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Optimization-Based Multi-Tree Multicast Algorithms for Content Distribution

Permanent Link: http://ufdc.ufl.edu/UFE0043536/00001

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Title: Optimization-Based Multi-Tree Multicast Algorithms for Content Distribution
Physical Description: 1 online resource (123 p.)
Language: english
Creator: Cho, Chunglae
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: algorithm -- allocation -- content -- distribution -- multicast -- network -- optimization -- rate
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: Massive content distribution has become one of the most important applications on the Internet. In this work, we present three algorithms for utility maximization problems using multiple multicast trees for content distribution. All of the algorithms are much decentralized and local. The operations are well distributed throughout the network components and there is no global exchange of control messages. We first consider the problem over a single-layer infrastructure network where infrastructure nodes are strategically placed by the providers and are well managed and relatively static. On this setting, we first present a virtual-queue-based multicast backpressure algorithm and provide a rigorous analysis of the performance including the primal optimality and the bound on real queue sizes using the Lyapunov optimization technique and the convex optimization techniques together. Next, we present a multicast backpressure algorithm using the real queue backlogs and provide a rigorous analysis to prove the optimality of the algorithm, using a fluid limit model. As an important byproduct, we show that the backpressure algorithm with real queues is exactly a subgradient algorithm in the fluid limit model. As our second problem, we consider the problem of content distribution among data servers over a two-layer network consisting of the overlay and underlay networks, where the control variables can be adjusted at both networks. We propose an optimal backpressure algorithm for content distribution over the layered network. The algorithm is well separated into the overlay and underlay operations such that the communication between the two layers only occurs locally at their interface. We show that our algorithm achieves optimality even when the two network layers operate under different timescales and without time synchronization.
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 Chunglae Cho.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Xia, Ye.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043536:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043536/00001

Material Information

Title: Optimization-Based Multi-Tree Multicast Algorithms for Content Distribution
Physical Description: 1 online resource (123 p.)
Language: english
Creator: Cho, Chunglae
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: algorithm -- allocation -- content -- distribution -- multicast -- network -- optimization -- rate
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: Massive content distribution has become one of the most important applications on the Internet. In this work, we present three algorithms for utility maximization problems using multiple multicast trees for content distribution. All of the algorithms are much decentralized and local. The operations are well distributed throughout the network components and there is no global exchange of control messages. We first consider the problem over a single-layer infrastructure network where infrastructure nodes are strategically placed by the providers and are well managed and relatively static. On this setting, we first present a virtual-queue-based multicast backpressure algorithm and provide a rigorous analysis of the performance including the primal optimality and the bound on real queue sizes using the Lyapunov optimization technique and the convex optimization techniques together. Next, we present a multicast backpressure algorithm using the real queue backlogs and provide a rigorous analysis to prove the optimality of the algorithm, using a fluid limit model. As an important byproduct, we show that the backpressure algorithm with real queues is exactly a subgradient algorithm in the fluid limit model. As our second problem, we consider the problem of content distribution among data servers over a two-layer network consisting of the overlay and underlay networks, where the control variables can be adjusted at both networks. We propose an optimal backpressure algorithm for content distribution over the layered network. The algorithm is well separated into the overlay and underlay operations such that the communication between the two layers only occurs locally at their interface. We show that our algorithm achieves optimality even when the two network layers operate under different timescales and without time synchronization.
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 Chunglae Cho.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Xia, Ye.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043536:00001


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OPTIMIZATION-BASEDMULTI-TREEMULTICASTALGORITHMSFORCONTENTDISTRIBUTIONByCHUNGLAECHOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011ChunglaeCho 2

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Tomywife,Sunmiandmyson,Hyunsung 3

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ACKNOWLEDGMENTS Thisworkwouldnothavedonewithoutthehelpofmanypeoplearoundme.Firstofall,Iwouldliketothankmyadvisor,Dr.YeXia,forhisguidance,patience,andencouragement.HeguidedmetotherightdirectionwheneverIlostmywayintheresearch.HehastaughtmewithpatienceandencouragedmetoovercomealotofdifcultiesIhaveencountered.Iamalsothankfultomyothercommitteemembers:Dr.ArunavaBanerjee,Dr.ShigangChen,Dr.AhmedHelmyandDr.JohnM.Sheafortheirprecioustimeandadvice.IamthankfultothefriendsImetattheUniversityofFlorida.Especially,IwouldliketothankXiaoyingZhengandMarcusMiguezwhohaveworkedwithmeonthecontentdistributionprojects.SpecialthankstoYoungInYeo,DuckkiLee,JaeWoongLee,andSungwookMoonfortheirhelp.IwouldalsoliketothankallmyKoreanfriends.ManythankstoYeonjeongJeongandMinsooKimfortheirsupport.SpecialthankstoJinohKimforhisinterestingtalk.Iwouldliketogivemydeepestgratitudetomyparentsandparents-in-lawfortheirlovelysupport.Thisworkwouldnothavebeennishedwithouttheirlove.Lastbutnotleast,thisdissertationisdedicatedtomywife,Sunmi,andmyson,Hyunsung.Duringthelastsixyears,Icouldnotbethebesthusbandandthebestfather,respectively.Morethanhalfoftheperiod,weweregeographicallyseparatedfromeachother.Withouttheirlovelysupportandsacrice,Icouldnothavenishedthiswork. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 7 ABSTRACT ......................................... 8 CHAPTER 1INTRODUCTION ................................... 10 1.1Virtual-Queue-BasedMulti-TreeMulticastAlgorithm ............ 12 1.2Real-Queue-BasedMulti-TreeMulticastAlgorithm ............. 13 1.3Multi-TreeMulticastAlgorithmonTwo-LayerNetworks ........... 15 2VIRTUAL-QUEUE-BASEDMULTI-TREEMULTICASTALGORITHMFORCONTENTDISTRIBUTION ................................... 19 2.1ProblemDescription .............................. 20 2.1.1NetworkModel ............................. 20 2.1.2ProblemFormulation .......................... 21 2.2BackpressureAlgorithm ............................ 24 2.3Long-TimeAveragePerformanceAnalysis .................. 28 2.4AlternativeWaytoProvetheAlgorithmPerformance ............ 36 2.5FurtherResultsonQueueBoundedness ................... 38 2.5.1DualOptimalityandBoundednessofVirtualQueues ........ 38 2.5.2BoundednessofRealQueues ..................... 42 2.6SimulationResults ............................... 44 2.6.1SimulationSetup ............................ 44 2.6.2SingleSession ............................. 44 2.6.3MultipleSessions ............................ 50 2.7RelatedWork .................................. 52 3REAL-QUEUE-BASEDMULTI-TREEMULTICASTALGORITHMFORCONTENTDISTRIBUTION ................................... 55 3.1ProblemDescription .............................. 55 3.1.1DiscreteSystemModel ......................... 55 3.1.2Notation ................................. 57 3.1.3ProblemFormulation .......................... 57 3.2BackpressureAlgorithm ............................ 60 3.3FluidLimitAnalysis ............................... 62 3.3.1FluidLimitModel ............................ 62 3.3.2PropertiesofFluidLimitModel .................... 63 3.3.3DualOptimalityoftheAlgorithm .................... 74 3.3.4PrimalOptimalityoftheAlgorithm ................... 77 5

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3.4RelatedWork .................................. 78 4MULTI-TREEMULTICASTALGORITHMFORCONTENTDISTRIBUTIONONTWO-LAYERNETWORKS ........................... 79 4.1ModelandProblemFormulation ....................... 79 4.1.1NetworkModel ............................. 80 4.1.2UtilityFunctions ............................. 81 4.1.3OverlayTransmissionDecisionsandLinkCapacities ........ 82 4.1.4OtherNotations ............................. 83 4.1.5ProblemDescription .......................... 83 4.2AsynchronousAlgorithm ............................ 86 4.2.1TimeStructureforAsynchronousOperations ............ 86 4.2.2Algorithm ................................ 87 4.2.3IllustratingExample ........................... 90 4.2.4PerformanceAnalysis ......................... 92 4.3SimulationExperiments ............................ 104 4.3.1SimulationSetup ............................ 104 4.3.2SingleSessionCase .......................... 105 4.3.3CaseofMultipleSessions ....................... 109 4.3.4ExampleofInstableRealQueuesWhen=1.0 .......... 111 4.4RelatedWork .................................. 112 5CONCLUSIONS ................................... 116 REFERENCES ....................................... 118 BIOGRAPHICALSKETCH ................................ 123 6

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LISTOFFIGURES Figure page 1-1Twolayersofnetwork. ................................ 16 2-1Anexamplenetwork. ................................. 21 2-2Sessionrateandreceivingrates. .......................... 45 2-3Timeaveragetreerates. ............................... 47 2-4Aggregatequeuesizesofthetrees. ........................ 47 2-5Effectofonachievablerates. ........................... 48 2-6Effectofonthequeuesizes. ........................... 49 2-7Effectofthenumberoftrees. ............................ 50 2-8Timeaverageratesofmultiplesessions. ...................... 51 4-1Timestructureforasynchronousoverlayandunderlayoperations. ....... 87 4-2Howthetrafcishandledinadataserver. ..................... 90 4-3Timeaveragerates. ................................. 106 4-4Effectofontheachievedrateandonthetimeaverageaggregatequeuesize. 107 4-5Trajectoryoftheaggregaterealqueuesize. .................... 108 4-6Objectivevaluesofthecasesofmultiplesessions. ................ 109 4-7Timeaverageratesofmultiplesessionsundersynchronoustimescales. .... 110 4-8Timeaverageratesofmultiplesessionsunderasynchronoustimescales. ... 111 4-9Effectofonrateallocation. ............................ 112 4-10Effectofonaggregatequeuesize. ........................ 112 4-11Trajectoryoftheaggregaterealqueuesize. .................... 113 4-12Exampleofinstableaggregaterealqueuesizewhen=1.0. .......... 113 7

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyOPTIMIZATION-BASEDMULTI-TREEMULTICASTALGORITHMSFORCONTENTDISTRIBUTIONByChunglaeChoDecember2011Chair:YeXiaMajor:ComputerEngineering MassivecontentdistributionhasbecomeoneofthemostimportantapplicationsontheInternet.Inthiswork,wepresentthreealgorithmsforutilitymaximizationproblemsusingmultiplemulticasttreesforcontentdistribution.Allofthealgorithmsaremuchdecentralizedandlocal.Theoperationsarewelldistributedthroughoutthenetworkcomponentsandthereisnoglobalexchangeofcontrolmessages. Werstconsidertheproblemoverasingle-layerinfrastructurenetworkwhereinfrastructurenodesarestrategicallyplacedbytheprovidersandarewellmanagedandrelativelystatic.Onthissetting,werstpresentavirtual-queue-basedmulticastbackpressurealgorithmandprovidearigorousanalysisoftheperformanceincludingtheprimaloptimalityandtheboundonrealqueuesizesusingtheLyapunovoptimizationtechniqueandtheconvexoptimizationtechniquestogether.Next,wepresentamulticastbackpressurealgorithmusingtherealqueuebacklogsandprovidearigorousanalysistoprovetheoptimalityofthealgorithm,usingauidlimitmodel.Asanimportantbyproduct,weshowthatthebackpressurealgorithmwithrealqueuesisexactlyasubgradientalgorithmintheuidlimitmodel. Asoursecondproblem,weconsidertheproblemofcontentdistributionamongdataserversoveratwo-layernetworkconsistingoftheoverlayandunderlaynetworks,wherethecontrolvariablescanbeadjustedatbothnetworks.Weproposeanoptimalbackpressurealgorithmforcontentdistributionoverthelayerednetwork.Thealgorithm 8

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iswellseparatedintotheoverlayandunderlayoperationssuchthatthecommunicationbetweenthetwolayersonlyoccurslocallyattheirinterface.Weshowthatouralgorithmachievesoptimalityevenwhenthetwonetworklayersoperateunderdifferenttimescalesandwithouttimesynchronization. 9

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CHAPTER1INTRODUCTION MassivecontentdistributionhasbecomeoneofthemostimportantapplicationsontheInternet.Someoftheexamplesarethedeliverysystemsoftelevisionprogramsorvideocontent,le-sharingsystems,e-sciencenetworks,communityDVRormedianetworks.Oneimportantclassofcontentdistributiontechniqueisswarming.Inaswarmingsession,aleisbrokenintomanychunksatthesourcenode,whicharethendistributedtothereceiversthroughvariouspaths. Despiteitsoriginintheend-system-basedpeer-to-peer(P2P)community[ 2 14 18 33 35 48 ],swarmingisalsoattractiveforcontentdistributionservicesoverinfrastructurenetworksprovidedbynetworkorserviceproviders.Forthoseservices,infrastructurenodesarestrategicallyplacedbytheprovidersandarewellmanagedandrelativelystatic.Thesecharacteristicsofinfrastructurenetworksareverydifferentfromthoseofend-systempeer-to-peernetworks.Inthissetting,ithasbeenshownthatitisbenecialtoviewswarmingasdistributionovermultiplemulticasttrees[ 61 62 ].Thisviewallowsustofocusonaformalapproachbasedonoptimizationtheorytodevelopoptimalsolutionssystematically.Furthermore,itisofteneasiertorstdevelopsophisticatedalgorithmsunderthestaticassumption,andthen,adaptthemtopracticalsituations. Inthiswork,westudytheproblemofutilitymaximizationofmulticastsessionswithmultipletrees.Wesupposethateachsessionhasaninnitedatabackloginitssourceandisgivenasmallnumberoftreesoverwhichitsdataisdividedandthendistributedtoitsreceivers.Theobjectiveistomaximizethesumofsessionutilitieswhicharefunctionsoftheiradmittedrates.Weassumethattheutilityfunctionsaregeneralconcavefunctions,whichcanreectvariousfairnesscriteriaamongsessions.Themajorconstraintsarelinkcapacityconstraints.Then,ourproblemistondtheoptimaltreerateallocationwhilethequeuesinthenetworkremainnite. 10

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Weareinterestedindevelopingdistributedandlocalalgorithmswhereratecontrolandlinkschedulingaremadelocallyateachnode.1Toachievethegoal,wegetinspirationfrombackpressurealgorithmsfortheunicastnetworkowproblems[ 5 6 23 36 44 46 47 57 60 ].Inbackpressurealgorithms,eachnodemakesatransmissiondecisionbasedonthedifferencesofqueuesizesbetweenthenodeitselfanditsneighboringnodes.Thealgorithmscanbenotonlydecentralizedbutalsolocal:Thereisnoglobalexchangeofcontrolmessages. However,applyingthebackpressureapproachtomulticastproblemsisnotstraightforward.Mostbackpressurealgorithmsforunicastarederivedbyrelaxingtheowconservationconstraintsinthestaticoptimizationformulation([ 24 36 46 ]forrepresentativeexamples).Ontheotherhand,itisnotobviouswhatconstraintsrepresentsuchowconservationinmulticastproblemssinceitisnottrueinmulticastthattheamountofincomingowintoanodeisequaltotheamountofoutgoingowfromthenode,whichistheowconservationconditioninunicastowproblems. Tocircumventthedifculty,weintroduceaproblemformulationwithtree-owconservationconstraints,bywhichwemeanthattheamountofowonalinkonatreeshouldbenolessthantheamountofowonitsparentlinkwithrespecttothetree.Then,wepresentasubgradientalgorithmbyrelaxingthetree-owconservationconstraints.Weassumethatwehaveasmallnumberoftreesforeachsession.Wemayusetheresultsofpreviouswork[ 19 20 27 62 ]tondagoodcandidatesetofmulticasttrees. Inthiswork,wepresentthreemulti-treemulticastalgorithmsforcontentdistributionoverinfrastructurenetworks.Thersttwoalgorithmsareforasingle-layerinfrastructure 1Here,bylinkschedulingwemeanthedecisionatanodeaboutwhichpacketitservesforitsdownstreamlinksateachtime. 11

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networkandthelastoneisforatwo-layerinfrastructurenetwork.Allofthemarebackpressurealgorithms.Theyarenotonlydecentralizedbutalsolocal. 1.1Virtual-Queue-BasedMulti-TreeMulticastAlgorithm InChapter 2 ,wepresentavirtual-queue-basedbackpressurealgorithmfortheutilitymaximizationproblemofmulti-treemulticast.Thecontrolisdistributedoverthenetworkcomponentsandthecommunicationoverheadislowsinceeachnodeonlyneedstoknowthequeuesizesatitselfandatitsneighboringnodes.Thealgorithmismorelocalthanearliermulticastalgorithmsin[ 13 20 61 62 ].Thosealgorithmsaredistributedbutnotmuchlocalbecausetheyneedglobalexchangeofcontrolmessages. Ourcontributionsareasfollows.First,wedevelopabackpressure-basedadaptive-controlalgorithmfortheutilitymaximizationproblemofmulti-treemulticast,whichisamenabletoimplementationinlargenetworksystems.Whileitisrelativelystraightforwardtodevelopsuchanalgorithmforunicastproblems,itislesssointhemulticastcase.Thekeytoderivingourbackpressurealgorithmisanovelproblemformulationthatspecicallyincorporatesthetree-owconservationconstraints:Theamountofowonalinkonatreeshouldbenolessthantheamountofowonitsparentlinkwithrespecttothetree.Wethenrelaxthetree-owconservationconstraintsandwritedownasubgradientalgorithmforthecorrespondingstaticoptimizationproblem,whichthenleadstothebackpressurealgorithm.Thus,wehaveasystematicmethodfordevelopingbackpressurealgorithmsformulti-treemulticast. Oursecondcontributionistechnical,anditalsomarksamoresubstantialdistinctionofthisworkfromtheclosestrelatedwork(e.g.,[ 52 ][ 11 ]).Wehaveobtainedmorecompleteresultsaboutthealgorithmoptimalityandnetworkstability,andwehavebeenabletodosobyusingacombinationoftwosetsofanalyticaltechniques.Theresultsincludebothprimalanddualoptimalityandtheboundednessofboththevirtualqueuesandtherealqueues,undergeneral(non-strictly)concaveutilityfunctions. 12

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Wewillbrieydiscussthetwosetsoftechniquesandwhattheycanaccomplishforus.Itisdifculttoapplytheconventionalconvexoptimizationtheoryandtechniquesforsubgradientalgorithmstoshowtheprimaloptimalityofouralgorithm.Thisismainlyduetotheassumptionofgeneralconcavityratherthanstrictconcavityontheutilityfunctions.Theconsequenceisthelackofacontinuousmapfromthedualtotheprimalvariables,whichiscrucialtotheproofofprimalconvergenceasdescribedin[ 37 38 ].Evenifthedualvariablesconverge,theprimalvariablesmayoscillate.Instead,weshowtheprimaloptimalityofthealgorithminthelong-timeaveragesenseusingtheLyapunovoptimizationtechnique([ 24 46 47 ]).Fornetworkstability,however,theLyapunovoptimizationtechniquecanonlyshowtheboundednessofthelong-timeaveragevirtualqueuesizes2,whichisnotenoughtoconcludenetworkstability.Weusetheconvexoptimizationtechniquestoprovethedualoptimality,andthen,weusethatresulttoshowthatboththevirtualandrealqueuesareboundedforalltimeslots.Thetwosetsofanalyticaltechniquescomplementeachotherinouranalysis. ThethirdcontributionisatechnicalpointwithrespecttotheapplicationoftheLyapunovoptimizationtechnique.ToapplytheLyapunovoptimizationtechnique,oneofthemostimportantstepsistoconstructsomefeasiblesolutionforthepurposeofcomparisonwiththeproposedalgorithm.Thisusuallyinvolvesdeningan-tightened(or-modied)problem.Weshowtwowaysofdoingthisbymodifyingdifferentconstraints.Theygivedifferentperformancebounds.Explorationsofthiskindcanbeusefulforndingtighterperformancebounds. 1.2Real-Queue-BasedMulti-TreeMulticastAlgorithm Inanimplementationofthevirtual-queue-basedbackpressurealgorithmintroducedinChapter 2 ,everynodeshouldmanagevirtualqueues,whichmaybeaburden.Itisanaturalquestionwhetherwecanuserealqueuesinsteadofvirtualqueuesin 2Ouralgorithmusesthevirtualqueuesizesforcontrol. 13

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thealgorithm.Ithasbeenshownthatusingrealqueuesintheunicastbackpressurealgorithmsstillprovidesguaranteedperformancebounds[ 24 46 47 ].InChapter 3 ,westudythesameproblemasinChapter 2 andpresentabackpressurealgorithmbasedonrealqueuebacklogs. Ourrstcontributionistodevelopamulti-treemulticastbackpressurealgorithmbasedontherealqueuesizes.Backpressurealgorithmsaredesirablesincetheycanbenotonlydecentralizedbutalsolocal:Theexchangeofcontrolmessagesisrestrictedtobeamongtheneighboringnodesandthereisnoglobalexchangeofcontrolmessages.Intheunicastsetting,backpressurealgorithmsofvariousformshavebeenwidelyapplied[ 5 6 23 36 44 46 47 57 60 ].However,multicastbackpressurealgorithmsarerare[ 11 15 52 ],inpartbecauseitisdifculttodevelopthemcorrectlywithrespecttothedesignobjectives. Thealgorithmin[ 15 ]isrelatedtobutdifferentfromtheoneinChapter 2 .Thenewalgorithmusesrealqueuesizesforratecontrolandlinkschedulingwhereasvirtualqueuesareusedin[ 15 ].Morespecically,thebackpressureonalinkonatreeisdenedtobethedifferenceintherealqueuesizesforthelinkandforthesetofitschildlinksonthetree.Thelinkmakesatransmissiondecisionbasedonthebackpressuresofthetreesusingthelink.Itisadesirablefeaturethatthereisnoneedtomaintainadditionalvirtualqueuelengthinformationatnodestorunthealgorithm.Sarkaretal.[ 52 ]alsointroducedasimilaralgorithmusingrealqueues,buttheyconsideredastabilityproblemwhereasweconsideranutilitymaximizationproblem,whichismoregeneral.Buietal.[ 11 ]alsointroducedasimilarproblemtoours,buttheyonlyconsideredasingletreepersessionandmainlyfocusedonmulti-ratemulticast.Moreover,theirprooftechniquetoshowtheoptimalityissubstantiallydifferentfromours. Thechallengeistoshowthatthebackpressurealgorithmbasedontherealqueuesworkscorrectlywithrespecttotheobjectivefunction.ItisshowninChapter 2 thatthebackpressurealgorithmbasedonthevirtualqueuesisasubgradientalgorithm,which 14

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achievestheoptimizationobjectiveoptimally.However,itdoesnotfollowimmediatelythatthebackpressurealgorithmusingtherealqueuesisalsoasubgradientalgorithmorisoptimal.Therearesomediscussionsin[ 24 45 ]abouttheconnectionbetweenthebackpressurealgorithmwithrealqueuesandtheconvexoptimizationproblemthatcharacterizesthedesignobjectiveandconstraints.But,thediscussionsareinformalandtheexactrelationshipbetweenthetwoisyettoberevealed. Therefore,oursecondandmoreimportantcontributionistoprovideaprooffortheasymptoticoptimalityofthereal-queue-basedbackpressurealgorithmusinguidlimitanalysis.Weshowthatintheuidlimit,everylimitpointoftheadmittedratesisprimaloptimalandthequeuevectorapproachesanoptimaldualsolution.Weshowthatthebackpressurealgorithmwiththerealqueuesisexactlyasubgradientalgorithmintheuidlimit. Althoughtheanalyticalstepsforouralgorithmappearsimilartosomeofthepreviousworkusingtheuidlimittechnique[ 4 21 25 39 54 56 ](especially[ 21 54 ]),theproofsinthosepapersdonotapplytoouralgorithmmainlyduetothedifferenceofthecontrolstructures.Itisnotstraightforwardtoprovemostofourresults. 1.3Multi-TreeMulticastAlgorithmonTwo-LayerNetworks Traditionally,InternetServiceProviders(ISPs)andcontentproviders(CPs,includingthecontentdistributionserviceproviders)areindependententities.ContentdistributionserviceisperformedbyonlytheCPswithoutexplicitparticipationoftheISPs.TheISPsonlyprovideconnectivitytotransportcontent.However,astheserviceproliferates,theISPshavethemotivationtoactivelyparticipateintheserviceforthefollowingreasons.First,theycanimprovetheutilizationoftheirnetworkresourcesbycooperatingwiththeCPs.Ithasbeenshownthatnetwork-obliviousend-to-endcontentdistributiontrafchasasignicantlynegativeimpactonthenetworkefciency[ 31 ].CooperationbetweentheISPsandCPsnotonlyleadstomoreefcientnetworkresourceutilizationbutalsoimprovesthecontentdistributionperformance[ 28 59 ].Second,theISPscangenerate 15

