingforthesimplexmethod(forthelinearMin-Congestionproblem),whichtakesmorethan44;000iterationsandalotofcomputingpowertoreachtheoptimum.Fig. 2-10 (b)showsthatthebarrierfunctionf2(x)boundstheaggregateserversendingratesverywell,asitissupposedto.Intheexampleofthegure,thesendingrateisclosetobutbelowthemaximumallowedsendingrate,10. ManyP2Plesharing/distributionsystemshavebeenproposed,mostofwhichareend-userbased.Inthesesystems,peersusuallyformameshnetworkandexchangelechunkscollaboratively.Examplesofmedia-streamingsystemsincludePROMISE[ 55 ],GridMedia[ 56 ],PRO[ 57 ],andPRM[ 58 ].ExamplesofbulkdatadistributionsystemsincludeBitTorrent[ 1 ],FastReplica[ 59 ],SplitStream[ 60 ],Bullet0[ 61 ],ChunkCast[ 62 ],andCoBlitz[ 63 ].Withafewexceptions,theseP2Psystemsaremostlyconcernedwithperformanceexperiencedbyindividual 48

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Theliteratureonserver/nodeselectionisvast.WewillfocusonrecentworksinthecontextofP2Porcontentdistributionsystems,whichhandlenodeselectioninavarietyofways 60 ]andFastReplica[ 59 ],theselectionisessentiallyrandom.Othersystemsemployapeerrankingfunction,andeverynodetriestoselectthosepeerswithhighranking.Atypicalstrategytoaccomplishthisisbasedonsampling:Eachnodeinitiallyselectssomerandompeers,butdynamicallyprobesotherpeersandgraduallyswitchestothosewithbetterrankingoverthecourseoftransmission.BitTorrent[ 1 ],Bullet0[ 61 ],ChunkCast[ 62 ]andCoBlitz[ 63 ]allusesomeformofthisstrategy.Anoftenusedrankingfunctionisnodalload,suchasinCoBlitz.Otherrankingfunctionsincludetheround-triptime(RTT)suchasinChunkCast,andthedegreeofcontentoverlapsuchasinBullet[ 64 ].Morerelevanttothischapter,BitTorrent,Bullet0andSlurpie[ 65 ]usethesendingorreceivingbandwidthtoorfromapeerastherankingfunction.Theperformanceofsuchaschemeisdifculttounderstand.But,ithasbeenshownthatBitTorrentworkswellforaccess-limitednetworks[ 66 ]. Thenexttwoarecontentdistributionsystemsthatpayattentiontonetworkperformance.Julia[ 67 ]assumestheunderlyingP2Pnetworkislocality-aware.Eachpeerexchangeslechunkswithneighborsindifferentialamount,morewithcloserneighbors.Thisreducesthetotalwork,whichisdenedastheweightedsumofthetotalnetworktrafcduringtheentiredistributionprocess,wheretheweightsarethedistancestravelledbybits.In[ 68 ],theserver-to-clientassignmentisdeterminedbyminimizingtheweightedcostofthetotaltrafcduringamedia-streamingprocessunderxedserverandclientbandwidthconstraints,wheretheweights 49

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In[ 69 ],severalsingle-clientserver-selectionproblemsareformulatedundertheoptimizationframework.Theservercapacityisthelimitedresourceintheseproblems.Inmoretraditionalnode-selectionliterature,[ 70 ]presentsadynamicselectionschemebasedoninstantaneousmeasurementoftheRTTandavailablebandwidth.Similarapproachesarealsoreportedin[ 71 73 ]. Thereexisttwoclassesofsolutionstotheoptimizationproblem,thesubgradientalgorithmandaspecialgradientprojectionalgorithm.Thechapterfocusesondevelopingthegradientprojectionalgorithm,whichistechnical.Wecomparethetwosolutionscarefully,mainlywithrespecttotheconvergencespeed.Thegradientprojectionalgorithmachievesalocallylinearconvergencerateintheory;thesimulationexperimentshavedemonstratedthatitconvergesfasterthanthesubgradientalgorithminpractice.However,neitherofthemworkswellenoughonlargenetworksduetothefactthatourproblemisill-conditioned.Hence,wehavedevelopeda 50

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Ourapproachmakestheassumptionthatthenetworknodescanactuallyimplementthefunctionalitiesrequiredbythedistributedalgorithm.Inpractice,itiscertainlydifculttodeploythesefunctionsateverynetworknode,especiallyatthecorerouters.However,foranISPnetworkoramanagedcontentdistributionnetwork,thedeploymentdifcultyisdrasticallyreducedsincethenetworkisusuallymuchsmaller(upto10,000sofnodesinsteadofmillions)andthenetworkoperatorhastheauthoritytodeploywhateverfeaturestheydesiretothenetworknodes.Furthermore,wecanseeatleastthreewaystomakethedeploymenteasier.First,thedistributedalgorithmmaybeimplementedatasubsetofthenetworknodesthatarenotthecorerouters,forinstance,thegatewaynodesofedgenetworks,orthenodesthatinterfacewithslowerlinks.Thesenodesaremorelikelytobecomebandwidthbottlenecks;theyarealsomorelikelytohavesufcientcomputingpowertoimplementthealgorithm,sincetheydonothandlethelargesttrafcvolume.Second,overlayserverscanbeattachedtothenetworknodeswherethealgorithmneedstobeimplemented.Theserverscanimplementthealgorithmattheapplicationlayeronbehalfofthenetworknodes.Third,iftrafcisroutedontheoverlaynetwork,asisusuallythecaseforcontentdistributionnetworks,thenetworknodesarethemselvesoverlaynodes,whichcanimplementthealgorithm.Ouralgorithmcanalsobeusefulforspecializednetworkssuchasresearchandeducationnetworksandwirelessmeshnetworks.Itiseasiertodeploynewalgorithmsinthesenetworksbecausetheytendtobesmall,andhence,moreupgradeable. Wecancompareourserver-selectioncriterion(basedontherst-derivativepathcost)withthemetricsconsideredbyotherapproachesdiscussedinSection 2.7 .Inourformulation,thecapacitylimitationsoftheserversandclientsarenaturallycapturedbytheconstraintsoftheoptimizationproblem.OurformulationdoesnotconsidertheRTT,whichmatterstotheresponsetimeofshorttransactionsbutdoesnotnecessarilyaffectlong-timeaveragebandwidth,amuch 51

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Theproblemformulatedinthischapterisstillamongthesimplest.Inthefuture,onemayconsidertheproblemwithheterogeneouscontents,possiblyunderenvironmentaluncertainties.Onemayalsoinvestigatethecaseofmultiplepathsperserver-clientpairinsteadofthexedshortestpathrouting.Finally,inthecaseofstreamingcontent,futureformulationmayalsoneedtotakeintoaccountthetimingandsequencingconstraints. 2 )-( 2 )convergesinthespaceofpathowstoanelementxofX,thesetofoptimalowrates,andachievesalocalgeometricconvergencerateintheneighborhoodofx(undersometechnicalconditions). Theproofforthegeometricconvergenceratewillcloselyfollow[ 53 ].Onekeydifferenceisthat,in[ 53 ],itrequiresthatthediagonalentriesofr2^f1(G1x)andr2^f2(G2x)areboundedawayfromzerobysomepositivescalarsforallx2X0.Ifthisweretrue,[ 53 ]actuallyshowsglobalgeometricconvergence.Inourcase,whiler2^f2(G2x)satisesthisrequirementforallx2X0,butr2^f1(G1x)doesnot.However,undertheconditionA1inSection 2.4.3 ,wecanproveLemma 1 .Thekeyisthatthediagonalentriesofr2^f1(G1x)areboundedawayfromzerobysomepositivescalarinsomeneighborhoodofanoptimalx.Withsomemoremodicationoftheargumentin[ 53 ],thiswillleadtotheconclusionofalocalgeometricconvergencerate. 52

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1 )Lety=G1x,wherexisanoptimum,andlety=G1x.Assumeq>2.ByA1,r2^f1(y)>0.Bycontinuity,thereisaclosedball^XX0aroundx,suchthatr2^f1(G1x)>0forallx2^X.Let^1>0beaminimumofr2^f1(G1x)on^X. Let^Y=fG1xjx2^Xg.^Yisaclosedandconvexsetsince^Xisclosedandconvex.Furthermore,r^f1(y)^1forally2^Y.Letx2^X.Bythemeanvaluetheorem,thereexistssomet2[0;1]suchthat 53

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and 2 )canbewrittenas 1 .^2isapositivescalarsuchthatthediagonalentriesofr2^f2(G2x)areboundedbelowby^2forallx2X.Whenq=2,theaboveinequalityholdsforallx2X. 2 .^Lisapositivescalarsuchthatthediagonalentriesofr2^f(Fx)areboundedaboveby^Lforallx2^X.

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53 ].[ 53 ]requiresthatthediagonalentriesofr2^f(Fx)areboundedawayfromzerobysomepositivescalarforallx2X0.Itthenusesthispropertyofr2^f(Fx)toshowthat 2 ,weareabletohavetheaboveinequalityinaneighborhoodx.Therestoftheproofisnearlyidenticaltothatin[ 53 ]. Thebounds^Land^arewelldenedforourspecic^f1and^f2inthecompactset^X.Furthermore,ifq=2orifwecanmodifyour^f1tomakethediagonalentriesofitsHessianboundedbelowbyapositivescalarforallx2X0(SeethecommentattheendofSection 2.4.3 onhowtodothis.),Theorem 5 willholdforallx2X0insteadofonlyforxnearanoptimum. Let1=a =(LjSj),wehavethefollowingthreelemmas.Theonlychangefromtheircounterpartsin[ 53 ]involvessubstitutionofappropriateconstants. jSjL 2(x(t))2+(3L 22)jjx(t)x(t)jj2:(2) UponcombiningtheprecedingthreelemmasandTheorem 5 ,Theorem 2 canbeshownsimilarlyasin[ 53 ]. 1 55

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2 ,for0<1andforalltthatx(t)2X0,wehave jSjL =LjSj,theright-handsideoftheaboverelationisnonpositive.Hence,iffx(t)ghasalimitpoint,theleft-handsidetendsto0.Thealgorithm( 2 )-( 2 )canbedenotedasafunctionA(x)(i.e.,x(t+1)=A(x(t))).Therefore,jjx(t)x(t)jj!0,whichimpliesthatforeverylimitpoint~xoffx(t)gwehave~x=A(~x).Itiseasytoshowthat,if~x=A(~x),then,foreveryi2Sandj2C,wehave whichisexactlytheoptimalityconditionin( 2 ).(Proposition2.3.2andExample2.1.2in[ 47 ]). 56

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Aserver-clientdependencygraph.Thesolidanddashedlinesindicateserver-to-clientrelationship.Thesolidlinesindicatethenalserverselectionresults.Forinstance,node6hastheserversetS6=f1;3;4g.Itendsupselectingservers1and3.Node1hastheclientsetC1=f3;4;5;6g. Figure2-2. Ratioofworst-caselinkutilization:randomvs.optimalowassignment.Thenetworkshave1000nodes,16144linksand500clients. 57

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(b) Figure2-3. ParalleldownloadunderTCP-Renocongestioncontrolvs.optimalowassignment.Thenetworkshave50nodes,496links,10serversand40clients.Ki=4:0,Qj=1:0:(a)Worst-caselinkutilization;(b)Networkthroughput. Figure2-4. Networkthatleadstoanill-conditionedproblem:q=10. Figure2-5. Convergenceofthesubgradientalgorithm. 58

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(b) Figure2-6. Flowrateconvergenceofthesubgradientalgorithm:Ki=6:0;Qj=1:0. (a) (b) Figure2-7. Convergenceofgradientprojectionalgorithm:(a)Optimalutilizationwithdifferentq;(b)Convergenceofthealgorithm.q=10. (a) (b) Figure2-8. Convergenceofdiagonallyscaledgradientprojectionalgorithm:(a)50nodes,496links,10serversand40clients.q=10;(b)100nodes,1606links,20serversand60clients.q=10. 59

