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Complexity and Delay Characterization of Wireless Scheduling Algorithms

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Complexity and Delay Characterization of Wireless Scheduling Algorithms
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Boyaci, Cem
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Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Computer Engineering
Computer and Information Science and Engineering
Committee Chair:
Xia, Ye
Committee Co-Chair:
Ungor, Alper
Committee Members:
Kahveci, Tamer
Chen, Shigang
Fang, Yuguang
Graduation Date:
8/11/2012

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Algorithms ( jstor )
Approximation ( jstor )
Computational complexity ( jstor )
Harmonic functions ( jstor )
Mathematical vectors ( jstor )
Optimal solutions ( jstor )
Polynomials ( jstor )
Scheduling ( jstor )
Sufficient conditions ( jstor )
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Computer and Information Science and Engineering -- Dissertations, Academic -- UF
coloring -- delay -- graph -- networking -- scheduling -- throughput -- wireless
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Computer Engineering thesis, Ph.D.

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Abstract:
Recently, escalating number of wireless network users and popularity of  multimedia-driven applications give rise to an increasing demand  for high throughput and low delay that is to be met by wireless  data networks. The performance of wireless communication is  strongly tied to the underlying medium access control (MAC)  schemes that coordinate scheduling decisions  under interference constraints. In last two decades,  a few number of algorithms are proposed to address efficient  utilization of wireless medium. Among them there is a class  of throughput optimal algorithms which illustrate a diverse  spectrum of requirements on computational resources.  In this work our primary objective is to build a formal relation  between throughput optimality and corresponding  resource demands. Our secondary objective is to derive tight  packet delay bounds for optimal and sub-optimal wireless  scheduling algorithms. Under the protocol model, wireless scheduling problem can be  related to the weighted fractional chromatic number  of the underlying interference graph. In the second chapter  we formalize this relation, and consequently  show that a deterministic version of wireless scheduling  problem is computationally intractable, given queue sizes  are to be kept within practically implementable bounds, unless \NP=\ZPP. In the third chapter, we investigate the delay bounds of  $k$-throughput optimal wireless scheduling algorithms  under stationarity. We reveal a simple result that governs  the manipulation of the negative drift condition, and  tighten the previously provided results. It is also shown that  the same framework can be used to bound the moments  of queue size distribution at stationarity under a popular  suboptimal algorithm called Longest Queue First.  It is possible to derive delay bounds using a single  parameter or a multi-parameter approach. In the fourth  chapter we introduce a new single parameter characterization  of the delay bound under Maximum Weight Scheduling (MWS)  policy. We provide a performance comparison with the previously derived  bounds and underline the significance of our improvement. Next we show  , when the traditional quadratic potential function is employed in the  analysis, the new delay parameter is optimal. Our study reveals the  new parameter is an ideal choice in practice for deriving delay bounds  when a single parameter delay characterization of MWS is desired.  In the fifth chapter we generalize MWS policy using functions  that are integrable and differentiable. The generalization  enables us to use the functions in the weight calculations  and change the activation vectors accordingly. We refine the  sufficient conditions in literature for throughput optimality by  adding new requirements.and show there are additional requirements  for stability. The sixth chapter summarizes our results and a provides  a list of related open problems outlining immediate research  pursuit in the field of wireless scheduling. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Xia, Ye.
Local:
Co-adviser: Ungor, Alper.
Statement of Responsibility:
by Cem Boyaci.

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Copyright Boyaci, Cem. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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COMPLEXITYANDDELAYCHARACTERIZATIONOFWIRELESSSCHEDULINGALGORITHMSByCEMBOYACIADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012CemBoyaci 2

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Tomymother 3

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ACKNOWLEDGMENTS IwouldliketoexpressmyappreciationforallthehelpIreceivedfrommyadvisorDr.YeXia.Theenlighteningdiscussionswehad,hisobservations,correctionsandcommentswereallintegraltomywork.IalsowouldliketothankDr.AlperUngorforhissupport,guidance,andpatiencethroughoutmyresearch.Iwouldalsoliketothankmycommitteemembers,Dr.YuguangFang,Dr.TamerKahveci,and,Dr.ShigangChenforalloftheirhelpandvaluablesuggestions.Myfamily'spersistentsupportprovidedmethestrengthandendurancetocontinueatthetimesofmiscellaneousobstacles.Aheartfulthankyougoesto:CanBoyac,CemilBoyac,Nabi,andAytenBalkanl.Finally,thepeoplewhowerealwaysthereforme,myfriends.OnurDursun,Gunes.Yakupoglu,Eris.Aslan,KaganAbidin,BekirArslan,OguzhanTopsakal,HaleErten,FerhatAy,MehmetOnurBaykan,CihanS.ahin,GunhanGulsoy,BradleyNolan,GoktugC.nar,BesimSolakandmanymoreIcannotlisthere.AdditionallyIamthankfultoBoLiforhisfriendshipandforbeingmyprimarycomradeintheprocess. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 7 LISTOFFIGURES ....................................... 8 ABSTRACT ........................................... 9 CHAPTER 1INTRODUCTION .................................... 11 1.1WirelessSchedulingProblem ............................ 11 1.2ProposedModelsandRelatedChallenges ..................... 12 1.2.1WirelessInterferenceModels ........................ 12 1.2.2SystemModel ................................ 13 1.3ResearchOverview ................................. 15 1.3.1OptimalAlgorithms ............................. 15 1.3.2SuboptimalAlgorithms ........................... 16 1.4Contributions .................................... 17 1.4.1EstablishingHardnessofAchievingLowDelay .............. 17 1.4.2BoundsonSumofHigherOrderQueueDistributionMomentsatSta-tionarity ................................... 18 1.4.3SingleParameterCharacterizationofDelayBounds ............ 18 1.4.4GeneralizedMWSPolicy .......................... 18 2COMPLEXITYOFACHIEVINGLOWDELAY .................... 20 2.1Background ..................................... 20 2.2Preliminaries .................................... 22 2.2.1WirelessLinkScheduling .......................... 23 2.2.2FractionalColoringProblem ........................ 24 2.2.3MinimumSpanPacketScheduling ..................... 25 2.3RelevanceofFractionalColoringtoScheduling .................. 27 2.4ComplexityofSchedulingProblem ......................... 31 2.5QueueDynamicsunderDeterministicArrival ................... 33 2.6ACloserLooktotheCapacityRegionunderProtocolModel ........... 36 2.7CapacityRegionandLocalPooling ........................ 38 2.8EffectsofRedeningLoad-slackofArrivalVectors ................ 42 2.9ResultsandDiscussion ............................... 47 3STEADYSTATEDELAYCHARACTERISTICSOFAPPROXIMATEMWSPOLICY 48 3.1Background ..................................... 48 3.2SystemModel .................................... 49 5

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3.3GraphTheoreticalResults ............................. 51 3.4DelayBoundsofApproximateMWSPolicy .................... 54 3.5LQFasanApproximateMWSPolicy ....................... 57 3.6MiscellaneousResultson-LocalPooling .................... 60 3.6.1GraphswithArbitrarilySmallSet-LocalPoolingFactors ........ 61 3.6.2ComputationalComplexityofCalculatingG ............... 63 3.6.3SpecialStructuresThatPreserveG .................... 65 3.7ResultsandDiscussion ............................... 67 4CHARACTERIZINGTHEEXPECTEDQUEUE-SUMBOUNDOFTHEMWSPOLICY .......................................... 68 4.1Background ..................................... 68 4.2DelayCharacterizationUsingaSingleParameter ................. 68 4.2.1ChallengesinComputing() ........................ 70 4.2.2PerformanceComparisonBetween()and() .............. 73 4.3TightnessofDelayParameters ........................... 75 4.3.1DerivingDelayBoundsforaGeneralizedMWSPolicyUsingaQuadraticPotentialFunction .............................. 75 4.3.2EstablishinganIntuitiveMeaningfor():AStepTowardsSinglePa-rameterDelayCharacterizationofMWSPolicy .............. 78 4.4ResultsandDiscussion ............................... 80 5GENERALIZINGMWSPOLICY ............................ 82 5.1Background ..................................... 82 5.2ChoosingWeightFunctioninMWSforStability .................. 83 5.3ResultsandDiscussion ............................... 88 6CONCLUSIONS ..................................... 90 6.1SummaryofResults ................................. 90 6.2FutureResearch ................................... 91 REFERENCES ......................................... 92 BIOGRAPHICALSKETCH .................................. 97 6

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LISTOFTABLES Table page 2-1Atabularcomparisonofnegativeresultsfrom[ 49 ]andTheorem 2.4 .......... 44 7

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LISTOFFIGURES Figure page 1-1Networkgraph,interferencegraphandthematrixrepresentationoftheinterferencerelation .......................................... 14 2-1Interferencegraphforasix-cyclenetworkanditsschedulevectors ........... 37 2-2Capacityregionandarrivalratevectorsindifferentsections ............... 38 3-1IllustrationontightnessofLemma 17 .......................... 53 3-2Hypercubesofdifferentdimensions ........................... 63 3-3IllustrationofGandG' .................................. 65 4-1Anasymmetricnetworkrealizingthequeue-sumboundthrough()beinganim-provementovertheboundthrough() .......................... 71 4-2Vertextransitivitydoesnotrequireorimplymembershipinclass ........... 74 4-3IllustrationofK-cliqueandanodelooselyconnectedforK=6 ............ 74 5-1Conictgraphofathreelinknetworkillustratingthelinetopology ........... 88 8

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCOMPLEXITYANDDELAYCHARACTERIZATIONOFWIRELESSSCHEDULINGALGORITHMSByCemBoyaciAugust2012Chair:YeXiaCochair:AlperUngorMajor:ComputerEngineeringRecently,escalatingnumberofwirelessnetworkusersandpopularityofmultimedia-drivenapplicationsgiverisetoanincreasingdemandforhighthroughputandlowdelaythatistobemetbywirelessdatanetworks.Theperformanceofwirelesscommunicationisstronglytiedtotheunderlyingmediumaccesscontrol(MAC)schemesthatcoordinateschedulingdecisionsunderinterferenceconstraints.Inlasttwodecades,afewnumberofalgorithmsareproposedtoaddressefcientutilizationofwirelessmedium.Amongthemthereisaclassofthroughputoptimalalgorithmswhichillustrateadiversespectrumofrequirementsoncomputationalresources.Inthisworkourprimaryobjectiveistobuildaformalrelationbetweenthroughputoptimalityandcorrespondingresourcedemands.Oursecondaryobjectiveistoderivetightpacketdelayboundsforoptimalandsub-optimalwirelessschedulingalgorithms.Undertheprotocolmodel,wirelessschedulingproblemcanberelatedtotheweightedfractionalchromaticnumberoftheunderlyinginterferencegraph.Inthesecondchapterweformalizethisrelation,andconsequentlyshowthatadeterministicversionofwirelessschedulingproblemiscomputationallyintractable,givenqueuesizesaretobekeptwithinpracticallyimplementablebounds,unlessNP=ZPP.Inthethirdchapter,weinvestigatethedelayboundsofk-throughputoptimalwirelessschedulingalgorithmsunderstationarity.Werevealasimpleresultthatgovernsthemanipulationofthenegativedriftcondition,andtightenthepreviouslyprovidedresults.Itisalsoshownthatthesameframeworkcanbeusedtoboundthemomentsofqueue 9

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sizedistributionatstationarityunderapopularsuboptimalalgorithmcalledLongestQueueFirst.Itispossibletoderivedelayboundsusingasingleparameteroramulti-parameterapproach.InthefourthchapterweintroduceanewsingleparametercharacterizationofthedelayboundunderMaximumWeightScheduling(MWS)policy.Weprovideaperformancecomparisonwiththepreviouslyderivedboundsandunderlinethesignicanceofourimprovement.Nextweshow,whenthetraditionalquadraticpotentialfunctionisemployedintheanalysis,thenewdelayparameterisoptimal.OurstudyrevealsthenewparameterisanidealchoiceinpracticeforderivingdelayboundswhenasingleparameterdelaycharacterizationofMWSisdesired.InthefthchapterwegeneralizeMWSpolicyusingfunctionsthatareintegrableanddifferentiable.Thegeneralizationenablesustousethefunctionsintheweightcalculationsandchangetheactivationvectorsaccordingly.Werenethesufcientconditionsinliteratureforthroughputoptimalitybyaddingnewrequirements.andshowthereareadditionalrequirementsforstability.Thesixthchaptersummarizesourresultsandaprovidesalistofrelatedopenproblemsoutliningimmediateresearchpursuitintheeldofwirelessscheduling. 10

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CHAPTER1INTRODUCTION 1.1WirelessSchedulingProblemInwirelessmediumsimultaneouslinktransmissionscauseinterferenceinthelackofcoor-dination.Inpracticethismeanstwodevicescannotconductsuccessfulwirelesscommunicationatthesametime,interferingeachotherstransmission,iftheirdistanceisbelowahardwaredenedthreshold.Inasmallwirelessnetworkwithfewdevicestheimpactofusinganoptimalmediumaccessschememightnotbeimmediate.Howeverasthenumberofdevicesandlinksinthenetworkincreasestheperformanceofwirelesscommunicationsystemsprimarilyrelyontheefcientutilizationofthesharedtransmissionmedium.Today'stypicalwirelessrouterscansupporthundredsofuserssimultaneously.Additionallythedemandonhighperformancewire-lesscommunicationisstillincreasingasthewirelesslocalareanetworks(WLAN)becomingthemostpopularmethodofaccessingtheInternet.Mediumaccesscontrol(MAC)protocolsaddresstheefcientmediumutilizationproblembydevisingchannelaccessmechanismsthatoptimizeobjectivesofchoice.MostMACalgorithmstargetavoidingtransmissioncollisionswhilemakingsensibledecisionsonthesetoflinkstoactivate.Throughput,whichingeneralmeansaveragerateofsuccessfuldelivery,hasbeencon-sideredasoneofthemostprevalentnetworkperformancemetricsinthelasttwodecades.Accordingly,thereisamassivebodyofresearchstrivingtodevisethroughputoptimalMACpoliciesthatareresourceefcientaswell.Moreprecisely,lowcomputationalcomplexityandlowdelayarethedesiredfeaturesofawirelessschedulingalgorithmtodeemitapplicableinpractice.Inthisdissertation,ourmainobjectiveistounderstandlimitationsonachievingthedesiredfeaturesofawirelessschedulingalgorithm.Towardsthisend,werstshowthatitisthe-oreticallyhardtoachievecomputationallyefcientalgorithmswithlowdelay.Nextweanalyzemomentsofthequeuesizedistributionatstationarityunderapproximatemaxweightscheduling(WMS)togainfurtherinsightonthedelaycharacteristicsofawiderangeofwirelessschedulingalgorithms. 11