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Figure1-1. Twolayersofnetwork. revenuebyhostingtheserviceanddistributingcontenttotheircustomers,ortheycangainadditionalprotsbyprovidinghigh-qualityresourcestotheCPs[ 17 ]. Giventheabovecontext,weconsidertheproblemofcontentdistributionamongdataserversoveratwo-layernetworkconsistingoftheoverlayandunderlaynetworks,asshowninFig. 1-1 .TheoverlaynetworkismanagedbyaCPandtheunderlayphysicalnetworkisadministeredbyanISP(orISPs)3.Theoverlaynetworkconsistsofdataserversandoverlaylinks.TheCPmanagesmultiplesessionswhereeachsessiondistributesdatafromasingledataserver(source)toasubsetofthedataservers(receivers)ontheoverlaynetwork.Weassumethattheoverlaynetworkishighlyconnectedsuchthatmultipledistributiontreescanbeestablishedforeachsession.Theunderlaynetworkconsistsofrouters/switches,dataservers,andunderlaylinks,whichmaybeeitherphysicallinksorvirtuallinks. 3AlthoughtheCPisnormallyanindependentcontent(ordistributionservice)provider,anISPcanbeboththeISPandtheCP. 16

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InChapter 4 ,westudyhowtodistributecontentoptimallyusingmultiplemulticasttreesontheoverlaynetworkwhenthecontrolvariables(e.g.,routingandrateallocations)canbesetatboththeoverlayandunderlaynetworks.Beingabletocontrolatbothnetworksamountstomoreefcientuseofthenetworkresourcesandimproveddistributionperformance.Thisisincontrastwiththevastmajorityofthepriorworkwhereeitherthereisonlyoneatnetworkinvolvedorthecontrolcanbeadjustedonlyatonelayerandtheotherlayermustbeheldasgiven.Themainchallengeistondasimpleoptimalalgorithmthatdoesnotrequireexcessivecoordinationbetweentheindependentoperatorsofthetwonetworks.Hence,ourfocusistondanoptimalalgorithmthatcanbecleanlypartitionedintotheoverlaypartoperatedbytheCPandtheunderlaypartoperatedbytheISP(orISPs)withonlyminimalcoordinationandwithouttimesynchronization.Specically,weassumethateachsessionhasinnitedatabacklogatitssourceandisgivenasmallnumberoftrees.Thedatawillbedividedandthendistributedoverthetreestothereceivers.Theproblemweformulateistondanoptimaltreerateallocationsubjecttotheunderlaylinkcapacityconstraints.Theoptimizationobjectiveistomaximizethesumofthesessionutilities,eachofwhichisaconcavefunctionoftheadmittedtrafcrate. Ourcontributionsareasfollows.First,wedevelopasimplebackpressurealgorithmthatisidealforcontentdistributionoveratwo-layernetwork.Thealgorithmiswellseparatedintotheoverlayandunderlayoperationssuchthatthecommunicationbetweenthetwolayersonlyoccursattheinterfacebetweenthelayers.Thealgorithmiswelldistributedandlocal.Ratecontrolandoverlaylinkschedulingaredonelocallyateachdataserverandarebasedontheneighboringqueuesizes.Second,weshowthatouralgorithmachievesoptimalityevenwhenthetwonetworklayersoperatewithouttimesynchronizationand/orunderdifferenttimescales.Thisisimportantbecauseinpracticethetwolayersarelikelytobeoperatedbyindependentoperators.Third,weanalyzethealgorithmunderastochasticnetworksettingwheretheunderlaylinkcapacities 17

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mayvaryrandomlywithsomestationarydistributions.UsingtheLyapunovoptimizationtechnique,weprovetheoptimalityofthealgorithminthelong-timeaveragesenseandtheboundednessofthevirtualqueuesizes.Weexpectthattherealqueuesarealsobounded,althoughtheclaimhasnotbeenprovedyet.Weusesimulationexperimentstosupporttheconjecture.Overall,ouralgorithmisabletoachievehigherapplicationperformanceandbetternetworkresourceefciencythanmostoftheearlieralgorithmsthatdonotoffercontrolatbothnetworklayers;itissimplerandmorepracticalthantherestofthealgorithmsthatmayachievecomparableperformance. Theremainderoftheworkisorganizedasfollows.InChapter 2 ,wepresentavirtual-queue-basedbackpressurealgorithmfortheutilitymaximizationproblemofmulti-treemulticastoverasingle-layernetwork.InChapter 3 ,westudythesameproblemasinChapter 2 andpresentabackpressurealgorithmbasedonrealqueuebacklogs.InChapter 4 ,weconsidertheproblemofcontentdistributionoveratwo-layernetworkconsistingoftheoverlayandunderlaynetworksandproposeanoptimalbackpressurealgorithmforcontentdistributionoverthelayerednetwork.TheconcludingremarksareinChapter 5 18

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CHAPTER2VIRTUAL-QUEUE-BASEDMULTI-TREEMULTICASTALGORITHMFORCONTENTDISTRIBUTION Thischapteraddressestheproblemofutilitymaximizationofmulticastsessionswithmultipletreesforcontentdistributionoverstaticinfrastructurenetworks1.Theutilityfunctionsaregeneralconcavefunctions.Inspiredbythederivationofthequeue-length-basedbackpressurealgorithms,weintroduceaproblemformulationwithtree-owconservationconstraintsandderiveamulticastbackpressurealgorithm.Thealgorithmismuchmoredistributedandlocalthanprevioustree-basedalgorithmsforcontentdistribution.WeprovidearigorousanalysisoftheperformanceincludingtheprimaloptimalityandtheboundonrealqueuesizesusingtheLyapunovoptimizationtechniqueandtheconvexoptimizationtechniquestogether.ToapplytheLyapunovoptimizationtechnique,weneedtoconstructsomefeasiblesolutionforcomparisonbydeningan-modiedproblem.Weshowthattherearealternativewaysofdoingthisbymodifyingdifferentconstraintsandtheygivedifferentperformancebounds.Wealsoprovidesimulationresultstoevaluatethealgorithmperformance. Theremainingchapterisorganizedasfollows.Section 2.1 describesthenetworkmodelandproblemformulation.InSection 2.2 ,wepresentthebackpressurealgorithm.InSection 2.3 ,weshowthatouralgorithmachievesanoptimalsolutionandthevirtualqueuesareboundedinthelong-timeaveragesense.WealsodiscussanalternativewaytoprovethealgorithmperformanceinSection 2.4 .WeprovidestrongerresultsonqueueboundednessinSection 2.5 .WegivesimulationresultsinSection 2.6 .WediscusstherelatedworkinSection 2.7 1Somepartoftheworkinthischapterhasappearedinthe49thIEEEConferenceonDecisionandControlin2010[ 15 ]. 19

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2.1ProblemDescription 2.1.1NetworkModel Consideranetworkwhichisrepresentedbyadirected,edge-capacitatedgraphG=(V,E),whereVisthesetofnodesandEisthesetofdirectedlinks.EachlinkeinEhasanitecapacityce>0.Letcmax,maxe2Ecebethemaximumlinkcapacityoveralllinks.LetSbethesetofallmulticastsessionsinthenetworkandjSjbethenumberofsessions.Eachsessionisassociatedwithasourceandasetofdestinations,whichisasubsetofV. EachsessionsisgivenasetoftreesTsthatitusesfordatatransmission2.LetjTsjbethenumberoftreesinTs.LetTbetheunionofTs.Notethatift12Ts1andt22Ts2havethesametopology,weregardthemasdifferenttrees.Letrtebethetransmissionrateassignedfortreetonlinke.LetEtbethesetoflinksontreet.Denoteb(e)andd(e)tobethetransmitting(tail)nodeandthereceiving(head)nodeoflinke,respectively.LetVtbethesetoftransmittingnodesb(e)foralleinEt.Vtrepresentsthesetofnodesontreetwhicharenotleaves.Leto(t)betherootnodeoftreet.Letp(t,e)betheparentlinkoflinkeontreet.Let(t,n)bethesetofchildlinksatnodenontreet.Letxsbetheadmittedrateforsessions,andytnbetheadmittedrateatnodenontreet.Weassumethateachsourcehasaninnitebacklogofdata.LetQtebetherealqueuesizeatlinkefortreet. Fig. 2-1 illustratesthenetworkmodelandthenotationsbyshowingapartofanetworkconsistingofsomelinksusedbytwotrees,t1andt2,ofsessions.Thereisareservoirforsessionsatitssourcenodeb(e1),whichistherootnodeofthetrees.Sessionspushesitsdataintothenetworkatratexs.Thedataisdistributedintotwo 2Weassumethateachsessionisgivenasmallnumberoftrees.Onemayusetheresultsofpreviouswork[ 19 20 27 62 ]tondagoodcandidatesetofmulticasttrees.Ouralgorithmdoesnotscalewellwhenallpossibletreesareallowedforeachmulticastsession. 20

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Figure2-1. Anexamplenetwork. streamsatrateyt1o(t1)andyt2o(t2),respectively,andtheadmitteddataarepushedintoqueuesQt1e1andQt2e1,respectively.Queuesatalinkaremaintainedatthetransmittingnodeofthelink.Forexample,queuesatlinke1areactuallyatnodeb(e1).Onlinke1,thedataforthetwotrees,whichareinqueuesQt1e1andQt2e1,aretransmittedatratesrt1e1andrt2e1,respectively,assumingtherearesufcientbacklogsinthequeues.Thetransmitteddatafortreet1onlinke1isreplicatedintothequeuesQt1e2andQt1e3,whichareatthechildlinks(t1,d(e1))=fe2,e3gofthetreeatnoded(e1).Similarly,fortreet2,thetransmitteddataone1isreplicatedintothequeuesat(t2,d(e1)).Thelinke1istheparentlinkoflinkse2ande3fortreet1,andtheparentlinkoflinkse3ande4fortreet2. 2.1.2ProblemFormulation Denote(ai)tobethevectorwithentriesai.Letx,yandrbethevectors(xs)s2S,(ytn)t2T,n2Vt,and(rte)t2T,e2Et,respectively.Thenotationsuchas(a)t2Tmeansrepeatingthevalueainallentriesofthevector.Denote1tobethevectorwhoseentriesareall1's. Letxs(k)bethetimeaverageoftheadmittedtrafcrateforsessionsuptotimek,i.e.,xs(k),1 kk)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0xs(). 21

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Letxsbethelimitofxs(k)3,i.e.,xs,limk!1xs(k). Wedeneytn(k),ytn,rte(k)andrtesimilarly. OurproblemPisasfollows.P:maxXs2SUs(xs)s.t.rte)]TJ /F4 11.955 Tf 11.58 0 Td[(rtp(t,e))]TJ /F4 11.955 Tf 12.24 0 Td[(ytb(e)0,8t2T,8e2Et, (2)Xt2Tsyto(t)=xs,8s2S, (2)Xt:e2Etrtece,8e2E, (2)0xsXmax,8s2S, (2)ytn0,8t2T,8n2Vt, (2)rte0,8t2T,8e2Et. (2) Theconstraintsin( 2 )implythatforeverytreet,theallocatedlinkrateonalinkeonthetreeshouldbenolessthanthesumofthelinkrateofitsparentlinkp(t,e)andtheexogenousarrivaltothetailnodeb(e).Itcanbeconsideredasarelaxedformoftheowconservationconstraintsfortrees.Weassumethatrtp(t,e),0ifp(t,e)isnull.Notethatthetreeratevariablesytnfortreetaredenedforallnon-leafnodesn2Vt.Thevariablesytnforn6=o(t)appearunnecessarybecausetreeratesonlyneedtobeassignedtotherootnodesoftrees.Infact,ouralgorithmalwaysassigns0tothosevariables.However,weusethemfortheconvenienceintheperformanceanalysisinSection 2.3 3Wetemporarilyassumethatthelimitexists.Weshallreplacelimwithliminforlimsupincasethatthelimitdoesnotexist. 22

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Theconstraintsin( 2 )meanthattheaggregateratetransmittedbythetreesforasessionmustbeequaltothesessionrate.Theconstraintsin( 2 )arethelinkcapacityconstraintsthatthesumofthetransmissionratesonalinkcannotexceedthelinkcapacity.In( 2 ),weassumethesessionratesmustbeboundedbyXmaxfromabove,whereXmaxisaconstant. Letbe,f(x,y,r)j(x,y,r)isfeasibletoproblemP.g. Wehavethefollowingassumptionon. Assumption1. ThereexistsafeasiblesolutionofPsuchthattheconstraintsin( 2 )holdwithstrictinequality. Itishighlyprobablethattheaboveassumptionisvalidinrealnetworksanditisusedforshowingthedualoptimalityofouralgorithm. Wealsodenethesets(x,y)andrasfollows,respectively.(x,y),f(x,y)j(x,y)satisestheconstraintsin( 2 ),( 2 )and( 2 ).g,r,frjrsatisestheconstraintsin( 2 )and( 2 ).g. Thesenotationswillbeusedtosimplifylaterexpressions. WehavethefollowingassumptionsontheutilityfunctionUs(xs). Assumption2. Usisaconcavefunction.Usiscontinuousanddifferentiable.ThederivativeofUsisboundedon[0,Xmax]. NotethatUsisasomewhatgeneralconcavefunction.Itcouldbelinearornon-strictlyconcave.Wedonotevenassumethatitismonotonicallyincreasing.Underthisassumption,whichismoregeneralthanthatin[ 36 37 ],wedonothaveacontinuousmapfromthedualvariablestotheprimalvariablesinthesubgradientalgorithm.Suchcontinuityisavailablein[ 36 37 ]andisusedtoprovetheprimaloptimality.TheassumptionalsoimpliesthatUsisboundedon[0,Xmax]. 23

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Differentutilityfunctionscanreectvariousobjectivesofnetworkorserviceprovidersoperatingthecontentdistributionservice.Forexample,iftheywanttogetproportionalfairnessamongsessions,theycanuseUs(xs)=log(xs+1)astheutilityfunctionsforsessions4.Ontheotherhand,iftheywanttomaximizetheweightedthroughput,theycanuseUs(xs)=wsxswherewsistheweightforsessions.Moreexamplescanbefoundin[ 43 53 ]. 2.2BackpressureAlgorithm Let[]+and[]badenotetheprojectionontothenon-negativedomainandtheintervalof[a,b],respectively.Lettebethenon-negativeLagrangemultipliersassociatedwiththeconstraintsin( 2 ).Letbethevector(te)t2T,e2Et.FromproblemP,werelaxtheconstraintsin( 2 )andwritetheLagrangianasfollows.L(x,y,r;)=Xs2SUs(xs)+Xt2TXe2Ette(rte)]TJ /F8 11.955 Tf 11.95 0 Td[(rtp(t,e))]TJ /F8 11.955 Tf 11.96 0 Td[(ytb(e))=Xs2SUs(xs))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2Tsyto(t)Xe2(t,o(t))te)]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs2SXt2TsXe2Et:e62(t,o(t))ytb(e)te+Xe2EXs2SXt2Ts:e2Etrte(te)]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))te0). Thelastequalityholdsbecause,forallt,Xe2Etytb(e)te=Xe2(t,o(t))ytb(e)te+Xe2Et:e62(t,o(t))ytb(e)te, whereb(e)=o(t)fore2(t,o(t)). 4Strictlyspeaking,wehavetouselogxstoachieveexactproportionalfairness.Weuselog(xs+1)tohavethederivativeoftheutilitybounded. 24

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Then,thedualfunctionisgivenbyD()=max(x,y)2(x,y),r2rL(x,y,r;)=max(x,y)2(x,y)Xs2SUs(xs))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2Tsyto(t)Xe2(t,o(t))te+max(x,y)2(x,y)Xs2S)]TJ /F16 11.955 Tf 12.89 11.35 Td[(Xt2TsXe2Et:e62(t,o(t))ytb(e)te+maxr2rXe2EXs2SXt2Ts:e2Etrte(te)]TJ /F16 11.955 Tf 27.56 11.36 Td[(Xe02(t,d(e))te0) (2)=max(x,y)2(x,y)Xs2SUs(xs))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2Tsyto(t)Xe2(t,o(t))te+maxr2rXe2EXs2SXt2Ts:e2Etrte(te)]TJ /F16 11.955 Tf 27.56 11.36 Td[(Xe02(t,d(e))te0), wherethesecondequalityholdsbecausetheconstraintsofthemaximizationcanbeseparatedoverthethreeterms,andthelastequalityholdsbecausethemaximumofthesecondtermin( 2 )isalwayszero.Then,thedualproblemisasfollows.minD()s.t.te0,8t2T,8e2Et. Followingthestandardsubgradientmethod[ 7 ],wehaveasubgradientalgorithmasfollows.(x(k),y(k),r(k))=argmax(x,y)2(x,y),r2rL(x,y,r;(k)), (2)te(k+1)=[te(k))]TJ /F3 11.955 Tf 11.95 0 Td[((rte(k))]TJ /F8 11.955 Tf 11.95 0 Td[(rtp(t,e)(k))]TJ /F8 11.955 Tf 11.95 0 Td[(ytb(e)(k))]+,8t2T,8e2Et, (2) where>0isastepsize. Letqte(k)=(1=)te(k),whichrepresentsavirtualqueuesizeatlinkefortreetattimeslotk.Letq(k)bethevector(qte(k))t2T,e2Et.Then,wecanrewritethealgorithmasinAlgorithm 2-1 25

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Algorithm2-1Virtual-queue-basedalgorithm 1. Sessionandtreeratecontrol:Ateachtimeslotk,sessionssolvesthefollowingoptimizationproblemandassignsanoptimalsolutiontoxs(k)andyto(t)(k)fort2Ts.max1 Us(xs))]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2Tsyto(t)Xe2(t,o(t))qte(k) (2)s.t.Xt2Tsyto(t)=xs,0xsXmax,yto(t)0,8t2Ts. 2. Linkscheduling:Ateachtimeslotk,eachlinkesolvesthefollowingoptimizationproblemandassignsanoptimalsolutiontorte(k)fortreestsuchthate2Et.maxXt:e2Etrte(qte(k))]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))qte0(k)) (2)s.t.Xt:e2Etrtece,rte0,8t:e2Et. 3. Virtualqueueupdate:Ateachtimeslotk,eachlinkeupdatesthevirtualqueuesfortreestsuchthate2Etasfollows:qte(k+1)=[qte(k))]TJ /F8 11.955 Tf 11.96 0 Td[(rte(k)+rtp(t,e)(k)+ytb(e)(k)]+. (2) Thesubproblems( 2 )and( 2 )canbeeasilysolved.First,wecansolvesubproblem( 2 )asfollows.Letts(k)beatreewiththeminimumtotalqueuebacklogatthesourcefortheoutgoinglinksonthetree:ts(k)2argmint2TsXe2(t,o(t))qte(k). Iftherearemorethanonesuchtrees,wepickoneofthemarbitrarilybutdeterministically.Then,weconsiderthefollowingexpression:s(xs),1 Us0(xs))]TJ /F16 11.955 Tf 42.05 11.35 Td[(Xe2(ts(k),o(ts(k)))qts(k)e(k). 26

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Ifthereexistsxson[0,Xmax]satisfyingtheequalitys(xs)=0,letxs(k)denoteit.Otherwise,wesetxs(k)asfollows:xs(k)=8><>:Xmax,ifs(xs)>0forallxs2[0,Xmax],0,otherwise. Then,forallt2Ts,wesetyto(t)tobeyto(t)(k)=8><>:xs(k),ift=ts(k),0,otherwise. NotethatsinceUs0maynotbeone-to-one,xs(k)canoscillateovertimeevenifqte(k)stabilizes.Oncesourcesdeterminesitssessionratexs(k)andtreets(k),itpushesxs(k)amountofdatafromitsreservoirtotheoutgoinglinksattherootoftheselectedtree.Tosummarize,sourcesselectsonlyonetreethathastheminimumtotalqueuebacklogatthesourcefortheoutgoinglinksonthetree.Then,itsendsxsamountofdataontotheselectedtree. Subproblem( 2 )canbesolvedasfollows.First,eachlinkendsthetreewiththemaximumdifferentialbacklog,e(k),maxt:e2Etfqte(k))]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))qte0(k)g. (2) Ife(k)0,thenlete(k)bethetreethatsolves( 2 )withtiebrokendeterministically.Ife(k)<0,thenwesete(k)tobenull.Next,linkeassignsrte(k)foreachtreetonlinkeasfollows:rte(k)=8><>:ce,ift=e(k),0,otherwise. Ife(k)isnull,allrte(k)atlinkeareassignedzero. Afterlinkedeterminesthetreee(k)whichusesthelinkcapacityexclusively,thetailnodeofthelinktransmitsceamountofdataifithassufcientdatainthe 27

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(real)queue.Otherwise,ittransmitsonlythedatainthequeueanddoesnotusetheremaininglinkcapacity5.Then,thereceivingnodeduplicatesthereceiveddataintothequeuesofthechildlinksoflinkeonthetree. Itisnotdifculttocheckthatateachtimeslotk,theabovealgorithmndsasolutionthatmaximizestheLagrangianLgiventhedualvariablesq(k).Thispropertyisimportantinthelaterperformanceanalysis. Theparametercanbeusedtoadjusttheperformancebound.Rewritingthealgorithmfrom( 2 )-( 2 )into( 2 )-( 2 )hasthebenetthatonlythesourcesneedtoadjustiftheperformanceboundneedstobechanged. Thealgorithmiswelldistributedandlocalbecauseallthenecessaryinformationcanbeobtainedfromtheneighboringnodes.Inarealimplementation,eachsourcenodeperformsthesessionandtreerateallocationandeachnodeperformsthelinkschedulingandvirtualqueueupdatesforitsoutgoinglinks.Theycanexchangethevirtualqueuelengthoftheiroutgoinglinkswiththeirneighborsateachtimeslot.Furthermore,sincethetotalnumberoftreesisreasonablysmall,eachnodecankeepthetreetopologyinformation.Therefore,eachpacketonlyneedstocarryitstreeidentier. 2.3Long-TimeAveragePerformanceAnalysis Weshowinthissectionthatwiththealgorithm( 2 )-( 2 ),theachievedutilitycanbearbitrarilyclosetotheoptimumandthevirtualqueuesizesareboundedinthelong-timeaveragesense.TheanalysisfollowstheLyapunovoptimizationtechniquein[ 24 46 47 ]. 5Intheanalysisofrealqueueboundedness,weneedtoassumethattheremaininglinkcapacityisnotused. 28

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DeneYtobeY,fyj9(x,r)suchthat(x,y,r)isfeasibletoP.g. LetYbethelargestadmittedrateonytnsuchthatthevectorysym,(Y)t2T,n2Vt,whoseentriesareallY's,isinY.Thatis,ysymisobtainedbypushingthesameratesasmuchaspossiblenotonlyintotherootnodesofthetreesbutalsointoallothernon-leafnodesofthetrees. Weconsiderthefollowing-tightenedproblemP().P():maxXs2SUs(xs)s.t.rte)]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,e))]TJ /F8 11.955 Tf 11.96 0 Td[(ytb(e),8t2T,8e2Et, (2)andtheconstraints( 2 )-( 2 ), where0<
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Remark:Theoptimalsolutionofthe-tightenedproblemisusedintheanalysis.Togetaproperperformancebound,itisimportanttodecidewhichconstraintsaretightened.Here,wetightentheconstraintsthatwererelaxedwhenwederivedthealgorithm.Itallowsustoeasilyassociatewitheachofthevirtualqueues,whichiscrucialtogetthevirtualqueuesizebound.ThepointwillbecomeclearerintheproofofTheorem 2.1 .InSection 2.4 ,wewillshowotherpossibilities. DeneUmaxtobeUmax,Xs2Smax0xsXmaxUs(xs). NotethatUmaxiswelldened.Letfandf()betheoptimalobjectivevaluesofproblemPandP(),respectively.Then,wehavethefollowinglemma. Lemma1. f()!f,as!0. Proof. LetMUbethemaximumderivativeofUsoverallsessions,i.e.,MU,maxs2S,0xsXmaxUs0(xs). Wewillshowthatthefollowinginequalitieshold.ff()f)]TJ /F3 11.955 Tf 11.95 0 Td[(MUXmaxjSj Y. (2) Therstinequalityistrivialbecause(). SinceYisconvexandbothyandysymareinY,wehave(1)]TJ /F3 11.955 Tf 15.86 8.09 Td[( Y)y+ Yysym2Y, whichisequivalentto(1)]TJ /F3 11.955 Tf 15.86 8.08 Td[( Y)y+12Y. 30