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(b) Figure2-9. Poorperformanceofsubgradientandunscaledgradientprojectionalgorithms.100nodes,1606links,20serversand60clients.Ki=4:5,Qj=1:0:(a)Subgradientalgorithm;(b)Gradientprojectionalgorithm.q=10. (a) (b) Figure2-10. Diagonallyscaledgradientprojectionmethod.Thenetworkhas1000nodes,16036links,150servers,750clients:(a)Convergenceofthealgorithm;(b)Convergeofsendingrate.Ki=10. 60

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1 ],FastReplica[ 59 74 ],Bullet[ 61 64 ],Chunkcast[ 62 ],CoBlitz[ 63 ],andJulia[ 67 ].ThemostpopularoneamongthemistheBitTorrentprotocol. Swarmingenablescontentproviderswithpoorcapacitytoreachalargenumberofaudience,allowsrapiddeploymentofalargedistributionsystemwithminimuminfrastructuresupport,andisincreasinglyusedinmainstreamandcriticalnetworkservices.Onthenegativeside,swarmingtrafchasmadecapacityshortageinthebackbonenetworksagenuinepossibility,whichwillbecomemoreseriouswithber-basedaccess 46 ]. 75 ].Thespeedoftheaccessberiscurrentlyat100Mbpsorlower,headingto1Gbpsby2020andislikelytoreach10Gbpsthereafter[ 76 ]. 61

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77 ]),networkcachesystemsandexistingcontentdistributionnetworks(e.g.,Akamai[ 78 ]).Aswillbeseen,swarmingcanbethoughtasdistributingcontentonmultiplemulticasttrees.Whendoneproperly,itprovidesthemostefcientutilizationofthenetworkcapacity,aremedyforbackbonecongestion,orgivesthefastestdistribution. Forillustrationofthemainideasinthischapter,considerthetoyexampleinFig. 3-1 .Thenumbersassociatedwiththelinksaretheircapacities.Supposealargeleissplitintomanychunksatsourcenode1.Wewishtondthefastestwaytodistributeallchunkstoreceivers2and3. 3-2 .Thequestionbecomeshowtoassignthechunkstodifferentdistributiontreessothatthedistributiontimeisminimized,subjecttothelinkcapacityconstraint.Forthissimpleexample,itiseasytoseethatdistributingthechunksin1:2ratioonthesecondandthirdtree,whileleavingtherstunused,isoptimal. Theideascontainedinthetoyexamplearealsogivenin[ 79 ].Thatworkfocusesonhowtocomputethemaximumthroughput,whichleadstothefastestdistribution.Ourcontributionin 62

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OurapproachillustratedbythetoyexamplecanbecontrastedwithexistingP2Pdistributionsystemsortechniques[ 1 59 61 64 67 74 ].Beingoriginallydesignedforend-systemle-sharingapplications,thecurrentswarmingsystemsleavemuchroomforimprovement.Mostsystemsonlyselectandsolvepartoftheproblemanddosowithpoorlyunderstoodheuristics.Notenoughisknownonhowwelltheyworkorhowmuchimprovementremainspossible.Theyhaveuser-centricperformanceobjectives,butmostlyignoretheimportantnetwork-centricobjectives,suchasminimizingnetworkcongestion.Thebandwidthbottleneckisoftenassumedattheaccesslinks,ratherthanthroughoutthenetwork.Oursolutionwillbeabletoautomaticallyadapttocapacityconstraintanywhereinthenetwork.Furthermore,withinourframework,weareabletosolvefourseparateproblemsjointlyandoptimally:peerselection(fromwhichpeertodownload),chunkselection(whichchunkstodownloadfromapeer),distributiontreeselectionandbandwidthallocation.Incontrast,otherP2Psystemssolveoneortwooftheseproblemsinisolation,typicallyusingad-hocapproaches. Thechapterisorganizedasfollows.ThemodelsandproblemformulationsaregiveninSection 3.2 .ThedistributedalgorithmisgiveninSection 3.3 .InSection 3.4 ,wepresentthecolumngenerationapproach,combineitwiththedistributedalgorithmdescribedinSection 3.3 .InSection 3.5 ,wediscusspracticalissuesinapplyingouralgorithmtorealisticsettings,suchasscalabilityandcopingwithnetworkdynamicsandchurn.InSection 3.6 ,weevaluate 63

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3.7 ,wediscussadditionalrelatedwork.TheconclusionisdrawninSection 3.8 3.5.1 .)andwecallthosememberssourcesofthesession.Areasonableassumptionaboutasessionisthatallmembersinthesessionareinterestedinthele,andattheendofledistribution,everymemberinthesessionwillhaveacompletecopyofthele. LetMbethesetofallmulticastsessions.Foreachsessionm2M,letV(m)Vrepresentthesetofmembersinsessionm,andletS(m)V(m)bethesetofsourcesinsessionm.Foreachsources2S(m),letL(m)sbethetotalsizeofthelechunksatsourcesforsessionm.Letthesetofallpossiblemulticasttreesspanningallmembersinthesessionrootedatsources2S(m)bedenotedbyT(m)s.Amulticasttreemaycontainnodesnotinthesession,inwhichcasethetreeiscalledaSteinertree.Inthecasewhereallnodesonthetreebelongtothesession,themulticasttreeiscalledaspanningtree,meaningitspansthemulticastsession(ratherthanthewholenetworkV).Fortheithtreet(m)s;i2T(m)s,wheretheorderofindexingisarbitrary,denotez(m)s;itobethesendingrateontreet(m)s;ifromtheroots. 64

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Theoptimizationproblemisasfollows.mint Condition( 3 )saysthat,ifonelooksatallthemulticasttreesrootedatasourcesforasessionm,thesumofthedistributionratesonallthesetrees,multipliedbythedistributiontime,shouldbeequaltothetotalsizeofallthelechunksstoredatsourcesforsessionm.Thismeansthateverybitofthelestoredatsmustbetransmittedexactlyonce.Amomentofthinkingrevealsthatnothingisgainedbytransmittingthesamebitmorethanonce.Condition( 3 )isthelinkcapacityconstraint.Ateachlinke,theowrateonthelinkshouldbenogreaterthanthelinkcapacity,ce. Itturnsouttheaboveproblemisequivalenttoaminimizing-congestionproblem.Thisisimmediateifwedeney(m)s;i=tz(m)s;iandmakethesubstitutionofvariables.But,wewilldothisalittledifferentlyforeaseofinterpretation.Letz(m)s=PjT(m)sji=1z(m)s;ibethetotalsendingrateatasourcenodesofsessionm.Selectasetofconstantsfr(m)sg,eachbeingproportionaltoL(m)s

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3 ),z(m)sisproportionaltoL(m)s,thetotalsizeofthechunksatsforsessionm.Then,z(m)s=r(m)s,forsomeconstant>0.Next,dene=1=.Wethenmakethesubstitutionofvariablesbyt=L(m)s Intheaboveformulation,r(m)scanbeunderstoodasthedemandedrate,andisthemaximumlinkutilization,whichalsomeasurestheworstlinkcongestion.Theproblemistominimizetheworstlinkcongestionsubjecttothefulllmentofalldemandedrates. Thus,wehavetwoequivalentviewsofoptimalmulticasttreepacking.Intherstview,theobjectiveistominimizetheoveralldistributiontime(ormaximizethedistributionthroughput)whilesatisfyingthelinkcapacityconstraint.Inthesecondview,theobjectiveistobestbalancethenetworkloadwhilesatisfyingtheratedemandforallsourcesandallsessions. Anotherminorreformulationwillbehelpfullater.Letestandfortheutilizationoflinke,andlet~denotethevectorofeoveralllinks.Letjj~jj1denotethemaximumnorm(i.e.,jj~jj1=maxe2Ee).Theaboveminimizing-congestionformulationisequivalenttothefollowing.minjj~jj1 66

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TheoptimizationproblemproposedsofarisequivalenttotheproblemofpackingSteinertrees[ 80 81 ],whichiscomputationallyintractable.Fortunately,forP2Pcontentdistribution,theproblembecomessimpler.Themainreasonisthattheoverlaynetworkofeachsessionconsistsofexactlythosenodesinthesession.Givensuchanoverlaynetwork,anySteinertreethatcoversallnodesofthesessionisinfactaspanningtree.Wewillshowlaterthatthealgorithmusedtosolvetheoptimizationprobleminvolvesaminimum-costspanningtreeproblemineachiteration.ShouldsomeSteinernodeexist,itwouldhaveinvolvedaminimum-costSteinertreeproblem.TheformerisfarmoretractablethanthelatterNP-hardproblem.Oneshouldberemindedthat,althoughthecomputationfortheoverlaynetworkcaseisfareasier,theachievableperformanceissub-optimalsincetheoverlayedgesaredeterminedbythexedunderlayrouting. Unliketheinexibleunderlayrouting,thebandwidthoftheoverlayedgemayhavealternatives.Next,weconsidertwocases,bothofwhichcanbeuseful. 82 ].Weassumetheoverlaynodesknowaboutthebandwidthoneachoverlaylink.Forinstance,inthecaseofTCP,theoverlaylinkbandwidthcanbemeasured.Whentheoverlaylinkbandwidthisxed,differentdistributionsessionsbecomedecoupled.WethenhavejMjtotallyindependentoverlaynetworks.TheoriginaloptimizationproblembecomesseparatedtojMjidenticalbutindependentoptimization 67

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Weillustratethisbyfocusingononeoftheoverlaynetworks,whichcorrespondstoonesession.Let^G=(^V;^E)representtheoverlaynetwork.Forallothernotations,sincethereisnodangerofconfusingthemwithearlierdenitions,wewillre-denethem.Thebandwidthassociatedwitheachoverlaylinke2^Eisce,whichisallocatedalreadyandisaconstant.Theutilizationofoverlaylinkeisdenotedbye.AssumeS^Visthesetofsources.LetTsrepresentthesetofallpossible(overlay)multicasttreesrootedatsourcesspanningalloverlaynodes.Letts;i2Tsbetheith(overlay)multicasttreeandzs;ibetheassociatedsendingrateontreets;i.Therateontheoverlaylinke2^Eis Theoptimizationproblemisnowminjj~jj1 68

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Theoptimizationproblemisexactlywrittenasin( 3 )-( 3 ). 3 )andkeepthesameconstraints.~servesasaregularizationterm.Thestrictlypositivevector~(i.e.,>0)guaranteesaglobalgeometricconvergencerateofourgradientprojectionalgorithm;if~=~0,wecanonlyclaimthatourgradientprojectionalgorithmconvergestooneoptimalsolutionglobally. 3 )becomesminXe2^E(xe 69

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3 ),theoptimalityconditionisespeciallysimple[ 47 ].Ithasbeenshownin[ 52 ]and[ 51 ]thatthereexistsaspecialgradientprojectionalgorithm.Forourcase,thegradientprojectionalgorithmcanalsobeeasilyextendedtoanequallysimplescaledversion.Thelatterovercomestheissuethatourproblemmaybeill-conditioned,andhence,drasticallyimprovesthealgorithm'sconvergencetime.Ourcomputationalexperienceshaveshownthatthescaledgradientalgorithmismuchfasterthantheunscaledalgorithmorthesubgradientalgorithm.Thelatterisoftenusedinnetworkoptimizationproblems. Anotherdifcultyisthelargenumberofpossiblespanningtrees,andhence,thelargenumberofvariables.Fortunately,thealgorithmdoesnotmaintainallpossiblespanningtrees.Thefollowingstepstakeplaceforeverysourceattheoverlaynetworklevel.Thealgorithmstartsoutwithoneorfewspanningtrees.Ineachiteration,acostisassignedtoeach(overlay)linktoreectthecurrentlinkcongestion.Then,aminimum-costspanningtreecanbecomputed.Thealgorithmshiftsanappropriateamountoftrafc(rate)fromeachcurrentlymaintainedspanningtreetotheminimum-costtree.Thenewminimum-costtreeentersthecurrentcollectionofspanningtrees.Somepreviousspanningtreemayleavethecollectionifitsdistributionrateisreducedtozero. Wenextillustratesomedetails.LetT=Ss2STsbethecollectionofallmulticasttreesrootedatanysource.Letzbethevector(zs;i)wheres2S;i=1;:::;jTsj,withanarbitraryindexingorderforthesources.Inproblem( 3 ),letthefeasiblesetdenedby( 3 )and( 3 )bedenotedbyZ. Foreachoverlaylinke2^E,recallthatxeistheaggregateowrateitcarries.Letxbethevector(xe)e2^E.LetHdenotethej^EjjTjlink-treeincidencematrixassociatedwiththetreesinT(i.e.[H]et=1iflinkeliesontreet;and[H]et=0otherwise).Obviously,x=Hz.Nowdene^fe(xe)=(xe=ce+)qand^f(x)=Pe2^E^fe(xe).Theobjectivefunction,denotedbyf(z),isgivenby 70