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1.2ProposedModelsandRelatedChallenges 1.2.1WirelessInterferenceModelsWirelessinterferenceisprimarilymodeledusingtwodifferentapproaches.Physicalinterferencemodelcapturespowercontrol,andfeaturesadjustabletransmissionratesthataffectsinterference.Inthismodeltwointerferinglinkscantransmitsimultaneouslygiventheyhavereducedthetransmissionratessufciently.Theinterferenceinthismodelactslikeacontinuousentitythatisdeterminedbythetransmissionrateswhichareingeneralafunctionofthepowerallocation.Protocolmodelontheotherhandassumesapairwiseinterferenceinteractionbetweenlinksanddoesnotallowsimultaneousactivationofinterferinglinks.Inotherwords,protocolmodelsimpliesthecomplexinteractionsdrivenbypowerallocationviasimplyassumingmaximumpowerisusedineachtransmission.Theprotocolmodelhasdifferentvariantswhichcapturegeneralizationsofsimplelimita-tions.Onewaytogeneralizethemodelistoconsiderthepathbetweentwowirelessdevicesthathastheminimumnumberoflinksastheconstrainingfactor.IntheK-hopinterferencemodelthelinksinthenetworkinterfereifandonlyiftheshortestpathbetweentwolinksislessthanorequaltoKinthenetworkgraph.Asanexamplethe2-hopinterferencemodelsuccessfullycapturesthelimitationsimposedbyRTS/CTSmechanismofIEEE802.11,oneofthemostpopularWLANcommunicationstandardstoday.Intheprotocolmodel[ 16 ],interferencebetweenlinkscouldbecapturedbyasymmetric0-1matrix.ForalinksetE,the(i;j)thentryofthejEjjEjinterferencematrixis1iflinksiandjcannottransmitsimultaneouslyandbothsucceed,and0otherwise.Equivalently,theinterferencerelationshipcanalsoberepresentedbyaninterferencegraph(orconictgraph),inwhichanoderepresentsaphysicallinkandanedgerepresentstheexistenceofinterferencebetweentwophysicallinks.Figure 1-1 illustratesawirelessnetworkgraphwithunit-discgraphtopology,itscorrespondinginterferencegraphforthe2-hopinterferencemodelandtheassociatedsymmetric0-1matrix,insequence. 12

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Itisimportanttonotethatindependentsetsofinterferencegraphrepresentfeasiblesetsoflinksthatcanbeactivatedsimultaneouslybywirelessschedulers.Howeverfor1-hopinterferencemodel,usinganinterferencegraphisnotdesiredsincethematchingsintheoriginalnetworkgraphmaptoindependentsetsintheinterferencegraphand,forthemostpart,workingwithmatchingsiseasierthanworkingwithindependentsets.HoweverwhenK>1usinganunderlyinginterferencegraphisjustiedin[ 51 ]byshowingsimilarityinthestructuresofK-hopmatchingsofthenetworkgraphandindependentsetsoftheinterferencegraph.Moreoverwhentheinterferencegraphisapplicable,theeldofgraphtheoryoffersmanyresultsthatarereadilyapplicable.Asanexample;hardnessofschedulingisinvestigatedfor2-hopinterferencemodelin[ 2 ]usinginterferencegraphs.Laterin[ 51 ]thecomplexityresultsaregeneralizedforK2. 1.2.2SystemModelTheinterferencemodelisingeneralappliedtoanunderlyingsystemmodelthatcapturesnetworktrafcwhichconsistsofarrivals,storageandserviceofnetworkpackets.Overallthewirelessnetworkismodeledasatime-slottedqueuedsystem.Weconsiderasetofqueuesoperatingindiscrete-time.Ateachtimeslot,anumberofpacketsfromdifferentqueuescanbeserved,accordingtoascheduleconstrainedtoliewithinaprespeciedsetwhichcapturesinterferenceandschedulingconstraintssomeofwhichareexplainedinthepreviousdiscussion.Whenaqueueisactivatedforserviceapacketfromthequeue,ifthereisone,leavesthenetwork.Newpacketsmayalsoarrivetoeachofthosequeues,accordingtoanexternalarrivalprocesswith,generallyunknowntothescheduler,niterate.Thenetworkdeparturesareassumedtobe0-1,meaningapacketisservedonlyifitsqueueisactivatedbytheschedulerandthereisatleastonepackettoserveinthatqueue.Itisalsosafetoassumearrivalsarenormalizedaccordingtotheservicerates.Itisimportanttonotethatthemodeldescribedaboveisnotonlyapplicabletowirelessnetworks.Indeedthisgeneralmodelhasbeenusedtodescribevarioussettings,includinginput-queuedswitches,akeyfunctionalcomponentofInternetrouters,wheresimultaneouspackettransferfromingressportstoegressportsareconstrainedbyunderlyinghardware.Accordingly 13

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ANetworkgraph BInterferencegraph 2666640111010110110111110100110377775CMatrixrepresentationFigure1-1. Networkgraph,interferencegraphandthematrixrepresentationoftheinterferencerelation.A)Networkgraphwithunit-disktopology.B)Theinterferencegraphpertainingto2-hopinterferencemodel.C)Matrixrepresentationofthewirelessinterference. 14

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theresearchoninput-queuedswitches,whichincorporatewirednetworksetup,isthepredecessorofthewirelessschedulingresearch. 1.3ResearchOverviewIntheirseminalwork[ 52 ],Tassiulasetal.denedthestabilityofthesystemasthepositiverecurrenceofunderlying(irreducible)Markovchain.ThestatespaceofthatMarkovchainisformedbythepossiblequeuesizesandiscountableassumingthediscretearrivalanddepartureprocesses.Therearealsostrongerdenitionsofstabilitysuchastheexpectedvalueofqueuesizesumtobeniteinlimit.Initsintuitiveform,stabilityreferstoqueuesizesbeingboundedonaverage.Anoptimalalgorithmwithagivenstabilitynotionsetstheschedulingdecisionsateachtime-slot(inatime-slottedsystem)suchthat,looselyspeaking,allaveragearrivalratevectorsthatcouldbestabilizedbysomealgorithmarestableundertheoptimalone. 1.3.1OptimalAlgorithmsIn[ 52 ],Tassiulasetal.proposedathroughputoptimalschedulingpolicythat,whenappliedtoprotocolmodel,activatesaweightedmaximumindependentset(WMIS)ofthequeue-weightedinterferencegraphateachtime-slot.HowevertheiralgorithmisnotapplicabletolargenetworkssinceWMISproblemisNP-hardingeneral.Laterin[ 53 ],alineartimeprobabilisticalgorithmwasprovedtoachievethroughputoptimality.Althoughfromthecomplexityperspective,thetwoalgorithmsdiffersignicantlyin[ 27 ]itisshownthatthelattertriestotracktheformerusingarandomizedsearchscheme.ThefollowingresearchrevealedaperiodiccalculationofWMISwouldbesufcienttoachievestabilitywithoutaddressingtheaveragequeuesizebehaviorunderthatsetting[ 47 ].Thealgorithmin[ 52 ]iswidelyreferredasWMISschedulingpolicyinliterature.AnothercommonnameforthesamealgorithmisMaximumWeightScheduling(MWS)policy.Bothtermsareusedinterchangeablyinthiswork.ItisnotnecessaryforanoptimalalgorithmtosolveWMISproblem.Therearerandomizedoptimalalgorithmswhich,giventheaveragearrivalrates,computeactivationprobabilitiesforapolynomialnumberofindependentsetssuchthatthearrivalratesarecoveredbytheactivations.Inasense,thiscorrespondstosolvingtheoptimizationversionofrate-weightedfractional 15

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chromaticnumberproblemderivedusingtheinterferencegraph.HoweverthisapproachofsolvingtheWMISproblemrequirescentralcontrolmechanismsoverthecommunicationnetworkaswellastheknowledgeofaveragearrivalrates.Thelatestresearchstrivestocreatedistributedschemestoachievethroughputoptimality.In[ 21 ]authorsproposedafullydistributedadaptivecarriersensemultipleaccess(CSMA)algorithmtoachievingthatobjective.Thealgorithmachievesthroughputoptimalitywhilekeepingthemessagecomplexitysufcientlylow.Howeverthequestionwhetherthealgorithmproposedin[ 21 ]canreplacetraditionalMAClayerprotocolslikeIEEE802.11dependsonvariouscharacteristicsofthemethodsuchas:transientandstationaryexpectedqueuesizes,momentsofqueuesizedistributionatstationarity,andthecomplexityofimplementingthescheme.Tothatend,thisdissertationconcentratesonalloftheconcernslistedabove. 1.3.2SuboptimalAlgorithmsThehardnessofWMISproblemledresearcherstodevisesuboptimalalgorithmsthatperformwellinpractice.Typicallythosealgorithmsareveryefcientintermsofcomputationalresourcesandtheyareeasytoimplementinpracticalsettings.Variousattemptstocomeupwithsimpleyetefcientalgorithmsconcentrateonmakingsimplelocaldecisionswithoutbeingtoofarawayfromtheoptimalchoice.Thealgorithmslistedin[ 7 ],[ 56 ],[ 25 ],and[ 23 ]areexamplesofthoseattempts.Thosealgorithmstypicallystabilizeascaledthroughputregion.AmongtheclassofsuboptimalalgorithmstheLongestQueueFirst(LQF),whichalsoknownas,GreedyMaximalScheduling(GMS)[ 23 ],isdistinguishedduetoitssimplicityandaptnessfordistributedimplementation.PerformanceofLQFisstudiedin[ 8 ]and[ 24 ]meticulously.In[ 8 ]theauthorsintroducetheconceptoflocal-poolinggraphs,aclassofnetworkinterferencegraphsunderwhichthroughputoptimalityofLQFalgorithmisguaranteed.In[ 24 ]thenotionoflocalpoolingisgeneralizedforarbitrarygraphsviatheconceptof-localpooling.Weprovidedmiscellaneousresultsontopologicalfeaturesof-localpoolingfactorinChapter 3 .Thereisabodyresearchonthroughputcharacterizationofthesuboptimalalgorithms,how-evertothebestofourknowledgetwoimportantaspectsofwirelessschedulingalgorithms:the 16

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delaycharacteristicsandhighermomentsofthequeuelengthdistributionatstationarityarenotwellstudied.WederiveboundsforhighermomentsofqueuelengthdistributionatstationarityunderapproximateMWSpolicyinChapter 3 .AdditionallywehighlighteffectivenessofLQF,asasuboptimalalgorithm,whenqueue-summomentsatstationarityareconcerned.Combinedwithitssimplicity,ourresultdeemsLQFtobeanidealsuboptimalalgorithminpracticalsettings. 1.4ContributionsAsaresultofourmotivation,westudydelaycharacteristicsofschedulingalgorithmsinwirelesscommunicationnetworks.Thefollowingsectionssummarizeourcontributionsinthisdissertation. 1.4.1EstablishingHardnessofAchievingLowDelayThewiderangeofcomputationalresourcerequirementsofthroughputoptimalalgorithmshasbeenoneofthepuzzlingaspectsofthewirelessschedulingresearchinlasttwodecades.TherstthroughputoptimalalgorithmhadanNP-Completesub-problem,whilesubsequentdevelopmentsrevealedalineartimerandomizedalgorithmprovidingmaximumthroughput.Weanalyzedthetraditionalqueuebased,time-slottedsystemutilizingthecomputationalcomplexitytheory.Ourapproachisparalleltothemethodologyprovidedin[ 1 ]sothatanetworkowproblemisattackedbysimulatingaqueueingnetworkandconductingbinarysearchusingthesimulationresult.Tothebestofourknowledgethisworkistherstattempttogetherwithsimultaneousindependentworkprovidedin[ 49 ]torevealanticipatedtrade-offbetweenper-time-slotcomplexityandqueuesizes.Ourwork'simmediateconsequenceisapointerforfutureresearchersintheeldofwirelessscheduling:athroughputoptimalalgorithmcouldonlybeeffectiveeitherintermsofcomputationalresourcesorintermsthetransientexpectedqueuesizes.AquesttosearchforefcientalgorithmsisidenticaltothequesttondapolynomialtimealgorithmforanNP-Completeproblem.Accordinglyonecanprefertodesignalgorithmstotargetspecialtopologieswherethenegativeresultnolongerapplies.Additionallyitmightbepossibletoabandonthroughputoptimalitytoworkaroundthislimitation.HoweverthatshouldbedonecarefullysinceadeeperanalysisofmaintheoreminChapter 2 revealsanalgorithm 17

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whichisefcientonlyfortheuniformarrivalvectorsstillfacesthelimitationsthatarementionedabove. 1.4.2BoundsonSumofHigherOrderQueueDistributionMomentsatStationarityInChapter 3 weprovideaframeworktostudyhighermomentsofthequeuelengthdistri-butionatstationarityofthesystemwhenanapproximateWMISschedulerisutilized.DelayboundsofWMISschedulingisstudiedin[ 26 ].Ourresults,buildingonthesameframeworkofquadraticpotentialfunction,improvetheboundpresentedthereinandextendtheresultsforapproximatealgorithmsincludingtheoriginalwhichhasanapproximationratioof1.Weprovidetheboundsontheimprovementfactorusinggraphtheoreticalresults.Werelatetheproductofweightedfractionalchromaticnumberandweightedindependencenumbertotheinnerproductoftheweightvectorsthroughaninequality.ThatinequalityplaysakeyroleinderivationofthedelayboundsforapproximateWMISscheduling.Usingthatrelationandhigherorderpotentialfunctionssuchascubicfunction,weendupachievingboundsforsumofhighermoments.Thosebounds,inturn,canbeusedtoboundsumofqueuevariances.Webelieveabetterunderstand-ingofthequeuevarianceatstationaritycanbeintegraltodesigninghigherefciencywirelessnetworkhardware. 1.4.3SingleParameterCharacterizationofDelayBoundsDelayboundsofgeneralizedMWSschedulingisstudiedin[ 15 ].Buildingontheirframe-workweintroduceanewboundforMWSpolicyandprovethattheweightselectionin[ 15 ]isoptimalforminimizingdelay,whenthequadraticpotentialfunctionisbeingutilizedintheanalysis.Wealsoproveournewboundistightthroughthesamemethodology.ThoseresultstogetherestablishaframeworktocharacterizeandassesstightnessonthedelayboundofMWSalgorithm:asingleparameteronefornativeMWSpolicyandamulti-parameteroneforthegeneralizedversionconsideredin[ 15 ]. 1.4.4GeneralizedMWSPolicyVariationsofMWSalgorithmcanprovidethroughputoptimalitywhilealsooptimizingparametersofchoice.Resultsgivenin[ 15 ]motivateustofurtherinvestigatealessconstrained 18