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Now,weshowthat(1)]TJ /F3 11.955 Tf 12.47 0 Td[(=Y)y2Y().First,since(1)]TJ /F3 11.955 Tf 12.47 0 Td[(=Y)y+12Y,thereexist~xand~rsuchthat(~x,(1)]TJ /F3 11.955 Tf 11.98 0 Td[(=Y)y+1,~r)arefeasibletoproblemP.Then,itiseasytoseethat(1)]TJ /F3 11.955 Tf 12.19 0 Td[(=Y)yand~rsatisfytheconstraintsin( 2 )ofproblemP().Next,let^xs=Pt2Ts(1)]TJ /F3 11.955 Tf 12.3 0 Td[(=Y)yto(t)foralls2S.Then,^xand(1)]TJ /F3 11.955 Tf 12.3 0 Td[(=Y)ysatisfytheconstraintsin( 2 )ofproblemP().Next,wehave^xs=Pt2Ts(1)]TJ /F3 11.955 Tf 12.24 0 Td[(=Y)yto(t)=(1)]TJ /F3 11.955 Tf 12.24 0 Td[(=Y)xsforalls.Since0xsXmax,itisclearthat0(1)]TJ /F3 11.955 Tf 12.25 0 Td[(=Y)xsXmaxforalls,whichshows^xsatisestheconstraintsin( 2 )ofproblemP().Sinceallotherconstraintsaretriviallysatised,(1)]TJ /F3 11.955 Tf 11.96 0 Td[(=Y)x,(1)]TJ /F3 11.955 Tf 11.95 0 Td[(=Y)y,and~rarefeasibletoproblemP(). Sincex()isapartofanoptimalsolutionofproblemP()and(1)]TJ /F3 11.955 Tf 11.99 0 Td[(=Y)xisapartofafeasiblesolutionoftheproblem,wehaveXs2SUs(xs())Xs2SUs((1)]TJ /F3 11.955 Tf 15.86 8.09 Td[( Y)xs). WealsohaveXs2SUs((1)]TJ /F3 11.955 Tf 15.86 8.09 Td[( Y)xs)=Xs2SUs(xs)]TJ /F3 11.955 Tf 15.86 8.09 Td[( Yxs)Xs2SUs(xs))]TJ /F8 11.955 Tf 11.95 0 Td[(MU YxsXs2SUs(xs))-222(jSjMUXmax Y. Therefore,theinequalitiesin( 2 )hold.Then,f()!fas!0. LetPs,t,e()betheabbreviatednotationofPs2SPt2TsPe2Et().DenetheLyapunovfunctionV(k)andtheLyapunovdrift(k)asfollows:V(k),Xs,t,e(qte(k))2,(k),V(k+1))]TJ /F8 11.955 Tf 11.96 0 Td[(V(k). (2) Then,wecangetaboundfortheLyapunovdriftasfollows. 31

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Lemma2. Foranynite>0and0<
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Denefunctions(k)and(k)asfollows:(k),Xs2S1 Us(xs(k)))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2TsXe2Etytb(e)(k)qte(k),(k),Xs,t,eqte(k)rte(k))]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,e)(k)=Xe2EXs2SXt2Ts:e2Etrte(k)qte(k))]TJ /F16 11.955 Tf 27.56 11.35 Td[(Xe02(t,d(e))qte0(k). Sincethealgorithm( 2 )-( 2 )greedilymaximizes(k)and(k),wehavethefollowinginequalitiesforouralgorithm.(k)Xs2S1 Us(~xs))]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2TsXe2Et~ytb(e)qte(k),(k)Xs,t,eqte(k)~rte)]TJ /F4 11.955 Tf 11.57 0 Td[(~rtp(t,e), where~x,~yand~rareanynonnegativevectorssuchthat(~x,~y)2(x,y)and~r2r. Since(x(),y(),r())isfeasibletoproblemP,wehave(k)Xs2S1 Us(xs()))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2TsXe2Etytb(e)()qte(k). (2) Moreover,since(x(),y(),r())isfeasibletoproblemP(),wehaverte())]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,e)())]TJ /F8 11.955 Tf 11.95 0 Td[(ytb(e)(),8t2T,8e2Et. Then,wehave(k)Xs,t,eqte(k)rte())]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,e)()Xs,t,eqte(k)(ytb(e)()+). (2) Applyingtheaboveinequalitiesin( 2 )and( 2 )tothedriftexpressionin( 2 )andcancelingthecommonterms,wegettheinequalityin( 2 ). 33

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Next,wewillshowthelong-timeaverageoftheachievedutilityislower-boundedbytheoptimalsolutiontothe-tightenedproblemP().Wewillalsoshowthatthelong-timeaverageoftheaggregatevirtualqueuesizeisbounded. Lemma3. Foranynite>0and0<
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rearrangingtheaboveinequalityin( 2 ),weget1 KK)]TJ /F12 7.97 Tf 6.59 0 Td[(1X=0Xs,t,eqte())]TJ /F8 11.955 Tf 13.15 8.09 Td[(V(0) 2KB+2Umax= 2. TakingthelimsupasK!1yieldstheboundonthelong-timeaveragevirtualqueuesizeasin( 2 ). Usingtheabovelemmas,wecanderivethefollowingtheorem. Theorem2.1. Foranyparameter>0,thealgorithm( 2 )-( 2 )satisesthefollowingperformancebounds.liminfk!1Xs2SUs(xs(k))f)]TJ /F8 11.955 Tf 13.15 8.09 Td[(B 2, (2)limsupk!11 kk)]TJ /F12 7.97 Tf 6.59 0 Td[(1X=0Xs,t,eqte()B+2Umax= 2Y. (2) Proof. Theinequalitiesin( 2 )and( 2 )holdforanysuchthat0<
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2.4AlternativeWaytoProvetheAlgorithmPerformance Inthissection,weconsideranalternativewaytoprovethealgorithmperformancebymodifyingdifferentconstraints.Althoughthealternativemethoddoesnotnecessarilyleadtoimprovedperformancebounds,itisinterestingtoseethattighteningtheconstraintsin( 2 )isnottheonlywaytoshowthealgorithmperformance. Weconsideranalternativewaytoprovethealgorithmperformancebyreducingsessionowratesbyamount.LetXsymbethelargestadmittedsessionratexssuchthatthevector(Xsym)s2SisapartofafeasiblesolutiontoP.Wemayconsiderthefollowing-modiedproblemPA1().PA1():maxXs2SUs(xs)s.t.Xt2Tsyto(t)=xs+,8s2S,andtheconstraints( 2 ),( 2 )-( 2 ), where0<0,thealgorithm( 2 )-( 2 )satisesthefollowingperformancebounds.liminfk!1Xs2SUs(xs(k))f)]TJ /F8 11.955 Tf 13.15 8.08 Td[(B 2, (2)limsupk!11 kk)]TJ /F12 7.97 Tf 6.59 0 Td[(1X=0Xs,t,eqte()(B+2Umax=)maxs2SjTsj 2Xsym, (2) whereBisthesameconstantasinLemma 2 36

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Proof. Wewillshowtheoutlinetoderivetheperformanceboundbyconsideringthe-modiedproblemPA1().Usingthefactthat(x(),(ytn())]TJ /F3 11.955 Tf 13.16 0 Td[(=jTsj),r())and(x()+1,y(),r())arefeasibletoproblemP,wehave(k)Xs2S1 Us(xs()))]TJ /F16 11.955 Tf 12.89 11.35 Td[(Xt2TsXe2Et(ytb(e)())]TJ /F3 11.955 Tf 20.71 8.08 Td[( jTsj)qte(k), and(k)Xs,t,eqte(k)rte())]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,e)()Xs,t,eqte(k)ytb(e)(). UsingthesameargumentasintheproofofLemma 2 ,weget(q(k)))]TJ /F4 11.955 Tf 13.15 8.09 Td[(2 Xs2SUs(xs(k))B)]TJ /F4 11.955 Tf 13.16 8.09 Td[(2 Xs2SUs(xs()))]TJ /F4 11.955 Tf 11.95 0 Td[(2Xs2S jTsjXt2TsXe2Etqte(k)B)]TJ /F4 11.955 Tf 13.16 8.08 Td[(2 Xs2SUs(xs()))]TJ /F4 11.955 Tf 11.95 0 Td[(2 maxs2SjTsjXs,t,eqte(k). Then,usingthesameargumentasintheproofofLemma 3 ,wegetthefollowingresult:Foranynite>0and0<
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Notethat,comparedwithTheorem 2.1 ,thetermmaxs2SjTsjmayloosentheboundonthelong-timeaverageofthevirtualqueuesizes,whereasthetermXsymmaytightenit. Remark:Inconstructingthe-tightened(or-modied)problem,theusualprocedureistomodifythelinkcapacityconstraintsasfollows.FromtheproblemformulationofP,considertighteningthelinkcapacityconstraintsas:Xt:e2Etrtece)]TJ /F3 11.955 Tf 11.96 0 Td[(,8e2E, forsatisfying0<
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Proof. SincetheproblemPismaximizingaconcavefunctionwithlinearconstraintsandfisnite,thestrongdualityholds.Therefore,thereisnodualitygap,i.e.,f=D(),forall2)]TJ /F11 7.97 Tf 6.78 4.34 Td[(.Italsoimpliesthat)]TJ /F11 7.97 Tf 6.78 4.34 Td[(isnon-empty. Now,supposethat)]TJ /F11 7.97 Tf 6.77 4.34 Td[(isnotbounded.RecallthatD()=max(x,y)2(x,y),r2rXs2SUs(xs)+()TJ(y,r) Let(^x,^y,^r)bethefeasiblesolutiontoproblemPsuchthatJ(^y,^r)>0,wherethestrictinequalityistakenentrywise.SuchsolutionexistsduetoAssumption 1 .Considerthefollowingexpression.Xs2SUs(^xs)+()TJ(^y,^r). Then,wecanmakeD()arbitrarilylargebychoosing2)]TJ /F11 7.97 Tf 6.78 4.34 Td[(withalargeenoughnormkk.However,thiscontradictsthatfisnite.Therefore,)]TJ /F11 7.97 Tf 6.77 4.34 Td[(isbounded. Now,weshowthat)]TJ /F11 7.97 Tf 6.77 4.33 Td[(isclosed.ThedualfunctionD()isconvexandhencecontinuous.)]TJ /F11 7.97 Tf 6.78 4.34 Td[(istheinverseimageofthesinglepointsetfD()g,whichisclosed.Hence,)]TJ /F11 7.97 Tf 6.77 4.34 Td[(isclosed. Fornite>0and2)]TJ /F11 7.97 Tf 6.78 4.34 Td[(,let\(),f0jD()D()+g. Lemma6. \()isbounded. Proof. Supposethat\()isnotbounded,whichmeansthatthereexists~2\()witharbitrarilylargenormk~k.Then,givenassumption 1 ,wecanmakeD(~)arbitrarilylargebychoosingsuch~withalargeenoughnormand(x,y,r)2suchthatJ(y,r)>0.ThiscontradictsthatD()isniteandD(~)D()+.Therefore,\()isbounded. Lemma7. ThevectorJ(y(k),r(k))isasubgradientofDatq(k). 39

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Proof. Forall0,D()=max(x,y)2(x,y),r2rL(x,y,r;)L(x(k),y(k),r(k);)=Xs2SUs(xs(k))+TJ(y(k),r(k))=Xs2SUs(xs(k))+q(k)TJ(y(k),r(k))+()]TJ /F3 11.955 Tf 11.95 0 Td[(q(k))TJ(y(k),r(k))=D(q(k))+()]TJ /F3 11.955 Tf 11.95 0 Td[(q(k))TJ(y(k),r(k)), whichshowsthatJ(y(k),r(k))isasubgradientofDatq(k)([ 7 ],page731). Now,wearereadytoshowthatthescaledvirtualqueuevectorsq(k)canbearbitrarilycloseto)]TJ /F11 7.97 Tf 6.78 4.34 Td[(bychoosingsmallenough>0.Letd(,)]TJ /F11 7.97 Tf 12.26 4.34 Td[(),min2)]TJ /F18 5.978 Tf 4.82 2.27 Td[(k)]TJ /F3 11.955 Tf 12.24 0 Td[(kbethedistancebetweenandthenearestoptimalsolutionofthedual.Fornite>0,let(),max2\()d(,)]TJ /F11 7.97 Tf 12.25 4.34 Td[()+. Lemma8. Forany>0,thereexist>0andasufcientlylargeK0<1suchthat,withanyniteinitialfeasibleq(0),forallkK0,d(q(k),)]TJ /F11 7.97 Tf 17.14 4.34 Td[()<. Proof. Forany2)]TJ /F11 7.97 Tf 6.77 4.33 Td[(,letuswriteq,(1=).Then,by( 2 )andbythenon-expansivepropertyofprojection[ 7 ],wehavekq(k+1))]TJ /F3 11.955 Tf 11.96 0 Td[(k2=k[q(k))]TJ /F8 11.955 Tf 11.96 0 Td[(J(y(k),r(k))]+)]TJ /F3 11.955 Tf 11.96 0 Td[(qk2k(q(k))]TJ /F8 11.955 Tf 11.95 0 Td[(J(y(k),r(k))))]TJ /F3 11.955 Tf 11.96 0 Td[(qk2=kq(k))]TJ /F3 11.955 Tf 11.96 0 Td[(k2+2kJ(y(k),r(k))k2)]TJ /F4 11.955 Tf 11.95 0 Td[(22(q(k))]TJ /F8 11.955 Tf 11.95 0 Td[(q)TJ(y(k),r(k)). (2) 40

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SincethevectorJ(y(k),r(k))isasubgradientofDatq(k)asshowninLemma 7 ,wehaveD()D(q(k))+()]TJ /F3 11.955 Tf 11.96 0 Td[(q(k))TJ(y(k),r(k)),)]TJ /F4 11.955 Tf 29.89 0 Td[((q(k))]TJ /F8 11.955 Tf 11.95 0 Td[(q)TJ(y(k),r(k))1 (D())]TJ /F8 11.955 Tf 11.95 0 Td[(D(q(k))). (2) Applyingtheinequality( 2 )to( 2 ),wegetkq(k+1))]TJ /F3 11.955 Tf 11.96 0 Td[(k2kq(k))]TJ /F3 11.955 Tf 11.96 0 Td[(k2+2kJ(y(k),r(k))k2+2(D())]TJ /F8 11.955 Tf 11.96 0 Td[(D(q(k))). (2) Fix>0andconsider\().Pick=MJ.Then,aslongasq(k)62\(),i.e.,D(q(k)))]TJ /F8 11.955 Tf 11.96 0 Td[(D()>,from( 2 ),wegetkq(k+1))]TJ /F3 11.955 Tf 11.95 0 Td[(k2kq(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k2+2MJ)]TJ /F4 11.955 Tf 11.95 0 Td[(2kq(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k2+ MJMJ)]TJ /F4 11.955 Tf 11.95 0 Td[(2kq(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k2)]TJ /F3 11.955 Tf 11.96 0 Td[(. Theaboveinequalitymeansthatq(k)eventuallyenterstheset\(). Ontheotherhand,ifwepick=p MJ,thenonceq(k)2\(),wehavekq(k+1))]TJ /F3 11.955 Tf 11.96 0 Td[(kk(q(k))]TJ /F8 11.955 Tf 11.95 0 Td[(J(y(k),r(k))))]TJ /F3 11.955 Tf 11.96 0 Td[(kkq(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k+kJ(y(k),r(k))kkq(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k+, where,again,therstinequalityisdueto( 2 )andthenon-expansivepropertyofprojection.Thesecondinequalityisduetothetriangleinequality,andthelastinequalityholdsbypluggingintheupperboundsofandkJ(y(k),r(k))k. 41

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Sincetheaboveinequalitiesholdforany2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(,wegetd(q(k+1),)]TJ /F11 7.97 Tf 35.37 4.94 Td[()d(q(k),)]TJ /F11 7.97 Tf 17.13 4.94 Td[()+. Ifwechoosesuchthatminf=MJ,=p MJg,thenthereexistsatimeK0suchthatd(q(k),)]TJ /F11 7.97 Tf 17.14 4.34 Td[()()forallkK0. Notethat()!0as!0.Sincethedualfunctioniscontinuous,max2\()d(,)]TJ /F11 7.97 Tf 12.25 4.34 Td[()!0,as!0. Then,forany>0,bypicking>0sufcientlysmall,wehave()<.Therefore,thereexists0=minf=MJ,=p MJg>0,suchthatforany0andanyinitialfeasibleq(0),thereexistsatimeK0suchthatd(q(k),)]TJ /F11 7.97 Tf 17.14 4.34 Td[()K0. Inthenextlemma,weshowthatthevirtualqueuebacklogsateverytimeslotareboundedaboveinouralgorithm. Lemma9. Foranynite>0,thereexistsaniteMq>0suchthat,foreverylinkeandtreet,thevirtualqueuesizeqte(k)0,wechoosesuchthat=maxfMJ,p MJg.Sinceminf=MJ,=p MJg,bythesameargumentusedinLemma 8 ,thereexistsK0suchthatd(q(k),)().Then,bytheboundednessof)]TJ /F11 7.97 Tf 6.77 4.34 Td[(,supkkq(k)k<1.Since>0,supkkq(k)k<1. 2.5.2BoundednessofRealQueues Toarguethatthenetworkisstabilizedwiththealgorithm,weneedtoshowthattherealqueuesizesarebounded.In[ 11 46 ],theauthorsshowthestabilityoftheiralgorithmsbyclaimingthatthelong-timeaverageoftherealqueuesizesarebounded.In[ 36 ],theauthorsonlyshowthatthescaledvirtualqueuesizesarebounded.Inthis 42

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section,weshowthatwithouralgorithm,therealqueuesizesateverytimeslotarebounded,whichisastrongerresult. LetQte(k)betherealqueuesizeatlinkefortreetattimeslotkandQ(k)bethevector(Qte(k))t2T,e2Et.Thealgorithm( 2 )-( 2 )satisesthefollowingrealqueuebound. Theorem2.2. ThereexistsaniteMq>0suchthat,foreverylinkeandtreet,therealqueuesizeQte(k)
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2.6SimulationResults Inthissection,weverifythecorrectnessandevaluatetheperformanceofthebackpressurealgorithmwithsimulation.Weexaminehowthescalingparameterandthenumberoftreesinasessionaffecttheperformanceofthealgorithm.Wealsoshowtheeffectoftheutilityfunctionswhentherearemultiplesessions. 2.6.1SimulationSetup WesupposethatwerunthebackpressurealgorithmoveranISPnetworkwherethedataserversarestrategicallyplacedbythenetworkoperatorandlinkcapacitiesareexclusivelyallocatedtothecontentdistributionservice.Tomakethesetuprealistic,weusetheSprintlink'sISPnetworktopologyobtainedfromtheRocketfuelproject[ 1 ],whichconsistsof315backbonenodesand1944links.Then,weattach100dataserversrandomlytosomebackbonenodes,withatmostoneserverperbackbonenode.Weassignlinkcapacity1000tomostofthebackbonelinksexceptsomecriticallinks6.Weassignrelativelylargelinkcapacity4000tothosecriticallinkssothattheywillnotbecomebottlenecks.Weassignlinkcapacity3000tothelinksbetweenadataserveranditsattachedbackbonenode. Weusethefollowingheuristictoobtainthesetofmulticasttreesforeachsession.Werunthealgorithmin[ 61 ],whichcomputestheoptimalsetoftrees(approximately),aswellastheirrateallocationandcosts.Thenumberoftreesreturnedbythatalgorithmisusuallyverylargewhenthenetworksizeislarge.Wechoosetouseafewofthemintheincreasingorderofthetreecost. 2.6.2SingleSession Werstexaminehowthealgorithmworksforasinglesession.Thesessionconsistsofonesourceand99receivers.WechooseUs(xs)=xsastheutilityfunction.Therefore, 6Bycriticallinks,wemeanroughlythelinksthateasilybecomebottleneckiftheydonothavesufcientcapacity. 44

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(a)Timeaverage (b)Exponentialmovingaverage Figure2-2. Sessionrateandreceivingrates. theobjectiveistomaximizethethroughput.Thescalingparameterissettobe10)]TJ /F12 7.97 Tf 6.59 0 Td[(5andthenumberoftreesis10.Forcomparisonpurpose,ifthereisnorestrictiononthenumberoftrees,thealgorithmin[ 61 ]haseventuallyfound57treeswhichformtheoptimaltreesetandtheoptimalsessionrateis1846,whichisthemaximumrateachievablebythebackpressurealgorithm. ThesolidlineinFig. 2-2 (a)showstheconvergenceofthetimeaveragesessionrate(TA-S).Wealsoplotthetimeaverageofthereceivingratewithadottedline.Beforereachingthesteadystate,thesessionratecanbedifferentfromeachofthereceivingratesbecausesomepacketsaretemporarilyqueuedinthenetwork.Weshowthe 45

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timeaverageratesatevery5000iterations.Eachpointonthedottedlinerepresentstheaverageacrossthereceiversofthetimeaveragereceivingrates(TA-R)atthecorrespondingiteration.Theboxcenteredaroundeachpointmarksonestandarddeviationaboveorbelowtheaveragerate,andtheendpointsoftheverticallineoneachpointrepresentthemaximumandtheminimum(withrespecttotheratesamplescollectedacrossthereceiversatthecorrespondingiteration).Onecanseethattheaverageofthetimeaveragereceivingrateseventuallyapproachesthetimeaveragesessionrate.Thissuggeststhatthequeuesinthenetworkwillnotgrowindenitely.Moreover,thestandarddeviationofthetimeaveragereceivingratesalsodecreasesasconvergencetakesplace. Eventhoughtheconvergenceofthelong-timeaverageratesappearsslow,theshort-timeaverageratesexperiencedbythesourceandthereceiversapproachmorequicklytotheconvergentvalues.Weverifythisbyexaminingtheexponentialmovingaverageoftherates,whichiscomputedas~xs(k)=(1)]TJ /F3 11.955 Tf 11.95 0 Td[()~xs(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1)+xs(k), where~xs(k)istheexponentialmovingaverageofthesessionrateforsessionsattimek,andisaconstanton(0,1).Inoursimulationexperiments,issetat0.1.Wecomputetheexponentialmovingaverageofthereceivingratessimilarly.Althoughsuchanaveragingoperationretainstheeffectofthesamplesbacktotheverybeginning,thecontributionfromthepastsamplesdiminishesveryrapidlyfor=0.1.Theresultingaveragereectstheshort-timeaverage,i.e.,theaverageofthecurrentandrecentsamples. Fig. 2-2 (b)showstheexponentialmovingaveragesessionrate(EMA-S)andtheaverageoftheexponentialmovingaveragereceivingrates(EMA-R).Itshowsthattheshort-timeaverageratesapproachmorequicklytothesteadystatevaluesthanthe 46

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Figure2-3. Timeaveragetreerates. Figure2-4. Aggregatequeuesizesofthetrees. long-timeaveragerates.Thereasonisthattheratesassignedattheinitialiterationareusuallyfarfromtheoptimaandtheireffectisslowtodissipateinthelong-timeaverage. Fig. 2-3 showsthatthetimeaveragetreeratesalsoconverge.Someofthetreeshavetinyrates,whichimpliesthatwemayusefewertreeswithoutsignicantlossinthesessionrate.Wewillshowtheimpactofthenumberoftreesontheperformancelater. TheaggregatequeuesizesofthetreesareshowninFig. 2-4 .Theaggregatequeuesizeofatreeisthesumofthequeuesizesonallthelinksofthetree.Thequeuesizesgrowrapidlyinitiallyandthenbecomestabilized.Notethatthetimewhenthe 47