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3 )canbewrittenas s.t.z2Z: ce(xe Notethat@^f(xe) 47 51 ].Itreectsthecurrentcongestionlevelatlinke.@f(z) Foreachs2S,letisbetheindexofaminimum-costtreerootedats(i.e.,withsasthesource).Thatis, Iftherearemultipleminimum-costtrees,wechooseanarbitraryone.SincethefeasiblesetZisaconvexsetandtheobjectivefunctionisaconvexfunction,wecancharacterizeanoptimalsolutionztotheproblem( 3 )bythefollowingoptimalitycondition. Thisoptimalityconditioncanbeequivalentlywrittenas,foranysources2S, 71

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Itturnsoutthisisexactlywhatthegradientprojectionalgorithmdoes.Wewilldevelopthegradientprojectionalgorithmfollowingtheproposalin[ 51 ]tosolvetheproblem( 3 ).Butwewilladddiagonalscalingtospeedupthealgorithm'sconvergencetime. Theequalityconstraintin( 3 )impliesthat,foreachsource,oneofthevariablesdependscompletelyontherestofthevariables.Wecaneliminatethisvariableandhaveaproblemwithfewervariables.Tobeconcrete,atafeasiblevectorz,let'seliminatethevariablezs;isforeachs2S.Deneanewobjectivefunctiong(^z)onRjTjjSj,where^zconsistsoftheremainingzs;i'safterzs;isiseliminatedforeachs.Withoutlossofgenerality,suppose,foreachsources,iscorrespondstothetreewiththelargestindex(i.e.,is=jTsj).Alsosupposethesourcesareindexedfrom1tojSj.Then, 3 )-( 3 )isequivalenttoming(^z) 72

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where(k)isapositivestepsizeand[]+istheprojectionoperatoron^z0.Inthiscase,[y]+justmeansthat,ifyiisacomponentofy,wetakemax(yi;0)asthecorrespondingcomponentofthevector[y]+.Thekeyistocomputerg(^z(k)).Letis(k)beashorthandforis(z(k)).Itiseasytoshow,fors2Sandi6=is(k), Therstderivativesaregivenin( 3 ). Thealgorithmin( 3 )isactuallytheconstrainedsteepest-descentalgorithm.Itiswell-knownthatthesteepest-descentalgorithmcanbeslowiftheoptimizationproblemisill-conditioned.Ithappensthattheminimizing-congestiontypeofnetworkproblemsisoftenill-conditioned.Inourcase,theproblembecomesmoreill-conditionedwhentheparameterqintheqnormbecomeslarger.Anultimatesolutiontoanill-conditionedproblemisNewton'salgorithm.However,Newton'salgorithmisgenerallyverycomplexsinceitrequirestheinverseoftheHessianmatrixoftheobjectivefunction.Forlargeproblems,thiscomputationisgenerallyimpractical.Wewillnextdevelopthediagonallyscaledgradientalgorithm,whichisamuchsimpleralternativeandagoodapproximationofNewton'salgorithm.Thescaledgradientprojectionalgorithmcanbewrittenas whereD(k)isapositivedenitematrix.Fordiagonalscaling,D(k)ischosentobeadiagonalmatrix. 73

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Thisway,thematrixD(k)approximatestheinverseoftheHessianofgat^z(k).Foreachs2Sandi6=is(k),thesecondderivativeofgisfurthergivenby Thesecondderivativesaregivenin( 3 ). Wecannowcollectdifferentpiecesofthedevelopmentaboveandformallygivethediagonallyscaledgradientprojectionalgorithmintheoriginaldomainwherezlies.Aslightgeneralizationispresentin( 3 ). with In( 3 ),(k)isascalaron[a ;1],forsomea 3 )saysthatthenewratevectoratthe(i+1)thiteration,z(k+1),isonthelinesegmentbetweenz(k)andz(k). Themainpartofthealgorithmisexpression( 3 ),whichcomputestheendpointofafeasibledirection,z(k),entrybyentry.Therearethreecases. 74

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Notethatthedescriptionincase3ensuresthatz(k)isfeasible(inZ).Sincez(k)isalsofeasible,by( 3 ),thenewratevectorz(k+1)isfeasible.Hence,ifwestartwithafeasiblesolutioninZ,z(k)isinZforallk. 3 ),theratereductionisproportionaltothecostdifferencewiththeproportionalconstant(stepsize)s(k)>0. Thefactor(ds;i(k))1doesthediagonalscaling,whichcaneffectivelydealwithourill-conditionedproblem.Thescalingfactor(ds;i(k))1canbeunderstoodasallowingdifferentcomponentsofthevectorztousedifferentstepsizes.Notethat,intheexpressionfords;i(k)in( 3 ),whichcorrespondstotheithtree,thesumisoverthenon-overlappinglinksbetweentheithtreeandtheis(k)thtree,thelatterbeingtheminimum-costtree. Thealgorithmin( 3 )-( 3 )isadistributedone.Inordertocomputethetreecost,@f(z) 3 ),andthescalingfactor,(ds;i(k))1in( 3 ),eachlinkecanindependentlycomputeitscorrespondingtermbasedonthelocalaggregaterate,xe,passingthroughthelink.Then,thetreecostandthescalingfactorcanbeaccumulatedbythesourcesbasedonthelinkvaluesalongthetree.Tondtheminimum-costtreeis(k),eachsourceneedstocomputetheminimum-cost 3 )canbereplacedwithamoregeneralupdatez(k+1)=A(k)z(k)+(Iz(k))z(k),whereA(k)isaPs2SjTsjPs2SjTsjdiagonalmatrixwithdiagonalentriesintheinter-val[a ;1],forsomea 75

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83 ][ 84 ][ 85 ].BothachieveO(n2)timecomplexityforacompletegraphwithnnodes.Inthedistributedversion,theamountofinformationexchangedisalsoO(n2).Inourimplementation,eachsourcecollectsthe(overlay)linkcostsfromallthereceiversandusesacentralizedalgorithmtocomputetheMDSP.Otherthanthat,thegradientalgorithmiscompletelydecentralized. Inadditiontofastconvergence,anotherstrengthofthisgradientalgorithmliesinthatitavoidstheenumerationofallpossiblespanningtrees.Thesourceonlyneedstomanagethesetofactivemulticasttrees(i.e.,thosetreeswithpositiveows).Ateachiteration,thesourcecomputesanewminimum-costtree.Anon-activetreewillnotbecomeactiveunlessitistheminimum-costtree.Thesourceonlyadjuststheowratesamongthesetofactivetrees.Thesetofactivetreesusuallyisnotverylargeifthealgorithmconvergesfast,since,ateachiteration,atmostonemoretreebecomesactive.Fortheoriginallinearmodel( 3 ),thereareatmostj^Ej+jSjactivetreesinanyextremepointsolution.Butsincethisgradientalgorithmisakindofinteriorpointmethod,strictlyspeaking,j^Ej+jSjisnotreallyanupperbound.Nevertheless,itshouldgivearoughsenseonwhattheboundmightbe. Westressthatthereasontoapplythescalingfactoristocountertheill-conditionedproblemwhenqislarge.Inanill-conditionedproblem,single-unitchangesofdifferentvariableshavedisproportionateeffectsonthecost(e.g.,objective)change.Forconvergence,thestepsizeintheiterativealgorithmmustbetunedaccordingtothevariablesthatcauselargecostchanges.However,suchastepsizecanbetoosmallforothervariables,andasaresult,theyhardlychangefromiterationtoiteration.Thediagonallyscaledalgorithmessentiallyre-scalesthevariablessothatsingle-unitchangesinthescaledvariableshavesimilareffectonthecostobjective.Forourproblem,thescalinghasthesimpleinterpretationthatdifferenttreesusedifferentstepsizes,eachroughlybeingproportionaltoapoweroftheworstlinkutilizationonthetree.Theresultingscaledalgorithmisfarsuperiortotheplaingradientprojectionalgorithm. 76

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3.2.3 canbeworkedoutinasimilarway,leadingtoascaledgradientprojectionalgorithm.Substituteewithxe LetT=Ss2S(m)m2MT(m)sbethecollectionofallmulticasttreesrootedatanysourcesforanysessionm.Letzbethevector(z(m)s;i)wheres2S(m);m2M;i=1;:::;jT(m)sj,withanarbitraryindexingorderforthesources.Inproblem( 3 )-( 3 ),letthefeasiblesetdenedby( 3 )and( 3 )bedenotedbyZ. Let^E=Sm2M^E(m)bethecollectionofalloverlaylinksinallsessions.Let^Hdenotethej^EjjTjoverlaylink-treeincidencematrixassociatedwiththetreesinT(i.e.[^H]^et=1ifoverlaylink^eliesontreet;and[^H]^et=0otherwise).RecallEisthesetofunderlaylinks,letHdenotethejEjj^Ejunderlaylink-overlaylinkincidencematrixassociatedwiththeoverlaylinksin^E(i.e.[H]e^e=1ifunderlaylinkeliesonoverlaylink(underlaypath)^e;and[H]e^e=0otherwise). Foreachunderlaylinke2E,recallthatxeistheaggregateowrateitcarries.Letxbethevector(xe)e2E.Itiseasytoseex=H^Hz.Notethatanunderlaylinkemightcarrymultiplecopiesofthesamelechunkdistributedbyonetreet.Dene^fe(xe)=(xe=ce+)qand^f(x)=Pe2E^fe(xe).Theobjectivefunction,denotedbyf(z),isgivenby 77

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3 )canbewrittenas s.t.z2Z: ce(xe Foreachs2S(m)insessionm,leti(m)sbetheindexofaminimum-costtreerootedats(i.e.,withsasthesource).Thatis, Leti(m)s(k)beashorthandofi(m)s(z(k)). with Theresultingalgorithmisstillfullydistributed. 53 ]tondanupperboundonthestepsizethatguaranteestheglobalconvergenceofthesynchronousgradientprojectionalgorithmtoanoptimalsolution.Furthermore,withthestrictlypositive 78

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Intheoptimizationproblem( 3 ),weassumeq2,sothat^fe(xe)iscontinuousontheinterval[0;1),tendsto1asxeapproaches1,anditsderivativeandsecondderivativearecontinuousandpositiveon(0;1).Assumingthelinksareindexedfrom1toj^Ej,theHessianr2^f=diag[@2^f @x21;;@2^f @x2j^Ej]isanj^Ejj^Ejdiagonalmatrixwithnonnegativediagonalentries.Furthermore,if>0,thediagonalentriesofr2^farepositiveandboundedbelowbymine2^Efq(q1) Weassumethereisatleastonefeasiblesolution(i.e.,z(0)2ZsatisfyingHz(0)2Qe2^E[0;1)),anddeneacompactsetZ0=fz2Zjf(z)f(z(0))g.Sincethissetiscompact,fmustattainaminimumonthisset.Hence,thereisaz2Z0satisfyingf(z)=f,wheref=minz2Z0f(z).Wecallanysuchzanoptimalsolution,andwedenotebyZthesetofoptimalsolutions.(Theremaybemorethanoneoptimalsolutionssince,although^fisstrictlyconvex,fisnot.)Thatis 53 ][ 47 ]. Let1=a =(Lmaxs2SjTsj),whereL>0isanupperboundofthenormofr2foverZ0. maxs2SjTsjL TheproofLemma 5 followstheproofforasimilarlemmain[ 53 ].Theonlychangeinvolvessubstitutionofappropriateconstants. 3 )-( 3 )withz(0)2Z0isoptimal.