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generalizationofMWSpolicythroughwhichalargeclassoffunctionscanbeusedtocomputelinkweightsinMWSalgorithm.Werenethesufcientconditionsforstabilitythatwasgivenin[ 27 ].Asaresult,newrequirementsareaddedtotheprevioussufcientconditions.Weconsiderasetofexamplesandillustrateaspectrumoffunctionsmovingfrominadequacyforestablishingstabilitytotheabilitytoprovidestability. 19

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CHAPTER2COMPLEXITYOFACHIEVINGLOWDELAY 2.1BackgroundGivenaninterferencemodel,thischapterinvestigateshowtoschedulethetransmissionsofthelinkssothatsomechosenobjectivecanbeachieved.Forsimplicity,weconsideratime-slottedsystemwheretimeisdividedintoxed-sizedslots,thepacketsareallidenticalinsize,andeachlinkhasthecapacityoftransmittingonepacketinatimeslotifaccessisgranted.Undertheprotocolmodel,afeasiblescheduleisasubsetofthelinkssuchthatnotwolinksinthesetinterferewitheachother.Suchasetoflinkscanbeactivatedforsimultaneoustransmissiononthesametimeslot.Notethatascheduleisfeasibleifandonlyifitcorrespondstoanindependentsetoftheinterferencegraph.Acanonicalschedulingproblemisasfollows.Supposethearrivalratesoftrafctothelinksfallintothecapacityregion[ 41 ],which,looselyspeaking,isthesetofarrivalratesthatareschedulablebysomealgorithm.Findaschedulingpolicy,whichcanproduceafeasiblescheduleineachtimeslot,sothatthenetworkqueuesremainstable.Variousdenitionsofstabilityhavebeen(ormaybe)developed,suchasthequeuesbeingboundedortheMarkovianqueueprocessesbeingpositiverecurrent.Themotivationofthischapteristounderstandtheinherentcomplexityofthescheduling-for-stabilityproblem.Manyauthorshavestudiedvariousversionsofthisschedulingproblem.Theproblemappearstobeverydifcultingeneral.TassiulasandEphremidesshowedthatifaschedulecorrespondingtoaWMISisusedineachtimeslot,thenthenetworkofqueuescanbestabilizedforanyarrivalratevectorinsidethecapacityregion(inwhichcase,wesaythecapacityregionisachieved),foraclassofMarkovprocesses[ 52 ].FindinganWMISisaverydifcultprobleminitself(e.g.,stronglyNP-hardandcannotbeapproximatedbyaconstantratio).Thishasledtothebeliefthattheoverallschedulingproblemisdifcult.However,wehavefoundveryfewstudiesthataddresstheinherentdifcultyoftheoverallschedulingproblem(asopposedtotheone-stepsub-problemoneachtimeslot).Later,Tassiulasshowedthatthecapacityregion 20

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canbeachievedbyasimplerprobabilisticalgorithmrunningineachtimeslot[ 53 ].However,hedidcommentinformallythattheprobabilisticalgorithmtriestosolvetheWMISsub-problembutovermultipletimeslots.Thecommentwaslaterformalizedandgeneralizedin[ 57 ].Atthispoint,onemighthypothesizethattheindependentsetsub-problemisunavoidableandanyschedulingpolicythatemploysasimpleone-step(per-time-slot)algorithmmustpayfortheinherentcostofschedulinginotherways.Mostlikely,thecostismerelyamortizedovertime,andthismayleadtopoortransientperformanceofthepolicy,suchaslargequeuesorslownessinreachingsystemsteadystate.Recentdevelopmenthasmadeitevenmorerelevanttostudytheinherentcomplexityofscheduling.Muchofthelatestworkconcentratesondevisingevensimpleranddecentralizedone-stepalgorithmsthatstabilizethefullcapacityregion[ 46 ][ 9 ][ 42 ][ 6 ][ 20 ][ 37 ][ 45 ].However,ifthecomplexityoftheschedulingproblemisirreducible,thenonemaywonderwherethehiddencostisinthesealgorithms.Arelatedstreamofwirelessschedulingresearchconcentratesondevisingsimpleone-stepalgorithmsthatareguaranteedtoachievesomesubsetsofthecapacityregion,foranarbitrarynetwork[ 35 ][ 34 ][ 25 ][ 56 ][ 24 ][ 22 ].Again,acomplexityquestionexistshere:Istheschedulingprobleminherentlyeasyoneachofthesesubsets?In[ 8 ][ 58 ][ 5 ],theauthorsshowedthat,iftheinterferencegraphsatisesaconditioncalledlocalpooling,thentheLongestQueueFirst(LQF)policy(whichissimple)achievesthefullcapacityregion.But,itisunknownwhethertheschedulingproblemforthisspecialclassofnetworksiseasyornot.In[ 24 ],theauthorsdenedthe-localpoolingfactorforanarbitrarynetwork,where0<1,andshowedthatLQFachieves,whereisthecapacityregion.Severalstudieshaveimplicitlyaddressedsimilarconcernsofthischapter.Inparticular,theauthorsof[ 57 ]examinedthetradeoffsamongthecomplexityoftheone-stepalgorithm,theachievablerateregionandthedelay.There,theone-stepalgorithmwasassumedtobetheWMISscheduleoritsapproximations.Thedelayandcomplexityrelationshipwasalsobrieysurveyed.Otherworksthatexaminedthedelayissueofschedulinginclude[ 44 ],[ 43 ],[ 48 ].Theyinfactexaminedthequeuesizesinsteadoftheactualdelays.Theupperboundofthequeue 21

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sizesdeterminesthedelayproperty.Somelineartrade-offswereobservedbetweendelayandcomplexity.Thefollowingisasummaryofourmainresultsandcontributions.Westartwithformulatingthecapacityregionusingthefractionalcoloringproblem.Althoughin[ 17 ]asimilarstudyisconducted,weconcentrateonahardersettingwherea(maximal)feasiblescheduleateachstepisamaximalindependentsetoftheinterferencegraph.Wenextconsiderthescheduling-for-stabilityproblemunderdeterministicarrivalsandstudyitscomplexity.Weincorporatethequeuesizeboundintothecomplexityresults.Weshowthat,ingeneral,itisimpossibletohavesmallqueuesizesusingsimpleone-stepalgorithmsunlessZPP=NP.Tothebestofourknowledge,thisisoneofthefewformalconnectionstothetheoryofcomputationalcomplexityforthestabilityproblems.Wealsousetheapproachin[ 52 ]todemonstratethatanalgorithmthatusesanWMISsolverasaone-stepsubroutineisstorage-wiseefcient.Wethenexaminethegeometryofthecapacityregion.Finally,werelatethefractionalcoloringproblemandthenewinterpretationofthecapacityregiontothenotionofset-localpooling.Webelievethattheresultsofthischapterareusefulforcontinuedinvestigationontheinherentcomplexityofwirelessscheduling.Thischapterisstructuredasfollows.InSection 2.2 ,weprovidethenotationanddenitionsofrelatedproblems.InSection 2.3 ,wedenethecapacityregionintermsofthefractionalcoloringproblem,andhence,connectschedulingtocoloring.InSection 2.4 ,weanalyzethecomplexityofefcientlystabilizinganetworkunderdeterministicarrivals.WeshowthattheWMIS-basedalgorithmisstorage-wiseefcientinSection 2.5 .InSection 2.6 ,weinvestigatedifferentinterestingsectionsofthecapacityregion. 2.2PreliminariesWeconsiderawirelessnetworkwithone-hoptrafc.Thatis,anydataistransmittedonlyonce,andafterthat,itleavesthenetwork.WemodelthewirelesslinksetL'sinterferencerelationswithaninterferencegraph,G=(V;E),whereeachlinkl2Lintheoriginalnetworkcorrespondstoanodev2V.Twonodesv1;v22VareconnectedinG,ifandonlyifthecorrespondinglinksintheoriginalnetworkinterferewitheachother.Weassumeasymmetric 22

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interferencerelationthusGisundirected.Afeasiblescheduleisdenedtobeasetofnon-interferingnodesinG.Amaximalscheduleisafeasibleschedulethatcannotincludeanymorenodeswithoutviolatingthefeasibilityconstraint.AfeasiblescheduleinanetworkcorrespondstoanindependentsetinG.ThesetofallfeasibleschedulesisdenotedbyIL;thissetcanalsobeinterpretedasallindependentsetsofthegraphG.WedenotethesetofallmaximalschedulesbyML,andusem2MLtodenoteasinglemaximalschedule.Throughoutthischapter,weusethewordscheduletorefertoafeasiblemaximalschedule.Whenapplicable,MLisregardedasajVjjMLj0-1matrix.Eachcolumnofthematrixisa0-1vectorrepresentationofamaximalindependentsetofG,with1indicatingthatthecorrespondingnode(inG)isselected(andthelinkisactiveintheschedule)and0otherwise.Acolumnvectorthathas1'sinallentriesisdenotedby~e.Weadopttheconventionaldenitionofthecapacityregion:=fj0forsome2Co(ML)g:Theinteriorofthecapacityregionisdenedas:o=fj0
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westepbackalittleanddeneamorebasicversionoftheschedulingproblem.Itisourbeliefthatthisbasicversioncontainsessentialelementsaboutthecomplexityofvariousstabilityproblems.Wenowdeneastaticversionoftheschedulingproblem.Givenanarrivalratevector,theschedulingproblemistondaconvexcombinationoftheschedulessuchthattheresultingserviceratedominatesthearrivalrate.Formally,givena(rational)arrivalratevector,ndanon-negativetimesharevectorsuchthatMLandPjMLji=1i1(fasterthanunittime).Ourconjectureisthatthecomplexityofvariousscheduling-for-stabilityproblemsisdeterminedbytheshapeoftheconvexpolytopeoftheabovestaticfeasibilityproblem.Forinstance,ifisstrictlyinsidethecapacityregionandifwecannda(servicerate)vectorthatstrictlydominatesandsatisestheabovefeasibilityproblem,thenthearrivaltrafccanbestabilizedfornearlyalldenitionsofstabilityencounteredintheliterature.Hence,thefocusofthechapterisaboutthisstaticproblemanditsvariants.Weusetodenoteasolutiontotheaboveschedulingproblem.TheproblemandsolutionsbothdependonthegraphG.Here,wesuppressthedependenceonGinthenotation. 2.2.2FractionalColoringProblemThechromaticnumberofagraphG,(G),istheminimumnumberofcolorstopainttheverticessothattheconnectednodesdonotsharethesamecolor.Duetoitsrelationtothevertexcoveringproblems,thechromaticnumberproblemhasbeenextensivelystudiedinthegraphtheoryeld.ThechromaticnumberproblemisNP-complete.Itcanberepresentedwithanintegerprogramming(IP)formulation.Alinearprogramming(LP)relaxationofthisIP,whichisknownasthefractionalcoloringproblem,isalsocentraltothefractionalpackingandcoveringproblemsingraphs.Here,weprovideaformulationfortheweightedfractionalcoloringproblem. 24

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Denition1. Letw2QjVjbea(component-wise)non-negativeweightvectorandconsiderthefollowingoptimizationproblem:f(G;w)=minimizeXm2MLmsubjecttoMLw0:Theoptimalvaluef(G;w)iscalledtheweightedfractionalchromaticnumber.Weletw2QjVjdenoteasolutiontothefractionalcoloringproblem.Theweightedfractionalchromaticnumberofagiveninterferencegraphinthewirelessschedulingcontextcanbeinterpretedasthefastestwayofservingqueueddata(withoutaddi-tionalarrivals)intheamountgivenbywifwecanhopbetweenschedulesininnitesimaltimeslots.In[ 17 ],asimilarproblemisstudiedforthe1-hopinterferencemodel,wheretheschedulesaremaximalmatchings.Theauthorsshowedthat,underthatsetting,theschedulingproblemadmitspolynomialalgorithms.In[ 17 ],threeinterpretationsoftheoptimizationproblemaredis-cussed.Althoughthisworkconcentratesontheindependentsetpolytope,therstinterpretationin[ 17 ]andourperspectivearecloselyrelated. 2.2.3MinimumSpanPacketSchedulingGivenaninterferencegraphGandinitialqueuesizesatallthelinks,denotedbythevectorw,wedenetheminimumspanpacketscheduling(MSPS)problemtobethesmallestnumberoftimeslotstodrainthepacketsfromthequeuesundertheinterferenceconstraintsandunittransmissioncapacityateachlink.ThefollowingisanIPformulationofthisproblem: 25

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\(G;w)=minimizeXm2MLcmsubjecttoMLcwcm2Z+;whereZ+denotesnon-negativeintegers.Now,considerreningthetimeslotswhilekeepingtheinitialamountofqueueddataxed.Whenreducingthesizeofthetimeslots,wealsomaketheunitpacketlengthsmaller.IfwechoosethetimeslottobeRtimessmaller,thentheinitialqueuesizeswillbecomeRtimeslargerintermsofpacketcount.Theoptimizationproblemcanbewrittenasfollows.\(G;w;R)=minimizeXm2MLcmsubjecttoMLcRwcm2Z+:Thefollowinglemmaindicatesifwemakethetimeslotsizeinnitesimallysmallandscaletheoptimalvaluedownbythesamescalingfactor,thesolutionoftheaboveproblemwillconvergetothesolutionofthefractionalcoloringproblem. Lemma1. limR!1\(G;w;R) R=f(G;w). Proof. Letwbeasolutioncorrespondingtof(G;w).Sincew2QjVjandisnon-negative,wecanconcludew2QjMLjandisnon-negative.Wedenotethereducedformofeachrationalnumberwibyai biforsomeai;bi2Z+;bi6=0,whereaiandbiarerelativelyprime.Lettheleastcommonmultipleofthebi'sbeB.Similarly,denotethereducedformofeachwibyci diforsomeci;di2Z+;di6=0,wherecianddiarerelativelyprime.Denotetheleastcommonmultipleofthedi'sbyD.Foranyj2Z+andj1,theoptimizationproblemforderivingf(G;jBDw) 26