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(a)Timeaveragesessionrates (b)Timeaveragereceivingrates Figure2-5. Effectofonachievablerates. queuesizesbecomestabilizedismuchearlierthanthetimewhenthetimeaveragesessionrateconverges. Wenextshowhowthescalingparameteraffectstheperformanceofthealgorithm.Fig. 2-5 (a)showsthattheachievedtimeaveragesessionratetendstoincreaseasdecreases.Ofcourse,theamountofincreaseeventuallybecomesnegligibleas 48

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Figure2-6. Effectofonthequeuesizes. becomessufcientlysmall7.Thisrelationshipcanbeseenbetterwhenthereceivingratesarealsoexamined,asshowninFig. 2-5 (b). Thescalingparameteraffectstheconvergencespeedofthesessionrate.Asdecreases,ittakeslongerfortheratetoconverge.Forexample,for=510)]TJ /F12 7.97 Tf 6.58 0 Td[(5,thetimeaveragesessionratenearlystabilizesataroundthe2104thiteration,whereasfor=10)]TJ /F12 7.97 Tf 6.59 0 Td[(5,itdoessoataroundthe8104thiteration.Similarresultscanalsobefoundforthetimeaveragereceivingrates. Throughitseffectontheconvergencespeed,thescalingparameteralsoaffectsthequeuesizesinthenetwork.InFig. 2-6 ,weplotthetrajectoryofthetotalqueuesizeinthenetworkatevery1000iterationsforeach.Asdecreases,thetimeittakesfortheaggregatequeuesizetostabilizeincreasesandtheaggregatequeuesizeincreasesaswell. Next,weshowhowthenumberoftreesinthesessionaffectstheperformanceofthealgorithm,intermsoftheachievablerates.Inthisexperiment,wesettobe 7With=10)]TJ /F12 7.97 Tf 6.58 0 Td[(6,althoughthecurveisbecomingquiteatatthe105thiteration,thetimeaveragesessionratehasnotfullyconverged.Thisismadeclearbythecorrespondingreceivingrateatthattime,whichismuchlessthanthesessionrate. 49

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(a)Timeaveragesessionrates (b)Timeaveragereceivingrates Figure2-7. Effectofthenumberoftrees. 10)]TJ /F12 7.97 Tf 6.59 0 Td[(5.Fig. 2-7 showsthattheachievedtimeaveragesessionratetendstoincreaseasthenumberoftreesincreases.However,beyond5trees,furtherimprovementinperformanceisslight.Ontheotherhand,withonly2trees,theresultingsessionrateisonlyabout60percentofwhatisachievablewith5trees.Nevertheless,using2treesisstillmuchbetterthanusingasingletree. 2.6.3MultipleSessions Wenextexaminehowthealgorithmworksinthecasewithmultiplesessions.Theexperimentalsetupconsistsofvesessions,eachwithonesourceand99receivers.Thevesourcesaredistinct.Weusetwoobjectivefunctions,Ps2SxsandPs2Slog(xs+ 50

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(a)TimeaveragerateswhenUs(xs)=xs (b)TimeaveragerateswhenUs(xs)=log(xs+1) Figure2-8. Timeaverageratesofmultiplesessions. 1),respectively,toseehowdifferentobjectivefunctionsaffecttheresultingperformance.Therstobjectivefunctioncorrespondstomaximizingthethroughputandthesecondisknowntoleadtoproportionalfairness.Wesettobe510)]TJ /F12 7.97 Tf 6.58 0 Td[(6whenweusetheobjectivefunctionPs2Sxs,andsettobe1.610)]TJ /F12 7.97 Tf 6.59 0 Td[(8whenweusePs2Slog(xs+1). Theresultsindicatethatthealgorithmworkscorrectlyforthismultiplesessioncase.Fig. 2-8 showshowfastthetimeaveragesessionratesapproachtheoptimalratesforthetwoobjectivefunctions.Inaddition,Fig. 2-8 (a)showsthatthetimeaveragesessionratesbecomequitedifferentfromeachotherwhentheobjectivefunctionisPs2Sxs, 51

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whichmaybedeemedunfairinsomecases.Ontheotherhand,Fig. 2-8 (b)showsthatthevesessionratesarenearlythesameundertheobjectivefunctionPs2Slog(xs+1). 2.7RelatedWork Fewoftheexistingalgorithmsformulticastutilityoptimizationareofthebackpressuretype.Muchofthesubsequentdiscussionfocusesonhowourworkiscomparedtothefewexceptions.Priorworkeitherhassubstantiallydifferentproblemformulations,orusesdifferentclassesofalgorithms,orisnotascomprehensiveinperformanceanalysis.Werstcontrastourresultswiththeworkin[ 11 20 29 30 51 52 ],whichhaveintroducedsimilarproblemsandbackpressure-basedsolutions.In[ 52 ],theauthorspresentabackpressurealgorithmforastabilityproblemwithrandomarrivalstothesourceswhereasweconsideranoptimizationproblemwithinnitebacklogsatthesources.Theanalyticaltechniqueof[ 52 ]isLyapunovdriftanalysisforstabilityasin[ 57 ],whichisnotthesameastheLyapunovoptimizationtechnique.Neitherarethereanyconvexoptimizationresultsoranalysis.Theauthorsin[ 11 ]consideramulticastbackpressurealgorithm,butforawirelessnetwork8.Themainsimilaritybetween[ 11 ]andthisstudyisthatthebackpressurealgorithmsarederivedfromsubgradientalgorithms.Otherthanthat,therearesubstantialdifferencesintheoptimizationproblemsthemselves,thedetailsofthealgorithms,andtheanalyticalresultsandtechniques.Thedifferenceintheproblemformulationisthattheyrestricttothecaseofasingletreepersession,and,insteadofthewirelinelinkcapacityconstraints,theyhavethewirelessnetworkcapacityconstraint.TheproblemsandalgorithmsaredifferentenoughthatnewproofsareneededevenforthepartoftheresultsthatarederivedfromtheLyapunovoptimizationtechnique.Furthermore,[ 11 ]containsnoconvexoptimizationresultsoranalysis.Anothermajordifferenceisthatrealqueues 8Thediscussionon[ 11 ]isrestrictedtothepartaboutsingle-ratemulticast,whichismorerelevanttoourproblem.Thefocusof[ 11 ]isinfactonmulti-ratemulticast. 52

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areusedforcontrolin[ 52 ]and[ 11 ]whereasweusevirtualqueues.Becauseofthisdifference,theperformanceissuesweinvestigatehaveanewtwist:Althoughouralgorithmisbasedonthevirtualqueues,ultimately,wewishtoprovetheboundednessoftherealqueues.Asaresult,theprooftechniquesusedin[ 11 52 ]arenotsufcient,andnewonesareneeded.Wementioninpassingthat,whileusingrealqueuesmaybeassociatedwitheaseofimplementationundersomecircumstances,usingvirtualqueuesmayalsohavecertainadvantagesoverusingrealqueues.Forinstance,itmayleadtosignicantlysmallerrealqueuesizes,andhence,lowerdelay,asshownin[ 12 ]. In[ 20 29 30 51 ],theauthorsconsidermulti-ratemulticastproblemswithasinglemulticasttreeforeachsession.Theirobjectivesarewithrespecttothereceivers'utilitiesratherthanthesessions.Hence,thoseproblemsaredifferentfromours.Moreover,thesestudiesarenotconcernedwiththeboundednessoftherealqueues.In[ 51 ],theperformanceobjectiveismaxminfairness,anditisnotclearhowtoextenditsresultstoothertypesoffairness.In[ 20 29 30 ],theauthorsassumethattheutilityfunctionsarestrictlyconcave,whichisakeyconditionusedintheirproofs.Therefore,theirprooftechniquescannotbeappliedtoourcasewithnon-strictconcaveutilityfunctions. Wenextdiscusssomelessrelatedwork.Theuseoftheoptimizationapproachonmulticastproblemshasbeenreportedin[ 13 19 40 41 61 62 ].However,theresultingalgorithmsarenotofthebackpressuretypeandallofthemrequireglobalexchangeofcontrolmessages,i.e.,thesourcenodesandlinksbelongingtothesamesessionexchangeinformationaboutthesessionratesandlinkprices.In[ 19 61 ],specialobjectivesareconsideredsuchasmaximizingthroughputorminimizingnetworkcongestion.In[ 13 40 41 ],theauthorsassumethatthenetworkbottlenecksareatthenodes'uplinkcapacitieswhereasweassumethatthebottleneckscanbeanywhereinthenetwork.In[ 61 62 ],althoughanoptimalsetoftreeswithanoptimalrateallocationisfoundamongallpossibletrees,thealgorithmsarelesslocalandthecomputationrequirementateachiterationismuchhigherthanthealgorithmproposedinthischapter. 53

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Theunicastmulti-pathutilitymaximizationproblemshavebeenstudiedintheliterature[ 26 32 36 38 58 ].In[ 38 58 ],theauthorsattempttosolvetheoscillationproblemoftheprimalvariablesduetothelackofstrictconcavitybyusingtheproximaloptimizationmethod[ 8 ].Thealgorithmsaredistributedbutlesslocalthanours.In[ 36 ],abackpressuresubgradientalgorithmispresentedforunicast.Sincetheauthorsassumestrictlyconcaveutilityfunctions,theycanshowtheprimaloptimalityusingtheconvexoptimizationtechniques. 54

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CHAPTER3REAL-QUEUE-BASEDMULTI-TREEMULTICASTALGORITHMFORCONTENTDISTRIBUTION Thischapteraddressesthesameproblemastheoneintroducedinthepreviouschapter.Weconsidertheproblemofutilitymaximizationofmulticastsessionswithmultipletreesoverstaticinfrastructurenetworks.Weassumetheutilityfunctionsareconcavebuttheconcavityisnotnecessarilystrict.Inthischapter,wepresentamulticastbackpressurealgorithmusingtherealqueuebacklogs.Thealgorithmisdistributedandlocal.Weprovetheoptimalityandconvergenceofthealgorithm,usingauidlimitmodel.Weshowthatthebackpressurealgorithmwithrealqueuesisexactlyasubgradientalgorithmintheuidlimitmodel. Theremainingchapterisorganizedasfollows.Section 3.1 describesthesystemmodelandtheproblemformulation.InSection 3.2 ,wepresentthebackpressurealgorithmusingtherealqueuesizes.InSection 3.3 ,wederiveauidlimitmodelofthesystemandanalyzeitsproperties.Weshowthatthealgorithmisexactlyasubgradientalgorithmintheuidlimitanditisasymptoticallyoptimal.WediscusstherelatedworkinSection 3.4 3.1ProblemDescription Inthissection,wepresentadiscretesystemmodelandformalformulationsoftheproblemanditsdual.ThesystemmodelandproblemintroducedherearesimilartothoseinChapter 2 3.1.1DiscreteSystemModel Thenetworkisrepresentedbyadirected,edge-capacitatedgraphG=(V,E),whereVisthesetofnodesandEisthesetofdirectedlinks.LetSbethesetofallmulticastsessionsinthenetwork.Eachsessionisassociatedwithasourceandasetofdestinations,whichisasubsetofV.EachsessionsisgivenasetoftreesTsalongwhichittransmitsitsdata.LetTbetheunionofTs.Notethatweregardthetreesofdifferentsessionsasdifferenttreeseveniftheyhavethesametopology.LetEtbethe 55

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setoflinksontreet.Denoteb(e)andd(e)tobethetransmitting(tail)nodeandthereceiving(head)nodeoflinke,respectively.LetVtbethesetoftransmittingnodesb(e)foralleinEt.Vtrepresentsthesetofnodesontreetwhicharenotleaves.Leto(t)betherootnodeoftreet.Letp(t,e)betheparentlinkoflinkeontreet.Let(t,n)bethesetofchildlinksatnodenontreet.Weassumethateachsourcehasaninnitebacklogofdata. Weconsideradiscretetime-slottedsystem.Weidentifyanintegertimetwiththeunittimeinterval[t,t+1).Weassumethatallthepacketshavethesamesize.Eachlinkcapacityisrepresentedasthenumberofpacketsthatcanbeservedinonetimeunit.Eachlinkecanserveanintegernumberceofpacketsinonetimeslot.LetCmax,maxe2Ecebethemaximumlinkcapacityoveralllinks. LetQte(k)bethenumberofpacketsfortreetqueuedatlinkeattimek.Letxs(k)bethenumberofpacketsinjected(admitted)intothenetworkforsourcesattimek.Letytn(k)bethenumberofpacketsadmittedfortreetandthroughnodenattimek.Notethatytn(k)0ifnodenisnottherootoftreetbecausethedataisonlyinjectedintotherootnodeofthetree.Wedenethenotationytn(k)forallnodesjusttosimplifytheproblemformulation.Letrte(k)bethenumberofpacketsoftreetallowedtobetransmittedonlinkeattimek.Letzte(k)bethenumberofpacketsoftreetactuallytransmittedonlinkeattimek.Notethatzte(k)cannotexceedrte(k)orQte(k).Thatis,zte(k)isequaltominfrte(k),Qte(k)g. LetXs(k)bethecumulativenumberofpacketsforsourcesthathavebeenadmittedbytimek.Similarly,letYtn(k),Rte(k),andZte(k)bethecumulativeamountsofpacketsassociatedwithytn(k),rte(k),andzte(k),respectively.Thatis,Xs(k)=k)]TJ /F12 7.97 Tf 6.58 0 Td[(1Xl=0xs(l),Ytn(k)=k)]TJ /F12 7.97 Tf 6.58 0 Td[(1Xl=0ytn(l),Rte(k)=k)]TJ /F12 7.97 Tf 6.58 0 Td[(1Xl=0rte(l),Zte(k)=k)]TJ /F12 7.97 Tf 6.58 0 Td[(1Xl=0zte(l). 56

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Wehavethefollowingqueueevolutionfortherealqueues:Qte(k+1)=Qte(k)+ytb(e)(k)+ztp(t,e)(k))]TJ /F8 11.955 Tf 11.95 0 Td[(zte(k), (3) fort2Tande2Et.ThequeueevolutioncanbealsoexpressedasQte(k)=Qte(0)+Ytb(e)(k)+Ztp(t,e)(k))]TJ /F8 11.955 Tf 11.96 0 Td[(Zte(k), fort2Tande2Et.WeassumethatQte(0)isniteforallt2Tande2Et. 3.1.2Notation WedenotebyR,R+,Z,andZ+thesetsofreal,non-negativereal,integer,andnon-negativeintegernumbers,respectively.LetkkbetheEuclideannorm.Wedenotebyd(a,b),ka)]TJ /F8 11.955 Tf 13.14 0 Td[(bkthedistancebetweenvectorsaandbinRn,andbyd(a,B),infb2Bd(a,B)thedistancebetweenavectora2RnandasetBRn.Ifa(k)isavectorfunctiona:R+!RnandBisasetinRn,theconvergencea(k)!Bmeansthatd(a(k),B)!0ask!1. Denote(ai)tobethevectorwithentriesai.Letx,yandrbethevectors(xs)s2S,(ytn)t2T,n2Vt,and(rte)t2T,e2Et,respectively.Thenotationsuchas(a)t2Tmeansrepeatingthevalueainallentriesofthevector.Foranytwovectorsa,b2Rn,denoteabtobethescalarproductofthetwovectors. Whendiscussingtheconvergenceoffunctions,theabbreviationu.o.c.meansuniformoncompactsets.Thetermalmosteverywhere(a.e.)meansalmosteverywherewithrespecttotheLebesguemeasure.DenotebyP(S)theprobabilityoftheeventS. 3.1.3ProblemFormulation Letxs(k)bethetimeaverageoftheadmittedtrafcrateforsessionsuptotimek,i.e.,xs(k),1 kXs(k). 57

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Letxsbethelimitofxs(k),i.e.,1xs,limk!1xs(k). Wedeneytn(k),ytn,rte(k)andrtesimilarly. OurproblemPisasfollows.P:maxXs2SUs(xs)s.t.rte)]TJ /F4 11.955 Tf 11.58 0 Td[(rtp(t,e))]TJ /F4 11.955 Tf 12.24 0 Td[(ytb(e)0,8t2T,8e2Et, (3)Xt2Tsyto(t)=xs,8s2S, (3)Xt:e2Etrtece,8e2E, (3)0xsXmax,8s2S, (3)ytn0,8t2T,8n2Vt, (3)rte0,8t2T,8e2Et. (3) Notethatallthevariablesarelong-timeaverages.Theobjectiveistomaximizethesumofthesessionutilitiesandeachsessionutilityfunctionisaconcavefunctionoftheadmittedsessionrate.Theobjectivefunctioncanreectvariousfairnesscriteriaamongthesessionssuchasweightedthroughputmaximization,proportionalfairnessormax-minfairness[ 43 ].Theconstraintsin( 3 )implythatforeverytreet,theallocatedlinkrateonalinkeonthetreeshouldbenolessthanthesumofthelinkrateofitsparentlinkp(t,e)andtheexogenousarrivaltothetailnodeb(e).Theconstraintsarenecessaryforqueuestoremainbounded.Itcanbealsoconsideredasarelaxedform 1Wetemporarilyassumethatthelimitexists.Weshallreplacelimwithliminforlimsupincasethatthelimitdoesnotexist. 58

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oftheowconservationconstraintsfortrees.Weassumethatrtp(t,e),0ifp(t,e)isnull.Recallthatthetreeratevariablesytn,0ifnodenisnottherootoftreet. Theconstraintsin( 3 )meanthattheaggregateratetransmittedbythetreesforasessionmustbeequaltothesessionrate.Theconstraintsin( 3 )arethelinkcapacityconstraintsthatthesumofthetransmissionratesonalinkcannotexceedthelinkcapacity.In( 3 ),weassumethesessionratesmustbeboundedbyaconstantXmaxfromabove. Let,f(x,y,r)j(x,y,r)isfeasibletoproblemPg. Wemakethefollowingassumptionon. Assumption3. ThereexistsafeasiblesolutionofPsuchthattheconstraintsin( 3 )holdwithstrictinequality. Assumption1isusedforshowingthedualoptimalityofouralgorithm.Itishighlyprobablethattheassumptionisvalidinrealnetworks. Wealsodenethesets(x,y)andrasfollows.(x,y),f(x,y)j(x,y)satises( 3 ),( 3 )and( 3 )g,r,frjrsatises( 3 )and( 3 )g. Thesenotationswillbeusedtosimplifylaterexpressions. WehavethefollowingassumptionsontheutilityfunctionUs(xs). Assumption4. Usisconcaveanddifferentiable.ThederivativeofUsisboundedon[0,Xmax]. NotethatUscanbelinear,non-strictlyconcave,andevennon-increasing. Lettebethenon-negativeLagrangemultipliersassociatedwiththeconstraintsin( 3 ).Letbethevector(te)t2T,e2Et.FromproblemP,werelaxtheconstraintsin( 3 ) 59

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andwritetheLagrangianasfollows.L(x,y,r;)=Xs2SUs(xs)+Xt2TXe2Ette(rte)]TJ /F8 11.955 Tf 11.95 0 Td[(rtp(t,e))]TJ /F8 11.955 Tf 11.96 0 Td[(ytb(e))=Xs2SUs(xs))]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2Tsyto(t)Xe2(t,o(t))te+Xe2EXs2SXt2Ts:e2Etrte(te)]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))te0). Thelastequalityholdsbecause,forallt,Xe2Etytb(e)te=Xe2(t,o(t))ytb(e)te+Xe2Et:e62(t,o(t))ytb(e)te, whereb(e)=o(t)fore2(t,o(t))andytn,0ifnodenisnottherootoftreet.Then,thedualfunctionisgivenbyD()=max(x,y)2(x,y),r2rL(x,y,r;)=max(x,y)2(x,y)Xs2SUs(xs))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2Tsyto(t)Xe2(t,o(t))te+maxr2rXe2EXs2SXt2Ts:e2Etrte(te)]TJ /F16 11.955 Tf 27.56 11.35 Td[(Xe02(t,d(e))te0). Then,thedualproblemisasfollows.minD()s.t.te0,8t2T,8e2Et. Let)]TJ /F11 7.97 Tf 6.78 4.34 Td[(,f0jD()=min0D()gbethesetofoptimaldualsolutionsofproblemP.)]TJ /F11 7.97 Tf 6.78 4.34 Td[(isnon-empty,closedandbounded[ 15 ]. 3.2BackpressureAlgorithm Inthissection,weintroduceabackpressurealgorithmforcontrollingthesessionandtreeratesandforschedulingthetransmissionsonthelinks. 60

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Letmbeapositiverealnumber.Later,misusedasascalingparametertoderiveauidlimitmodelofthesystem.ThealgorithmisinAlgorithm 3-1 Algorithm3-1Real-queue-basedalgorithm 1. Ateachtimeslotk,eachsourcessolvesthefollowingsubproblem.(~xs(k),(~yto(t)(k))t2Ts)2argmaxPt2Tsyto(t)=xs,0xsXmax,yto(t)0nUs(xs))]TJ /F4 11.955 Tf 15 8.09 Td[(1 mXt2Tsyto(t)Xe2(t,o(t))Qte(k)o. (3) 2. Ateachtimeslotk,eachlinkesolvesthefollowingsubproblem.(~rte(k))t:e2Et2argmaxPt:e2Etrtece,rte0nXt:e2Etrte)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(Qte(k))]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))Qte0(k)o. (3) 3. Notethat~xs(k),~ytn(k),and~rte(k)areingeneralrealnumbers.Weneedtoconvertthemintointegers.Letxs(k),ytn(k),andrte(k)benon-negativeinteger-valuedrandomvariableswithmeans~xs(k),~ytn(k),and~rte(k),respectively.Weensurethatxs(k),ytn(k),andrte(k)havethefollowingproperties.P(0xs(k)Xmax)=1,P(0ytn(k)Xmax)=1,P(0rte(k)ce)=1. Wealsoensurethatxs(k),ytn(k),andrte(k)areeachindependentovertime.Thealgorithmalwaysassigns0to~ytn(k)andytn(k)forn6=o(t). 4. Eachsourcesadmitsanintegeramountyto(t)(k)ofpacketstotreetfort2Tsandeachlinketransmitsanintegeramountminfrte(k),Qte(k)gofpacketsfortreet. 5. Ateachlinke,thedifferentialbacklogfortreetattimekisdenedtobeQte(k))]TJ /F16 11.955 Tf 25.17 8.96 Td[(Pe02(t,d(e))Qte0(k).Thealgorithmalwaysassigns0to~rte(k)andrte(k)fortreetwithanegativedifferentialbacklog. Itiseasytosolvethesubproblems( 3 )and( 3 )[ 15 ].Thealgorithmisdistributedandlocalbecauseallthenecessaryinformationcanbeobtainedfromtheneighboringnodes.Notethat(~x(k),~y(k),~r(k))maximizestheLagrangianLwith(1=m)Q(k)astheLagrangianmultipliers. 61

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3.3FluidLimitAnalysis Inthissection,wederiveauidlimitmodelofthesystemandanalyzethepropertiesoftheuidlimitmodel. 3.3.1FluidLimitModel LetF=fF(k),k=0,1,...gbetheprocessdescribingtheevolutionofthenetworksystem,whereF(k)=(X(k),Y(k),R(k),Z(k),Q(k)).WeextendthedenitionofQte(k)tocontinuoustimek2R+byadoptingtheconventionthatQte(k)isconstantwithineachtimeslot[k,k+1).WedoanalogousextensionfortheothercomponentsofprocessF(k)aswell. Wedene^Fm=(^Xm,^Ym,^Rm,^Zm,^Qm)tobethescaledprocesswhereeachcomponentof^FmisobtainedbytimeandspacescalingfromthecorrespondingcomponentofF.Thatis,^Xm(k),1 mX(mk),^Ym(k),1 mY(mk),^Rm(k),1 mR(mk),^Zm(k),1 mZ(mk),^Qm(k),1 mQ(mk). Notethateachcomponentof^Fmispiecewiseconstantontheinterval[k=m,(k+1)=m)fork2Z+. Wethengetasequenceofprocesses^Fm=(^Xm,^Ym,^Rm,^Zm,^Qm)asm!1.Theinitialstate^Qm(0)isniteforallm2M. Denition1. AxedsetoffunctionsF=(X,Y,R,Z,Q)issaidtobeauidlimitofthesystemifthereexistasequenceM0=fmj,j=1,2,...gsuchthatmj>0foreachjandmj!1asj!1,andasequenceofscaledprocessesf^Fmgsuchthat,asm!1alongthesequenceM0,^Fm!Falmostsurelyinthetopologyofuniformconvergenceovercompactsets(u.o.c.). Lemma10. Thereexistsauidlimitofthesystem. 62