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Lmaxs2SjTsj,theright-handsideoftheinequality( 3 )isnon-positive.Hence,iffz(k)ghasalimitpoint,theleft-handsidetendsto0.Thealgorithm( 3 )-( 3 )canbedenotedasafunctionA(z)(i.e.,z(k+1)=A(z(k))).Therefore,jjz(k)z(k)jj!0,whichimpliesthatforeverylimitpoint~zoffz(k)gwehave~z=A(~z).Itiseasytoshowif~z=A(~z),foranys2S,wehave~zsi>0onlyif@f(~z) 3 ).So~zisstationary(Proposition2.3.2andExample2.1.2in[ 47 ]). Whentheregularizationvector~isstrictlypositive,thediagonalentriesofr2^farepositiveandboundedbelowbymine2^Efq(q1) 53 ]aresatised.Wecanstatetheglobalgeometricconvergencerateforalgorithm( 3 )-( 3 ). Suppose>0.Letsatisfy0<1.Thesequencefz(k)ggeneratedbythesynchronousgradientprojectionalgorithm( 3 )-( 3 )convergestoanelementofZwithaninitialfeasiblez(0)andtheconvergencerateislinear(i.e.,geometric)inthesensethatforallk, Theconstantsandparametersareasfollows.D5=a =(D4+1),D4=((5L+1)(D3)2+1+21+6L(1)2=a

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3.3 avoidtheenumerationofallpossiblespanningtrees,theycomputeaminimum-costspanningtree(cf.( 3 )and( 3 ))ateachiteration,whichisexpensive.Inthissectionwewillintroducethecolumngenerationmethodtoreducethenumberoftimeswhenthecomputationofminimum-costtreesisinvoked.Wewilldescribethecolumngenerationmethodontheproblemwithxedoverlaylinkbandwidth;thecolumngenerationmethodworksontheproblemwithoptimallyallocatedoverlaybandwidthinasimilarway,andforsimplicity,wedonotdetailitinthischapter. Letuscalltheproblem( 3 )themasterproblem(MP).Wecallthesub-problemofndinganewminimum-costspanningtreein( 3 ),aglobaltreesearchingproblem,sinceitinvolvesndingatreefromallpossibleones,thesolutiontothissub-problemaglobalminimum-costtree,andtheachievedminimumcosttheglobalminimumtreecost.Wedenotethisglobalminimumcostunderaxedzforanysources2Sby Thevalueofeachcomponentof~wisusuallysmallandthetreesofZ(~w)inthe~w-RMPareenumerable. 81

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Theoptimizationistakenoverthewscurrentlyknowntreesforeachsources2S.Theproblemin( 3 )iscalledalocaltreesearchingproblem,thesolutiontothissub-problemiscalledalocalminimum-costtree,andtheachievedminimumcostiscalledthelocalminimumtreecost.Wedenotethislocalminimumcostunderzby Iftherearemorethanonetreeachievingthelocalminimumcost,thetieisbrokenarbitrarily. LetzdenoteoneoftheoptimalsolutionsoftheMP,andz(~w)denoteoneoftheoptimalsolutionsofthe~w-RMP.Sinceanyoptimalsolutiontothe~w-RMPisfeasibletotheMPandthe~w-RMPismorerestrictedthantheMP,wegetthefollowingupperboundfortheoptimalobjectivevalueoftheMP. 82

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Foreachsources2S,letIsdenotethesetofindicesoftheminimum-costtreesattheoptimumz(~w)ofthe~w-RMP,thatis,Is=fi:@f(z(~w)) Therstinequalityisduetothepropertyofconvexfunctions.Thesecondequalityisexpandingtheinnerproduct.Thethirdequalityholdsbecausez(~w)s;i=0foranyithtreets;i2Tsandi62Is.ThefourthequalityholdsbythedenitionofIsandre-arrangingtheterms.ThefthinequalityholdsbecausePi2Isz(~w)s;i=rsbytheoptimalitycondition( 3 ),thedenitionofIsandthefeasibilityofz(~w);andz0.ThelastequalityisduetoPTsi=1zs;i=rs. By( 3 )and( 3 ),thegapbetweentheupperandlowerboundsfortheoptimalobjectivevalueoftheMPisPs2Srs((w)s(z(~w))s(z(~w))).Sincers((~w)s(z(~w))s(z(~w)))0andtheequalityholdsonlyif(~w)s(z(~w))=s(z(~w))foranys2S. 83

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Proof. 3 )and( 3 ),if(~w)s(z(~w))=s(z(~w))foranysources2S,f(z(~w))=f(z)andz(~w)isoptimaltotheMP. Ifz(~w)isoptimaltotheMP,accordingtotheoptimalityconditionoftheMP( 3 ),foranysources2S,z(~w)s;i>0onlyif[@f(z(~w)) 6 says,atthecurrentsolutionz(w),ifnoneofthetreesthatachievetheglobalminimumtreecostareinthesubsetZ(~w),thenthecurrentoptimalsolutionofthe~w-RMPisnotoptimaltotheMP.Inthiscase,therearereasonstoprefertheintroductionofthegloballyoptimaltreespeciedby( 3 )asthenewtreetotherestrictedmasterproblem.ThisstrategyisalocalgreedyapproachtoimprovetheupperboundoftheoptimalvalueoftheMP.Infact,itcanbeviewedasaconditionalgradientmethodforoptimizingtheupperbound,whentheupperboundisviewedasafunctionofz[ 7 ]. 3 andmakeseveralcommentsregardingthisalgorithm. 7 ]. 84

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3 )-( 3 )forseveral(anitenumber)timesontheRMP. 3 )underthecurrentrstderivativecost@f(z)foreachsources2S. 0. Ifforeachsources2S,thetreecorrespondingtothesolutionof( 3 )isalreadyinthecurrentcollectionoftrees,gotoStep1; 0. otherwise,introducethistreeintothecurrentcollectionoftrees,increase~w,andgotoStep1. 3 infactdescribesawholeclassofalgorithms.Inoneendofthespectrum,ifthediagonallyscaledgradientprojectionalgorithminstep1runsonlyonceontheRMP,thealgorithmbecomesapurediagonallyscaledgradientprojectionalgorithmasinSection 3.3 .Intheotherendofthespectrum,ifthediagonallyscaledgradientprojectionalgorithmrunsontheRMPuntilconvergence,thealgorithmbecomesapurecolumngenerationmethodwiththediagonallyscaledgradientprojectionalgorithmasabuildingblockforsolvingtherestrictedproblemsbetweenconsecutivecolumngenerationsteps.Bychoosingdifferentnumbersoftimestorunthediagonallyscaledgradientprojectionalgorithminstep1,wehavemanyalgorithms,representingdifferentperformance,convergencespeedandcomplextradeoffs. Theorem9. 3 convergestooneoptimalsolutionofthisparticular~w-RMP(i.e.,z(~w)).Furthermore,afterAlgorithm 3 convergestoz(~w),s(z(~w))=(~w)s(z(~w))foranys2S. Proof. 3 willstopintroducingnewtrees.Hence,thereexistsa~w,1wsjTsjforanysources2S,suchthat,afterAlgorithm 3 stopsintroducingnewtrees,thenumberoftreesthathavebeenintroducediswsforeachsources2S.Lettheconvexhullformedbythese~wtreesbedenotedbyZ(~w).AfterAlgorithm 3 nolongerintroducesnewtrees,itbehavesjustlikethediagonallyscaledgradientprojectionalgorithmbutontherestrictedsetZ(~w).AccordingtotheTheorem 7 ,thediagonallyscaledgradientprojectionalgorithmconverges.Thus,Algorithm 3 convergestoz(~w)onthisparticular~w-RMP. 85

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3 convergestoz(~w),wehaves(z(~w))=(~w)s(z(~w))foranys2S.First,notethats(z)(~w)s(z)bydenition.Next,itmustbetruethats(z(~w))(~w)s(z(~w))foranys2S.Otherwise,thetreewhosecostiss(z(~w))mustnothavealreadybeeninZ(~w)andwillbeselectedtoenter.Thisviolatestheassumptionthatthealgorithmneverselectsmorethan~wtrees. ByTheorem 9 andLemma 6 ,nallywehave, 3 convergestoanoptimumoftheMP. 3.2.2 and 3.2.3 ,weseetwocontentdistributionscenarioswitheitherxedoroptimally-allocatedoverlaybandwidth.Thelattershouldachievebetterdownloadingtimethantheformer.However,thelatterrequiresthedeploymentofouralgorithmstoallnetworkelement(i.e.routers),whichisalmostcertainlyimpossible.Thereisanalternativeframeworkinwhichsomeroutersordevicesattachedtotheroutersaredeployedwithouralgorithm,whileothersnot.Forinstance,ouralgorithmcanbedeployedatthecross-ISPlinksandaccesslinks,wherethe 86

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Ifanysourceowninguniquechunksleavesbeforeitnishesdisseminatingthem,thosechunksarenolongeravailableinthenetwork.Tominimizesuchrisk,intheoptimizationformulation,wecanadjusttherequestedsendingratesrsofthesources.Ifanysourceisexpectedtoleavethenetworksoon,itmayrequest(orbeassigned)ahighersendingratesothatitcanspreaditschunkstonetworkmorequickly. Othertypesofnetworkandmemberdynamicsincludelinkfailures,thechangeoflinkcapacities,thearrivalofnewsources,andthedepartureandarrivalofreceivers.Asarguedin[ 22 ],adistributedalgorithmhasbuilt-inabilitytoadapttovariations.Adistributedalgorithmcanreactrapidlytoalocaldisturbanceatthepointofdisturbancewithslowernetuningintherestofthenetwork.Suchadaptiveabilityisintimatelyconnectedwiththealgorithm'sspeedofconvergenceinthestaticcase.Sinceouralgorithmistheresultofconsciousefforttoimprovetheconvergencespeed(bydiagonalscalingofthegradientalgorithm),webelieveitissuperiorincopingwithnetworkandmemberdynamicscomparedtoothersimilardistributedalgorithms.Inaddition,ourdistributedalgorithmisnaturallyrobustbecauseofthelackofrelianceona 87

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3.6 ,wewillshowasmallexampleofhowouralgorithmsuccessfullyadaptstothedepartureandarrivalofreceivers. 85 ].Butthepricetopayisthepotentiallyslowerspeedduetothecoordinationoverheadofdistributedoperation.).Thus,ourgradientalgorithmcanonlydealwithdistributionsessionswithlimitedsize,sayseveralthousandsofmembersineachsession.Inordertoimprovethescalabilityofouralgorithm,weshallpartitioneachsessionandrunthealgorithmhierarchically,asmostscalablenetworkalgorithmswoulddo.Thoughcurrentlywedonothaveawell-denedwaytopartitionthesession,wewillshowonenaiveapproachtopartitionalargesessioninSection 3.6 3 )-( 3 )(or( 3 )-( 3 ),respectively)couldbedevelopedandthecorrespondingconvergenceresultcouldbestatedfollowingtheapproachin[ 54 ][ 53 ]. 74 ].WeselectBitTorrentbecauseitstechniquesareinterestinganditisthemostpopularP2Papplication.WeselectAFRbecauseitcanbethoughtasusingmultiplemulticasttreesfordistribution.Butonlyasubsetofthetreesareallowed,whichwecalltwo-leveltwo-phasetrees.Ineachofthesetrees,thesourceisconnectedtoonereceiveratlevel1,andthenthelevel1receiverisconnectedtoallotherreceiversatlevel2.Itmightappearthatsuchacollectionoftreesisquiteenoughforachievingnearoptimalperformance.Inanaccess-constrainednetwork,thisisindeedtrue.However,we 88