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hasanintegeroptimalsolution,andhence,f(G;jBDw)=\(G;w;jBD).NowsupposekBDR<(k+1)BDforsomek2Z+.Notethat \(G;w;kBD)\(G;w;R)\(G;w;(k+1)BD):Hence, f(G;kBDw)\(G;w;R)f(G;(k+1)BDw):DividingeachtermabovebyRandusingthelinearityoff(G;w)inw,wehave f(G;kBDw R)\(G;w;R) Rf(G;(k+1)BDw R):Usingthecontinuityoff(G;w)inw,asR!1,bothf(G;kBDw R)andf(G;(k+1)BDw R)approachf(G;w). Asimilarresultforthenon-weightedcasewasderivedin[ 28 ]wheretheauthorsstudiedhowtominimizethespectrumusageinFDMAnetworks. 2.3RelevanceofFractionalColoringtoSchedulingUndertheprotocolmodel,theschedulingproblemassumesastrongcombinatorialstructure.Inthissection,wewillbridgeschedulingwiththefractionalcoloringproblemtohighlightitscombinatorialnature.Theresultswillbeusefulforthefollowingsections.Theorem 2.1 statesthatthecapacityregioncanbedenedintermsoftheweightedfractionalchromaticnumber,wheretheweightvectoristhearrivalratesonthelinks. Theorem2.1. =fj0f(G;)1g: Proof. Takeanyarrivalvector2.Bydenition,thereexists2Co(ML)suchthat.Since2Co(ML),thereexistssuchthatML=,where0andPjMLji=1i=1.Hence,isafeasiblesolutiontotheoptimizationproblemforf(G;)anditscomponentssumto1.Therefore,f(G;)1.Forthereversedirection,takeanarrivalvectorsuchthat0f(G;)1.Assumeisasolutioncorrespondingtof(G;).Thatis,MLand 27

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B=PjMLji=1i1.IfB=0,then=0,whichisin.IfB6=0,thenML BandML B2Co(ML).Thisindicates2. BytheproofforTheorem 2.1 ,weconcludethatasolutiontothefractionalcoloringproblemcanbethoughtasatime-sharingvectorofschedulesfortheschedulingproblem.Similarly,wecandenetheboundaryofthecapacityregionintermsoftheweightedfractionalchromaticnumber.But,weneedatechnicalfactrst.Fact1ForanyJL,thereexists2Co(ML)suchthatj>0forallj2J.Theproofissimpleandisomitted.Next, Theorem2.2. 2()]TJ /F7 11.955 Tf 11.96 0 Td[(o)()f(G;)=1. Proof. Werstshowif2()]TJ /F7 11.955 Tf 12.14 0 Td[(o),thenf(G;)=1.SincebyTheorem 2.1 ,f(G;)1forall2,weonlyneedtoshowwhen0f(G;)<1,2o.Iff(G;)=0,then=0andhence2o.Considerthecase0jjforj=2J.Now,takeany^2Co(ML)withtheproperty^j>0forj=2J.ThisispossiblebyFact1.ConstructanewvectorinCo(ML):~=(1)]TJ /F3 11.955 Tf 12.29 0 Td[()+^,where0<<1.Whenissufcientlysmall,wecanhave~>0and~>.Hence,2o.Next,supposef(G;)=1.ByTheorem 2.1 ,2.Supposeisino.Then,thereexists2Co(ML)suchthat>.Thisinturnimpliesthatthereexists>0suchthat>(1+),or1 1+>.SupposesatisesML=,where0andPjMLji=1i=1.Then,1 1+isafeasiblesolutiontotheoptimizationproblemforndingf(G;),butitscomponentssumtolessthanone.Thiscontradictstheassumptionf(G;)=1.Therefore=2o. Corollary1. o=fj0f(G;)<1g. 28

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Aninterpretationofthetheoremsandcorollaryaboveisasfollows.Thecapacityregionconsistsofvectorsthatcanbeservicedinunittimeorlessbysomemixingofschedules.Theinteriorcontainsvectorsthatareschedulableinstrictlylessthanunittime.Theboundaryofthecapacityregion,)]TJ /F7 11.955 Tf 12.64 0 Td[(o,consistsofvectorsthatleavethecapacitypolytopewhenmultipliedbyascalaroftheform1+forany>0.Thefollowingformalizesthisinterpretationoftheboundary. Lemma2. 2()]TJ /F7 11.955 Tf 11.95 0 Td[(o)()2and(1+)=2forall>0. Proof. ByTheorem 2.2 ,2()]TJ /F7 11.955 Tf 12.26 0 Td[(o)impliesf(G;)=1.Suppose(1+)2forsome>0.Bythelinearityoff(G;w)inw,f(G;(1+))=(1+)f(G;).ByTheorem 2.1 ,0(1+)f(G;)1,whichimpliesf(G;)1 1+,contradictingtheassumption.Conversely,suppose2andthereisno>0suchthat(1+)2.Suppose2o.Then,thereexists2Co(ML)suchthat>.Hence,thereexists>0suchthat>(1+).Then,(1+)2,whichisacontradiction. Avectorinsidethecapacityregionbecomesincreasinglydifculttoserviceasitapproachestheboundary.Typicalsuboptimalschedulingalgorithmsintheliteratureachieveapartofthecapacityregionderivedbyscalingthecapacityregionuniformlybysomeless-than-1constant(e.g.,[ 24 35 56 ]).Motivatedbythisfact,weassociateanumberwitheveryarrivalratevector. Denition2. Foranynon-negativeratevector,let ,supf j 2g:(2)Notethat =1ifandonlyifisontheboundaryofthecapacityregion.If2,alargervalueof indicatesisdeeperinsidethecapacityregion.Wenextdemonstratetherelationshipbetweenthisnumberandtheweightedfractionalchromaticnumber. Lemma3. Foranon-negativevector,f(G;)=1 Proof. Bylinearity,wehavef(G; )= f(G;).ByLemma 2 isontheboundaryofthecapacityregion.ByTheorem 2.2 ,f(G; )=1.Hence, f(G;)=1. 29

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Thisresultassociatesthefractionalcoloringproblemwiththedifcultyofservicinganarrivalratevector.Notethatasolutiontothefractionalcoloringproblemisalsoasolutiontotheschedulingproblem(whichisafeasibilityproblem)butnotviceversa.Accordingly,weturntothequestionofhowscheduling(andhopefully,achievingstability)andtheweightedfractionalcoloringproblemarerelatedwhenanarrivalratevectorisclosetheboundary.Thefollowinglemmaindicatesthatasolutiontotheschedulingproblemapproachesasolutiontothefractionalcoloringproblemastheratevectorgetssufcientlyclosetotheboundary. Lemma4. Foranarrivalvector2,any(seethedenitionofthisnotationinSection 2.2.1 )isa approximationtotheoptimizationproblemforf(G;). Proof. ByLemma 3 ,f(G;)=1 .Nowtakeanysolutionoftheschedulingproblem,,bydenitionisafeasiblesolutiontotheproblemforndingf(G;)andPjMLji=1i1=f(G;) .Thisconcludestheproof. Oneimplicationisthatanalgorithmthatstabilizesanarrivalratevectoryieldsanapproxi-mationoftheweightedfractionalcoloringproblem2.Furthermore,theapproximationratiotendsto1asthearrivalratevectorgetsclosertotheboundary.Theresultestablishesapartialequiva-lencebetweenthetwoproblems.Theyarethesameforthevectorsnearorontheboundary,butarepotentiallyverydifferentforvectorsfurtherfromtheboundary.The(unweighted)fractionalcoloringnumberofagraphG,denotedbyf(G),isthesolutiontotheweightedversionwhentheweightvectorconsistsofallones,i.e.,f(G)=f(G;~e).Considertheproblemofachievingthemaximumidenticalrateonalllinks.Wedenote 2Strictlyspeaking,weonlyshowedthatthesolutiontothestaticschedulingproblemgivessuchanapproximation. 30

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thisrateby(G).ThefollowingistheLPformulationfornding(G).(G)=maximizezsubjecttoMLz~ez;0: Lemma5. (G)=1 f(G). Proof. ByLemma 2 andtheoptimality,(G)~eisontheboundaryof.ByTheorem 2.2 ,weconcludef(G;(G)~e)=1.Observef(G;(G)~e)=(G)f(G;~e)=1. 2.4ComplexityofSchedulingProblemInSection 2.3 ,weestablishedaconnectionbetweenthefractionalcoloringandtheschedul-ingproblems.Next,wewillusethederivedresultstorevealcomplexitycharacteristicsoftheschedulingproblemandstablealgorithms.Wewillfocusonadeterministicsystem,wherethearrivalspertimeslotareconstantforeverylink.ThearrivalratesareassumedtobeinQjLjandalllinkshaveunittransmissioncapacity.Astablealgorithminthissettingisdenedtobeanyalgorithmthat,giveninitiallyemptyqueues,keepsthequeuesizesboundedthroughouttheexecutionforallarrivalratesintheinteriorofthecapacityregion.Forastablealgorithm,theupperboundonthequeuesizesmightbeafunctionofthearrivalrateandtheencodinglengthoftheproblem.Later,wewillrelatetheupperboundtothedistancetotheboundaryforeacharrivalrate.Theexistenceofastablealgorithmimpliestheexistenceofavectorthatsolvesthewirelessschedulingproblem.However,givenasolutiontotheschedulingproblem,applyingitasanalgorithmateachtimeslotmightnotbestraightforward.Inadditiontothefrequenciesoftheschedules,whentoactivateeachschedulealsoplaysakeyrole.Additionally,althoughthetwoarerelated,thenotionofstabilityrequirestheconditionthatthesystemrunsforever.Todealwiththeseissuesandtocontinueourinvestigationofcomputationalcomplexity,wewill 31

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investigatestablealgorithmsthatnotonlyboundthequeuesizesbutalsoensurethattheboundispolynomialintheinput'sencodinglength.Beforewestart,wedeneaquantitycalledload-slackdenotedbyforeacharrivalratevector,whichisameasureofdistancetotheboundary.Itwillbeusedtocharacterizethequeuesizeupperboundofastablealgorithm. ,supfj+~e2g:(2) Denition3. Analgorithmstabilizesanarrivalratevectorcomputationallyefcientlyifthealgorithmisstableandateachiteration,theoperationofthealgorithmtakesapolynomialnumberofstepsinjLj. Denition4. AnalgorithmstabilizesanarrivalratevectorP-storage-wiseefcientlyifthealgorithmisstableandthequeuesizesareupperboundedbyP(jLj;())]TJ /F6 7.97 Tf 6.59 0 Td[(1),wherePisapolynomialfunction. Theorem2.3. ThereisnoalgorithmthatstabilizesallarrivalratevectorsinocomputationallyandP-storage-wiseefcientlyforanypolynomialP,unlessZPP=NP. Proof. Assumethereexistsanalgorithmthatstabilizesanydeterministicarrivalratevector2ocomputationallyandP-storage-wiseefciently.Then,allqueuesareupper-boundedbyP(jLj;())]TJ /F6 7.97 Tf 6.58 0 Td[(1)duringtheoperationofthenetwork.LetusdenotethisnumberwithPforshort.Also,eachtimeslottakesatmostP(jLj)computationsteps,forsomepolynomialfunctionP.Givenanapproximationfactor,wecanusethealgorithmasanoracletocreateafullypolynomialtimeapproximationscheme(FPTAS)forthefractionalcoloringproblemf(G).(Asimilartechniqueisusedfortheconcurrentowproblemin[ 1 ].)AssumewerunthealgorithmforRtimeslots.Then,thetotaltrafcinsertedintoqueueiisRiforeveryi.Ifallthequeuesareupper-boundedbyP,theamountofthetrafcservedatqueueiintheRtimeslotsisnolessthanRi)]TJ /F18 11.955 Tf 12.04 0 Td[(P.TheaverageservicerateatqueueiisatleastRi)]TJ /F19 7.97 Tf 6.59 0 Td[(P R.LettingR=P)]TJ /F6 7.97 Tf 6.59 0 Td[(1,weensurequeueiisservedatanaverageratei)]TJ /F3 11.955 Tf 11.5 0 Td[(~e.Therefore,wecanrunthestablealgorithm 32

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forP)]TJ /F6 7.97 Tf 6.59 0 Td[(1timeslotsandeitherachieveatimesharevectorsuchthatML)]TJ /F3 11.955 Tf 12.46 0 Td[(~eorconcludethatisnotinthecapacityregionthuscreatinganoracle.ByLemma 5 ,thearrivalratevector1 f(G)~eisontheboundaryofthecapacityregion.Bysettingtheprecisionto(6jLj))]TJ /F6 7.97 Tf 6.58 0 Td[(1,wecanconductbinarysearchandndanumberzsuchthatz1 f(G)z+(2jLj))]TJ /F6 7.97 Tf 6.58 0 Td[(1usingtheoracleforO(log()]TJ /F6 7.97 Tf 6.59 0 Td[(16jLj))times.Since1f(G)jLj,f(G)1=z(1+)f(G).Thus,thetotalrunningtimeofthealgorithmisP(jLj)O(log()]TJ /F6 7.97 Tf 6.58 0 Td[(16jLj))P(jLj;)]TJ /F6 7.97 Tf 6.59 0 Td[(16jLj))]TJ /F6 7.97 Tf 6.59 0 Td[(16jLj,whichispolynomialininputG.However,in[ 10 ],itisproventhatf(G)cannotbeapproximatedwithin(jLj1)]TJ /F5 7.97 Tf 6.58 0 Td[()forany>0,unlessZPP=NP.Therefore,devisingaFPTASherewouldequalizeNPandZPPaswell. ThefractionalcoloringproblemisknowntobeNP-complete.Grotscheletal.[ 13 ][ 14 ]establishedanequivalencebetweenthefractionalcoloringandWMISproblems.Formally,onecansolvefractionalcoloringbyusingasolverfortheWMISproblemapolynomialnumberoftimes.Additionally,anapproximatesolverfortheWMISproblemcanbeusedtoderiveanapproximationalgorithmforthefractionalcoloringproblem.TheapproximationratiooftheWMISsolverisdirectlycarriedtothelatterproblem'ssolution[ 18 ][ 19 ].Thisresult,however,shouldbementionedtogetherwiththefactthattheWMISproblemisNP-completeanddoesnotadmitapolynomialtimeapproximationalgorithmwithanapproximationratiojVjforsome>0[ 40 ]. 2.5QueueDynamicsunderDeterministicArrivalInthissection,wewilldemonstrateastorage-wiseefcientschedulingalgorithm:TheWMIS-basedalgorithmisP-storage-wiseefcientforP=3 2jLj())]TJ /F6 7.97 Tf 6.59 0 Td[(1+p jLj.Wedenotethequeuesizeoflinki2LattimetbyQi(t).ThearrivalanddepartureratesatlinkiandtimetareAi(t)andDi(t),respectively.NotethatAi(t)=iandDi(t)=min(1;Qi(t)+Ai(t))ui(t),whereui(t)isanindicatorfunction,equalto1iflinkiisscheduledfortransmissionattimetand0otherwise.Thesystempotentialisdenedtobethesumofallqueuesizesquares,(Q)=PjLj1Q2i.Wewillshowwhenthequeuesizesgrowbeyondathreshold,thesystempotentialdropsinthe 33