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Theproofisomittedforbrevity.See[ 21 54 ]fortheproceduretoobtainauidlimitforadiscretestochasticnetwork. Fromnowon,m!1meansthatmgoestoinnityalongthesequenceM0,unlessotherwisespecied. Lemma11. Theuidlimitofthesystemsatisesthefollowingproperties.Qte(k)=Qte(0)+Ytb(e)(k)+Ztp(t,e)(k))]TJ /F4 11.955 Tf 13.36 2.66 Td[(Zte(k),8t2T,8e2Et,Qte(k)0,Qte(0)<1,8t2T,8e2Et, forallk0.Moreover,X,Y,R,ZandQareLipschitzcontinuouson[0,1)andhaveproperderivativesforalmosteveryk0.Whenthederivativesexist,theysatisfy_Xs(k)0,8s2S,_Ytn(k)0,8t2T,8n2Vt,_Rte(k)0,_Zte(k)0,8t2T,8e2Et,_Qte(k)=_Ytb(e)(k)+_Ztp(t,e)(k))]TJ /F4 11.955 Tf 14.76 4.35 Td[(_Zte(k),8t2T,8e2Et. (3) Theproofisomittedforbrevity.LipschitzcontinuityofallthecomponentsofFfollowsfromthefactthateachcomponentfunctionofFhasauniformlyboundedincrementovereverytimeslot. Wecallatimeinstancek>0regularifallthecomponentfunctionsofFhaveproperderivativesatk. 3.3.2PropertiesofFluidLimitModel Inthissubsection,wederivenon-trivialandmoreinterestingpropertiesoftheuidlimit,whichwillbeusedforprovingtheoptimalityofthealgorithm. Wedenevectors_X(k),_Y(k),_R(k)and_Z(k)tobe(_Xs(k))s2S,(_Ytn(k))t2T,n2Vt,(_Rte(k))t2T,e2Etand(_Zte(k))t2T,e2Et,respectively. 63

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LetusdeneasetL(Q)tobeL(Q),argmax(x,y)2(x,y)r2rL(x,y,r;Q). NotethatL(Q)mayhavemorethanoneelementsinceUsmaybenotstrictlyconcave.WealsodeneasetV(Q,)tobeV(Q,),f(x,y,r)j(x,y)2(x,y),r2r,andD(Q))]TJ /F8 11.955 Tf 11.96 0 Td[(L(x,y,r;Q)g.V(Q,)isthesetofpoints(x,y,r)suchthatthecorrespondingLagrangianfunctionvaluesgivenQaresmallerthanthemaximumLagrangianvaluebynomorethan.NotethatD(Q))]TJ /F8 11.955 Tf 12.49 0 Td[(L(x,y,r;Q)0becauseD(Q)isthemaximumoftheLagrangiangivenQ.Then,wehavethefollowingpropertiesofL(Q)andV(Q,). Lemma12. ForanyfeasibledualvectorQandascalar>0,thereexists>0suchthatforanyfeasibledualvector~Qwherek~Q)]TJ /F8 11.955 Tf 11.96 0 Td[(Qk<,itimpliesthatL(~Q)V(Q,). Proof. LetQand~Qbefeasibledualvectors.NotethatthedualfunctionD(Q)iscontinuousandboundedinsomeneighborhoodofQ.Then,for0>0,thereexistssuchthatjD(~Q))]TJ /F8 11.955 Tf 12.51 0 Td[(D(Q)j<0whenkQ)]TJ /F4 11.955 Tf 14.5 2.66 Td[(~Qk<.SupposethatkQ)]TJ /F4 11.955 Tf 14.5 2.66 Td[(~Qk<.Let(~x,~y,~r)besuchthat(~x,~y,~r)2L(~Q).Deneavector"=Q)]TJ /F4 11.955 Tf 13.94 2.66 Td[(~Q.Notethatk"k<. Let~hbethevectorsuchthat~hte,~rte)]TJ /F4 11.955 Tf 11.75 0 Td[(~rtp(t,e))]TJ /F4 11.955 Tf 12.42 0 Td[(~ytb(e)fort2Tande2Et.Then,wehaveD(Q))]TJ /F8 11.955 Tf 11.96 0 Td[(L(~x,~y,~r;Q)=D(Q))]TJ /F8 11.955 Tf 11.96 0 Td[(L(~x,~y,~r;~Q+")=D(Q))]TJ /F16 11.955 Tf 11.96 9.68 Td[()]TJ 7.47 1.68 Td[(Xs2SUs(~xs)+~Q~h+"~h=D(Q))]TJ /F16 11.955 Tf 11.96 9.68 Td[()]TJ /F8 11.955 Tf 5.48 -9.68 Td[(D(~Q)+"~h. Notethat"~hcanbearbitrarilysmallifwemakesufcientlysmallbecause~hisbounded.Let"~h=1.As!0,1!0.SincejD(~Q))]TJ /F8 11.955 Tf 12.1 0 Td[(D(Q)j<0and"~h=1,we 64

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haveD(Q))]TJ /F8 11.955 Tf 11.96 0 Td[(L(~x,~y,~r;Q)<0+j1j. Sincewecanmake0arbitrarilysmallandas!0,j1j!0,wecanndsmallenoughsuchthatD(Q))]TJ /F8 11.955 Tf 11.95 0 Td[(L(~x,~y,~r;Q). Inthefollowinglemmas,weusethenotationL(x,y,r;Q)andL((x,y,r);Q)interchangeably. Lemma13. V(Q,)isconvex. Proof. Let(x,y,r)and(~x,~y,~r)beanytwovectorsinV(Q,).Lethbethevector(hte)t2T,e2Etsuchthathte,rte)]TJ /F8 11.955 Tf 12.1 0 Td[(rtp(t,e))]TJ /F8 11.955 Tf 12.09 0 Td[(ytb(e)fort2Tande2Et,anddene~hsimilarly.Then,wewillshowthat,forany01,((x,y,r)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()(~x,~y,~r))2V(Q,). Clearly,((x,y)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()(~x,~y))2(x,y)and(r+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()~r)2r.WealsohaveD(Q))]TJ /F8 11.955 Tf 11.95 0 Td[(L((x,y,r)+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()(~x,~y,~r);Q)=D(Q))]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs2SUs(xs+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()~xs))]TJ /F8 11.955 Tf 11.96 0 Td[(Q(h+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()~h)D(Q))]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs2S)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(Us(xs)+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()Us(~xs))]TJ /F8 11.955 Tf 11.96 0 Td[(Q(h+(1)]TJ /F3 11.955 Tf 11.96 0 Td[()~h)=(D(Q))]TJ /F8 11.955 Tf 11.95 0 Td[(L(x,y,r;Q))+(1)]TJ /F3 11.955 Tf 11.95 0 Td[()(D(Q))]TJ /F8 11.955 Tf 11.95 0 Td[(L(~x,~y,~r;Q)), wheretherstinequalityisduetotheconcavityofUs. Lemma14. Leti!0asi!1.Foranysequence(x,y,r)(i)!(x,y,r)suchthat(x,y,r)(i)2V(Q,i),(x,y,r)2V(Q,0). Proof. Suppose(x,y,r)62V(Q,0).ItimpliesthatD(Q))]TJ /F8 11.955 Tf 12.63 0 Td[(L((x,y,r);Q)>0.LetD(Q))]TJ /F8 11.955 Tf 11.96 0 Td[(L((x,y,r);Q)=20,where0>0. 65

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BythecontinuityofLwithrespectto(x,y,r),given0,wecannd>0suchthatk(x,y,r))]TJ /F4 11.955 Tf 12.68 0 Td[((x,y,r)k0,thereexists>0suchthatwhenk~Q)]TJ /F4 11.955 Tf 13.94 2.65 Td[(Q(k)k<,wehave,foranyvector(x,y,r)2L(~Q),D(Q(k)))]TJ /F8 11.955 Tf 11.95 0 Td[(L(x,y,r;Q(k)). (3) BythecontinuityofQ,forasufcientlysmall>0,wehavekQ(l))]TJ /F4 11.955 Tf 13.95 2.66 Td[(Q(k)k<=2, (3) 66

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forl2[k,k+].Bytheconvergenceof^QmtoQu.o.c.,forallscaledprocesses^Qmwithsufcientlylargem,wehavek^Qm(l))]TJ /F4 11.955 Tf 13.94 2.66 Td[(Q(l)k<=2, (3) forl2[k,k+]. From( 3 )and( 3 ),for>0denedin( 3 ),thereexistsasufcientlysmall>0suchthatforallscaledprocesses^Qmwithsufcientlylargem,wehavek^Qm(l))]TJ /F4 11.955 Tf 13.94 2.66 Td[(Q(k)k<, forl2[k,k+].Then,forthecorrespondingunscaledprocesses,wehavek1 mQ(l))]TJ /F4 11.955 Tf 13.94 2.65 Td[(Q(k)k<, forl2[mk,m(k+)]. LetX(l),X(l+1))]TJ /F8 11.955 Tf 12.23 0 Td[(X(l),wherel2Z+.Similarly,wedeneY(l)andR(l),respectively. Then,since(EX(l),EY(l),ER(l))2L(1 mQ(l))becauseofthecontrolalgorithmandL(1 mQ(l))V(Q(k),)forl2[mk,m(k+)]by( 3 ),wehaveD(Q(k)))]TJ /F8 11.955 Tf 11.96 0 Td[(L(E(X(l),Y(l),R(l));Q(k)), forallintegersl2[mk,m(k+)]. LetusdeneX(mk,m)tobeX(mk,m),bm(k+)cXl=bmkc+1X(l)=X(bm(k+)c))]TJ /F8 11.955 Tf 11.95 0 Td[(X(bmkc). 67

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WedeneY(mk,m)andR(mk,m),similarly.Then,bytheconvexityofV(Q(k),)(Lemma 13 ),wehaveEX(mk,m) bm(k+)c)-222(bmkc,Y(mk,m) bm(k+)c)-222(bmkc,R(mk,m) bm(k+)c)-222(bmkc2V(Q(k),). Letusdene^Xm(k,)tobe^Xm(k,),^Xm(k+))]TJ /F4 11.955 Tf 13.65 2.65 Td[(^Xm(k)=1 mX(m(k+)))]TJ /F4 11.955 Tf 15 8.09 Td[(1 mX(mk)=1 mX(bm(k+)c))]TJ /F4 11.955 Tf 14.99 8.09 Td[(1 mX(bmkc)=1 mX(mk,m), wherethesecondequalityisbecauseX(l)isdenedtobepiecewiseconstantontheinterval[l,l+1).Wedene^Ym(k,)and^Rm(k,),similarly. Dene(m),(bm(k+)c)-222(bmkc)=m.Fromtheaboveobservation,wehaveE)]TJ /F4 11.955 Tf 6.67 -1.59 Td[(^Xm(k,) (m),^Ym(k,) (m),^Rm(k,) (m)2V(Q(k),). (3) LetusdeneX(k,)tobeX(k,),X(k+))]TJ /F4 11.955 Tf 13.64 2.66 Td[(X(k). WedeneY(k,)andR(k,),similarly.Wehavelimm!1E)]TJ /F4 11.955 Tf 6.68 -1.6 Td[(^Xm(k,) (m)=limm!1E^Xm(k,) limm!1(m)=Elimm!1^Xm(k,) limm!1(m)=EX(k,) =X(k,) 68

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wherethesecondequalityisduetothedominatedconvergencetheoremandtheuniformboundednessof^Xm(k,)[ 9 ].Thelastequalityisbecausetheuid-limitprocessesaredeterministic.Similarly,wegetlimm!1E)]TJ /F4 11.955 Tf 6.68 -1.6 Td[(^Ym(k,) (m)=Y(k,) ,limm!1E)]TJ /F4 11.955 Tf 6.67 -1.6 Td[(^Rm(k,) (m)=R(k,) Fromtheaboveobservation,theclosednessofV(Q(k),),and( 3 ),weget)]TJ /F4 11.955 Tf 6.68 -1.59 Td[(X(k,) ,Y(k,) ,R(k,) 2V(Q(k),). Sincecanbechosenarbitrarilysmallforagivenxed,andtimekisregular,byletting!0,weget(_X(k),_Y(k),_R(k))2V(Q(k),). Finally,byLemma 14 ,letting!0,weget(_X(k),_Y(k),_R(k))2V(Q(k),0), whichimpliesthelemmaholds. Comment:Theproofoftheabovelemmaissubstantiallydifferentfromsimilarlemmasinthepreviouswork[ 4 25 39 54 56 ].Theproofsinthosepapersdonotapplytoouralgorithmmainlyduetothedifferenceofthecontrolstructures.Allthosepreviouspapersshareacommoncontrolstructureofselectingv(k)2argmaxv2~(k)v,wherev(k)istheassignedratesattimek,~isthefeasiblerateregion,and(k)isaxedparameterattimek.Ontheotherhand,ourcontrolistoselectv(k)2argmaxv2~L(v;(k)),whereL(v;(k))isaconcavefunctionofvwithaxedparameter(k).Thedifferencerequiresatechnicallydifferentproof. 69

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Inthefollowinglemma,weshowthat(_X(k),_Y(k),_Z(k))maximizestheLagrangianLwithQ(k)astheLagrangianmultipliers.Notethattheratesofchange_Q(k)oftherealqueuesizesaredeterminedby(_X(k),_Y(k),_Z(k))asin( 3 ).Therefore,thelemmaimpliesthat,atalmosteverytimek,thetrajectoryofQmovesinthedirectionwheretheLagrangianL(;Q(k))ismaximized. Lemma16. Foralmosteveryk2R+,(_X(k),_Y(k),_Z(k))2argmax(x,y)2(x,y)r2rL(x,y,r;Q(k)). Proof. Since(_X(k),_Y(k),_R(k))maximizestheLagrangianLwithQ(k)astheLagrangianmultipliersandwehaveL(_X(k),_Y(k),_R(k);Q(k))=Xs2SUs(_Xs(k))+Xt2TXe2EtQte(k)(_Rte(k))]TJ /F4 11.955 Tf 14.71 4.35 Td[(_Rtp(t,e)(k))]TJ /F4 11.955 Tf 15.28 4.35 Td[(_Ytb(e)(k)), andL(_X(k),_Y(k),_Z(k);Q(k))=Xs2SUs(_Xs(k))+Xt2TXe2EtQte(k)(_Zte(k))]TJ /F4 11.955 Tf 14.76 4.35 Td[(_Ztp(t,e)(k))]TJ /F4 11.955 Tf 15.28 4.35 Td[(_Ytb(e)(k)), itsufcestoshowthatforalmosteveryk0,Xt2TXe2EtQte(k)(_Rte(k))]TJ /F4 11.955 Tf 14.7 4.35 Td[(_Rtp(t,e)(k))=Xt2TXe2EtQte(k)(_Zte(k))]TJ /F4 11.955 Tf 14.75 4.35 Td[(_Ztp(t,e)(k)). Aftersomemanipulation,weneedtoshowXt2TXe2Et_Rte(k)(Qte(k))]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))Qte0(k))=Xt2TXe2Et_Zte(k)(Qte(k))]TJ /F16 11.955 Tf 27.56 11.35 Td[(Xe02(t,d(e))Qte0(k)), (3) foralmosteveryk0. 70

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Wewillshowthatforalmosteveryk0,_Rte(k)(Qte(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(Xe02(t,d(e))Qte0(k))=_Zte(k)(Qte(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(Xe02(t,d(e))Qte0(k)), (3) forallt2Tande2Et,whichissufcienttoprove( 3 )holds. Fixaregularpointk0.First,wewillshowthatifQte(k))]TJ /F16 11.955 Tf 12.15 8.97 Td[(Pe02(t,d(e))Qte0(k)<0,( 3 )holds. BythecontinuityofQte,thereexists>0suchthatXe02(t,d(e))Qte0(l))]TJ /F4 11.955 Tf 13.95 2.65 Td[(Qte(l)>0, foralll2[k,k+].Letb1=minl2[k,k+])]TJ 21.08 1.67 Td[(Xe02(t,d(e))Qte0(l))]TJ /F4 11.955 Tf 13.94 2.66 Td[(Qte(l). Notethatb1>0.Then,bytheconvergenceof(^Qte)m(l)toQte(l)u.o.c.,foralargeenoughm,wehaveXe02(t,d(e))(^Qte0)m(l))]TJ /F4 11.955 Tf 11.95 0 Td[((^Qte)m(l)b1 2>0, forl2[k,k+].Then,forl2[mk,m(k+)],1 mQte(l))]TJ /F16 11.955 Tf 27.56 11.35 Td[(Xe02(t,d(e))1 mQte0(l)<0. Itimpliesthatthedifferentialbacklogoftreetonlinkeisstrictlynegativeduring[mk,m(k+)]inthecorrespondingunscaledprocesses,andhence,duringtheinterval,treetcannotbeselectedfortransmissiononlinke.Therefore,inthescaledprocesses,wehave(^Rte)m(k+))]TJ /F4 11.955 Tf 11.95 0 Td[((^Rte)m(k)=0. (3) 71

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Takingthelimitasm!0,wehaveRte(k+))]TJ /F4 11.955 Tf 13.31 2.66 Td[(Rte(k)=0. Itimpliesthat_Rte(k)=0. From( 3 ),usingthefactthattheonetimeslotincrementoftheunscaledprocessZteisnogreaterthanRte,wealsoget(^Zte)m(k+))]TJ /F4 11.955 Tf 12.34 0 Td[((^Zte)m(k)=0,whichimpliesthat_Zte(k)=0.Therefore,( 3 )holds. Next,wewillshowthatifQte(k))]TJ /F16 11.955 Tf 12.73 8.96 Td[(Pe02(t,d(e))Qte0(k)>0,( 3 )holds.BythecontinuityofQte,thereexists>0suchthatQte(l))]TJ /F16 11.955 Tf 27.56 11.35 Td[(Xe02(t,d(e))Qte0(l)>0, foralll2[k,k+].Letb2=minl2[k,k+])]TJ /F4 11.955 Tf 7.46 -7.03 Td[(Qte(l))]TJ /F16 11.955 Tf 27.56 11.36 Td[(Xe02(t,d(e))Qte0(l). Notethatb2>0.Then,bytheconvergenceof(^Qte)m(l)toQte(l)u.o.c.,foralargeenoughm,wehave(^Qte)m(l))]TJ /F16 11.955 Tf 27.57 11.36 Td[(Xe02(t,d(e))(^Qte0)m(l)b2 2>0, forl2[k,k+].Then,intheunscaledprocesses,foralargeenoughm,wehaveQte(l))]TJ /F16 11.955 Tf 27.56 11.35 Td[(Xe02(t,d(e))Qte0(l)mb2 2>ce, forl2[mk,m(k+)].SinceQte(l)>ceforl2[mk,m(k+)],wehaveRte(m(k+)))]TJ /F8 11.955 Tf 11.95 0 Td[(Rte(mk)=Zte(m(k+)))]TJ /F8 11.955 Tf 11.95 0 Td[(Zte(mk), whichfollowsthat(^Rte)m(k+))]TJ /F4 11.955 Tf 11.95 0 Td[((^Rte)m(k)=(^Zte)m(k+))]TJ /F4 11.955 Tf 11.95 0 Td[((^Zte)m(k). 72

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Lettingm!1,wehaveRte(k+))]TJ /F4 11.955 Tf 13.31 2.66 Td[(Rte(k)=Zte(k+))]TJ /F4 11.955 Tf 13.36 2.66 Td[(Zte(k). Dividingbothsidesbyandletting!0,weget_Rte(k)=_Zte(k). Therefore,( 3 )holds. Lastly,ifQte(k))]TJ /F16 11.955 Tf 12.25 8.96 Td[(Pe02(t,d(e))Qte0(k)=0,then( 3 )triviallyholds.Therefore,thelemmaholds. Thefollowinglemmaimpliesthatthealgorithmisasubgradientalgorithmintheuidlimit. Lemma17. Foralmosteveryk2R+,)]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k)isasubgradientofDatQ(k). Proof. Forall0,D()=max(x,y)2(x,y),r2rL(x,y,r;)L(_X(k),_Y(k),_Z(k);)=Xs2SUs(_X(k))+Xt2TsXe2Ette(_Zte(k))]TJ /F4 11.955 Tf 14.75 4.35 Td[(_Ztp(t,e)(k))]TJ /F4 11.955 Tf 15.29 4.35 Td[(_Ytb(e)(k))=Xs2SUs(_X(k))+()]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k))=Xs2SUs(_X(k))+Q(k)()]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k))+()]TJ /F4 11.955 Tf 13.95 2.66 Td[(Q(k))()]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k))=D(Q(k))+()]TJ /F4 11.955 Tf 13.95 2.66 Td[(Q(k))()]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k)). Thisshowsthat)]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k)isasubgradientofDatQ(k)([ 7 ],page731). 73

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3.3.3DualOptimalityoftheAlgorithm Inthissubsection,weshowthatthealgorithmachievesdualoptimalityintheuidlimit.WeshowthatthetrajectoryofQ,theuidlimitoftherealqueuesizes,approachesanoptimaldualsolutionin)]TJ /F11 7.97 Tf 6.77 4.33 Td[(forproblemP. First,weshowthatthetrajectoryofQapproachesthesetofoptimaldualsolutions)]TJ /F11 7.97 Tf 6.77 4.34 Td[(. Lemma18. Ask!1,Q(k)!)]TJ /F11 7.97 Tf 6.77 4.34 Td[(. Proof. Consideraregulark0.Pickanarbitrary2)]TJ /F11 7.97 Tf 6.78 4.34 Td[(andletV(),k)]TJ /F3 11.955 Tf 12.39 0 Td[(k2whereisafeasibledualvector.NotethatViscontinuouslydifferentiableandboundedbelow.Wehave_V(Q(k))=d dkkQ(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k2=2()]TJ /F4 11.955 Tf 13.95 2.65 Td[(Q(k))()]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k)). (3) ByLemma 17 ,wehave,forall0,D()D(Q(k))+()]TJ /F4 11.955 Tf 13.95 2.66 Td[(Q(k))()]TJ /F4 11.955 Tf 12.68 4.35 Td[(_Q(k)). (3) Applying( 3 )into( 3 ),weget_V(Q(k))2(D())]TJ /F8 11.955 Tf 11.96 0 Td[(D(Q(k)))0, wherethelastinequalityholdsbecauseminimizesD.Therefore,_V(Q(k))0forallfeasibleQ(k).Notethat_V(Q(k))<0,ifQ(k)62)]TJ /F11 7.97 Tf 6.77 4.94 Td[(, (3) sinceD())]TJ /F8 11.955 Tf 11.96 0 Td[(D(Q(k))isstrictlynegative. Now,wearereadytoprovethelemmabycontradiction.SupposeQ(k)doesnotconvergeto)]TJ /F11 7.97 Tf 6.78 4.34 Td[(.Itimpliesthatthereexists>0suchthatforallK>0,thereexistsk0Ksuchthatd(Q(k0),)]TJ /F11 7.97 Tf 17.13 4.33 Td[()>.Thismeansthatthereisaninnitenumberoftime 74