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Althoughwearemoreinterestedinnetworkinteriorbottlenecks,ouralgorithmcanequallydealwithbottlenecksattheaccesslinks,attheISPbackboneoratthecross-ISPlinks.Hence,wewillconsiderallthesecases.ThecommercialISPbackboneandcross-ISPtopologiesareobtainedfromtheRocketfuelproject[ 86 ].IntheterminologyofBitTorrent,aseedisasource,andaleecherisareceiver.Intheprevioussections,ourobjectivefunctionistheworstnetworkutilizationjj~jj1.Intheevaluationpart,wewillfocusonthesourcethroughputRs=rs=jj~jj1,wherersisthescaledsendingrate,andthedownloadingtimet=Ls=Rs,sincethesearewhatBitTorrentexperimentsyielddirectly.However,recallthatthetwomeasuresarethetwosidesofthesamecoin. BitTorrentsimulation.WeusetheBittorrentsimulatordevelopedbyBharambeet.al.[ 87 ].Sincetheoriginalsimulatoronlysupportsaccesslinkconstraint,wemodiedthesimulatorsothatitsupportsgeneralphysicalnetworktopologies.Theoverlaylinkbandwidth,whichistheper-connectionbandwidthattheunderlay,isdeterminedbythemax-minbandwidthallocation[ 82 ].IntheBitTorrentsimulation,weusethefollowingsimulationenvironment. 87 ].Seedsarewithdistinctles. Table 3-2 summarizesthesimulationenvironmentforourtestcases. 74 ],the 89

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wheren0isthesource,ni,i=1;;marethereceivers,andcninjistheend-to-endpath(overlaylink)capacitybetweenniandnj. 88 90 ],researchershaveanalyzedamodelofP2Plesharingamongresidentialusersinlowaccess-speedenvironment.Eachparticipatingend-systemhasanuplink(tothenetwork)andadownlinkwithlimitedcapacity.Thecapacityofthenetworkisconsideredunlimited.ThesourceistodistributealetoLreceivers.Lettheuplink(downlink)bandwidthofreceiveribeui(di,respectively),fori=1;;L.Lettheuplinkcapacityofthesourcebeus.Then,themaximumdistributionspeedisshowntobe In( 3 ),thethreetermsaretheoptimalspeedswhenthebottleneckisatthesourceuploadlink,atadownloadlink,orduetotheaggregateuploadbandwidth,respectively. Bytwo-phasedistribution,wemeaneachdistributiontreehasadepthatmost2.Thefollowingfactisknowntobetrue. 88 ]achievesthe(overlay-network)routingcapacity[ 88 ],whichis( 3 ). 91 ][ 92 ]. 90

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InTable 3-3 ,theoptimaldownloadingtimeiscomputedfromby( 3 ).Theresultsindicatethatthegradientalgorithmobtainsnearthetheoreticallyoptimalsolution. 3-3 showsthetimewhen50%,95%and100%re-ceiversnishdownloadinginBitTorrent,respectively.BitTorrent'sperformanceisnotbadcomparedwiththeoptimalvalue.Thiswasexplainedin[ 66 ],whichmodelsthedownloadingtimeofBiTtorrent.Itshowsthat,inthecasethataashcrowdarrivesatthesametime,thebandwidthconstraintisattheaccesslinks,andthereceiversstayaftertheynishdownloading,BitTorrentachievesnearoptimaldistributionspeed.InFig. 3-3 and 3-4 ,weshowtheperfor-mancecomparisonofdifferentdistributionschemesunderprole1and4.Theresultsunderprole2and3areomittedforbrevity.Thedownloadpercentagereferstothetotalamountofdatadownloadedateachtimeinstancenormalizedagainstthetotaldatadownloadedattheendofthedistribution.Sincethetwolineshavedifferentslopes,wecanextrapolatethelinesandexpectthegradientalgorithmtodomuchbetterifthelesizebecomeslarger.Thisobservationseemstocontradicttheconclusionin[ 66 ]. 3 ).Table 3-3 showsthatAFR'stwo-phaseapproachachievestheoptimaldownloadingtimewhenthebottleneckiseitheratthedownloadlinkoratthesource.Butwhenthebottleneckisduetotheaggregateuploadbandwidth,AFRfailstoachievetheoptimum,althoughweknowthatsomeothertwo-phase 91

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WealsocomparethenumberofactivetreesAFRandthegradientalgorithmeventuallyuse.ThegradientalgorithmusesfewertreesthanAFR.Weinspectedtheactivetrees.Withtheaccess-linkconstraint,thegradientalgorithmusesthreekindsoftrees,thedepth-rstsearchtree(DFS),thebreadth-rstsearchtree(BFS),andatwo-phasetree.Fig. 3-5 showsthestructuresofthethreetypesoftreesontheoverlaynetwork.Ingeneral,thereexistsanoptimalsolutionthatusesonlytwo-phasetreesfornetworkswithsuchastartopology.Butitseemsthatthegradientalgorithmprefersthechain-likeDFStreeandthefatBFStree.Itmayappearcounter-intuitivethatsuchachain-likedistributionpathispreferredbecausethechainseemstoinvolvelargestdelay.However,thisisinfactnottruebecauseofouruidmodeloftrafcandbecausewedonotconsiderpropagationdelay.Thebitthatarrivesatnode1fromthesourcecanimmediatelybetransmittedtonode2,andtonode3,soon.Weleaveittofutureworkonhowtoincorporatethepropagationdelayinoptimaltreeselection.Table 3-1 showsthat,whenthebottleneckisateitherthedownloadlinksorthesource,thegradientalgorithmnaturallypreferstheDFStree.Whentheaggregateuploadbandwidthisthebottleneck,wehavetodistributethechunksoverthetwo-phasetrees.Inaddition,themoreheterogenousthereceiversare,themoretwo-phasetreesweneedandthemorebandwidthisallocatedtothetwo-phasetrees.Thekeyconclusionhereisthatthegradientalgorithmisabletondthebestdistributiontreesfortheparticularnetworkenvironment.Withoutthehelpofthegradientalgorithm,whattypesoftreesareselectedisnotalwaysobvious. 86 ].The 92

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Wedidseveralexperimentswiththelinkcapacitiesuniformlydistributedinsomerange.Theactuallinkcapacitydataisunavailable.Wendthegradientalgorithmoftengivestrivialoptimalsolutions.Afterinspectingthesolutionsandthenetworkgraphs,itturnsoutthattheISPbackboneispoorlyconnected.Therearemanylinksthatliesonalltheroutingpathsbetweenonepeerandallotherpeers,whichmeansifanyoneofthesecriticallinksisremoved,atleastonepeerwillbedisconnected.Ifthesecriticallinksdonothavemuchlargercapacitythanotherlinks,theyarelikelytobecomethebottleneck.Thegradientalgorithmisabletolocatethebottleneckimmediately.OtherveISPbackbonesshowthesameproperty.Presumablyinreality,theISPsareawareofsuchlinksandwouldensuretheyhaveverylargebandwidthsothattheyareneverthebottleneck.Inordertotestouralgorithminthisnon-trivialscenario,weassignthesamebandwidth,1000,toallbackbonelinks.Then,wescaleupthebandwidthofallcriticallinks(thoselinksthat,whenremoved,willleavesomepeersdisconnectedintheoverlay)tobelargeenoughsothattheyarenotthebottleneck. WedidthreetestsonProle5. 93

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3-6 showsthat,inTesta,thedownloadingtimeinthegradientalgorithmisonly30%ofthatinBitTorrent.BitTorrentisunabletogivegoodperformancewhenthecorenetworkiscongested.ButinTestb,aftertheoverlaybandwidthisxed,theoptimaldownloadingtimeismuchhigherthanthatofTesta,andalmostequaltoBitTorrent'stime.Inthegure,thedownloadpercentagereferstothetotalamountofdatadownloadedateachtimeinstancenormalizedagainstthetotaldatadownloadedattheendofthedistribution.Sincethelineshavedifferentslopes,wecanextrapolatethelinesandexpectthegradientalgorithmtodomuchbetterifthelesizebecomeslarger. 3-6 alsoshowsthat,inTestc,thegradientalgorithmap-proachesthemax-owlimitwhileAFRachievessomethingfarfromtheoptimum.Whenthecongestionhappensatthecorenetwork,thetwo-phasetreesalonefailtogivegoodsolution. 3-7 showstheconvergenceofthealgorithminTesta,bandcrespectively.ThetimespentononeiterationisaboutoneroundtriptimeplusthetimetocomputetheMDSP.Itseemsthatthealgorithmthatoptimallyallocatestheoverlaybandwidthconvergesmuchfasterthanthealgorithmwithxedoverlaybandwidth.Thishastodowiththefactthat,inthenalsolution,Testahas92activetrees,whileTestbhastotally4746activetrees.ItispossibleTestbhasanotheroptimalornearlyoptimalsolutionthathasmuchfewertrees.Findingsolutionswithfewertreesshouldimproveconvergencespeed,andisanimportantdirectiontopursue. 94

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Notethat,inthecaseofasinglesource,congestionatthecross-ISPlinksandeachISPcontainingsomepeers,themax-owlimitisachievable. Inbothproles6and7,thegradientalgorithmapproachesthemax-owlimitandbeatsBitTorrentandAFRbyalargeamount,uptoafactorof10(Tabel 3-4 :50%,95%and100%arethepercentageofthepeersthathavenisheddownloading.AlsoseeFig. 3-8 and 3-9 .).Wefound,withmorepeersperISP,BitTorrent'sperformancedeteriorates.FastReplica'sperformanceisfarworsethanboththegradientalgorithmandBitTorrent,andwedidnotevenshowitinFig. 3-8 and 3-9 Usually,cross-ISPtrafcismoreexpensive.Weinvestigatedthetrafcredundancyoverthecross-ISPlinks.WithneitherinsideISPcongestionnoraccessspeedconstraint,ideally,eachdestinationISPshouldreceiveonlyonecopyofeachchunkfromotherISPsandthesourceISPshouldnotreceiveanycopyfromotherISPs.InProle6,weinspectedtheactive(optimal)treesthealgorithmconstructedandfoundthateachdestinationISPindeedonlyreceivedonecopyfromotherISPs,butthesourceISPmightreceivesomecopiesfromotherISPs.Supposethenormalizedcross-ISPtrafcundertheidealdistributionis1:0.Wefoundthecross-ISPtrafcwas1:103inthegradientalgorithmand5:982inBitTorrent.ButinProle7,wefoundthedestinationISPsreceivedmultiplecopiesfromotherISPsinthegradientalgorithm.Thisisbecause,inProle6,eachmax-owbetweenthesourceISPandthedestinationISPhasthesamevalue.ThisisnotthecaseinProle7.Thus,theoptimalsolutiondoesallowmultiplecopiestobesenttoonedestinationISP.Again,ifthenormalizedcross-ISPtrafcundertheidealdistributionis1:0,thenthetrafcis2:22inthegradientalgorithmand9:72inBitTorrent. 95

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3.4 ,wecanintroducenewtreesatvaryingdegreeoffrequency.WedidexperimentswiththeProle5.Intheexperiments,wewillusethreefrequencies:fast,mediumandslow.Withthefastfrequency,wetrytointroducetreesbysolvingtheglobaltreesearchingproblem( 3 )ateachupdateof( 3 )-( 3 ),inwhichcase,Algorithm 3 degeneratesintothepuregradientprojectionalgorithm.Withtheslowfrequency,wetrytointroduceanewtreeafterevery20updatesof( 3 )-( 3 ).Inthiscase,Algorithm 3 ismoreclosetothepurecolumngenerationmethod.Withthemediumfrequency,weintroduceanewtreeevery5updates. WeevaluatethealgorithmontheoverlaynetworkwiththesinglesourceobtainedfromProle5Testc.InFig. 3-10 ,thefastschemealwaysimprovesthethroughputmorequicklyatthebeginning,whiletheslowschemeimprovesitmuchmoreslowlythantheothertwoschemes.Thereasonisthat,withthefastscheme,plentyoftreesareintroducedquickly.Theslowschemealwaystriestotakefulladvantageofthecurrentcollectionoftrees.Butlater,theslowschemecatchesupthefastscheme.Thismotivatestheuseofthemediumscheme.InFig. 3-10 ,weseethatthemediumschemeincreasesthethroughputnearlyasquicklyasthefastschemeatthebeginninganditsurpassesthefastschemesoonafter. InTable 3-5 ,wecomparethethreeschemesfortheircomputationcosts.Sincethemostexpensivecomputationisforsolvingtheglobaltreesearchingproblem( 3 ),thetotalcompu-tationtimeismainlycharacterizedbythenumberoftimestheglobaltreesearchingproblemissolved.Oneexpectsthatloweringthefrequencyofintroducingnewtreesiscorrelatedwithfewercomputationsfortheglobaltreesearchingproblem.But,weknownotheoreticalreasonswhythismustbetrue.Theresultconrmstheexpectation:Thenumberofsuchcomputationsis3500,700,and300,forthefast,mediumandslowschemes.Thereductionisdramatic. Wealsond,withalowerfrequency,thealgorithmusuallyproducesasolutionwithfeweractivetrees.Feweractivetreesmaybedesirablesinceitiseasiertomanageandcontrolthem,whichmayreducethesystemcomplexityandcontroloverhead.Theslowschemeonlyuses 96