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nexttimeslot.Thisinturnprovesthatthesystempotentialisboundedandconsequentlythequeuesizesareboundedaswell.Thequeueevolutionisgivenby:Qi(t+1)=Qi(t)+Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t):Thepotentialdifferencebetweentimestandt+1isdenotedby(t),(Q(t+1)))]TJ /F7 11.955 Tf 11.8 0 Td[((Q(t)).Then,(t)=Xi2LQ2i(t+1))]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xi2LQ2i(t)=Xi2L[Qi(t+1)+Qi(t)][Qi(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(Qi(t)]=Xi2L[2Qi(t)+Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t)][Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t)]=Xi2L[2Qi(t)(Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t))]+Xi2L[Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t)]2:Thesecondtermaboveisbounded:Pi2L[Ai(t))]TJ /F3 11.955 Tf 12.37 0 Td[(Di(t)]2jLj.Wenextinvestigatetherstterm.Xi2L[2Qi(t)(Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t))]=2Xi2L[Qi(t)Ai(t)])]TJ /F7 11.955 Tf 11.96 0 Td[(2Xi2L[Qi(t)Di(t)]:ThedeparturevectorD(t)=(D1(t);D2(t);:::;DjLj)Tisdeterminedbythechosenscheduleattimet,m(t)=argmaxm2MLQ(t)Tm.Notethatthedepartureisdominatedbythebinaryrepresentationofthechosenschedule,D(t)m(t). Lemma6. 0Q(t)Tm(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Q(t)TD(t)jLj. Proof. SinceQ(t);m(t);D(t)0andD(t)m(t),Q(t)TD(t)Q(t)Tm(t).Asusual,wedenotethei-thcomponentofm(t)bymi(t).RecallDi(t)=min(1;Qi(t)+Ai(t))ui(t)= 34

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min(1;Qi(t)+Ai(t))mi(t).Hence,Qi(t)mi(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qi(t)Di(t)=Qi(t)mi(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Qi(t)min(1;Qi(t)+Ai(t))mi(t):WhenQi(t)=0orQi(t)1ormi(t)=0,wehaveQi(t)mi(t))]TJ /F3 11.955 Tf 12.29 0 Td[(Qi(t)Di(t)=0.Theonlyinterestingcaseiswhen00and2Co(ML)suchthatA(t)+~e.SinceQ(t)0,Q(t)TA(t)+Q(t)T~eQ(t)T.Moreover,Q(t)Tm(t)Q(t)Tforall2Co(ML).Therefore,Q(t)TA(t)+Q(t)T~eQ(t)Tm(t).Xi2L[2Qi(t)(Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t))]=2Xi2L[Qi(t)Ai(t)])]TJ /F7 11.955 Tf 11.95 0 Td[(2Xi2L[Qi(t)Di(t)]2[Q(t)Tm(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Q(t)T~e)]TJ /F7 11.955 Tf 11.95 0 Td[((Q(t)Tm(t))-222(jLj)])]TJ /F7 11.955 Tf 26.9 0 Td[(2Q(t)T~e+2jLj:Hence,thepotentialdifferencecanbeboundedas(t))]TJ /F7 11.955 Tf 23.91 0 Td[(2Q(t)T~e+3jLj:Fortimet,wedenotethetotalqueuesumasQ(t)=Pi2LQi(t).WhenQ(t)>3jLj 2,(t)<0.Also,if(Q(t))>(3jLj 2)2,thenQ(t)>3jLj 2.Foranytimet,themaximumpotential 35

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increaseisboundedby2Q(t)+jLj(whichiswheneachqueueisincreasedby1).Ifthepotentialismorethan(3jLj 2)2atthebeginningoftimet,itwilldropafteronetimeslot.Themaximumpotentialthatthesystemcaneverreachisboundedby(3jLj 2)2+23jLj 2+jLj(3jLj 2+p jLj)2.Aboundedpotentialtranslatesintoboundedqueuesizes.Themaximumqueuesizeduringtheoperationofthesystemisboundedby3jLj 2+p jLj.Inconclusion,byusingasolverfortheWMISproblem,wecanachieveastorage-wiseefcientalgorithm. 2.6ACloserLooktotheCapacityRegionunderProtocolModelInthissection,wewillcharacterizethedifferentsectionsofthecapacityregionandprovidesamplearrivalratevectorsthatarerepresentativeofeachsection.Ingeneral,thecapacityregionconsistsofarrivalratevectorsthatcanbeservedbysomealgorithm.Theboundarycanbeconsideredasthesetofarrivalratevectorsthatarethemostchallengingtoserve.Inearliersections,weprovidetwocharacterizationsoftheboundary.Therstonerelatestothefractionalchromaticnumberandthesecondonestatesthatthevectorsontheboundarycannotbeextendedfurtherandyetremaininthecapacityregion.Bytheseconddenition,ifavectorisontheboundary,noothervectorinthecapacityregioncandominateitinitsdirection.However,itispossiblethat,fortwovectorsontheboundary,onedominatestheother.Therefore,itisworthtosingleoutthepartoftheboundarywhereavectorisnotdominatedbyanyothervectorinthecapacityregion.Thispartcontainsthemostchallengingarrivalratevectorstoserve.Wecallthisregiontheupper-boundaryofthecapacityregion,anddenoteitbyU().Thedenitionoftheupper-boundaryexcludesthosevectorsonthetraditionalboundarythataredominatedbysomevectorsinthecapacityregion.Formally,U()=f2jTforallT2andT6=g:Thenotationmeansnot. 36

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~M1T=(1,0,1,0,1,0) ~M2T=(0,1,0,1,0,1) ~M3T=(1,0,0,1,0,0) ~M4T=(0,1,0,0,1,0) ~M5T=(0,0,1,0,0,1) Figure2-1. Interferencegraphforasix-cyclenetworkanditsschedulevectors. Toillustratethesignicanceofdifferentregions,wewilluseasmallnetworkandexamineafewinterestingarrivalratevectors.Considerthe6-cyclenetworkwhoseinterferencegraphisgiveninFigure 2-1 .Theschedulesofthisnetworkarelistedonitsleft.Consideranarrivalratevectorontheupper-boundary,(1 2;1 2;1 2;1 2;1 2;1 2)T.Thisvectorcanberepresentedasaconvexcombinationoftheschedules,1 2~M1+1 2~M2.Itisimpossibletodominatethisvectorbyanyotherconvexcombinationoftheschedules.Next,considertheratevector(1 2;1 2;1 2;1 2;1 2;0)T.Itisontheboundary.However,thepreviousvectordominatesthisone.Therefore,thisvectorsitsontheboundarybutnotontheupper-boundary.Nowmoveinsidethecapacityregionandconsidertheratevector(1 3;1 3;1 3;1 3;1 3;1 3)T.Wecanprovidetwodifferentrepresentationsofthisvector:1 3~M1+1 3~M2or1 2~M3+1 2~M4+1 2~M5.Notethattheformerrepresentationismoreefcientthanthelattersincethecoefcientshaveasmallersum.Thisvectorisintheconvexhullofthe(maximal)schedulesbutnotontheboundary.InFigure 2-2 ,weprovideasimpliedillustrationofacapacityregiontodemonstratetheconcepts.Thesolidthickerlineindicatestheupper-boundaryandthedashedlineisfortherestoftheboundary.1isontheboundarybutalsodominatedbysomevectorinthecapacityregion.Therefore,1isnotontheupper-boundary.3cannotbedominatedbyanyothervectorinthecapacityregionanditisontheupper-boundary.2isaconvexcombinationoftheschedules,butitcanstillbescaledupwithoutleavingthecapacityregion;hence,itisbothinCo(ML)andin 37

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Figure2-2. Capacityregionandarrivalratevectorsindifferentsections. theinteriorofthecapacityregion.4isalsointheinteriorofthecapacityregionbutitisnotinCo(ML).4hasrelativelylowarrivalrates;itmaybeeasytoschedulesuchanarrivalratevector. 2.7CapacityRegionandLocalPoolingTheshapeofthecapacityregioncanaffectthedifcultyoftheschedulingproblem.Sincealltheothervectorsinthecapacityregionaredominatedbytheupper-boundary,aninvestigationoftheupper-boundarycanbecrucial.Foranyarrivalratevector,wecandeneanorm-likefunctionwithrespecttothecapacityregion,whichwewilllooselycallschedulenorm. Denition5. GivenalinksetL,theschedulenormofanarrivalratevector,denotedjjjjs,isgivenbyjjjjs=inffkj92Co(ML);kg:Here,isinterpretedasaserviceratevector.Ifthelinksareendowedwithqueueddataproportionaltoandtherearenofurtherarrivals,thenjjjjsistheshortesttimetodrainalldataamongallpossibleschedules.Wenextshowthattheschedulenormisequaltotheweightedfractionalchromaticnumber. 38

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Lemma7. Foranyarrivalratevector,jjjjs=f(G;). Proof. Thecaseof=0istrivial.Weonlyconsiderthecasewhere6=0.SinceCo(ML)isaclosedset,itcanbeshowneasilythesetinDenition 5 ,fkj92Co(ML);kg,isclosed,andtheinmumisachieved.Hence,foranyarrivalratevector,thereisa2Co(ML)suchthatjjjjs.Since2Co(ML),=MLforsome0andPjMLji=1i=1.Leti=jjjjsiforeachi;isfeasiblefortheoptimizationproblemtondf(G;).Hence,jjjjsf(G;).Toshowtheotherinequality,welet=ML f(G;),whereisanoptimalsolutiontotheproblemofndingf(G;).Then2Co(ML)andf(G;).Therefore,jjjjsf(G;). Corollary2. Thecapacityregionisthesetcontainingallthevectorswhoseschedulenormsarelessthanorequalto1.Theinteriorofthecapacityregionisthesetcontainingallthevectorswhoseschedulenormsarelessthan1.In[ 34 ],thelengthofavectorrespecttoaconvexregionisdenedas:jjxjj=1 supfkjk0;kx2g: (2) Lemma8. ThedenitionoftheschedulenormisidenticaltothedenitioninEquation 2 ,ifwelet=,i.e:jjxjjs=jjxjj: Proof. Notethatjjxjj=1= xbyDenition 2 .ByapplyingLemma 3 andLemma 7 ,weobtainthedesiredequality. Set-localpoolingisstudiedin[ 32 ].Ithasmanyinterestingpropertiesandisrelatedto-localpoolingdenedin[ 24 ].Wewillshowtherelationshipbetweentheset-localpoolingfactorandtheschedulenorm.LetLbeanynon-emptylinkset.Forconvenience,letL=fjLL;forallL;L2Co(ML)g: 39

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ThecomplimentofLiscL=fjL>L;forsomeL;L2Co(ML)g: Denition6. Givenanon-emptysetoflinksL,wesayLhasaset-localpoolingfactorLifthefollowingholds.L:=supfj2Lg:=inffj2cLg:Ithasbeenshownthattheset-localpoolingfactorisequaltotheoptimalsolutionofthefollowingproblem.min;L;LsubjecttoLLL;L2Co(ML):LetKL.ForanyjKj-dimensionalvector,wecandeneanextendedjLj-dimensionalvectorLbysettingalltheextendedcomponentstobe0.Conversely,foranyjLj-dimensionalvector,wecancreateajKj-dimensionalvectorbyrestriction,andwedenoteitby[]K. Lemma9. Theset-localpoolingfactorsatises:K=minfjjLjjsj2Co(MK)g. Proof. Forany2Co(MK),0
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Thelemmashowsthattheset-localpoolingfactorisequaltotheminimumschedulenormofallvectorsinCo(MK).Theset-localpoolingfactorforsetKcontainsinformationabouttheshapeofCo(MK),asillustratedbyCorollary 3 Corollary3. WhenL=1,any2Co(ML)satisesjjjjs=1. Lemma10. ForagivensetLandanytwovector1;22Co(ML),wehaveL(jj1jjs+jj2jjs)jj1+2jjs. Proof. Supposethereexisttwovectors1;22Co(ML)andL(jj1jjs+jj2jjs)>jj1+2jjs.Then,2L>jj1+2jjsorL>jj1=21+1=22jjs.Since1=21+1=222Co(ML),thiscontradictsLemma 9 Lemma11. Theupper-boundarysatisesUCo(ML). Proof. BythedenitionofU,any2Usatises2.Then,forsome2Co(ML):Thus,=,whichimpliesthat2Co(ML)andthelemmaholds. Lemma12. Theupper-boundarysatisesU=Co(ML)ifandonlyifL=1. Proof. First,wewillprovethatifU=Co(ML)thenL=1.SupposeL<1.Thereexisttwovectors;2Co(ML)suchthatL.Then,62U,whichisacontradiction.Next,wewillshowthatifL=1,thenU=Co(ML).FromLemma 11 ,wealreadyknowthatUCo(ML).WeonlyneedtoshowCo(ML)U.SupposeCo(ML)6U,thenthereexistsa2Co(ML)and62U.Since2,theremustbea2Co(ML)suchthatand6=.Then,thereisapartitionofsetLinto,sayPandQ,suchthatP\Q=;,P[Q=Land[]Q=[]Q;[]P>[]P.IfQ=;,thentheremustbea<1suchthat.ThiscontradictsL=1.SupposeQ6=;.Ifforalli2Q,i=i=0,thenagain,theremustbea<1suchthat.Wenextconsidertheremainingcase.Pickani2Qsuchthati=i>0.Forthisi,theremustbeaschedulesinwhichlinkiisnotactivated.Wechooseasmallenoughandlet0=(1)]TJ /F3 11.955 Tf 12.22 0 Td[()+s.Since;s2Co(ML),wehave02Co(ML).Next,setP P[figand 41