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instanceswhenQisawayfrom)]TJ /F11 7.97 Tf 6.78 4.34 Td[(bymorethan.SinceQ(k)isLipschitzcontinuouswithrespecttotimek,thereexists>0suchthatd(Q(l),)]TJ /F11 7.97 Tf 17.14 4.94 Td[()>=2, forlsuchthatl2[k0,k0+].Notethatwecanndwhichisindependentofk0sinceQ(k)isLipschitzcontinuous.Then,by( 3 ),thereexists1>0suchthat_V(Q(l))<)]TJ /F3 11.955 Tf 9.3 0 Td[(1, forallregularlsuchthatl2[k0,k0+].Sincetimekisregularalmosteverywhere,wehaveZk0+k0_V(Q(l))dl<)]TJ /F4 11.955 Tf 9.3 0 Td[(1. (3) Then,sincethereisaninnitenumberofsuchk0wheninequality( 3 )holds,wehavelimk!1V(Q(k))=V(Q(0))+Z10_V(Q(l))dl=. ThiscontradictsthatV(Q(k))isboundedbelow.Therefore,thelemmaholds. NotethattheabovelemmadoesnotimplythatQ(k)convergestoasinglepointin)]TJ /F11 7.97 Tf 6.77 4.33 Td[(.ThefollowingtheoremshowsthatQ(k)actuallyapproachesasinglepoint2)]TJ /F11 7.97 Tf 6.77 4.33 Td[(,whichisastrongerresultthanLemma 18 Theorem3.1. Ask!1,Q(k)!forsome2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(. Proof. First,supposeQ(0)2)]TJ /F11 7.97 Tf 6.77 4.33 Td[(.BytheargumentintheproofofLemma 18 ,forany2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(,wehaved dkkQ(k))]TJ /F3 11.955 Tf 11.95 0 Td[(k20, 75

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forallk0.Then,itholdsthatd dkkQ(k))]TJ /F4 11.955 Tf 13.94 2.66 Td[(Q(0)k20, forallk0,whichimpliesthatQ(k)=Q(0)forallk0. Next,supposeQ(0)62)]TJ /F11 7.97 Tf 6.78 4.34 Td[(.BytheargumentintheproofofLemma 18 ,forany2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(andQ(k)62)]TJ /F11 7.97 Tf 6.77 4.34 Td[(,wehaved dkkQ(k))]TJ /F3 11.955 Tf 11.96 0 Td[(k2<0. DeneB(Q(k)),fpjkp)]TJ /F3 11.955 Tf 12.42 0 Td[(kkQ(k))]TJ /F3 11.955 Tf 12.41 0 Td[(kgtobetheclosedballcenteredaroundwithradiuskQ(k))]TJ /F3 11.955 Tf 11.8 0 Td[(k.Sinced dkkQ(k))]TJ /F3 11.955 Tf 11.8 0 Td[(k2<0,Q(k+)shouldbeintheinteriorofB(Q(k))forsmallenough>0.Therefore,B(Q(k+))B(Q(k)),wheremeansstrictinclusion. LetA(Q(k)),T2)]TJ /F18 5.978 Tf 4.82 2.27 Td[(B(Q(k)).NotethatA(Q(k))isclosedandbounded.A(Q(k))hasatleastonepointin)]TJ /F11 7.97 Tf 6.78 4.34 Td[(.Let~bethenearestpointin)]TJ /F11 7.97 Tf 6.78 4.34 Td[(fromQ(k).Since)]TJ /F11 7.97 Tf 6.77 4.34 Td[(isconvex,wehavekQ(k))]TJ /F3 11.955 Tf 12.16 0 Td[(k>k~)]TJ /F3 11.955 Tf 12.17 0 Td[(kforall2)]TJ /F11 7.97 Tf 6.78 4.34 Td[(,whichimplies~isintheinteriorofB(Q(k))forall2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(.Therefore,A(Q(k))containsatleast~2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(.NotethattheinteriorofA(Q(k))representsthefeasibleregiontowardwhichQ(k)canmove. Foranon-emptysetA,letSA=fx2Rjx=kp)]TJ /F8 11.955 Tf 12.16 0 Td[(qk,p,q2Ag.Denediam(A),supSA.Letfkngbeasequencesuchthatkn
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3.3.4PrimalOptimalityoftheAlgorithm ThefollowingtheoremdemonstratesthatoursolutionisasymptoticallyoptimaltoproblemP. Theorem3.2. Alimitpointof(_X(k),_Y(k),_Z(k))k0existsandeverysuchlimitpointisaprimaloptimalsolution. Proof. Itiseasytocheck(_X(k),_Y(k),_Z(k))isbounded.Hence,alimitpointexists.Let(_X,_Y,_Z)beonesuchlimitpoint.ByTheorem 3.1 ,ask!1,Q(k)!forsome2)]TJ /F11 7.97 Tf 6.77 4.34 Td[(.Weonlyneedtoshow(_X,_Y,_Z)andsatisfytheoptimalityconditions.ByLemma 16 ,L(_X,_Y,_Z;)=D()(theLagrangeoptimalitycondition).By( 3 ),_Yand_Zsatisfy( 3 )withequality.Thisalsoimpliesthatthecomplementaryslacknessconditionissatised. Letbethesetofoptimalprimalsolutions,i.e.,,f(x,y,r)j(x,y,r)isanoptimalsolutiontoproblemP.g. Then,wehavethefollowingcorollary. Corollary1. d((_X(k),_Y(k),_Z(k)),)!0ask!1. Proof. Supposethatitdoesnothold.Namely,thereexistssome>0suchthatforanyKn>0,thereexistsknKnsuchthatd((_X(kn),_Y(kn),_Z(kn)),)>.Then,wehaveasequencef(_X(kn),_Y(kn),_Z(kn))gwherekn!1suchthatliminfkn!1d((_X(kn),_Y(kn),_Z(kn)),)>.Sincethesequenceisbounded,thereexistsalimitpoint,whichisnotaprimaloptimal.However,itcontradictsTheorem 3.2 Theabovecorollaryimpliesthat,intheuidlimit,therateallocationapproachesthesetofoptimalprimalsolutions.Notethattheconceptoftheasymptoticoptimalityofouralgorithmisdifferentfromtheonesinthepreviouswork[ 4 25 39 54 56 ].Inourcase, 77

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theremaybemultipleoptimalsolutionsandthealgorithmmayjumpfromoneoptimalsolutiontoanother. 3.4RelatedWork WehavealreadycontrastedourworkwiththecloselyrelatedpriorstudiesinSection 1.2 .Wenextmakesomeremarksaboutotherrelatedwork.Theoptimizationapproachisusedforothermulticastproblemsin[ 13 20 29 40 41 51 61 62 ].Mostofthem,except[ 29 51 ],endupwithalgorithmsthatrequireglobalexchangeofcontrolmessages.In[ 20 29 51 ],theauthorsconsidermulti-ratemulticastproblemswithasinglexedmulticasttreeforeachsession.In[ 13 40 41 ],theauthorsassumethatthenetworkbottlenecksareatthenodes'uplinkswhereas,inourcase,thebottleneckscanbeanywhereinthenetwork.In[ 61 62 ],theobjectiveistondbothanoptimalsetoftreesamongallpossibleonesandanoptimalrateallocation.Thealgorithmsaredifferent,lesslocalandmorecomplexintermsofthecomputationrequirements. 78

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CHAPTER4MULTI-TREEMULTICASTALGORITHMFORCONTENTDISTRIBUTIONONTWO-LAYERNETWORKS Traditionally,contentdistributionserviceisperformedbyonlythecontentproviderswithoutexplicitparticipationoftheInternetServiceProviders(ISPs).However,asmassivecontentdistributionhasbecomeanimportantapplicationontheInternet,theISPshavebeenmotivatedtoactivelyparticipateinthecontentdistributionservice.Inthischapter,weconsidertheproblemofcontentdistributionamongdataserversoveratwo-layernetworkconsistingoftheoverlayandunderlaynetworks,wherethecontrolvariablescanbeadjustedatbothnetworks.Weproposeanoptimalbackpressurealgorithmforcontentdistributionoverthelayerednetwork.Thealgorithmiswellseparatedintotheoverlayandunderlayoperationssuchthatthecommunicationbetweenthetwolayersonlyoccurslocallyattheirinterface.Weshowthatouralgorithmachievesoptimalityevenwhenthetwonetworklayersoperateunderdifferenttimescalesandwithouttimesynchronization. Theremainderofthechapterisorganizedasfollows.Section 4.1 describesthenetworkmodelandtheproblemformulation.Section 4.2 presentsthemainalgorithmandtheproofofitsoptimality.InSection 4.3 ,wefurtherevaluatetheperformanceofthealgorithmandshowtherealqueuesareboundedbysimulations.WediscusstherelatedworkinSection 4.4 4.1ModelandProblemFormulation Inthissection,weintroduceaproblemformulationwiththepropertythattheunderlaybandwidthallocationismuchseparatedfromtherateallocationontheoverlaynetwork.Itservesasthebasisforreducingtheinteractionbetweentheoverlayandtheunderlaytolocalcommunicationatthedataservers.Throughout,weconsideradiscrete-timesystem. 79

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4.1.1NetworkModel Weconsideratwo-layernetworkconsistingoftheoverlaynetworkandtheunderlaynetwork.Theoverlaynetworkisrepresentedbyadirectedgraph^G=(^V,^E),where^Visthesetofdataserversand^Eisthesetofoverlaylinks.TheunderlaynetworkisrepresentedbyadirectedgraphG=(V,E)whereVisthesetofnodes,whicharedataserversandrouters/switches,andEisthesetofunderlaylinks. Foreachunderlaylinke2E,letce(k)denotethecapacityoflinkeattimek,whichisgenerallytime-varyingduetothebackgroundtrafc.Weassumethat,foreache2E,fce(k)gkisani.i.d.randomprocessandwithameance.Inaddition,theseprocessesaremutuallyindependent.Wealsoassumethat0ce(k)cmaxe,whereeachcmaxeisapositiveconstant.LetCmax=maxe2Ecmaxe. Anoverlaylinkisanend-to-endunicastconnectionintheunderlaynetworkbetweentwodataservers.Weconsiderstatic,single-pathroutingfortheunderlaynetworkinthischapter.However,thisrestrictioncanbeeasilyrelaxedsuchthateachoverlaylinkcanbeassociatedwithmorethanoneunderlaypathsoritmaynotbeassociatedwithanypossiblepathsbetweenthecorrespondingnodepairifhop-by-hopdynamicroutingisadopted. LetSbethesetofallmulticastsessionsinthenetwork.Eachsessionisassociatedwithasourceandasetofreceivers,whichisasubsetof^V.Weassumethatthesourceofeachsessionhasaninnitebacklogofdatathatneedstobetransmittedtoallthereceiversinthesession.EachsessionsisgivenasetoftreesTsthatitusesfordatatransmission.Withoutlossofgenerality,weassumeTs'saredisjointacrossthesessions.LetTbetheunionofTs.Let^Etbethesetofoverlaylinksontreet.Letb(^e)andd(^e)denotethetransmitting(tail)nodeandthereceiving(head)nodeofoverlaylink^e,respectively.Let^Vtbethesetoftransmittingnodesb(^e)forall^ein^Et;itisthesetofnodesontreetwhicharenotleaves.Leto(t)betherootnodeoftreet.Letp(t,^e)be 80

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theparentlinkofoverlaylink^eontreet.Let(t,^n)bethesetofchildlinksatnode^nontreet.Let(^n)bethesetofalloutgoingoverlaylinksfromnode^n. Oneverytimeslot,thesourceofeachsessiondecideshowmanypacketsitadmitstothenetwork.Wedenotebyxs(k)theamountofadmitteddataofsessionsattimek.WeassumethatthereexistsapositiveconstantXmaxsuchthat0xs(k)Xmaxforalls.Theadmittedtrafcxs(k)shouldbetransmittedoverthetreesinTsforsessions.Letyt(k)betheadmitteddatathatistransmittedovertreet,t2Ts,attimek.Wehavexs(k)=Pt2Tsyt(k). 4.1.2UtilityFunctions Letxs(k)bethetimeaverageoftheadmittedtrafcrateforsessionsuptotimek,i.e.,xs(k),1 kk)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0xs(). Letxsbethelimitofxs(k),1xs,limk!1xs(k). EachsessionsisassociatedwithautilityfunctionUs(xs).WeassumethatUsisconcave,monotonicallyincreasing,andnon-negativeover[0,Xmax].Wealsoassumethatitiscontinuouslydifferentiable,andhence,thederivativeofUsisboundedon[0,Xmax].Differentutilityfunctionscanreectvariousfairnesscriteria.Forexample,settingUs(xs)=log(xs+1)forallsleadstotheproportionalfairnessamongsessions.Ontheotherhand,iftheobjectiveistomaximizetheweightedthroughput,onecansetUs(xs)=!sxswhere!s>0istheweightforsessions.Moreexamplescanbefoundin[ 43 53 ]. 1Wetemporarilyassumethatthelimitexists.Weshallreplacelimwithliminforlimsupwhenthelimitdoesnotexist. 81

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4.1.3OverlayTransmissionDecisionsandLinkCapacities Foreachxedtreet,overlaydataforwardingisdecidedhop-by-hopbythedataserversonthetree.Ahopheremeansanoverlaylinkonthetree.Specically,oneverytimeslotk,eachdataserver^ndeterminestheamountofdataitintendstotransmit,denotedbyrt^e(k),foreveryoutgoingoverlaylink^e2(^n)andeverytreetsuchthat^e2^Et.WeassumethatthereisapositiveconstantRmaxsuchthat0rt^e(k)Rmaxforall^eandt.Wewillcallrt^e(k)theoverlaytransmissionrate.But,notethatitisactuallytheamountofnewlyscheduleddatafortreetattimektobetransmittedovertheunicastconnectionassociatedwith^e.Onecanimaginethat,attimek,rt^e(k)amountofdataentersasharedqueueassociatedwith^e,waitingtobetransmittedacross^e;thisappliestoalltreescontaining^easabranch. Theamountofactualtransmissiononanoverlaylinkattimekdependsonthecapacityoftheoverlaylink,whichisdeterminedbytheunderlaybandwidthallocation.Let^e(k)bethecapacityoftheoverlaylink^eattimek.Letrt^eand^ebethelong-timeaveragesforrt^e(k)and^e(k),respectively,similarlydenedasforxs.Forthequeuetobestable,itisnecessarythatthelong-timeaveragearrivalrateisnomorethanthelong-timeaverageservicerate.Hence,werequirePt:^e2^Etrt^e^e. LetUbethesetofallfeasibleoverlaylinkcapacityvectorsthatcanbesupportedbytheunderlaynetwork.Udependsontheunderlayroutingpolicy.Forsingle-pathrouting,Ucanbewrittenasfollows.U,f0jX^e:e2E^e^ece,8e2Eg, (4) whereE^eisthesetofunderlaylinkscorrespondingtotheoverlaylink^e(whichisanunderlaypath).Notethatthetrafconanoverlaylinkcanbeconsideredasaunicastowontheunderlaynetwork.TheconstraintsinthedenitionofUimplythattheaggregateamountofunicastowsonanunderlaylinkcannotexceeditscapacityinthelong-timeaveragesense. 82

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4.1.4OtherNotations LetIfcgbetheindicatorfunctionsuchthatIfcg=1ifconditioncissatied,andIfcg=0otherwise.Let[]+denotetheprojectionontothenon-negativedomain.Denote(ai)tobethevectorwithentriesai.Letx,yandrbethevectors(xs)s2S,(yt)t2T,and(rt^e)t2T,^e2^Et,respectively.Givenavectorx2Rn,letkxk1betherstnormofx,i.e.,kxk1,Pn1jxijwherexiisthei-thcomponentofx.GivenasetA,letjAjbethenumberofelementsoftheset.LetPs,t,^e()betheabbreviationofPs2SPt2TsP^e2^Et(). 4.1.5ProblemDescription Inthissubsection,wepresenttheproblemformulation,introduceitsdualproblem,andderiveasubgradientalgorithm.ThesubgradientalgorithmisusedasareferencealgorithmtoderiveourmainalgorithminSection 4.2 Ourproblemformulationisasfollows.P:maxXs2SUs(xs)s.t.rt^e)]TJ /F4 11.955 Tf 11.57 0 Td[(rtp(t,^e))]TJ /F4 11.955 Tf 12.24 0 Td[(ytIf^e2(t,o(t))g0,8t2T,8^e2^Et, (4)Xt:^e2^Etrt^e^e,8^e2^E, (4)Xt2Tsyt=xs,8s2S, (4)2U, (4)0xsXmax,8s2S, (4)yt0,8t2T, (4)0rt^eRmax,8t2T,8^e2^Et. (4) Allthevariablesarethelong-timeaverages.Theobjectiveistomaximizetheaggregateutilityofallmulticastsessions.Theconstraintsin( 4 )arearelaxedformofthetreeowconservationconstraints.Theyimplythatforeverytreet,theoverlaytransmission 83

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rateonanoverlaylink^eonthetreeshouldbenolessthanthesumoftherateofitsparentlinkp(t,^e)andtheexogenousarrivaltothetree.Weassumethatrtp(t,^e)0ifp(t,^e)isnull.Theconstraintsin( 4 )saythatthesumoftheoverlaytransmissionratesonanoverlaylinkshouldnotexceedtheoverlaylinkcapacityassignedbytheunderlaybandwidthallocationalgorithm. Lett^eand^ebethenon-negativeLagrangemultipliersassociatedwiththeconstraints( 4 )and( 4 ),respectively.Then,byrelaxingtheconstraints( 4 )and( 4 ),wehavethefollowingLagrangianfunction.L(x,y,r,;,)=Xs2SUs(xs)+Xt2TX^e2^Ett^e)]TJ /F8 11.955 Tf 5.47 -9.68 Td[(rt^e)]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,^e))]TJ /F8 11.955 Tf 11.95 0 Td[(ytIf^e2(t,o(t))g+X^e2^E^e)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(^e)]TJ /F16 11.955 Tf 15.84 11.36 Td[(Xt:^e2^Etrt^e=Xs2S)]TJ /F8 11.955 Tf 5.47 -9.69 Td[(Us(xs))]TJ /F16 11.955 Tf 12.89 11.36 Td[(Xt2TsytX^e2(t,o(t))t^e+X^e2^EXt:^e2^Etrt^e)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t^e)]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))t^e0)]TJ /F3 11.955 Tf 11.96 0 Td[(^e+X^e2^E^e^e. LetthedualfunctionDbeD(,),max(x,y,r,)satises( 4 )-( 4 )L(x,y,r,;,). Then,thedualproblemis:minD(,)s.t.0,0. Then,wehaveasubgradientalgorithminAlgorithm 4-1 .Somefeaturesofthealgorithmareasfollows.Pros:Theoperationsofthealgorithmarewelldividedbetweentheoverlayandunderlaynetworks.Theoverlaynetworkrunstheoperations( 4 ), 84

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Algorithm4-1Subgradientalgorithm 1. Oneachtimeslotk,eachsourcessolvesthefollowingsubproblem.(xs(k),(yt(k))t2Ts)2argmaxPt2Tsyt=xs,0xsXmax,yt0nUs(xs))]TJ /F16 11.955 Tf 12.89 11.36 Td[(Xt2TsytX^e2(t,o(t))t^e(k)o. (4) 2. Oneachtimeslotk,eachoverlaylink^esolvesthefollowingsubproblem.((rt^e(k))t:^e2^Et)2argmax0rt^eRmaxnXt:^e2^Etrt^e)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(t^e(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))t^e0(k))]TJ /F3 11.955 Tf 11.95 0 Td[(^e(k)o. (4) 3. Oneachtimeslotk,theunderlaybandwidthallocationalgorithmsolvesthefollowingproblem,whichistondaratevector(k)maximizingtheweightedthroughput.maxX^e2^E^e(k)^e (4)s.t.2U. 4. Oneachtimeslotk,eachoverlaylink^eupdatest^eand^easfollows.t^e(k+1)=ht^e(k))]TJ /F3 11.955 Tf 11.96 0 Td[()]TJ /F8 11.955 Tf 5.48 -9.68 Td[(rt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,^e)(k))]TJ /F8 11.955 Tf 11.96 0 Td[(yt(k)If^e2(t,o(t))gi+, (4)^e(k+1)=h^e(k))]TJ /F3 11.955 Tf 11.95 0 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(^e(k))]TJ /F16 11.955 Tf 15.85 11.36 Td[(Xt:^e2^Etrt^e(k)i+. (4) ( 4 ),( 4 ),and( 4 )onthedataservers.Theunderlaynetworksolvestheunderlaybandwidthallocationproblem( 4 )onthedataserversandtherouters.Thecommunicationbetweenthetwonetworksoccursonlyatthedataservers,whichareattheboundariesbetweenthetwonetworks.Theunderlaynetworkatadataserverneedstheweights^e(k)oftheoutgoingoverlaylinksfromtheserver;theoverlaynetworkataserverneedstheoverlaylinkcapacities^e(k)oftheneighboringlinks.ThisfeatureisdesirablebecausetheoperationsforthetwonetworkscanberunseparatelybytheCPandtheISP,withasmallcommunicationcost.Cons:Theglobalunderlaybandwidthallocationproblem( 4 )needstobesolvedcompletelyoneverytimeslot,whichmay 85

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beunrealisticforalargenetworkorasmalltime-slotsize.Moreover,thealgorithmrequiresthemeanlinkcapacitiesce. 4.2AsynchronousAlgorithm Inthissection,weintroduceourmainalgorithm,whichallowsthetwonetworkstooperateasynchronouslyandpossiblyondifferenttimescales.Itisalsomoredistributedandlocal.ThealgorithmisdistinguishedfromthesubgradientalgorithminSection 4.1 bythefollowingfeatures.First,ratherthanrequiringtheunderlaynetworktosolvetheglobalunderlaybandwidthallocationproblemcompletelyoneverytimeslot,theunderlaynetworknowonlyperformsverysimplecomputationtoupdatetheoverlaylinkratesbasedonthelocaloverlayandunderlaylinkprices.Asaconsequence,theoverlaylinkrates^e(k)arenolongerconsideredastheguaranteedoverlaylinkcapacitiessupportedbytheunderlaynetworkattimek.Theseratesmaysometimesviolatetheunderlaylinkcapacityconstraintsandqueueswillform.Second,ratherthanrequiringtheoverlayandunderlayoperationstobeperformedsynchronouslyoneverytimeslot,thetwosetsofoperationscanbedoneasynchronouslyatdifferentupdatefrequencies.Asaresult,eachnetworkmayuseratheroutdatedinformationproducedbytheothernetwork.Thisismoreusefulinpracticesincethetwonetworksmaybeoperatedbydifferentoperators,i.e.,theCPandtheISP,andsynchronizedoperationswouldbedifculttoachieve.Third,thenewalgorithmdoesnotneedtohavetheknowledgeofthemeanlinkcapacitiesce. 4.2.1TimeStructureforAsynchronousOperations Weassumethattheoverlayandunderlayoperationsareperformedasynchronously.Fortheoverlayoperations,weconsiderasetofconsecutiveoverlaytimeframes,fFO0,FO1,FO2,...g,wheretheoverlayrateassignmentsoccuratthebeginningofeveryframe.AframeFOiconsistsofasetofconsecutivetimeslotsfromb(FOi)tob(FOi+1))]TJ /F4 11.955 Tf 12.85 0 Td[(1,whereb(FOi)isthersttimeslotofframeFOi.Similarly,weconsiderasetofconsecutiveunderlayframes,fFU0,FU1,FU2,...g,fortheunderlayoperations.The 86