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BasedonthestudyofFig. 3-10 andTable 3-5 ,weconcludethatthepuretwo-timescale(fast)orthepurecolumngeneration(slow)algorithmshavebothprosandcons.Anintermediatealgorithm(medium)mayachieveamoredesirablebalanceamongfactorssuchasoptimizationperformance,thecomputationalcost,andsystemcomplexityandoverhead. 3-11 showsthealgorithmadaptstothedynamicsquickly. 31 ]andLowet.al.[ 32 ]onoptimalowcontrol/bandwidthallocation.Manyrecentpapersextendedthisapproachandsolvednetworkingproblemsbycollectiveactionstakenacrossnetworkinglayers, 97

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8 9 35 93 ]).Severalotherrelatedstudies,eitherintopicsormethods,are[ 94 97 ]. 98

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Distributionofbandwidthallocatedfordifferenttrees. Prole1Prole2Prole3Prole4 DFS99.4%99.999%98.91%1.93%BFS0.26%00.754%0.95%Two-phase0.33%0.001%0.333%97.1% Table3-2. BitTorrentsimulationparameters. #SeedsFileSizeperSeedNeighborhoodSize Proles1-3162.765MB38-80Prole41128MB18-40Prole5264MB18-40Prole61128MB38-80Prole7132MB38-80 Table3-3. Comparisonofdownloadingtime(minutes)andnumberofactivetrees. Prole1Prole2Prole3Prole4 Optimum23.830.642.4331.4BT(50%)27.641.044.9264.8BT(95%)28.541.349.7428.6BT(100%)30.441.551.0441.8AFR23.830.642.7337.9GP23.930.643.5333.1GP#trees32353AFR#trees299299299100 99

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Downloadingtime(minutes)comparisonofProle6and7. BT(50%)BT(95%)BT(100%)GPAFRMFL P61619.519.63.8142.73.4P75.8326.5908.74131.28.72 Table3-5. Performancecomparisonofthefamilyofalgorithms(Prole5). FastMediumSlow #Iterations350035006000#TreesComputed3500700300#ActiveTrees1815313244#TreesIntroduced2169319244 100

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Node1sendstheletonode2and3. Figure3-2. AllpossibledistributiontreesfortheexampleinFig. 3-1 (a) (b) Figure3-3. Performancecomparison(Prole1):(a)numberofleechersthathavecompleteddownloadovertime;(b)downloadpercentageovertime. 101

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(b) Figure3-4. Performancecomparison(Prole4):(a)numberofleechersthathavecompleteddownloadovertime;(b)downloadpercentageovertime. Figure3-5. Structuresoftreesontheoverlaynetwork. (a) (b) Figure3-6. Performancecomparison(Prole5):(a)numberofreceiversthathavecompleteddownloadovertime;(b)downloadpercentageovertime. 102

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(b) Figure3-7. Convergenceofthroughput(Prole5).Twosourceswithrs=1:0:(a)Testa;(b)Testbandc. (a) (b) Figure3-8. Performancecomparison(Prole6):(a)numberofreceiversthathavecompleteddownloadovertime;(b)downloadpercentageovertime. (a) (b) Figure3-9. Performancecomparison(Prole7):(a)numberofreceiversthathavecompleteddownloadovertime;(b)downloadpercentageovertime. 103

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Convergenceofthefamilyofalgorithms(Prole5). Figure3-11. Dynamicdepartureandarrivalofreceivers. 104

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7 19 ].Theproblemcanbeformulatedasthemaximizationoftheaggregatesourceutilityoverthenetworkcapacityconstraints.Unlikethesimilarprobleminthewirednetwork,theessentialnatureoftheprobleminthewirelesssettingisthatthenetworkcapacityitselfisadecisionvariable.Duetowirelessinterference,notalltransmissioncongurationsareallowedateachtimeinstance.Forinstance,inthewell-knownmodelofthemultipleaccessschemeforthe802.11network,anallowedcongurationisasubsetofthelinkswhosetransmissionsdonotinterferewitheachother.SchedulingattheMAClayeristodecidewhichoftheallowedcongurationsshouldbeusedandhowtheyshouldbeused(e.g.,timeshared).Theresultofschedulingimplicitlydeterminesthenetworkcapacity. Thestandardsubgradientalgorithmisagoodcandidateinsolvingsuchaproblem.Bythesubgradienttechnique,theratecontrolandthewirelessresourceallocationaredecoupled:Thesourcesadapttheirsourceratesaccordingtothepathcongestioncosts,whereastheMAC-layerschedulingadjuststhetimeshareofdifferentallowedtransmissioncongurations,thusvaryingthelinkcapacitiesaccordingtothelinkcostssoastosupporttheowrates.However,thestandardsubgradienttechniquehasitsownlimitation,whichwillbediscussed. Weproposeatwo-timescale,column-generationapproachwithimperfectglobalschedulingtosolvetheaboveproblem.Comparedwiththesubgradienttechniqueandothers,ourapproachoffersthefollowingfeatures. 105

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10 11 ].Inourapproach,weallowtheintroductionofasub-optimalextremepoint,whichisoftenfareasiertoobtain.Thisopensthedoorfortheapplicationofmanyheuristicalgorithmsinsolvingthehardcombinatorialproblem.Importantly,weshowthat,ifthesub-optimalextremepointisa-approximationsolutiontothecombinatorialoptimizationproblem,thentheoverallutility-maximizationproblemalsoachievesapproximation. Wesubsequentlycallthesub-problemofndinganewextremepointinthecolumngenerationalgorithm,globalscheduling,sinceitinvolvesndinganallowedtransmissionschedulefromallpossibleones.Thissub-problemisacombinatorialoptimizationproblemonanexponentialnumberofpossibilities.Aperfectschedulereferstoanoptimalsolutiontothesub-problem;animperfectschedulereferstoasub-optimalsolutiontothesub-problem.Otheralgorithmsusuallyalsocontainthissub-problem.Howtoavoidglobalschedulingasmuchaspossibleandhowtosolveitfastwhenneededaretwokeyissues.Thischaptermakescontributionsinboth. Wenowgiveabriefsummaryofpriorworkonthejointdesignofcongestioncontrol,routingandschedulinginwirelessnetworks.Asurveyofresourceallocationandcross-layercontrolinwirelessnetworkscanbefoundin[ 19 ].Varbrandetal.[ 14 ]proposethecolumngenerationmethodtosolvetheresourceallocationprobleminwirelessadhocnetworks.JohanssonandXiao[ 7 ]extendtheuseofthecolumngenerationmethodtosolvethesameproblemundermorecomprehensivewirelessinterferencemodels.Butboth[ 14 ]and[ 7 ]givecentralizedsolutions,wheretherestrictedmasterproblemsaresolvedbysomelinear/nonlinearsolvers(Weareinterestedindistributedalgorithms.);andtheyonlyconsiderthecasewhereperfectschedulingisused.[ 17 ]alsogivesacentralizedcolumngenerationsolution.Bohacekand 106

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12 ]implicitlyapplythecolumngenerationmethodandtheirapproachiscentralized.In[ 10 11 ],theauthorsproposeawaytosolvethisproblembyadistributedsubgradientalgorithmwithimperfectscheduling.TheirapproachandconclusionaredifferentfromoursandwewilldetailthedifferencesinSection 4.4 .In[ 9 15 ],theauthorsformulatesimilarproblemsasoursanddevelopsubgradientalgorithms.[ 18 ]discussestheframeworkofcross-layeroptimizationinwirelessnetworks.Anotherrelatedchapteris[ 13 ].Thetwo-timescaleadaptivemethodisproposedin[ 22 ],andusedin[ 98 99 ]fortheproblemofmulti-pathrouting.Toourbestknowledge,nopriorworkhascombinedthethreeelementstogether,twotimescales,columngenerationandimperfectscheduling. Thechapterisorganizedasfollows.ThenetworkmodelandproblemformulationaregiveninSection 4.2 .Thetwo-timescalealgorithmanditsconvergenceproofaregiveninSection 4.3 .InSection 4.4 ,wepresentthecolumngenerationapproach,combineitwiththetwo-timescalemethod,andstudytheimpactofimperfectscheduling.Weshowtheperformancewithimperfectschedulingisbounded.InSection 4.5 ,wegivetheexperimentalexamples.TheconclusionisdrawninSection 4.6 11 ].Therate-powerfunctionisdeterminedbytheinterferencemodel. 107

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AccordingtoShannon'scapacitytheorem,themaximumdatarateoflinkeisce=Wlog(1+!e(P)),whereWisthesystembandwidth.Inpractice,thelinkrateisusuallylowerthantheShannoncapacity.Typicalwirelesssystemsallowanitesetoflinkrates,sayc1e;;cke,whichareassociatedwithasetofthresholdsfortheSINR,(c1e)e;;(cke)e.Thisisusuallyduetothenitenumberofmodulation/codingschemesbuiltintothewirelesstransceiver.Alinkecanusethetransmissionratecje,if!e(cje)e. Tosummarize,atanytimeinstance,thenumberofpossibleratevectorsisnite.Eachoftheseallowedratevectorswillbecalledaschedule.LetQdenotethetotalnumberofschedules.Letc(i)=(c(i)e)denotetheithschedule(ratevector)inthesetoffeasibleschedules,fori=1;;Q,wheretheorderisarbitrary.ThoughQisnite,itmightbeexponentialinthenumberoflinks.Bytimesharingofthesefeasibleschedules,theachievabletime-averagelink-rateregionistheconvexhullofc(i),i=1;;Q.DenotethisconvexhullbyC.Thus,Cisaconvexpolytope.Withslightabuseofterminology,wecallc(i),i=1;;Q,theextremepointsofC.Infact,someofthemmaynotbeextremepointsofthepolytope.Foranyc2C,itcouldberepresentedbythefollowingconvexcombinationoftheextremepointsofC, whereidenotesthetime-sharefractionoftheschedulethatusestheschedulec(i).Onecanndmorediscussiononwirelessinterferencemodelsin[ 7 ]. 108

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Theoptimalresourceallocationandschedulingproblemisformulatedas s.t.Ps:e2psxsce;8e2Ec2C 4 )withtheequivalentexpressionin( 4 ),were-writetheaboveproblemasfollows. MP:maxPs2SUs(xs) s.t.Ps:e2psxsPQi=1ic(i)e;8e2E 109

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4 ).TheLagrangianfunctionofMPis s.t.PQi=1i=1xs0;8s2Si0;8i=1;;Q: Dual-MP:min() s.t.0: 4.4 ,wewillcombinethistwo-timescalealgorithmwithacolumngenerationalgorithmandderiveafamilyofalgorithms. Werstconsidertheratecontrolproblemwithxedtimefractionvector. MP-A:():=maxxPs2SUs(xs) s.t.Ps:e2psxsPQi=1ic(i)e;8e2E

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MP-B:max() s.t.PQi=1i=1i0;8i=1;;Q: 4 ).TheLagrangianfunctionofMP-Ais 47 ].SincethereisnodualitygapattheoptimumofMP-Aunderaxed,wecanre-write()astheoptimalobjectivefunctionvalueofthedualproblemofMP-A, Dual-MP-A:()=min0A(;):(4) 111