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Q Q)-175(fig.ThisoperationcanberepeateduntileitherQ=;or[]Q=[0]Q=0.Then,theremustbea<1suchthat0,whichcontradictsL=1. ThislemmaimpliesthatwheneverL=1,theupper-boundaryisconvexandequaltoCo(ML).Then,anyrayfromtheorigincanonlyintersectCo(ML)atonepoint. 2.8EffectsofRedeningLoad-slackofArrivalVectorsInthissectionwederiveresultsforamodiedload-slackdenition.Thenewdenitionprovidesamoreintuitiveperspective,asitcomesclosertorepresentingthegapbetweenthenormalizedservicerateandthesystemloadincorrespondencetotheinterferencegraph.Westartwithjointlydeningtheconceptsofstorageefciencyandtheload-slack,().Weusearrivalvectorfortheaveragearrivalratevectorthroughoutthesection. Denition7. AschedulingalgorithmhasstorageefciencyforaveragearrivalratevectorifitsatisesthefollowinginequalitywiththeinitialconditionQ(0)=0:E[Q(t)Te]P(N;())]TJ /F6 7.97 Tf 6.58 0 Td[(1)8t20;1;::: (2)where()=1)]TJ /F3 11.955 Tf 12.24 0 Td[(f(G;),Pisapolynomiallyboundedfunction,andNistheencodinglengthoftheinterferencegraph(equivalentlyprobleminput). Denition8. Wecallanarrivalratevectoruniformifitisintheformk~efork>0.Thefollowinglemmawillprovideitselfusefulinderivingapproximationbounds: Lemma13. If00then1 zisan-approximationtothef(G),i.e.:f(G)1 z(1+)f(G). Proof. 1 f(G)z+kf(G)1 z+k(1+)f(G)(1+) z+k(1+) z+z=1 z: 42

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Nowwearereadytostatethemaintheoremwiththenewdenitionoftheload-slack: Theorem2.4. ThereisnoschedulingalgorithmthathasstorageefciencyandcomputationefciencyforuniformarrivalvectorsunlessP=NP. Proof. AssumewearegivenagraphGandanapproximationratio2(0;1].AtransientqueueboundP=P(N;(0))]TJ /F6 7.97 Tf 6.59 0 Td[(1)ondeterministicarrivals,where0= 4jLj,canbeusedtocompute-approximationtothefractionalchromaticnumber,Xf(G),asfollows.RuntheefcientschedulingalgorithmforP)]TJ /F6 7.97 Tf 6.59 0 Td[(1roundswhilekeepingaveragearrivalrateuniformamongqueues.TherearetwopossibilitieseitherallqueueswillstaybelowPthroughouttheexecutionoratleastonequeuepassesthethreshold.Nextwegoovereachcaseindetail:Case1:Qi(t)P8i;t:ThequeueiwillhaveP)]TJ /F6 7.97 Tf 6.58 0 Td[(1iamountofowinsertedandhasatleastservedtheamountP)]TJ /F6 7.97 Tf 6.59 0 Td[(1i)]TJ /F3 11.955 Tf 12.61 0 Td[(P.Ifyourunthealgorithmwithidenticalarrivalsatqueuesi.e.=ke;k>0andqueuesdonotexceedPduringP)]TJ /F6 7.97 Tf 6.59 0 Td[(1rounds;theaveragescheduleactivationswillformavector,,thatalsocouldbeinterpretedastime-sharingfactorofschedulessuchthatML()]TJ /F3 11.955 Tf 11.49 0 Td[()~e.Whichinreturnguarantees()]TJ /F3 11.955 Tf 11.49 0 Td[()~eisinthecapacityregionor)]TJ /F3 11.955 Tf 11.49 0 Td[(1 f(G).Case2:Qi(t)>P9i;t:IfanytimeduringtheprocedurethequeuesizesaretoexceedPitimplies1 f(G))]TJ /F3 11.955 Tf 11.38 0 Td[(0orotherwisethequeuebound,P,shouldhavebeenobserved.Bydiscretizingtheinterval(1 jLj;1]usingtheprecisionparameter,abinarysearchproce-durecanbeconducted.Leta;btobepositiverealnumberswherefortheidenticalarrivalvectora~eCase1holdsandfortheidenticalarrivalvectorb~eCase2holds.Additionallyletb=a+.Nowwehave: 43

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Table2-1. Atabularcomparisonofnegativeresultsfrom[ 49 ]andTheorem 2.4 .Therstcolumnrepresentstheregiontowherearrivalvectorsbelong;thesecondcolumnrepresentsthequeueboundachievementofwhichleadstoP=NP. RegionQueueBound Shahetal.[ 49 ]2p logN NoP(N)Theorem 2.4 oP(N;())]TJ /F6 7.97 Tf 6.59 0 Td[(1) a)]TJ /F3 11.955 Tf 11.96 0 Td[(1 f(G)b+0a)]TJ /F3 11.955 Tf 11.96 0 Td[(1 f(G)a++0a)]TJ /F3 11.955 Tf 11.96 0 Td[(1 f(G)(a)]TJ /F3 11.955 Tf 11.96 0 Td[()+++0:Theprocedurewilllocateanumberzsuchthat:f(G)1=z(1+)f(G)whichisthedenitionof-approximationforaminimizationproblemwithoptimalvaluef(G).Setz=a)]TJ /F3 11.955 Tf 12.32 0 Td[(,=0 2,=0 2andk=1 2jLjinLemma 13 ,andtheratioisachieved.Thepresentedmethodisapolynomial-time-approximationalgorithmtothefractionalcoloringproblem.Howevercombiningresultsfrom[ 54 ]and[ 49 ],weconcludesuchanalgorithmwillimplyP=NP. Corollary4. ThereisnoschedulingalgorithmthathasstorageefciencyforallvectorsintheinteriorofthecapacityregionwithpolynomialPandcomputationefciencyunlessP=NP.Authorsin[ 49 ]providedasimilarresultwhichdisplaysintractabilityofachievinglowqueuedelayswithsimpleper-time-slotcomputationswithinalinearlyscaled-downsubsetofthecapacityregion. Denition9. ([ 49 ])Apolicyhasthepoly-queuepropertyifthereexistsapolynomialp(n)suchthatforeveryn-nodegraph,andeveryarrivalratevectorforwhich()1 (n),theresulting 44

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queuesizessatisfysupE[hQ();1i]p(n); (2)for(n)=n 2p logn.Notethat()=f(G;),wheretherstfunctionusesgraphGimplicitly. Theorem2.5. (Theorem3.1in[ 49 ])Forthecaseofdeterministicarrivalsthereexistsnopoly-queuedeterministicpolicythatrunsinpolynomialtime,unlessP=NP.BothTheorem 2.4 andTheorem 2.5 suggestthatachievingpolynomiallyboundedqueuesizeswithpertime-slotsimple(polynomial)algorithmsischallenging.Table 2-1 summarizestheresults.Next,withthenewload-slackinmind,wewillinvestigatetheconsequencesofhavingapolynomialtimeoracleforWMISproblem.OurprimarymotivationistodemonstratethatwithaWMISsolver,evenifitsapproximate,polynomiallyboundedqueuesizes,whicharetransientinnaturefordeterministicarrivals,canbeachieved.AnalyzingperformanceofMWSviaLyapunovfunctionstypicallyrequiresrelatingtwoinnerproducts:A(t)TQ(t)andD(t)TQ(t).Lemma 14 achievesthistaskbycomparinginnerproductoftwonon-negativevectorsbytheassociatedgraphtheoreticalfunctions. Lemma14. Tf(G;)(G;)8;2RjVj+. Proof. Let2RjMLj+(mi2ML),tobeanoptimalsolutiontof(G;)i.e.,PjMLji=1imiandPjMLji=1i=f(G;),thenwehaveT(jMLjXi=1imi)T=jMLjXi=1imTijMLjXi=1i(G;)=f(G;)(G;): Thislemmacanbeinterpretedasanextensionofcorollary7.5.3in[ 12 ]toarbitraryweightvectors.Inthefollowingdiscussionwe'llillustratehowLemma 14 canbecentraltoderivingthedriftconditionandinSection 3.4 itwillbeusedforextendingtheboundonexpectedWMISsizetothequeuesum. 45

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Westartwithdeningthewirelessschedulingpoliciesink-WMISclass.Forthisclass,schedule(activationvector)m(t)attimetsatises:m(t)TQ(t)k(G;Q(t))where(G;Q(t))=maxm2MLmTQ(t).ConsiderthequadraticLyapunovfunction:RjVj!Rsuchthat(Q(t))=PjVji=1Q2i(t).Let(Q(t))=(Q(t+1)))]TJ /F7 11.955 Tf 11.95 0 Td[((Q(t)).(Q(t))=Q(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(Q(t)TQ(t+1)+Q(t)=A(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t)T2Q(t)+A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)=2A(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t)TQ(t)+A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)TA(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t): (2)SecondtermintherighthandsideofEquation 2 canbebounded:A(t))]TJ /F3 11.955 Tf 9.47 0 Td[(D(t)TA(t))]TJ /F3 11.955 Tf -457.95 -23.91 Td[(D(t)jLj.Nextweboundtherstterm:2A(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t)TQ(t)2TQ(t))]TJ /F3 11.955 Tf 11.95 0 Td[(k(G;Q(t))2f(G;)(G;Q(t)))]TJ /F3 11.955 Tf 11.95 0 Td[(k(G;Q(t))2(f(G;))]TJ /F3 11.955 Tf 11.96 0 Td[(k)(G;Q(t)): (2)CombiningEquation 2 andEquation 2 wehave:(Q(t))2(f(G;))]TJ /F3 11.955 Tf 11.95 0 Td[(k)(G;Q(t))+jLj: (2)Letk()=k)]TJ /F3 11.955 Tf 11.57 0 Td[(f(G;)then(Q(t)))]TJ /F7 11.955 Tf 21.92 0 Td[(2k()(G;Q(t))+jLj.When(G;Q(t))jLj 2k(),(Q(t))willdropinthenextround.NotethatbyLemma 14 ;Q(t)Te>f(G)jLj 2k()implies(G;Q(t))jLj 2k().Alsoif(Q(t))(f(G)jLj 2k())2thenQ(t)Te>f(G)jLj 2k().Thatimplieswhenpotentialexceeds(f(G)jLj 2k())2itwilldropinthenextround.Potentialincreaseforanytime-slottisboundedfrom 46

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aboveby2Q(t)+jLj.Themaximumpotentialthatthesystemcaneverreachisboundedby(f(G)jLj 2k())2+2f(G)jLj 2k()+jLj.Boundedpotentialfunctiontranslatesintoboundedqueuesizes:themaximumqueuesizeduringtheoperationofthesystemisboundedbyf(G)jLj 2k()+p jLj. 2.9ResultsandDiscussionInthischapter,weinvestigatedthecapacityregionofawirelessnetworkinameticulousmanner.Inparticular,wedenedthecapacityregionusingtheweightedfractionalcoloringproblemandusedthederivedrelationstoaddressthecomplexityofschedulingdeterministicarrivals.Thechoiceofdeterministicarrivalswasduetoitsimmediateconnectiontotherandom-izedcounterpartandtheeaseofutilizingthecomplexitytheory.Underthedeterministicsetting,weshowedthattheWMIS-basedalgorithmstabilizesthequeueswithefcientbounds.Wealsoinvestigatedinterestingsectionsofthecapacityregionanddenedtheupper-boundary.Were-latedthe-localpoolingfactortotheweightedfractionalcoloringproblem.Wecharacterizedtheupper-boundaryunderthesetlocalpoolingcondition.Wealsoderivedresultsforanewlydenedload-slack,whichmeasurestheconceptofsystemloadmoreconventionally.ItisdisplayedthatwithanapproximateWMISsolverthepolynomialqueueboundscanbeachievedwithinascaledportionofthecapacityregion.Therearemanyremainingquestions.First,intheschedulingcontext,thenecessityoftheWMISproblemincomputingfractionalcoloringnumberhasnotbeenestablishedyet.Second,theapplicabilityoftheresultstothestateofartcontrolalgorithmsrequiresconnectingdeterministicarrivalswithprobabilisticones.Third,webelievethatsetlocalpooling(andlocalpoolingingeneral)isrelatedtothecomplexityoftheWMISproblemingraphs.Webelievethosequestionsformtheimmediateconcernssurroundingthischapter. 47

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CHAPTER3STEADYSTATEDELAYCHARACTERISTICSOFAPPROXIMATEMWSPOLICY 3.1BackgroundPerformanceofwirelesscommunicationsystemsrelyontheefcientutilizationofthesharedtransmissionmedium.Simultaneouslinktransmissionsisknowntocauseinterferenceinthelackofcoordination.Mediumaccesscontrol(MAC)protocolsaddressthisproblembydevisingchannelaccessmechanismsthatoptimizeobjectivesofchoice.Throughputhasbeenconsideredasoneofthemostprevalentnetworkperformancemetricsinthelasttwodecades.Nevertheless,recentlythereisagrowinginterestonthedelayboundsofwirelessschedulingalgorithms[ 26 ][ 15 ].Thischapterstudiesthedelayboundsoftheapproximatemaximumweightscheduling(k-MWS)policy.Intheirseminalwork[ 52 ],Tassiulasetal.proposedathroughputoptimalschedulingpolicythat,whenappliedtoprotocolmodel,activatesaWMISofthequeue-weightedinterferencegraphateachtime-slot.(Alsocommonlyknownasmaximumweightscheduling(MWS)policy).HowevertheiralgorithmisnotapplicabletolargenetworkssinceWMISproblemisNP-hardingeneral.Laterin[ 53 ],Tassiulasprovidedalineartimeprobabilisticalgorithmachievingthroughputoptimality.Althoughfromcomplexityperspective,twoalgorithmsdiffersignicantly,looselyspeaking,thelattertriestosimulatetheformerusingarandomizedsearchscheme[ 27 ].Howeverwhenthelatteralgorithmisusedqueuesizesaresubjecttogrowexponentially,leadingtoanexorbitantmemoryusage.Accordinglyrecentresearchfocusesondelayboundsofthroughputoptimalalgorithms[ 26 ][ 15 ].Tothebestofourknowledgethereislittleprogressinderivingboundsfortransientexpectedqueuelengths.Asaresultresearchersconcentrateonthesteady-stateboundsofexpectedqueuesizes.Theprecedingresearchonswitchschedulingprovidedmanyresultsthatareapplicabletothewirelessschedulingwithminormodications.In[ 48 ]authorsinvestigatedtheexpectedqueuesizeboundsatstationarityforvirtualoutputqueuedswitches.Theirframeworkcorrespondstoaspecialcaseofthegeneralinterferencemodel,wherenodeexclusiveinterferencemodelisusedonacompletebipartitegraph.Later 48