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Figure4-1. Timestructureforasynchronousoverlayandunderlayoperations. underlayrateassignmentsoccuratthebeginningofeveryunderlayframe.AframeFUiconsistsofasetofconsecutivetimeslotsfromb(FUi)tob(FUi+1))]TJ /F4 11.955 Tf 12.45 0 Td[(1.WeassumethatthelengthofaframeisboundedsuchthateveryframeconsistsofatmostFmaxtimeslots.LetFO(k)andFU(k)betheoverlayandunderlayframestowhichthetimeslotkbelongs,respectively.Wedenotebyk2FOiork2FUiiftimeslotkbelongstotheframeFOiorFUi,i.e.,b(FOi)k
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queuesizecorrespondingtotheconstraintP^e:e2E^e^eceinthedenitionofUin( 4 ),andletwbethecorrespondingvector.Let>0beaparameter,whichwillbeusedtoadjusttheperformancetradeoffofthealgorithm.OurmainalgorithmtosolvetheproblemPisinAlgorithm 4-2 through 4-4 Algorithm4-2Asynchronousalgorithm:overlaypart 1. Oneachtimeslotk2KO,eachsourcessolvesthefollowingsubproblem.(xs(k),(yt(k))t2Ts)2argmaxPt2Tsyt=xs,0xsXmax,yt0n1 Us(xs))]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2TsytX^e2(t,o(t))qt^e(k)o,fork2KO. (4) 2. Oneachtimeslotk2KO,eachoverlaylink^esolvesthefollowingsubproblem.((rt^e(k))t:^e2^Et)2argmax0rt^eRmaxnXt:^e2^Etrt^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.95 0 Td[(p^e(k)o,fork2KO. (4) 3. Oneachtimeslotk62KO,therateallocationbytheoverlaypartisunchanged,i.e.,x(k)=x(k)]TJ /F4 11.955 Tf 11.96 0 Td[(1),y(k)=y(k)]TJ /F4 11.955 Tf 11.96 0 Td[(1),r(k)=r(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1),fork62KO, (4) Hence,duringanoverlayframe,therateallocationbytheoverlayisthesame;thatis,forallkintheframe,wehavex(k)=x(bO(k)),y(k)=y(bO(k)),r(k)=r(bO(k)). 4. Oneachtimeslotk,eachoverlaylink^eupdatesqt^easfollows.qt^e(k+1)=hqt^e(k))]TJ /F16 11.955 Tf 11.95 9.69 Td[()]TJ /F8 11.955 Tf 5.48 -9.69 Td[(rt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,^e)(k))]TJ /F8 11.955 Tf 11.95 0 Td[(yt(k)If^e2(t,o(t))gi+. (4) Notethattheupdateofthequeuesizeqt^ecanbeperformedlocallyduringanoverlayframeoncetheinformationofrateallocationrtp(t,^e)(bO(k))istransmittedfromtheparentlinkatthestartoftheframe.Notealsothattheupdateofthelinkweightscanbeperformedlocallyduringanunderlayframeoncetheinformationabouttheoverlaylinkrates(bU(k))istransmittedatthestartoftheframe. 88

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Algorithm4-3Asynchronousalgorithm:interfacebetweenthetwoparts 1. Oneverytimeslotk,eachoverlaylink^eupdatesp^easfollows.p^e(k+1)=hp^e(k))]TJ /F16 11.955 Tf 11.95 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(^e(k))]TJ /F16 11.955 Tf 15.84 11.36 Td[(Xt:^e2^Etrt^e(k)i+. (4) Thequeuesizep^e(k)isusedlocallybyboththeoverlayandunderlayparts.Sincetheupdateofp^e(k)requiresonlythelocalinformation,thereislittleburdentoupdateitoneverytimeslot. Algorithm4-4Asynchronousalgorithm:underlaypart 1. Oneachtimeslotk2KU,foreachoverlaylink^e,theunderlaypartupdatestheoverlaylinkrate^e(k)asfollows.^e(k)2argmax0^emax(p^e(k))]TJ /F16 11.955 Tf 12.97 11.36 Td[(Xe2E^ewe(k))^e,fork2KU. (4) Notethat^e(k)isboundedbymaxwheremaxisaconstant. 2. Oneachtimeslotk62KU,theoverlaylinkratesareunchanged,i.e.,(k)=(k)]TJ /F4 11.955 Tf 11.95 0 Td[(1),fork62KU. (4) Hence,duringanunderlayframe,theoverlaylinkratesremainconstant. 3. Oneachtimeslotk,foreachunderlaylinke,theunderlaypartupdatesthelinkweightwe(k)asfollows.we(k+1)=hwe(k))]TJ /F16 11.955 Tf 11.95 9.68 Td[()]TJ /F8 11.955 Tf 5.48 -9.68 Td[(ce(k))]TJ /F16 11.955 Tf 16.43 11.36 Td[(X^e:e2E^e^e(k)i+. (4) Remark.Inthepartsofthealgorithmthatsolvethesubproblems( 4 ),( 4 )and( 4 ),atieisbrokenrandomly.Weassumethatthevirtualqueuesareupdatedoneverytimeslotforeaseofpresentation.Infact,theyonlyneedtobeupdatedonceineveryframesincetheratesarenotchangedinaframeandthequeueinformationisusedinthenextframe. 89

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Figure4-2. Howthetrafcishandledinadataserver. 4.2.3IllustratingExample Wenextillustratehowthealgorithmworkswithanexample.Notingthatouralgorithmusesthevirtualqueuesizesqt^eandp^eforcontrol,wepayparticularattentiontohowthecorrespondingrealqueuesareorganizedandupdated.WeassumethatalltherealqueuesareFIFO. Consideradataserverwithtwodownstreamoverlaylinksandoneupstreamoverlaylink.Sessionstakesitasthesourceanddistributesitscontentovertwotrees,t1andt2.Thedataserverisalsoanintermediatenodeofanothersessionusingtreest3andt4.Fig. 4-2 showshowthetrafcishandledinthedataserver. Consideraxedoverlaylink^e.LetQt^ebetherealqueuesizeoftreet'sdatawaitingfortransmissiononto^e.Aftertheoverlaylinkscheduler,thepacketscomingoutofeachQt^ewillbeputintothequeueP^efor^e. Wesupposethatthetreest1,t2usedownstreamlinks^e1and^e2.Thesession/treeratecontrollerdeterminesthesessionratexsandthetreeratesfytgfort2ft1,t2gaccordingto( 4 )and( 4 ).Then,itadmitsYt(k)=yt(k)realpacketstothecorrespondingqueues.Theamountofadmittedrealpacketsforsessionsattimek 90

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isdenotedbyXs(k)andwehaveXs(k)=Pt2TsYt(k).Sincebothtreesusethetwodownstreamlinks,theadmittedpacketsshouldbeduplicatedandputintothequeuesQt^e1,Qt^e2wheret2ft1,t2g.Inanactualimplementation,wedonotneedtoduplicatethepacketsexplicitly;itisenoughtomaintainpointerstothepackets. LetL^e(k)betheamountofrealpacketsfromanupstreamoverlaylink^eandarrivingatthenoded(^e)attimek.LetLt^e(k)betheamountofrealpacketsoftreetarrivingatthenoded(^e)attimek.WehaveL^e(k)=Pt:^e2^EtLt^e(k).Oncethepacketsarrive,apacketclassierexaminesanddistributesthemintotheappropriatequeues. LetRt^e(k)betheamountofrealpacketstransferredfromqueueQt^etoqueueP^e.Theoverlaylinkschedulerforeachdownstreamoverlaylink^erstdeterminesfrt^e(k)gfortreestsuchthat^e2^Et,accordingto( 4 )and( 4 ).Then,itsetsRt^e(k)=minfQt^e(k),rt^e(k)g. LetD^e(k)betheamountofrealdeparturesfromqueueP^eontotheoverlaylink^e.Theoverlaylinkadmissionratecontrollerrstdetermines^e(k)accordingto( 4 )and( 4 ).Then,itsetsD^e(k)=minfP^e(k),^e(k)g. Then,therealqueuesQt^eandP^eevolveasfollows.Qt^e(k+1)=Qt^e(k)+Ltp(t,^e)(k)+Yt(k))]TJ /F8 11.955 Tf 11.96 0 Td[(Rt^e(k),P^e(k+1)=P^e(k)+Xt:^e2^EtRt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(D^e(k). ThetrafcleavingthequeueP^etraversesthroughtheunderlaylinkscorrespondingtotheoverlaylink^e.Weassumethateveryunderlaylinkhasasinglerealqueue.LetWebetherealqueuesizeattheunderlaylinke.LetAe(k)betheamountofrealpacketsarrivingatlinkeandDe(k)betheamountofdeparturesfromthelink.NotethatDe(k)=minfce(k),We(k)g.Then,thequeueWeevolvesasWe(k+1)=We(k)+Ae(k))]TJ /F8 11.955 Tf 11.95 0 Td[(De(k). 91

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4.2.4PerformanceAnalysis Inthissection,weshowtheoptimalityandvirtualqueuestabilityofthealgorithmusingtheLyapunovoptimizationtechniqueintroducedin[ 24 46 47 ]. Werstestablishtheboundsforafewterms.Theseareusedintheproofsofthefollowinglemmas.Forevery^e,theonetimeslotchangeofqt^e(k)isboundedsincetheoverlaytransmissionratert^e(k)isboundedbyRmaxandthetreerateyt(k)isboundedbyXmax.DeneQmaxtobeaconstantsuchthatforalltimeslotsk,kq(k+1))]TJ /F8 11.955 Tf 11.95 0 Td[(q(k)k1,Xs,t,^ejqt^e(k+1))]TJ /F8 11.955 Tf 11.96 0 Td[(qt^e(k)jQmax. (4) Similarly,foreverye,theonetimeslotchangeofwe(k)isboundedsincece(k)isboundedbycmaxeandP^e:e2E^e^e(k)isalsoboundedbyj^Ejmax.DeneWmaxtobeaconstantsuchthatforalltimeslotsk,kw(k+1))]TJ /F8 11.955 Tf 11.96 0 Td[(w(k)k1,Xejwe(k+1))]TJ /F8 11.955 Tf 11.95 0 Td[(we(k)jWmax. (4) Letusdenoteby(x(k),y(k))anoptimalsolutiontothefollowingproblemgiventhequeuesizesq(k)attimek,(x(k),y(k))2argmaxPt2Tsyt=xs,0xsXmax,yt0nXs2S1 Us(xs))]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2TsX^e2(t,o(t))qt^e(k)yto. Lemma19. Underthealgorithm,thereexistsaconstantB>0suchthatforalltimeslotsk,Xs2S1 Us(xs(k)))]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2TsX^e2(t,o(t))qt^e(k)yt(k)Xs2S1 Us(xs(k)))]TJ /F16 11.955 Tf 12.88 11.36 Td[(Xt2TsX^e2(t,o(t))qt^e(k)yt(k))]TJ /F8 11.955 Tf 11.95 0 Td[(B. Proof. Dueto( 4 )andtheboundednessofanoverlayframesize,wehavekq(k))]TJ /F8 11.955 Tf 11.95 0 Td[(q(bO(k))k1FmaxQmax, (4) 92

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foranyk. LetusdeneafunctionD(q)tobeD(q),maxPt2Tsyt=xs,0xsXmax,yt0Xs1 Us(xs))]TJ /F16 11.955 Tf 11.95 11.36 Td[(XtX^eqt^eyt. NotethatD(q)isacontinuousfunction.Then,foranyq0suchthatkq0)]TJ /F8 11.955 Tf 20.33 0 Td[(qk1FmaxQmax,thereexistsaconstantB0suchthatjD(q))]TJ /F8 11.955 Tf 11.96 0 Td[(D(q0)jB0. (4) Notealsothat(x(k),y(k))2argmaxPt2Tsyt=xs,0xsXmax,yt0nXs2S1 Us(xs))]TJ /F16 11.955 Tf 12.89 11.36 Td[(Xt2TsX^e2(t,o(t))qt^e(bO(k))yto. (4) Then,wehaveXs2S1 Us(xs(k)))]TJ /F16 11.955 Tf 12.89 11.35 Td[(Xt2TsX^e2(t,o(t))qt^e(k)yt(k)=Xs2S1 Us(xs(k)))]TJ /F16 11.955 Tf 12.89 11.36 Td[(Xt2TsX^e2(t,o(t))qt^e(bO(k))yt(k))]TJ /F16 11.955 Tf 11.96 11.36 Td[(Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(qt^e(bO(k))yt(k)D(q(bO(k))))]TJ /F8 11.955 Tf 11.96 0 Td[(FmaxQmaxXmaxD(q(k)))]TJ /F8 11.955 Tf 11.96 0 Td[(B0)]TJ /F8 11.955 Tf 17.9 0 Td[(FmaxQmaxXmax, wheretherstinequalityisdueto( 4 ),( 4 )andtheboundednessofyt(k),andthesecondinequalityisdueto( 4 ).LettingB,B0+FmaxQmaxXmax,thelemmaholds. 93

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Letusdenotebyr(k)anoptimalsolutiontothefollowingproblemgiventhequeuesizesq(k)andp(k)attimek,r(k)2argmax0rt^eRmaxnXs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.57 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k)rt^eo. Lemma20. Underthealgorithm,thereexistsaconstantB>0suchthatforalltimeslotsk>0,Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.56 11.35 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k)rt^e(k)Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k)rt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(B. Proof. Letht^e(k),qt^e(k))]TJ /F16 11.955 Tf 12.16 8.97 Td[(P^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 12.16 0 Td[(p^e(k).Foranytimeslotsk,k0suchthatjk)]TJ /F8 11.955 Tf 11.95 0 Td[(k0jFmax,Xs,t,^eht^e(k))]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs,t,^eht^e(k0)=Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.57 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k))]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k0))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))qt^e0(k0))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k0)Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(qt^e(k0)+Xs,t,^eX^e02(t,d(^e)))]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e0(k))]TJ /F8 11.955 Tf 11.96 0 Td[(qt^e0(k0)+Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(p^e(k0)FmaxQmax+Fmaxj^EjQmax+FmaxjTjPmax. 94

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LetHmax,FmaxQmax+Fmaxj^EjQmax+FmaxjTjPmax.Then,wehaveXs,t,^eht^e(k)rt^e(k)=Xs,t,^eht^e(bO(k))rt^e(k))]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(ht^e(bO(k)))]TJ /F8 11.955 Tf 11.96 0 Td[(ht^e(k)rt^e(k)Xs,t,^eht^e(bO(k))rt^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(HmaxRmaxXs,t,^eht^e(bO(k))rt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(HmaxRmax=Xs,t,^eht^e(k)rt^e(k))]TJ /F16 11.955 Tf 11.96 11.36 Td[(Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(ht^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(ht^e(bO(k))rt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(HmaxRmaxXs,t,^eht^e(k)rt^e(k))]TJ /F4 11.955 Tf 11.96 0 Td[(2HmaxRmax. LettingB,2HmaxRmax,thelemmaholds. Let(k)beanoptimalsolutiontothefollowingproblemgiventhequeuesizesp(k)andw(k)attimek,(k)2argmax0^emaxX^e2^E)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(p^e(k))]TJ /F16 11.955 Tf 12.97 11.36 Td[(Xe2E^ewe(k)^e. Lemma21. Underthealgorithm,thereexistsaconstantB>0suchthatforalltimeslotsk>0,X^e2^E(p^e(k))]TJ /F16 11.955 Tf 12.97 11.35 Td[(Xe2E^ewe(k))^e(k)X^e2^E(p^e(k))]TJ /F16 11.955 Tf 12.97 11.35 Td[(Xe2E^ewe(k))^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(B. Proof. Since,foranytimeslotk,^e(k)=^e(bU(k)),wehave,by( 4 ),(k)2argmax0^emaxX^e2^E)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(bU(k)))]TJ /F16 11.955 Tf 12.96 11.35 Td[(Xe2E^ewe(bU(k))^e. (4) 95

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Dueto( 4 ),( 4 ),andtheboundednessofaframelength,foranysuchthat0^emax,wehaveX^e2^E(p^e(bU(k)))]TJ /F16 11.955 Tf 12.96 11.36 Td[(Xe2E^ewe(bU(k)))^e)]TJ /F16 11.955 Tf 11.96 11.36 Td[(X^e2^E(p^e(k))]TJ /F16 11.955 Tf 12.97 11.36 Td[(Xe2E^ewe(k))^e=X^e2^E(p^e(bU(k)))]TJ /F8 11.955 Tf 11.95 0 Td[(p^e(k))^e+X^e2^EXe2E^e(we(k))]TJ /F8 11.955 Tf 11.96 0 Td[(we(bU(k)))^eFmax(Pmax+j^EjWmax)max. (4) Then,underthealgorithm,wehaveX^e2^E(p^e(k))]TJ /F16 11.955 Tf 12.96 11.36 Td[(Xe2E^ewe(k))^e(k)=X^e2^E)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(p^e(bU(k)))]TJ /F16 11.955 Tf 12.96 11.36 Td[(Xe2E^ewe(bU(k))^e(k))]TJ /F16 11.955 Tf 11.95 11.36 Td[(X^e2^E)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(bU(k)))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k)^e(k))]TJ /F16 11.955 Tf 11.95 11.35 Td[(X^e2^E)]TJ 8.48 1.67 Td[(Xe2E^ewe(k))]TJ /F16 11.955 Tf 12.97 11.35 Td[(Xe2E^ewe(bU(k))^e(k)X^e2^E)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(p^e(bU(k)))]TJ /F16 11.955 Tf 12.96 11.36 Td[(Xe2E^ewe(bU(k))^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(Fmax(Pmax+j^EjWmax)max,X^e2^E)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(p^e(bU(k)))]TJ /F16 11.955 Tf 12.97 11.36 Td[(Xe2E^ewe(bU(k))^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(Fmax(Pmax+j^EjWmax)max, (4) 96

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wheretherstinequalityisdueto( 4 )andthesecondoneisfrom( 4 ).Further,wehaveRHSof( 4 )=X^e2^E)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(p^e(k))]TJ /F16 11.955 Tf 12.97 11.36 Td[(Xe2E^ewe(k)^e(k))]TJ /F16 11.955 Tf 11.96 11.36 Td[(X^e2^E)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(bU(k))^e(k))]TJ /F16 11.955 Tf 11.96 11.36 Td[(X^e2^E)]TJ 8.48 1.68 Td[(Xe2E^ewe(bU(k)))]TJ /F16 11.955 Tf 12.96 11.36 Td[(Xe2E^ewe(k)^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(Fmax(Pmax+j^EjWmax)maxX^e2^E)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(k))]TJ /F16 11.955 Tf 12.97 11.36 Td[(Xe2E^ewe(k)^e(k))]TJ /F4 11.955 Tf 11.96 0 Td[(2Fmax(Pmax+j^EjWmax)max, wheretheinequalityisdueto( 4 ).LettingB,2Fmax(Pmax+j^EjWmax)max,thelemmaholds. Remark.Theabovelemmasimplythatouralgorithms( 4 )-( 4 )areconstantapproximationsolutionstothecorrespondingsubproblemsoneverytimeslot.Thedifferencebetweentheoptimalvalueandtheapproximationincreaseswiththemaximumframesize. 97

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Toanalyzetheperformanceboundofouralgorithm,weconsiderthefollowing-tightenedproblemP().P():maxXs2SUs(xs)s.t.rt^e)]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,^e))]TJ /F8 11.955 Tf 11.95 0 Td[(ytIf^e2(t,o(t))g,8t2T,8^e2^Et, (4)Xt:^e2^Etrt^e^e)]TJ /F3 11.955 Tf 11.96 0 Td[(,8^e2^E, (4)X^e:e2E^e^ece)]TJ /F3 11.955 Tf 11.96 0 Td[(,8e2E, (4)^e0,8^e2^E,andtheconstraints( 4 ),( 4 )-( 4 ), where0<
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Theorem4.1. Foranyparameter>0,thealgorithmsatisesthefollowingperfor-mancebounds.liminfk!1Xs2SUs(1 kk)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0Efxs()g)f)]TJ /F8 11.955 Tf 13.15 8.09 Td[(BA 2, (4)limsupk!11 kk)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0Efkz()k1gBA+2Umax= 2Y, (4) whereBA=B+2(B+B+B)andBissomeconstant. Proof. DenetheLyapunovfunctionV(k)andtheLyapunovdrift(k)asfollows:V(k),Xs,t,^e(qt^e(k))2+X^e(p^e(k))2+Xe(we(k))2,(k),EfV(k+1))]TJ /F8 11.955 Tf 11.96 0 Td[(V(k)jz(k)g. Bysquaringbothsidesofthequeueevolutionequationin( 4 )followedbysimplemanipulations,wehave(qt^e)2(k+1))]TJ /F4 11.955 Tf 11.96 0 Td[((qt^e)2(k)(rt^e(k))]TJ /F8 11.955 Tf 11.95 0 Td[(rtp(t,^e)(k))]TJ /F8 11.955 Tf 11.96 0 Td[(yt(k)If^e2(t,o(t))g)2)]TJ /F4 11.955 Tf 11.96 0 Td[(2qt^e(k)(rt^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,^e)(k))]TJ /F8 11.955 Tf 11.96 0 Td[(yt(k)If^e2(t,o(t))g). Summingtheinequalityoverallt2Tandall^e2^Etandtakingconditionalexpectations,wehaveEXs,t,^e(qt^e(k+1))2)]TJ /F16 11.955 Tf 11.95 11.36 Td[(Xs,t,^e(qt^e(k))2jz(k)B1)]TJ /F4 11.955 Tf 11.96 0 Td[(2Xs,t,^eqt^e(k)Ert^e(k))]TJ /F8 11.955 Tf 11.96 0 Td[(rtp(t,^e)(k))]TJ /F8 11.955 Tf 11.95 0 Td[(yt(k)If^e2(t,o(t))gjz(k). (4) whereB1,Ps,t,^e(Rmax+Xmax)2. 99

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Similarly,fromthequeueevolutionequationin( 4 ),wegetEX^e(p^e(k+1))2)]TJ /F16 11.955 Tf 11.96 11.36 Td[(X^e(p^e(k))2jz(k)B2)]TJ /F4 11.955 Tf 11.96 0 Td[(2X^ep^e(k)E^e(k))]TJ /F16 11.955 Tf 15.85 11.36 Td[(Xt:^e2^Etrt^e(k)jz(k), (4) whereB2,P^e(maxfjTjRmax,maxg)2. Similarly,fromthequeueevolutionequationin( 4 ),wegetEXe(we(k+1))2)]TJ /F16 11.955 Tf 11.96 11.35 Td[(Xe(we(k))2jz(k)B3)]TJ /F4 11.955 Tf 11.96 0 Td[(2Xewe(k)Efce(k))]TJ /F16 11.955 Tf 16.43 11.36 Td[(X^e:e2E^e^e(k)jz(k)g=B3)]TJ /F4 11.955 Tf 11.96 0 Td[(2Xewe(k))]TJ /F4 11.955 Tf 5.66 -9.68 Td[(ce)]TJ /F16 11.955 Tf 16.43 11.35 Td[(X^e:e2E^eEf^e(k)jz(k)g, (4) whereB3,Pe(maxfCmax,maxg)2. Combiningtheabovethreeinequalities( 4 ),( 4 ),( 4 ),addingtheterm)]TJ /F4 11.955 Tf 9.3 0 Td[(2=Ps2SEfUs(xs(k))jz(k)gtothebothsidesoftheinequality,andrearrangingtheterms,wehave(k))]TJ /F4 11.955 Tf 13.15 8.08 Td[(2 Xs2SEfUs(xs(k))jz(k)gB1+B2+B3)]TJ /F4 11.955 Tf 11.95 0 Td[(2Xs2SEf1 Us(xs(k))jz(k)g)]TJ /F16 11.955 Tf 12.88 11.35 Td[(Xt2TsX^e2(t,o(t))qt^e(k)Efyt(k)jz(k)g)]TJ /F4 11.955 Tf 11.95 0 Td[(2Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.69 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.95 0 Td[(p^e(k)Efrt^e(k)jz(k)g)]TJ /F4 11.955 Tf 11.95 0 Td[(2X^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(k))]TJ /F16 11.955 Tf 12.96 11.36 Td[(Xe2E^ewe(k)Ef^e(k)jz(k)g)]TJ /F4 11.955 Tf 11.95 0 Td[(2Xewe(k)ce. (4) 100