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4 )canbesolvedbythesubgradientmethodasinAlgorithm 4 ([ 47 100 ]),where(t)isapositivescalarstepsize,and[]+denotetheprojectionontothenon-negativedomain. Let(x();())denotetheoptimalprimal-dualsolutionsofMP-Aunderaxed. 4 )-( 4 ).ThenA(;(t))!A(;())andthereexistsasubsequenceTsuchthatf(t)gT!()[ 100 ]. 4 )-( 4 ),if(t)!(),thenx(t)!x(). Proof. 4 )-( 4 )worksundertheassumptionthatthetimefractionvectorremainsconstant.Nowwediscusshowtoadjusti;i=1;;Q,tosolvetheproblemMP-B.WeassumetheupdateofismuchslowersothattheminimizationofA(;)overcanberegardedasbeinginstantaneous.Here,wefollowtheapproachesin[ 22 98 99 ]. Letkindexthetimeslots(calledstages)oftheslowtimescale.Atstagek,giventhetimefractionvector(k),suppose(k)2argmin0A((k);)isanoptimaldualsolutiontoMP-A.Let'scalle(k)thepriceorcostoflinke.Therefore,e(k)c(i)eisthecostoflinkeundertheithschedule(i.e.,theithextremepointofC);andPe2Ee(k)c(i)eisthecostofthenetworkunder 112

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Ifthereisatie,anarbitrarymaximizingindexischosen.( 4 )maybecalledaschedulingproblem[ 11 ],sinceitaimsatndingaschedule.Because( 4 )isanoptimizationproblemoverallallowedschedules,1;;Q,wecall( 4 )aglobalschedulingproblem,andtheachievedmaximumcosttheglobalmaximumcostoftheschedule.Wedenotethisglobalmaximumcostunderaxedby ThetimefractionupdateisshowninAlgorithm 5 ,whichissimilartotheonein[ 22 98 99 ]. (4) Here,(k)isapositivestepsize.Notethati(k)0fori6=i(k)andi(k)0fori=i(k).Hence,thealgorithmincreasesthetimefractionofthemostcostlyschedulewhiledecreasesthetimefractionsofotheractiveschedules(i.e.,thosescheduleswithpositivetimefractionsi(k)).Furthermore,ifPQi=1i(k)=1,thenPQi=1i(k+1)=1.Hence,(k)willalwaysbevalidtimefractionvectorsforallkifPQi=1i(0)=1. 113

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Equalityin( 4 )occursifandonlyifi(k)=0foralli,whichisequivalentto Conditionsin( 4 )-( 4 ),andthosedescribedintheproceedingparagraphguaranteethatthetimefractionupdatealgorithmconvergestothecorrectoptimalvalue.Asin[ 99 ],weconsideracontinuous-time,differentiableversionofthealgorithm( 4 )-( 4 ).First,denetheset Thedifferentiableversionofthealgorithm( 4 )-( 4 )satisesthefollowingconditions,forany()2().QXi=1_i=0; (4) Theconditionin( 4 )isequivalentto 4 )-( 4 )convergestoanoptimalsolutionoftheproblemMP-B.

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47 ],page717).Thetheoremrequirestobeinacompactset.Inotherwords,itrequiresthatthereexistsacompactsetindependentofsuchthat()=min0A(;)=min2A(;).Wewillnextconstructonesuchcompactset.SinceUs()isconcave,wehaveU0s(0)U0s(xs)forallxs0.UnderassumptionA3,takesomeKmaxs2SU0s(0)>0.Let=f:0eK;8e2Eg.Forany62,thereexistsanon-emptysubsetE1E,wheree>Kforanye2E1andeKforanye62E1.LetdenoteasubsetofsourcesbyS1S,whereforanysources2S1,itsroutingpathpscontainssomelinksinthesetE1.Weconstructavector02where0e=Kforanye2E1,and0e=eforanylinke2EnE1.Foranys2S,ifitsaccumulatedpathcostisnolessthanK,thenthemaximumofUs(xs)xsPe2pseinthedenitionofA(;)isachievedatxs=0,whichmeansforanys2S1, ThenA(;)=QXi=1i(Xe2Eec(i)e)+Xs2S1maxxs0fUs(xs)xsXe2pseg+Xs2SnS1maxxs0fUs(xs)xsXe2pseg

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Thusforany,theminimumofA(;)over0occursin. TheconditionsrequiredbyDanskin'stheoremaremet.Let0(;_)denotethedirectionalderivativeof()inthedirectionof_.Let0A(;;_)bethedirectionalderivativeofA(;)atinthedirectionof_.Then,byDanskin'stheorem,0(;_)=min2()0A(;;_)=min2()QXi=1(Xe2Eec(i)e)_i=QXi=1(Xe2Ee()c(i)e)_i: where2()achievestheminimum. Then,by( 4 ),0(;_)0: BytheLasalleinvarianceprinciple[ 101 ],(t)convergestoaninvariantsetinsidef:0(;_)=0g.Takeatrajectoryinthisinvariantset,whichsatises0(;_)0.By( 429 ),PQi=1(Pe2Eec(i)e)_i0.Then,by( 4 ),_i0foralli.Hence,theinvariantsethasonlyonepoint,whichwillbedenotedby.Hence,(t)convergesto. Next,wewillshowthatsolvesproblemMP-B.Letx()and()betheoptimalsolutionofMP-Aunder.MP-Amaximizesastrictlyconcavefunctionwithlinearconstraints,andhence,theKKTconditionsarebothnecessaryandsufcientoptimalityconditionsforMP-A[ 47 ].Thus,attheoptimum(x();()),wehavethatx()isprimalfeasibleand()is 116

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At,wehave_=0.Hence,accordingto( 4 ),wehave Also,by( 4 ),ifweinitializetheupdateofatsome(0)satisfyingPQi=1i(0)=1,wewillhave whichimpliesthat Obviously,x();()andareallnon-negative.Thesenon-negativityconditions,thefactthatx()isprimalfeasibleforMP-A,andtheconditionsin( 4 )-( 4 )and( 4 )-( 4 )aretheoptimalityconditionsoftheMP.Hence,(x();;())isanoptimalprimal-dualsolutiontotheMP(alsotoMP-B). 4 )-( 4 ), 4 )-( 4 ). However,inmostwirelessinterferencemodels,problem( 4 )doesnotevenhaveacentralizedpolynomial-timesolution.Thishasbeenthemainobstacleindevelopingpracticalratecontrol/schedulingalgorithms.Innextsection,wewilltrytoovercomethisdifculty. 117

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4 )isusuallyanNP-hardcombinatorialproblem[ 7 10 11 ].Onefundamentalreasonisthattheconvexpolytope,C,usuallyhasanexponentialnumberofextremepointsintermsofthenumberoflinks.Thecolumngenerationmethodwithimperfectglobalschedulingcanbeintroducedtoovercomethisdifculty.Thecolumngenerationpartreducesthenumberoftimeswhentheglobalschedulingproblemisinvoked.Imperfectschedulingusesfastapproximationorheuristicalgorithmsforspeedup. s.t.Ps:e2psxsPqi=1ic(i)e;8e2E LetebetheLagrangemultiplierassociatedwiththeconstraint( 4 ).TheLagrangianfunctionoftheqth-RMPis

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s.t.Pqi=1i=1xs0;8s2Si0;8i=1;;q: 4 ). Theqth-RMPismorerestrictedthantheMP.Thus,anyoptimalsolutiontotheqth-RMPisfeasibletotheMPandservesasalowerboundoftheoptimalvalueoftheMP.Bygraduallyintroducingmoreextremepoints(columns)intoC(q)andexpandingthesubsetC(q),wewillimprovethelowerboundoftheMP[ 7 14 17 ]. Theoptimizationistakenovertheqcurrentlyknownschedules(extreme-pointlink-ratevectors).Theproblemin( 4 )iscalledthelocalschedulingproblem,andtheachievedmaximumcostiscalledthelocalmaximumcostoftheschedule.Wedenotethislocalmaximumcostunder0by Ifthereismorethanonelink-ratevectorachievingthelocalmaximumcost,thetieisbrokenarbitrarily. 119

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Let(x;;)denoteoneoftheoptimalprimal-dualsolutionsoftheMP,and(x(q);(q);(q))denoteoneoftheoptimalprimal-dualsolutionsoftheqth-RMP.Sincethestrongdualityholdsforbothproblems,wehave Sincetheqth-RMPismorerestrictedthantheMP,wehave Combining( 4 )and( 4 ),wegetthefollowinglowerboundfortheoptimalobjectivevalueoftheMP. Bytheweakduality[ 47 ],foranyfeasibletothedualproblemoftheMP,()isanupperboundfortheoptimalobjectivevalueoftheMP.Inparticular,consider(q),whichisoptimaltothedualoftheqth-RMPandfeasibletothedualoftheMP.((q))isanupperboundofPs2SUs(xs),thatis, Byinspectingthedualfunctions( 4 )and( 4 )oftheqth-RMPandtheMP,respectively,wenotethatx(q)istheuniqueLagrangianmaximizerat(q)forboth( 4 )and( 4 ).Bythe 120

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4 )and( 4 ).Hence,thegapbetweentheupperandlowerboundsfortheoptimalobjectivevalueoftheMPis((q))(q)((q)),whichisexactlythedifferencebetweentheglobalmaximumcostandthelocalmaximumcostofthescheduleunder(q).Therefore,weconcludethefollowingfact. 7 says,atthecurrentlinkcost(q),ifnoneoftheschedulesthatachievetheglobalmaximumcostofthescheduleareinthesubsetC(q),thenthecurrentoptimalsolutionoftheqth-RMPisnotoptimalfortheMP.Inthiscase,therearereasonstoprefertheintroductionofthegloballyoptimalschedulespeciedby( 4 )asthenewextremepointtotherestrictedmasterproblem.ThisstrategyisalocalgreedyapproachtoimprovethelowerboundoftheoptimalvalueoftheMP.Infact,itcanbeviewedasaconditionalgradientmethodforoptimizingthelowerbound,whenthelowerboundisviewedasafunctionofc[ 7 ]. 121

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4 )isusuallyNP-hard,whichmakesthestepofcolumngenerationverydifcult.However,accordingtoFact 14 ,wedonothavetosolveitprecisely.Instead,wemaysolveitapproximately,andthisisreferredtoasimperfectglobalscheduling[ 11 ]. 4 )withanapproximationratio1,thatis, where()isthecostoftheschedulegivenbytheapproximatesolution.Notethatboth()and()arenon-negativeforallvectors0. 4 )-( 4 )(whichwillcallthefasttimescalealgorithm)forseveral(anitenumber)timesontheqth-RMP. 4 )withapproximationratiounderthecurrentdualcost. 0. Iftheschedulecorrespondingtotheapproximatesolutionof( 4 )isalreadyinthecurrentcollectionofschedules,gotoStep1; 0. otherwise,introducethisscheduleintothecurrentcollectionofschedules,increaseqby1,andgotoStep1. WemakeseveralcommentsregardingAlgorithm 6 4 )canbeeasilysolvedpreciselysincethenum-berofextremepointsofC(q)isusuallysmall,andhence,enumerable. 122

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7 ]. 102 ]. 4 ),andbroadcasttheresults.Otherthanthat,thetwo-timescalealgorithm( 4 )-( 4 )and( 4 )-( 4 )ontheqth-RMPiscompletelydecentralized.Furthermore,iftheglobalschedulingproblem( 4 )canbesolvedapproximatelyinadecentralizedfashion,thenAlgorithm 6 iscompletelydecentralizedexceptthepartofthecontroller.InSection 4.5 ,wewillintroduceoneinterferencemodel,underwhich( 4 )canbesolvedinadecentralizedfashionapproximately[ 10 11 ]. 6 infactdescribesawholeclassofalgorithms.Toseethis,considerthespecialcasewhere=1(i.e.,thecaseofperfectglobalscheduling).Inoneendofthespectrum,iftheslow-timescalealgorithminstep1runsonlyonceontheRMP,thealgorithmbecomesapuretwo-timescalealgorithmasinSection 4.3 .Intheotherendofthespectrum,iftheslow-timescalealgorithmrunsontheRMPuntilconvergence,thealgorithmbecomesapurecolumngenerationmethodwiththetwo-timescalealgorithmasabuildingblockforsolvingtherestrictedproblemsbetweenconsecutivecolumngenerationsteps.Bychoosingdifferentnumbersoftimestoruntheslow-timescalealgorithminstep1,wehavemanyalgorithms,representingdifferentperformance,convergencespeedandcomplextradeoffs. Theorem15. 6 convergestooneoptimalprimal-dualsolutionofthisparticularqth-RMP(i.e.,(x(q);(q);(q))).Furthermore,afterAlgorithm 6 convergesto(x(q);(q);(q)),((q))=(q)((q)).