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in[ 26 ]ageneralizationforarbitrarygraphsisprovidedfortheMWSalgorithm.ThelatesteffortinthisdomainintroducedavariantofMWStoachievetightexpectedqueueboundsatstationarity[ 15 ].TocounterthedifcultyofcomputingWMISateachtime-slot,astreamofresearchconcentratesondevisingalgorithmsthataresimpleyetefcientinpractice.Thelongestqueuerst(LQF)policyisknownforitssimplicity,highperformance,andaptnessfordistributedimplementation,acharacteristicneededforschedulingalgorithmsinthelackofcentralcontrol[ 24 ].PerformanceofLQFisstudiedfromtheperspectiveofthroughputin[ 7 ],[ 8 ],and[ 24 ].However,tothebestofourknowledge,delaycharacteristicsofLQFpolicyinwirelessnetworksisstillnotaddressedformally.Thefollowingisthesummaryofourmainresultsandcontributionsinthischapter: Weprovideadelayboundforthek-MWSpolicy.Ourresultimprovestheanalyticalboundderivedin[ 26 ]whenk=1andgeneralizestheboundfor0
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relationswithaninterferencegraph,G=(V;E),whereeachlinkl2Lintheoriginalnetworkcorrespondstoanodev2V.Twonodesv1;v22VareconnectedinG,ifandonlyifthecorrespondinglinksintheoriginalnetworkinterferewitheachother.WeassumeasymmetricinterferencerelationthusGisundirected.Afeasiblescheduleisdenedtobeasetofnon-interferingnodesinG.Amaximalscheduleisafeasibleschedulethatcannotincludeanymorenodeswithoutviolatingthefeasibilityconstraint.AfeasiblescheduleinanetworkcorrespondstoanindependentsetinG.WedenotethesetofallmaximalschedulesbyML,andusem2MLtodenoteasinglemaximalschedule.Throughoutthechapter,weusethewordscheduletorefertoafeasiblemaximalschedule.Whenapplicable,MLisregardedasajVjjMLj0-1matrix.Eachcolumnofthematrixisa0-1vectorrepresentationofamaximalindependentsetofG,with1indicatingthatthecorrespondingnode(inG)isselected(andthelinkisactiveintheschedule)and0otherwise.Acolumnvectorthathas1'sinallentriesisdenotedbye.Weadopttheconventionaldenitionofthecapacityregion:=fj0forsome2Co(ML)g:Theinteriorofthecapacityregionisdenotedbyo.Weassumeatime-slottedsystem.Thepacketsarrivingateachlinkarequeued.Eacharrivalstreamisi.i.d.intime.Thedistributionofthenumberofpacketsarrivingtolinkiattimet,Ai(t),istimeinvariant.Assumesecondmoments,E(A2i),ofthearrivalprocessesareboundedbyC.ThearrivalvectorforallqueuesattimetisdenotedbyA(t).SimilarlythequeuelengthvectorisdenotedbyQ(t).Alinkcanbeactivatedinatimeslot,onlyifthequeueisnon-empty.Atmostonepacketcanbeservedataqueueinagiventimeslot.Foreachlinki,functionDi(t)indicateswhetherornotlinkireceivedserviceattimeslott.Notethat:Di(t)=8>><>>:1;ifQi(t)1andiisscheduled.0;otherwise. 50

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D(t)denotesthevectorofdeparturesattimet.Theevolutionofthequeueisasfollows:Q(t+1)=Q(t)+A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t):Inthischapterweinheritthedenitionofstabilityfrom[ 52 ],whereitisdenedastheunderlyingirreducibleDiscreteTimeMarkovChain(DTMC)beingpositiverecurrent.Athroughputoptimalalgorithmkeepsthequeuesstableforanyaveragearrivalratefromtheinteriorofo. 3.3GraphTheoreticalResultsInthissectionourprimaryfocuswillbeonbuildingsupportinglemmasfortheperformanceevaluationoftwoparticularalgorithms.Therstalgorithmisthek-MWSpolicy(k2(0;1]),whichpicksaschedule,denotedbymk(t),thatapproximatesthemaximuminnerproductwiththequeuelengthsatthecurrenttimesloti.e.,mk(t)TQ(t)kmaxm2MLmTQ(t).Itisknownthatthek-MWSwouldstabilizeaveragearrivalrateswithinko[ 57 ].ThesecondalgorithmistheLQFpolicywhichschedulesthelongestqueueamongtheavailablelinks,anddiscardstheneighboringlinks.Thisprocedureisrepeateduntilallthequeuesareeitherscheduledordiscarded.We'llcallanyschedulethatisproducedbythisprocedureaLQFschedule.Whenprotocolmodelisassumed,capacityregioncanbecharacterizedusingtheconceptoffractionalcoloring(seeDenition 1 ).Computingf(G;w)isNP-hardtoapproximateingeneral.Wedenotetheweightedindependencenumber(totalweightofWMIS)ofagraphwith(G;w),wherew2RjVj+istheweightvectorasusual.Wedenotef(G;e)and(G;e)withf(G)and(G)respectively.Let(G;;)bequeuelengthsumofaLQFschedule,where2RjVj+isthevectorrepresentingthequeuelengths.TheLQFscheduleischosenasgivenabovewithanadditionalparameter,apermutationofvertices,toprovidetie-breakingifnecessary.We'llbederivingresultsindependentof.Inordertoeliminatewe'lldeneafunctiontorepresentminimumqueue-sumLQFscheduleoverallpossiblepermutations,; 51

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Denition10. (G;)=min2(G;;).In[ 7 ]and[ 24 ]theconceptofinterferencedegreeisusedtocharacterizetheperformanceofLQF.Wedenotetheclosedneighborhoodofanodev2VbyN[v].AlsoforasubsetofnodesSV,G[S]denotesthevertexinducedsubgraphofG[ 55 ].Thefollowingisthedenitionoftheinterferencedegree: Denition11. =maxv2V(G[N[v]]).Inwords,interferencedegreeisthemaximumnumberofnodesthatcouldbeactivatedsimultaneouslyintheneighborhoodofanode.In[ 7 ],itisshownwhenanaveragearrivalratevectoriswithin1 o,LQFwillstabilizethenetwork.ThefollowinglemmarelatesLQFschedulingtoWMISandjustiestheunderlyingperformancebound: Lemma15. (G;)(G;)82RjVj+. Proof. Omittedforbrevity. ThegreedyscheduleselectionofLQFprovidesaapproximationtotheWMISproblem.We'lldenoteanLQFscheduleachieving(G;Q(t))withmLQF(t).Lemma 16 withCorollary 5 demonstratethataLQFscheduleproducedwithqueueweightsQ(t)isstillaapproximationtoWMISwhenthequeuesizesareraisedtoapowern2Z+: Lemma16. (G;Qn(t))mLQF(t)TQn(t)8Qn(t)2RjVj+;n2Z+. Proof. Firstwenotice,sinceexponentiationisanon-decreasingfunctionwhenQ(t)2ZjVj+andn2Z+,mLQF(t)isstillaLQFschedulepickedusingweightsQn(t)undersomepermutationofvertices.Combinethisfactwiththedenitionof(G;Qn(t))andtheresultfollows. Corollary5. 1 (G;Qn(t))mLQF(t)TQn(t).HowevertheboundprovidedinCorollary 5 forLQFisnotcommontoanyk-MWSpolicy.WeprovidethegeneralizedboundsinLemma 17 : Lemma17. kn ((G))n)]TJ /F6 7.97 Tf 6.59 0 Td[(1(G;Qn(t))mk(t)TQn(t)8Q(t)2ZjVj+;n2Z+;k2(0;1]. 52

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Proof. (G;Qn(t))((G;Q(t)))n(1 kmk(t)TQ(t))n(1 k)n(mk(t)TQ(t))n(1 k)n((G))n)]TJ /F6 7.97 Tf 6.58 0 Td[(1mk(t)TQn(t);wherethelastinequalityfollowsfromthefacts:forai;n;l2Z+,(Pli=1ai)n(l)n)]TJ /F6 7.97 Tf 6.58 0 Td[(1Pli=1(ai)nandcardinalityofanyscheduleisupperboundedby(G). Figure3-1. IllustrationontightnessofLemma 17 TheboundprovidedinLemma 17 couldbetightasillustratedinFigure 3-1 .InFigure 3-1 forthecurrenttime-slottthequeuesizesareasfollows;Q1(t)=6;Q2(t)=Q3(t)=Q4(t)=1.Fork=1 2,theactivationvectormk(t)=f0;1;1;1gcorrespondstoaschedulethatwouldbeactivatedbythek-MWSpolicy.TheschedulefromMWSpolicyontheotherhandistheactivationvectorm(t)=f1;0;0;0g.Also(G;Qn(t))=6n.Interferencedegree,forthegivengraphis3.Readercanverifyforn=2;3;:::,theboundgiveninLemma 17 istight.LQFpolicyontheotherhanddemonstratesaninvariantbehaviorwhenexponentiationisinvolved.ThatpropertywillproveitselftobeusefulinSection 3.5 whenwearetoboundthehighermomentsofthequeuesizedistributionatstationarity. 53

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3.4DelayBoundsofApproximateMWSPolicyInthissectionwederiveboundsonstationaryexpectedqueue-sumsintheguidanceofLemma 14 .ThedelayboundfollowsusingLittle'sLaw[ 36 ].Ourderivationfollowsfrom[ 26 ],however,weimprovetheboundsbyusingafractionalcoloringscheme.ConsiderthequadraticLyapunovfunction:RjVj!Rsuchthat(Q(t))=PjVji=1Q2i(t).(Q(t+1)))]TJ /F7 11.955 Tf 11.95 0 Td[((Q(t))=Q(t+1))]TJ /F3 11.955 Tf 11.95 0 Td[(Q(t)TQ(t+1)+Q(t)=A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)T2Q(t)+A(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t)=2A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)TQ(t)+A(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t)TA(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)=2A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)TQ(t)+A(t)TA(t)+D(t)TD(t))]TJ /F7 11.955 Tf 11.96 0 Td[(2A(t)TD(t): (3)Ifthenetworkisstableunderagivenpolicy;astimetendstoinnityQ(t);A(t);D(t)constitutesapositiverecurrentDTMC[ 52 ].Similarto[ 31 ],atstationarity,duetopolynomialmomentsofthequeuelengthsbeingnite[ 30 ],foralowerboundedpolynomialpotentialfunctionL:RjVj!Rwehavelimt!1E[L(Q(t+1)))]TJ /F3 11.955 Tf 12.35 0 Td[(L(Q(t))]=0.Sincesatisesthecondition,whenthesystemreachesstationarity:E[(Q(t+1)))]TJ /F7 11.955 Tf 11.96 0 Td[((Q(t))]=2E[(A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))TQ(t)]+E[A(t)TA(t)]+E[D(t)TD(t)])]TJ /F7 11.955 Tf 11.96 0 Td[(2E[A(t)TD(t)] (3)=0:NextweboundtheadditivetermsinEquation 3 .UsingLemma 14 wecanobtain: 54

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E[A(t)TQ(t)jQ(t)]=TQ(t)f(G;)(G;Q(t)):Bytakingexpectationonbothsideswehave;E[A(t)TQ(t)]f(G;)E[(G;Q(t))]: (3)SinceDi(t)2f0;1g;D2i(t)=Di(t).Additionallythesystemisatthestationarityi.e.,E[A(t)]=E[D(t)],thusE[D(t)TD(t)]=E[jVjXi=1Di(t)]=jVjXi=1i: (3)AlsoAi(t)isindependentofDi(t)leadingto;E[A(t)TD(t)]=E[A(t)]TE[D(t)]=jVjXi=12i: (3) Lemma18. Underk-MWSpolicy,foranarrivalrate2ko,atstationarityonehas:E[jVjXi=1Qi(t)]f(G) 2k)]TJ /F3 11.955 Tf 11.95 0 Td[(f(G;)B1whereB1=PjVji=1i+E[A2i(t)])]TJ /F7 11.955 Tf 11.96 0 Td[(22i. Proof. Underk-MWSpolicywehaveE[D(t)TQ(t)]kE[(G;Q(t))]:CombiningwithEquation 3 weget 55

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2E[A(t)TQ(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)TQ(t)]2(f(G;))]TJ /F3 11.955 Tf 11.96 0 Td[(k)E[(G;Q(t))]: (3)WeobtainthenextresultbyusingEquation 3 ), 3 ,and, 3 inEquation 3 :02(f(G;))]TJ /F3 11.955 Tf 11.96 0 Td[(k)E[(G;Q(t))]+jVjXi=1i+jVjXi=1E[A2i(t)])]TJ /F7 11.955 Tf 11.95 0 Td[(2jVjXi=12i:Notingf(G;)
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Ourresultimprovestheboundin[ 26 ]byafactorof(G) f(G),wheref(G)istheintegralcoloringnumberofG.Nextweprovideacomparisonbetweenmagnitudesof(G)andf(G).In[ 29 ]authorsprovedthedifferencebetween(G)andf(G)canbearbitrarilylarge: Lemma19. Foranyintegern2,thereexitsauniquelycolorable,vertextransitivegraphG,suchthat(G))]TJ /F3 11.955 Tf 11.96 0 Td[(f(G)>n)]TJ /F7 11.955 Tf 11.96 0 Td[(2.Howeverthatresultcountsonthegrowingsizeofthegraphconsidered.Lovaszin[ 39 ]providedaninequalitytoupperboundtheintegralcoloringnumberwhichrelatestothefractionalcounterpartasfollows: Lemma20. f(G)(G)<(1+ln((G)))f(G).Inotherwords,byLemma 19 ,ourboundcanbearbitrarilytighterthantheresultsprovidedin[ 26 ].HoweverLemma 20 statesthatwhentheboundsarenormalizedbythegraphsize,theimprovementfactorisatmost1+ln((G)). 3.5LQFasanApproximateMWSPolicyInthissectionwederiveanalyticalboundsforE[PjVji=1Q2i(t)],whenk-MWSpolicyisutilized,andtheaveragearrivalratesatises:2k2 (G)o(Thescalingcanbeconsideredasathroughputcompromisetoachievetighterbounds).CombinedwiththeresultfromSection 3.4 ,boundsonqueuelengthvarianceatstationaritycouldbeobtained.AdditionallywhenLQFschedulingpolicyisemployed,wendthat,upperboundsforhighermomentsofstationaryqueuesumdistributioncanbecomputedwithinasubsetofthestabilityregion,1 o,withoutcompromisingthroughput.ConsiderthecubicLyapunovfunction:RjVj!Rsuchthat(Q(t))=PjVji=1Q3i(t).Atstationarity,wehave: 57