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Denefunctions(k),(k)and(k)asfollows.(k),Xs2SEf1 Us(xs(k))jz(k)g)]TJ /F16 11.955 Tf 21.52 11.36 Td[(Xt2TsX^e2(t,o(t))qt^e(k)Efyt(k)jz(k)g,(k),Xs,t,^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(qt^e(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.96 0 Td[(p^e(k)Efrt^e(k)jz(k)g,(k),X^e)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(p^e(k))]TJ /F16 11.955 Tf 12.96 11.35 Td[(Xe2E^ewe(k)Ef^e(k)jz(k)g+Xewe(k)ce. LetA(k),A(k)andA(k)bethevaluesof(k),(k)and(k)underthealgorithm( 4 )-( 4 ).Let(k)bethemaximumof(k)where(x(k),y(k))satisestheconstraints( 4 ),( 4 ),( 4 )ofproblemP.Then,usingLemma 19 andthefactthat(x(),y(),r(),())isfeasibletoproblemPandP(),wehaveA(k)(k))]TJ /F8 11.955 Tf 11.96 0 Td[(BXs2S1 Us(xs()))]TJ /F16 11.955 Tf 12.89 11.36 Td[(Xt2TsX^e2(t,o(t))qt^e(k)yt())]TJ /F8 11.955 Tf 11.96 0 Td[(B. Let(k)bethemaximumof(k)wherer(k)satisestheconstraints( 4 )ofproblemP.Then,usingLemma 20 andthefeasibilityof(x(),y(),r(),())toproblemPandP(),wehaveA(k)(k))]TJ /F8 11.955 Tf 11.95 0 Td[(BXs,t,^e(qt^e(k))]TJ /F16 11.955 Tf 27.56 11.36 Td[(X^e02(t,d(^e))qt^e0(k))]TJ /F8 11.955 Tf 11.95 0 Td[(p^e(k))rt^e())]TJ /F8 11.955 Tf 11.95 0 Td[(B=Xs,t,^eqt^e(k)(rt^e())]TJ /F8 11.955 Tf 11.95 0 Td[(rtp(t,^e)()))]TJ /F16 11.955 Tf 11.95 11.35 Td[(X^ep^e(k)rt^e())]TJ /F8 11.955 Tf 11.95 0 Td[(BXs,t,^eqt^e(k)(yt()If^e2(t,o(t))g+))]TJ /F16 11.955 Tf 11.95 11.36 Td[(X^ep^e(k)rt^e())]TJ /F8 11.955 Tf 11.95 0 Td[(B. Let(k)bethemaximumof(k)where(k)satisestheconstraints0^emaxforall^ein^E.Then,usingLemma 21 andthefeasibilityof(x(),y(),r(),()) 101

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toproblemPandP(),wehaveA(k)(k))]TJ /F8 11.955 Tf 11.96 0 Td[(BX^e(p^e(k))]TJ /F16 11.955 Tf 12.96 11.35 Td[(Xe2E^ewe(k))^e()+Xewe(k)ce)]TJ /F8 11.955 Tf 11.95 0 Td[(B=X^ep^e(k)^e()+Xe2E^ewe(k)(ce)]TJ /F16 11.955 Tf 16.43 11.36 Td[(X^e:e2E^e^e()))]TJ /F8 11.955 Tf 11.96 0 Td[(BX^ep^e(k)^e()+Xe2E^ewe(k))]TJ /F8 11.955 Tf 11.95 0 Td[(B. Fromtheaboveinequalities,wehaveA(k)+A(k)+A(k)Xs2S1 Us(xs()))]TJ /F16 11.955 Tf 12.89 11.36 Td[(Xt2TsX^e2(t,o(t))yt()qt^e(k)+Xs,t,^eqt^e(k)(yt()If^e2(t,o(t))g+)+X^ep^e(k)(^e())]TJ /F8 11.955 Tf 11.96 0 Td[(rt^e())+Xe2E^ewe(k))]TJ /F4 11.955 Tf 11.95 0 Td[((B+B+B)Xs2S1 Us(xs())+Xs,t,^eqt^e(k)+X^ep^e(k)+Xe2E^ewe(k))]TJ /F4 11.955 Tf 11.95 0 Td[((B+B+B), wherethesecondinequalityisduetothecancelationofthecommontermsandthefeasibilityof(x(),y(),r(),())toproblemP(). LetBA=B1+B2+B3+2(B+B+B).Applyingtheaboveinequalitiestothedriftexpressionin( 4 ),weget(k))]TJ /F4 11.955 Tf 13.15 8.09 Td[(2 Xs2SEfUs(xs(k))jz(k)gBA)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2 f())]TJ /F4 11.955 Tf 11.96 0 Td[(2kzk1(k). (4) 102

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From( 4 ),takingtheexpectationoverthedistributionofz(k)andsummingtheinequalityoverk2f0,1,...,K)]TJ /F4 11.955 Tf 11.96 0 Td[(1g,wegetEfV(K))]TJ /F8 11.955 Tf 11.95 0 Td[(V(0)g)]TJ /F4 11.955 Tf 21.79 8.09 Td[(2 K)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0Xs2SEfUs(xs())gKBA)]TJ /F4 11.955 Tf 13.15 8.09 Td[(2K f())]TJ /F4 11.955 Tf 11.95 0 Td[(2K)]TJ /F12 7.97 Tf 6.59 0 Td[(1X=0Efkzk1()g. (4) UsingthefactthatremovingthetermsEfV(K)gand)]TJ /F4 11.955 Tf 9.3 0 Td[(2PK)]TJ /F12 7.97 Tf 6.58 0 Td[(1=0Efkzk1()gpreservestheinequality,rearranging( 4 ),andtakingtheliminfasK!1,wegetliminfK!11 KK)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0Xs2SEfUs(xs())gf())]TJ /F8 11.955 Tf 13.15 8.09 Td[(BA 2. UsingJensen'sinequalityandletting!0,weget( 4 ). Ontheotherhand,from( 4 ),usingthedenitionofUmaxandthefactthatremovingthetermsEfV(K)gand)]TJ /F4 11.955 Tf 9.3 0 Td[(2Kf()=preservestheinequality,rearranging( 4 ),andtakingthelimsupasK!1,wegetlimsupK!11 KK)]TJ /F12 7.97 Tf 6.58 0 Td[(1X=0Efkzk1()gBA+2Umax= 2. Letting!Y,weget( 4 ). Theorem 4.1 impliesthattheparametercanbeusedasaknobtotradeofftheoptimizationperformanceandthevirtualqueuesizebounds.Bychoosingtobesufcientlysmall,thelong-timeaverageofthesolutiongivenbythealgorithmcanproduceanobjectivevaluearbitrarilyclosetotheoptimalvalueofproblemP.However,theboundonthelong-timeaverageofthevirtualqueuesizesincreasesasdecreases.TheconstantBAincludestheconstantsB,B,andBduetotheasynchronousoperationsofthealgorithmbythetwonetworklayers.Hence,thelargerthemaximumframesizeis,whichimpliesahigherdegreeofasynchrony,thelesstighttheperformanceboundsare.Notethattheboundednessofthevirtualqueuesizes 103

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doesnotdirectlyimplynetworkstability.WesuspectthattherealqueuesizesinthenetworkarealsoboundedandwewillshowsimulationresultstosupporttheconjectureinSection 4.3 4.3SimulationExperiments Inthissection,weshowsimulation-basedperformanceevaluationofthealgorithminSection 4.2 .Weexaminetherateoptimalityandtherealqueueboundednessofthealgorithm. Forthestabilityoftherealqueues,weconsiderthefollowingslightlymodiedsystem.Whenasourcesinjectsrealpacketsintoatreet2Tsattimek,werequirethatthesourcesendspacketsataslightlysmallerratethanthevirtualtreerateyt(k)determinedbyourmainalgorithm.Specically,thesourceinjectsYt(k)realpacketswhereYt(k)isarandomvariablewithmeanyt(k)forsomeslightlylessthan1.Wecalltherealtrafcintensity.Ifyt(k)=0,thenYt(k)=0.WealsorequirethatYt(k)isboundedabovebyXmax.Usingtheconceptoftherealtrafcintensitytoshownetworkstabilitycanbeseenintheliterature[ 11 12 ]. WewishtoshowthatalltherealqueuesQt^eandP^einthedataserversandthequeuesWeattheunderlaylinksarebounded.TheaggregatequeuesizeattimekisdenedtobethesumofalltherealqueuesizesPs,t,^eQt^e(k)+P^eP^e(k)+PeWe(k). 4.3.1SimulationSetup WeusearealISPnetworktopologyobtainedfromtheRocketfuelproject[ 1 ],whichconsistsof41routersand136underlaylinks.Weattach100dataserversrandomlytothenetworkasfollows.Assumingwehavevesessions,weassign20dataserverspersession.Then,foreachsession,weattachits20dataserversrandomlytothenetworksuchthatnodataserversinthesessionareattachedtothesamerouter.Mostoftheunderlaylinksexceptsomecriticallinkshaveanaveragelinkcapacity1000each.Bycriticallinks,wemeanroughlythelinksthateasilybecomethebottleneckiftheydonothavesufcientcapacities.Weassignanaveragelinkcapacity3000toeach 104

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criticallink.Wealsoassignarelativelylargeaveragelinkcapacity200000toeachofthelinksbetweenadataserveranditsneighboringroutersothattheydonotbecomethebottleneck.Theunderlaylinkcapacitiesce(k)varyrandomlyfromtimeslottotimeslot.Theyarei.i.drandomprocesseswiththeuniformdistributionsandtheyaremutuallyindependent. Weusethefollowingheuristictoobtainthesetofmulticasttreesforeachsession.Werunthealgorithmin[ 62 ],whichcomputestheoptimalsetoftrees(approximately),aswellastheirrateallocationandcosts.Thenumberoftreesreturnedbythatalgorithmisverylarge2.Wechoosetouseonly10treesforeachsessionintheincreasingorderofthetreecost.WechooseUs(xs)=log(xs+1)astheutilityfunction. 4.3.2SingleSessionCase Inthissubsection,werunthealgorithmwithonlysinglesessionconsistingofonesourceand19receiversandexaminetheperformanceofthealgorithm.First,weshowhowthealgorithmworkswhentheoverlayandunderlaynetworksoperatesynchronously.Inthisexperiment,wesetboththeoverlayandunderlayframesizestobe1.Thevalueofis10)]TJ /F12 7.97 Tf 6.58 0 Td[(6andis0.99.Fig. 4-3 (a)showstheconvergenceofthetimeaverageratesatevery500iterations.Theachievedsessionrateis919,whichisquiteclosetotherate967obtainedbythealgorithmin[ 62 ].Notethatouralgorithmusesonly10treeswhereasthealgorithmin[ 62 ]uses22320trees.WealsoplotthetimeaverageofthereceivingrateinFig. 4-3 (a).Beforereachingthesteadystate,thesessionratecanbedifferentfromeachofthereceivingratesbecausesomepacketsaretemporarilyqueuedinthenetwork.Eachpointonthedottedlinerepresentstheaverageacrossthereceiversofthetimeaveragereceivingratesatthecorrespondingiteration.Theboxcenteredaroundeachpointmarksonestandarddeviationaboveorbelowtheaverage 2Thenumberoftreesreturnedbythatalgorithmis22320inthesinglesessioncaseanditis70595inthemultiplesessioncase. 105

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(a)Synchronouscase (b)Asynchronouscase Figure4-3. Timeaveragerates. rate,andtheendpointsoftheverticallineoneachpointrepresentthemaximumandtheminimum(withrespecttotheratesamplescollectedacrossthereceiversatthecorrespondingiteration).Onecanseethattheaverageofthetimeaveragereceivingrateseventuallyapproachesthetimeaveragesessionrate.Thissuggeststhatthequeuesinthenetworkwillnotgrowindenitely.Moreover,thestandarddeviationofthetimeaveragereceivingratesalsodecreasesasconvergencetakesplace. Next,werunthealgorithmasynchronouslytoshowhowtheperformanceisaffectedbytheasynchronousoperations.Inthisexperiment,theoverlayframesizesvaryrandomlybetween1and2andtheunderlayframesizesvarybetween2and3.All 106

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(a)Effectontheachievedrate (b)Effectonthetimeaverageaggregatequeuesize Figure4-4. Effectofontheachievedrateandonthetimeaverageaggregatequeuesize. theotherparametersarethesameasthesynchronouscase.Fig. 4-3 (b)showstheconvergenceofthetimeaveragerates.Theachievedsessionrateis756,whichis82%ofthesynchronouscase.Theperformancedegradationisbecausethealgorithmoftenusesoutdatedinformationforcontrolintheasynchronousoperations. Fig. 4-4 showshowtheparameteraffectsthealgorithmperformance.InFig. 4-4 (a),weshowthetimeaveragereceivingratesatthe2105thiterationforboththesynchronousandasynchronouscases,withvaryingbetween10)]TJ /F12 7.97 Tf 6.58 0 Td[(4and10)]TJ /F12 7.97 Tf 6.59 0 Td[(8.Asdecreases,theachievedreceivingratetendstoincrease.However,thequeuesizesin 107

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(a)=0.99 (b)=0.999 (c)=1.0 (d)=1.001 Figure4-5. Trajectoryoftheaggregaterealqueuesize. thenetworkalsotendtoincrease,asshowninFig. 4-4 (b).Theguresalsoshowthat,asdecreases,theachievedrateoftheasynchronouscaseapproachesthatofthesynchronouscase. Wenextexaminetheaggregaterealqueuesizeunderthealgorithm.Intheexperiment,wetestfourdifferentrealtrafcintensities:=f0.99,0.999,1.0,1.001g.Werunasynchronousoperationsandweuse=10)]TJ /F12 7.97 Tf 6.59 0 Td[(5.Fig. 4-5 showsthetrajectoryoftheaggregaterealqueuesizeupto106thiteration.Thegureindicatesthattherealqueuesarestableifislessthan13.Whenthequeuesarestable,theupperboundoftheaggregatequeuesizetendstoincreaseasincreases. 3Inthisexperiment,theaggregaterealqueuesizelooksstableevenwhen=1.0.However,itisnotalwaystrueasshowninthefollowingsubsection 4.3.4 108

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Figure4-6. Objectivevaluesofthecasesofmultiplesessions. 4.3.3CaseofMultipleSessions Nowweconsiderthecaseofmultiplesessions.Wehavevesessionsandeachsessionconsistsofonesourceand19receivers. Fig. 4-6 plotstheobjectivevaluesobtainedfromvarioustestcaseswithmultiplesessions.TestcaseSGisthecaseofrunningthesubgradientalgorithmusedin[ 62 ].TheachievedobjectivevalueinSGis26.6,whichisconsideredasthemaximumobjectivevaluewhichcanbeobtainedbyouralgorithm.TestcaseswithprexSarerununderthesynchronoustimescales,andthosewithprexAarerunundertheasynchronoustimescales.Thesufxnumberindicatesthevalueofparameter.Forexample,forthetestcaseS)]TJ /F4 11.955 Tf 12.63 0 Td[(4,wesettobe10)]TJ /F12 7.97 Tf 6.58 0 Td[(4.Thegureshowsthatasdecreases,ouralgorithmapproachestheoptimalobjectivevalue. Fig. 4-7 andFig. 4-8 showhowthesessionratesandreceivingratesofthevesessionsconvergeunderthesynchronousandasynchronoustimescales,respectively.Thevalueofis10)]TJ /F12 7.97 Tf 6.58 0 Td[(6andis0.99.Theguresshowthattheasynchronydegradestheachievedsessionratesasinthesinglesessioncase.Underthesynchronoustimescales,theachievedrateofthesessionexperiencingthehighestrateis245andtherateofthesessionexperiencingthelowestrateis107(43%ofthehighestrate).Underthealgorithmin[ 62 ],allthesessionshavenearlythesameachievedrate200.Itimplies 109

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(a)Realsessionrates (b)Realreceivingrates Figure4-7. Timeaverageratesofmultiplesessionsundersynchronoustimescales. thatundertheabovesetup,ouralgorithmresultsinlessfairrateallocationthanthealgorithmin[ 62 ].Undertheasynchronoustimescales,theachievedrateofthesessionexperiencingthehighestrateis229andtherateofthesessionexperiencingthelowestrateis85(37%ofthehighestrate).Itimpliesthatasynchronyalsoaffectsthefairness. Fig. 4-9 plotshowtherateallocationisachievedwithvariousparameterundereithersynchronousorasynchronoustimesscales.Thegureindicatesthatasdecreases,theoverallachievedrateincreasesandthefairnessamongsessionsimprovesaswell.However,theimprovementisobtainedwiththeincreaseoftheaggregatequeuesizeasshowninFig. 4-10 110

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(a)Realsessionrates (b)Realreceivingrates Figure4-8. Timeaverageratesofmultiplesessionsunderasynchronoustimescales. Wealsoexaminetheaggregaterealqueuesizewiththemultiplesessions.Fig. 4-11 showsthetrajectoryoftheaggregatequeuesizeupto106thiteration.Asinthesinglesessioncase,thegurealsoindicatesthattherealqueuesarestableifislessthan1. 4.3.4ExampleofInstableRealQueuesWhen=1.0 Intheaboveexperiments,therealqueueslookstableevenwhentherealtrafcintensityis1.0.However,wehavefoundthatinsomeexperiments,itisnottrue.Forexample,inthefollowingexperiment,theaggregaterealqueuesizeisnotstablewhenis1.0.Werunthealgorithmwithasinglesessionconsistingofsinglesourceand9 111

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Figure4-9. Effectofonrateallocation. Figure4-10. Effectofonaggregatequeuesize. receivers.Mostoftheparametersarethesameasthesinglesessioncaseinsubsection 4.3.2 .Fig. 4-12 showsthat,inthistestcase,theaggregaterealqueuesizecontinuestoincreasewhen=1.0whereasitisstablewhen=0.999.Itimpliesthattherealtrafcintensity1.0doesnotnecessarilyguaranteetherealqueuestability. 4.4RelatedWork Werstcontrastourworkwiththecloselyrelatedpriorstudies[ 11 20 29 30 51 52 ],whichhaveintroducedsimilarproblemsandbackpressure-basedsolutions.In[ 20 ],theauthorspresentanoptimalresourceallocationalgorithmforoverlaymulticastservice.Theirapproachisdifferentfromoursinthefollowingaspects.First,their 112

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(a)=0.99 (b)=0.999 (c)=1.0 (d)=1.001 Figure4-11. Trajectoryoftheaggregaterealqueuesize. (a)=0.999 (b)=1.0 Figure4-12. Exampleofinstableaggregaterealqueuesizewhen=1.0. problemisamultiratemulticastproblemwithasinglesessionusingasingletree,whereasoursisauniratemulticastproblemwithmultiplesessionsusingmultipletrees.Second,theyassumethatalltheoperationsarerunbytheCPandthereisnoparticipationfromanunderlaynetworkprovider.However,weassumethatboth 113

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partiesactivelyparticipateintheservice,andhence,wewishtodividetheoperationsaccordinglyandmaketheinteractionsbetweenthepartiesassimpleaspossible.Toachievethegoal,weformulateaverydifferentproblemand,asaresult,deriveaverydifferentalgorithm.Lastly,theprooftechniquesaredifferent.WeusetheLyapunovoptimizationtechniquetoprovetheoptimalityofouralgorithminastochasticsetting,whereastheyusetheconventionaltechniqueforstaticconvexoptimization,whichcannotbeappliedtostochasticproblems. Noneofthestudiesin[ 11 29 30 51 52 ]considertheoverlaynetworks,whichmakesthemmostlyinapplicabletothetwo-layernetworks.In[ 11 ],theauthorsintroducerateallocationalgorithmsformulticastsessionsinawirelessnetwork.In[ 29 30 51 ],theauthorsintroduceratecontrolalgorithmsformultiratemulticastproblemswiththeobjectiveofmaximizingthetotalutilityofallthereceivers.In[ 51 ],theperformanceobjectiveisthemaxminfairness,anditisnotclearhowtoextendtheresultstoothertypesoffairness.In[ 29 30 ],theauthorsassumethattheutilityfunctionsarestrictlyconcave,whichisakeyconditionusedintheirproofs.Therefore,theirprooftechniquescannotbeappliedtoourcasewithnon-strictlyconcaveutilityfunctions. In[ 59 ],theauthorsconsideranarchitecturecalledP4P,whichallowscooperativetrafccontrolbetweenthecontent-distributionapplicationsandthenetworkproviders.ThedifferencefromourapproachisthattheystartfromtheISP'sobjective,suchasminimizingthenetworkcongestion.Then,theapplicationsarerequiredtosolvethedualproblemderivedfromtheISP'sproblem.Ontheotherhand,ourutility-maximizationformulationismoregeneralanditalsomoredirectlyaddresseshowtoimprovetheperformanceofthecontentdistributionservicewithlimitednetworkresources. Therehavebeengame-theoreticstudiesabouttheinteractionbetweentheoverlaycontentdistributionandtrafcengineering(TE)attheunderlaynetworks[ 22 28 42 ].TheperspectiveisthattheISPandtheCPhavedifferentobjectivesandmayevencompetewitheachother.Anothermajordifferenceisthat,inthosestudies,the 114

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timescalesoftheTEandtheCPoperationsaresimilar,whereas,inourcase,thetimescaleoftheTEoperations(i.e.,adjustingtheunderlayroutes)canbemuchlargerthanthatoftheoverlayoperations. Peer-to-peersystemsarewidelyusedforcontentdistributionoveroverlaynetworks.Therehavebeenseveralproposals[ 3 10 16 49 ]toenhancetheenduserperformanceandnetworkresourceutilizationbytrafclocalization.Themainfocusisonselectingpeersincloseproximityeitherbyusingnetworkprobingfacilities[ 49 ]orbyinteractingwithunderlayoperators[ 3 10 16 ].Sincetheseproposalsrelyonheuristicmethodologiestoselectproperneighbors,theperformanceboundandfairnessamongsessionsarenotexplicitlyguaranteed. 115

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CHAPTER5CONCLUSIONS Wehaveintroducedthreemulti-treemulticastalgorithmsforcontentdistribution:twoalgorithmsforcontentdistributionoverasingle-layernetworkandonealgorithmforcontentdistributionoveratwo-layernetwork.Allofthemarebackpressurealgorithms.Theyarenotonlydecentralizedbutalsolocal:Theexchangeofcontrolmessagesisrestrictedtobeamongtheneighboringnodesandthereisnoglobalexchangeofcontrolmessages. InChapter 2 ,wehavepresentedabackpressurealgorithmfortheutilitymaximizationproblemformulti-treemulticast.Itisnotstraightforwardtoshowthatouralgorithmleadstobothprimaloptimalityandnetworkstabilityduetotheassumptionofthegeneralconcaveutilityfunctionsandthefactthatthealgorithmreliesonvirtualqueueupdates.Wehaveshownthatthetwosetsofanalyticaltools,theLyapunovoptimizationtechniqueandtheconvexoptimizationtechniques,cancomplementeachotherandcircumventthedifculty.Itisalsointerestingthattherearealternativewaystodenean-tightened(or-modied)problemintheproofofthealgorithmperformanceusingtheLyapunovoptimizationtechnique.Furtherexplorationofthealternativesmaybefruitful. InChapter 3 ,wehavepresentedabackpressurealgorithmusingtherealqueuesfortheutilitymaximizationproblemformulti-treemulticast.Wehavederivedauidlimitmodelforperformanceanalysisandexploredthebehaviorofthealgorithm.Wehaveprovidedarigorousproofofasymptoticoptimalityofthealgorithmusingtheuidlimitanalysis.Ouralgorithmisasymptoticallyoptimalinthesensethat,underappropriatescaling,everylimitpointoftherateallocationisanoptimalsolutiontotheproblem.Wehavealsoshownthat,intheuidlimit,thebackpressurealgorithmisexactlyasubgradientalgorithm. Lastly,inChapter 4 ,wehaveintroducedadistributedbackpressurealgorithmforcontentdistributionoveratwo-layernetwork,wherethecontrolvariablescanbe 116

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adjustedatbothlayers.Thealgorithmcanachieveoptimaldistributionperformancewithefcientuseofthenetworkresources.Inthemeantime,thealgorithmiswelldividedbetweentheoverlayandunderlaynetworkswithminimalinteractionandthetwonetworkscanoperateasynchronouslyorondifferenttimescales.WehaveproveditsoptimalityusingtheLyapunovoptimizationtechniqueandshowednetworkstabilitybysimulations.Theformalproofoftherealqueueboundednessislefttofutureresearch.Wecanextendouralgorithmtoallowmoreunderlaycontrols,suchasmulti-pathroutingordynamicroutingfortheunderlaynetwork,whichcanfurtherimprovetheapplicationperformanceand/orresourceutilization. 117

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BIOGRAPHICALSKETCH ChunglaeChoreceivedhisB.S.andM.S.degreesincomputersciencefromPusanNationalUniversity,Korea,in1994and1996,respectively.HeworkedasaresearchstaffmemberatElectronicsandTelecommunicationsResearchInstitute,Korea,between2000and2005.HereceivedhisPh.D.incomputerengineeringfromtheUniversityofFloridainthefallof2011.Hisresearchinterestsareinresourceallocation,loadbalancing,congestioncontrolandoptimizationincommunicationnetworks,peer-to-peernetworks,contentdistributionnetworks,wirelessnetworksandsensornetworks. 123