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6 willstopintroducingnewextremepoints.Hence,thereexistsaq,1qQ,suchthat,afterAlgorithm 6 stopsintroducingnewextremepoints,thenumberofextremepointsthathavebeenintroducedisq.LettheconvexhullformedbytheseqpointsbedenotedbyC(q).AfterAlgorithm 6 nolongerintroducesnewextremepoints,itbehavesjustlikethetwo-timescalealgorithmbutontherestrictedsetC(q).AccordingtothetheoremsinSection 4.3 ,thetwo-timescalealgorithmconverges.Thus,Algorithm 6 convergesto(x(q);(q);(q))onthisparticularqth-RMP. Wenextshowthat,afterAlgorithm 6 convergesto(x(q);(q);(q)),wehave((q))=(q)((q)).First,notethat((q))(q)((q))bythecommentafterAlgorithm 6 .Next,itmustbetruethat((q))(q)((q)).Otherwise,theschedulewhosecostis((q))mustnothavealreadybeeninC(q)andwillbeselectedtoenter.Thisviolatestheassumptionthatthealgorithmneverselectsmorethanqschedules. 15 saysthatthecolumngenerationmethodwithimperfectglobalschedulingconvergestoasub-optimumoftheMP.Next,wewillprovethattheperformanceofthissub-optimumisbounded. 124

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4 ),wehave 4 )isassumed.Thesecondequalityholdsbecause((q))=(q)((q))byTheorem 15 Sincethestrongdualityholdsontheqth-RMP,Ps2SUs(x(q)s)=(q)((q)),wehavethefollowing. 4 )holdswithequality,thenAlgorithm 6 isthecolumngenerationmethodwithperfectglobalscheduling,andthisalgorithmconvergestooneoptimumofMP. 4 saysthatthecolumngenerationmethodwithimperfectglobalschedulingconvergestoasub-optimumoftheMPandachievesthesameapproximationratioastheapproximatesolutiontotheglobalschedulingproblem.Finally, 6 ,whichcorrespondstoperfectglobalscheduling.Then,Algorithm 6 convergestoanoptimumoftheMP. 10 11 ],theauthorsproposeawaytosolvethisproblembyadistributedsubgradientalgorithmwithimperfectscheduling.Withperfectscheduling,theirapproachguaranteestheconvergenceofthelinkdualcostsandtheprimalsourcerates;butitdoesnotrecoverthetime 125

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4 )becomesthemaximumweightedmatching(MWM)problem[ 10 11 103 ].Thereisapolynomial-timealgorithmtosolveMWMprecisely[ 104 ]andagreedyalgorithmtosolveitapproximatelywithanapproximationratio=2.Thegreedyalgorithmismoreusefultoourproblembecauseitisdistributed[ 11 ].Underthismodel,ourcolumngenerationalgorithmwithimperfectschedulingwillconvergetoanapproximatesolutionfortheMPwithanapproximationratio=2,anditiscompletelydecentralized. Weremarkthatthenodeexclusiveinterferencemodelisasimpleinstanceoftheconict-graph-basedmodelsthatcapturethecontentionrelationsamongthelinks[ 9 12 ].Inaconictgraph,eachvertexrepresentsonewirelesslinkinthenetwork,andanedgerepresentscontentionbetweenthetwocorrespondinglinks,whicharenotallowedtotransmitatthesametime.Asetoflinksinthewirelessnetworkthatcantransmitdatasimultaneously(i.e.,aschedule),isanindependentsetinthecorrespondingconictgraph.Theschedulingproblem( 4 )becomesthemaximumweightedindependentset(MWIS)problem,wheretheedgeweightisece.Generally,MWIShasnoapproximatesolution.Someheuristicsorrandomsearchalgorithmsseemnecessarytocarryouttheimperfectscheduling. 126

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or wherewsaretheweightsfors2S,eisthebaseofthenaturallogarithmandas>0isasmallconstant,whichmaketheutilityfunctions( 4 )and( 4 )satisfytheassumptionsA2andA3.Theseutilityfunctionshavebeendiscussedin[ 42 ].Inthischapter,wewillusetheutilityfunctionin( 4 )withws=1:0foralls2S. AsdiscussedinSection 4.4 ,wecanintroducenewextremepointsatvaryingdegreeoffrequency.Intheexperiments,wewillusethreefrequencies:fast,mediumandslow.Withthefastfrequency,wetrytointroduceextremepointsbysolvingtheglobalschedulingproblem( 415 )ateachslow-timescaleupdate( 4 )-( 4 ),inwhichcase,Algorithm 6 degeneratesintothepuretwo-timescalealgorithm.Withtheslowfrequency,wetrytointroduceanewextremepointafterevery20slow-timescaleupdatesof( 4 )-( 4 ).Ourexperienceshaveshownthattherestrictedmasterproblemwithourexperimentsizesisoftenoptimizedwithin20slow-timescaleupdates.Ifso,Algorithm 6 becomesthepurecolumngenerationmethod.Withthemediumfrequency,weintroduceanewextremepointevery5slow-timescaleupdates. ThenetworkinFig. 4-1 hasbeenstudiedin[ 10 11 ].Thereare5classesofconnectionsasshowninFig. 4-1 .Thecapacityofeachlinkisxedat100units.Weinitializetheexperi-mentswithasetofschedules,whereeachcontainsexactlyonesingletransmittinglink.ThiscorrespondstothetraditionalTDMAscheduling[ 17 ].Fig. 4-2 showstheconvergenceoftheconnectionrateswithperfectschedulingandimperfectscheduling,respectively,wherebothareintroducingnewcolumnsatthefastfrequency.InFig. 4-2 (a),wehavetwogroupsofconnec-tions.Class4andClass5achievehigherratesbecausetheyinvolvelesswirelessinterferencecomparedwithothers.Fig. 4-2 (b)givesthesameorderoftheconnectionsintermsoftheirrates.But,theconnectionsarenotseparatedintoobviousrategroups.Thoughthetwoscheduling 127

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4 ),aminorchangeintheconnectionrateswillnotchangetheobjectivetoomuch.Fig. 4-3 showsthetwoschemesgetthecorrecttimefractionandthelongtimeaveragelinkcapacitiesareabletosupportthesourceowrates.Itmeansourtwo-timescalealgorithmsolvesboththeprimalanddualproblemsatthesametime. Wenextexperimentwithalargernetworkwith15nodes.Thenetworkisrandomlygeneratedand20end-to-endconnectionsareplacedonthisnetworkrandomly.Foreachconnection,theroutingisthexedshortestpathrouting.Intheexperiment,itturnsoutthese20connectionsuse28directedlinks.Thecapacityofeachlinkisxedat100units.Fig. 4-4 showsthe5connectionswiththehighestrates.Again,theperfectschedulingismorelikelytogroupconnections. Next,weevaluatethealgorithmwithdifferentfrequenciesofintroducingcolumnsonthelargenetwork.InFig. 4-5 ,withbothperfectandimperfectscheduling,thefastschemealwaysimprovestheobjectivefunctionvaluemorequicklyatthebeginning,whiletheslowschemeimprovesitmuchmoreslowlythantheothertwoschemes.Thereasonisthat,withthefastscheme,plentyofschedulesareintroducedquickly.Theslowschemealwaystriestotakefulladvantageofthecurrentcollectionofschedules.Butlater,theslowschemecatchesupthefastscheme,judgingfromthetrendofthecurves.Thismotivatestheuseofthemediumscheme.InFig. 4-5 ,weseethatthemediumschemeincreasestheobjectivefunctionvaluenearlyasquicklyasthefastschemeatthebeginninganditsurpassesthefastschemesoonafter.Thecurvesshowsomeoscillationsattheinitialphaseforthemediumandslowschemes.Thisisbecausethosetwoschemesspendmoreefforttoobtainbetterperformancefromthecurrentcollectionofschedules.Attheinitialphase,withfewerschedulesbutmoreoptimizedtimesharing,introducingonemorescheduleabruptlywilldecreasethefunctionvaluebyalittlebit. InTable 4-1 ,wecomparethethreeschemesfortheircomputationcosts.Sincethemostexpensivecomputationisforsolvingtheglobalschedulingproblem( 4 ),thetotalcomputation 128

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Wealsond,withalowerfrequency,thealgorithmusuallyproducesasolutionwithfeweractiveschedules. Fortheimperfectscheduling,wendthatboththefastandthemediumschemesgeneratemuchfewerschedulesthanintheperfectscheduling,althoughthenumberofcomputationsfortheschedulesremainthesame BasedonthestudyofFig. 4-4 andTable 4-1 ,weconcludethatthepuretwo-timescale(fast)orthepurecolumngeneration(slow)algorithmshavebothprosandcons.Anintermediate 129

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Next,weshowthat,inthepurecolumngenerationmethod,thegapbetweenthelowerandupperboundsfortheoptimalobjectvaluedecreasesastherestrictedmasterproblemexpands.Withtheimperfectscheduling,wecancomputetheupperboundby(q)((q))(q)((q))+((q))(q)((q))(q)((q))+((q)),where=2inourcase.Thelowerboundisobtainedfromthecurrentbestsolution.Fig. 4-6 showsthatthegapisquicklynarrowedafter10columnshaveentered.Italsoshowsthattheobjectivevaluesofboththeperfectschedulingandimperfectschedulingareinsidethetwobounds.Also,ourimperfectschedulingalmostachievestheglobaloptimumoftheoriginalproblem. Finally,wehavealsoappliedthesubgradientalgorithmsfortheseexperiments,andfoundthatitisverydifculttotunethealgorithmparameterstoreachconvergence. 130

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Performancecomparisonofthefamilyofalgorithms(largenetwork). #SchedulesComputed#ActiveSchedules#SchedulesIntroduced FastPer.3004956MediumPer.604952SlowPer.151515FastImper.3001922MediumImper.601930SlowImper.151515 Figure4-1. Smallnetworktopology. 131

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(b) Figure4-2. Convergenceinconnectionrates(smallnetwork):(a)fastfrequency,withperfectglobalscheduling;(b)fastfrequency,withimperfectGlobalScheduling. (a) (b) Figure4-3. Convergenceinlinkowrates(smallnetwork):(a)fastfrequency,withperfectglobalscheduling;(b)fastfrequency,withimperfectglobalscheduling. 132

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(b) Figure4-4. Convergenceinconnectionrates(largenetwork):(a)fastfrequency,withperfectglobalscheduling;(b)fastfrequency,withimperfectglobalscheduling. (a) (b) Figure4-5. Convergenceofthefamilyofalgorithms(largenetwork):(a)perfectglobalscheduling;(b)imperfectglobalscheduling. Figure4-6. BoundsfortheoptimalobjectivevalueoftheMP.Purecolumngenerationmethodwithimperfectglobalscheduling. 133

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XiaoyingZhengreceivedherbachelor'sandmaster'sdegreesincomputerscienceandengineeringfromZhejiangUniversity,P.R.China,in2000and2003,respectively.Herresearchinterestsareinthecomputernetworkingarea,includingapplicationsofoptimizationtheoryinnetworks,peer-to-peeroverlaynetworks,contentdistribution,wirelessadhocnetworks,andcongestioncontrol. 142