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E[(Q(t+1)))]TJ /F7 11.955 Tf 11.96 0 Td[((Q(t))]=3E[(A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))TQ2(t)]+3E[jVjXi=1((Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t))2Qi(t)]+E[jVjXi=1(Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t))3] (3)=0:NextweboundadditivetermsinEquation 3 .DeneB2=PjVji=1E[A3i(t)])]TJ /F7 11.955 Tf -407.63 -23.91 Td[(3iE[A2i(t)]+32i)]TJ /F3 11.955 Tf 11.96 0 Td[(ithen;E[jVjXi=1(Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t))3]=B2: (3)Rememberingsecondmomentofthearrivalprocessisboundedwehave;E[jVjXi=1((Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t))2Qi(t)]E[(C+1)jVjXi=1Qi(t)](C+1)f(G) 2k)]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;)B1: (3)SimilartothederivationinEquation 3 ,byLemma 14 ,E[jVjXi=1(Ai(t)Q2i(t)jQ(t))]=TQ2(t)(G;Q2(t))f(G;):Additionallyunderk-MWSpolicywehave;E[jVjXi=1Di(t)Q2i(t)jQ(t)]k2 (G)(G;Q2(t)): (3)Thusweobtain; 58

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E[jVjXi=1(Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t))Q2i(t)]E[(G;Q2(t))](f(G;))]TJ /F3 11.955 Tf 20.47 8.09 Td[(k2 (G)): (3)CombiningEquation 3 3 ,and, 3 inEquation 3 :E[(G;Q2(t))]3(k2 (G))]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;))3 2(C+1)f(G) k)]TJ /F3 11.955 Tf 11.95 0 Td[(f(G;)B1+B2:When2k2 (G)o;E[(G;Q2(t))](C+1)f(G)B1 2+B2(k)]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;)) 3 (k2 (G))]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;))(k)]TJ /F3 11.955 Tf 11.95 0 Td[(f(G;)):ByapplyingLemma 14 againwenallyget;E[jVjXi=1Q2i(t)]=E[eTQ2(t)]f(G)E[(G;Q2(t))]f(G)(C+1)f(G)B1 2+B2(k)]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;)) 3 (k2 (G))]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;))(k)]TJ /F3 11.955 Tf 11.95 0 Td[(f(G;)):NextweprovidetighterboundsforEPjVji=1Q2i(t)underLQFwithoutathroughputcompromise(within1 o).ConsiderCorollary 5 withk=1 andn=2i.e.,1 (G;Q2(t))mLQF(t)TQ2(t): 59

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When21 oandLQFschedulingpolicyisused,byusingthederivationprovidedaboveandconsideringLQFinEquation 3 weget:E[jVjXi=1Q2i(t)]f(G)(C+1)f(G)B1 2+B2(1 )]TJ /F23 10.909 Tf 10.91 0 Td[(f(G;)) 3 (1 )]TJ /F23 10.909 Tf 10.91 0 Td[(f(G;))2:FinallyitispossibletoderiveupperboundsonthequeuevarianceusingtheboundsonEPjVji=1Q2i(t)andEPjVji=1Qi(t).Theframeworkwe'vepresentedcanbeextendedtothehighermomentsofqueuesizedistributionatstationaritywithlittleeffort,givenallassociatedmomentsofthearrivalprocessarebounded. 3.6MiscellaneousResultson-LocalPoolingInthissectionwewillinvestigatethefollowingquestionsthatrelatetothestate-of-artperformancemetricofLQFpolicy,-localpoolingfactor.Adetaileddiscussiononthesignicanceof-localpoolingfactorcanbefoundin[ 32 ]. Isthereanylower-boundforthe-localpoolingfactoringraphs? Whatisthecomputationalcomplexityofcalculatingset-localpoolingfactor(whichformsthebasisforthedenitionoftheoverall-localpoolingfactor)? Whatkindofstructures,whentheyareaddedtoagraph,donotaffect-localpoolingfactoringraphs?Westartwithcitingalistofresultsfrom[ 33 ]thatwillbeusedforansweringquestionsgivenabove.Inresultslistedbelowusethefollowingnotation:xisaweightassignmentonthenodesoftheinterferencegraph,s2MLisamaximalindependentsetintheinterferencegraph,x(s)isthetotalweightofthesets,andG(V;E;I)istheinterferencegraphcomposedofvertexsetV,edgesetE,andacollectionofindependentsetsI. 60

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Lemma21. (Lemma6in[ 33 ])Foralinkl2E,listhesmallestLforallLEthatcontainsl,i.e.,l=minfLEjl2LgL (3) Lemma22. (Lemma1in[ 33 ])ForanetworkgraphG=(V;E;I),thefollowingholds.(G)=minl2El: (3)Equation9in[ 33 ]:L=w=maxx0minixTsi maxixTsi: (3) Lemma23. (Lemma5in[ 33 ])Listheoptimalvalueofthefollowingoptimizationproblem.maxx0;wws.t.xTMLeTc(L)xTMLweTc(L):Equation36in[ 33 ]:G=maxx0mins2MGx(s) maxs2MGx(s): (3) Lemma24. (Lemma24in[ 33 ])IfagraphGisvertex-transitive,thentheoptimizationproblemgiveninEquation 3 isachievedbyanequalweightassignmentonthenodes. 3.6.1GraphswithArbitrarilySmallSet-LocalPoolingFactorsWewillprovethatset-localpoolingfactorscanbearbitrarilysmallusingaspecialclassofgraphs,thehypercubes.Asimilarresultwasfoundindependentlyin[ 3 ]usingadifferentclassofgraphs.ThehypercubegraphQnisaregulargraphwith2nvertices.ThehypercubegraphQncanbeconstructedbylabelingthe2nvertices0;1;2;:::;2n)]TJ /F7 11.955 Tf 12.6 0 Td[(1andconnectingtwoverticeswhenevertheHammingdistanceofthebinaryrepresentationsofthelabelsisequalto1.ThisconstructionisillustratedinFigure 3-2 .ThevertexsetofQnisdenotedbyVQn. 61

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Lemma25. AmaximumcardinalityindependentsetofQnhasatleast2n)]TJ /F6 7.97 Tf 6.59 0 Td[(1elements. Proof. Wewillinductivelyconstructanindependentsetofthedesiredcardinality.Theinductionhypothesisis:Qnhasamaximalindependentset,ZVQn,suchthatVQn)]TJ /F3 11.955 Tf 12.55 0 Td[(Zisamaximalindependentsetaswell.Forn=1,thestatementholds(forn=0,thestatementwouldholdwithproperdenitions).ItisknownthatQn+1canbeconstructedbyusingtwohypercubesQnandconnectingcorrespondingverticestogether.TakeQ1nandQ2n,twocopiesofQn,toconstructQn+1.WewillshowQn+1hasamaximalindependentsetZ,whosecomplement,VQn+1)]TJ /F3 11.955 Tf 12.49 0 Td[(Z,isamaximalindependentsetaswell.Bytheinductionhypothesis,Q1ncanbepartitionedintotwomaximalindependentsets,Z1andVQ1n)]TJ /F3 11.955 Tf 12.08 0 Td[(Z1.LetZ2bethemaximalindependentsetofQ2ncorrespondingtoZ1.Now,observeZ=Z1[(VQ2n)]TJ /F3 11.955 Tf 12.53 0 Td[(Z2)isamaximalindependentsetanditscomplementVQn+1)]TJ /F3 11.955 Tf 12.39 0 Td[(Z=VQn+1)]TJ /F7 11.955 Tf 12.39 0 Td[((Z1[(VQ2n)]TJ /F3 11.955 Tf 12.39 0 Td[(Z2))=Z2[(VQ1n)]TJ /F3 11.955 Tf 12.39 0 Td[(Z1)isamaximalindependentsetaswell.Thustheresultfollows.WeconcludethatQncanbepartitionedintotwomaximalindependentsetsandoneofthemmusthaveatleast2n)]TJ /F6 7.97 Tf 6.58 0 Td[(1elements. Lemma26. AminimumcardinalitymaximalindependentsetofQnhasatmost2n=(n+1)elementswhenn=2k)]TJ /F7 11.955 Tf 11.95 0 Td[(1,wherek2Z+. Proof. In[ 38 ],itisshownwhenevernisintheformn=2k)]TJ /F7 11.955 Tf 12.53 0 Td[(1forsomepositiveintegerk,Qnhasaperfectdominatingset.Inperfectdomination,everyvertexcanonlybedominatedbyasinglenode.SinceeverynodeinQnhasdegreen,thereareatmost2n=(n+1)elementsinaperfectdominatingset.Notethatanindependentdominatingsetisamaximalindependentset. Lemma27. Letan=2n)]TJ /F7 11.955 Tf 11.96 0 Td[(1,thenQan21)]TJ /F5 7.97 Tf 6.59 0 Td[(n. Proof. Ahypercubeisvertex-transitive.ByLemma 24 ,Qancanbederivedunderanequalweightassignmentonthevertices.LetMQanbesetofallmaximalindependentsetsofQan.By 62

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Lemma 25 ,maxm2MQanjmj2an)]TJ /F6 7.97 Tf 6.58 0 Td[(1.ByLemma 26 ,minm2MQanjmj2an=(an+1).Since Qan=minm2MQanjmj maxm2MQanjmj2an=(an+1) 2an)]TJ /F6 7.97 Tf 6.59 0 Td[(1=21)]TJ /F5 7.97 Tf 6.59 0 Td[(n;thelemmaholds. Corollary6. Letan=2n)]TJ /F7 11.955 Tf 11.95 0 Td[(1.Then,limn!1Qan=0. Figure3-2. Hypercubesofdifferentdimensions. 3.6.2ComputationalComplexityofCalculatingGAgain,considertheinterferencegraph(orsubgraph)G=(V;E).ByLemma 23 ,theset-localpoolingfactor,G,istheoptimalvalueofthefollowingoptimizationproblem.maxw (3)subjecttox0MGe0 (3)x0MGwe0 (3)x;w0: (3) 63

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Here,MGisamatrixwherethecolumnsareallthemaximalindependentsetsofG,ande=(1;1;:::;1)0ofanappropriatedimension.WewillinvestigatethecomputationalcomplexityofcalculatingGbysolvingtheaboveoptimizationproblem.AseparationoraclefortheabovelinearprogramisaproceduretotestwhetheragivenvectorisintheconvexregiondenedbytheconstraintsgiveninEquation 3 throughEquation 3 ,andifnot,ndaviolatingconstraint.LetSEPORC(G)denotesuchaseparationoracle.WewillrstshowSEPORC(G)isNP-hardbygivingareductionfromtheminimumcardinalitymaximalindependentsetproblemtoSEPORC(G). Lemma28. SEPORC(G)isNP-hard. Proof. Thedecisionversionoftheminimumcardinalitymaximalindependentset(MCMIS,alsoknownastheminimumindependentdominatingset)problemaskswhetherthereisamaximalindependentsetofsizeKorless,where1KjVj.ItisknownthatthisproblemisNP-complete(page190,[ 11 ]).WewillprovideaTuringreductionfromtheMCMISproblemtoSEPORC(G).Givenavectorx2QjVjandw2Q,SEPORC(G)candecidethemembershipof(x;w)intheconvexregiondenedbytheconstraintsgiveninEquation 3 throughEquation 3 .Usingthisseparationoracle,wewillcreateasolverfortheMCMISproblem(thusprovidingaTuringreduction).Wesetxi=1 jVjfori=1;:::;jVjandsetw=(K+1)=jVj.Thevector(x;w)alwayssatisesconstraintinEquation 3 .Hence,feeding(x;w)toSEPORC(G)willtelluswhether(x;w)satisesconstraintinEquation 3 ornot.Ifyes,anMCMISisatleastofthesizejVjw=K+1;ifno,theMCMIShaslessthanorequaltoKelements. Remark:Intheproof,onlythefeasibilityaspectoftheseparationoracleisused.TheproofreallysaysthefeasibilityproblemisNP-hard.But,SEPORC(G)isatleastashardasthefeasibilityproblem. Lemma29. ComputingGbysolvingtheoptimizationproblemgiveninEquation 3 throughEquation 3 isNP-hard. 64

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Proof. In[ 13 ],theauthorshaveestablishedthatthecomplexityoftheseparationoracleandthatoftheoriginaloptimizationproblemarepolynomiallyequivalent.CombinethisfactwithLemma 28 Remark:ThestatementofLemma 29 shouldnotbeunderstoodasndingGisNP-hard.ItmeansthatsolvingtheoptimizationproblemgiveninEquation 3 throughEquation 3 isNP-hard,whichinvolvesndingbothanoptimalsolutionandtheoptimalvalue. 3.6.3SpecialStructuresThatPreserveGInagraph,wewillcallanoderasuper-nodeifitisconnectedtoallothernodes.Asuper-nodeinaninterferencegraphcorrespondstoalinkinthenetworkthatinterfereswithallotherlinks.SupposewestartwithaninterferencegraphG=(V;E).Wewillinvestigatetheeffectofinsertingasuper-nodetoG.Wewillshowthatthelocalpoolingfactor(forgraphs)isnotalteredbytheinsertion.Wedenotethegraphaftertheinsertionofthesuper-nodebyG0=(V0;E0).Withslightabuseofnotation,wedenotethegraphlocalpoolingfactorsby(G)and(G0)forGandG0,respectively(insteadofthenotationthatusesthenetworkgraph).GivenasubsetTV,wedenotetheset-localpoolingfactorcorrespondingTbyT. Figure3-3. IllustrationofGandG' Lemma30. (G)=(G0). Proof. Wewillconsidercomputingtheset-localpoolingfactorT,TV,accordingtoEquation 3 ,whichwere-writenextusingsomenewnotations.LetMTbethesetofmaximumindependentsetsfortheinterferencesubgraphassociatedwithT.Givenaweightassignmentxonthenodesandasetofnodessand,letx(s)bethetotalweightofthesets.We 65

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knowthatT=maxx0mins2MTx(s) maxs2MTx(s): (3)Alsonote(bycombiningLemma 22 andLemma 21 ) (G)=minTVT:(3)Letthesuper-nodebedenotedbyr.Then,V0=V[r.ConsiderthecasewhereT0V0andr2T0.LetT=T0)-245(frg.WewillshowTT0.LetxbeanoptimalweightassignmentonthenodesinTthatachievesT.Letabeanarbitraryvalueon[mins2MTx(s);maxs2MTx(s)],andassigntheweightatonoder.Thevectory=(x0;a)0isaweightassignmentonthenodesinT0.Next,notethatMT0=MT[fsg.Hence, mins2MT0y(s)=min(mins2MTx(s);a)=mins2MTx(s);and maxs2MT0y(s)=max(maxs2MTx(s);a)=maxs2MTx(s):Asaresult, T0mins2MT0y(s) maxs2MT0y(s)=mins2MTx(s) maxs2MTx(s)=T:(3)Next,(G0)=minT0V0T0=min(minT0V0;r2T0T0;minT0V0;r=2T0T0)=min(minT0V0;r2T0T0;(G)): (3)ByEquation 3 andtheone-to-onecorrespondencebetweenT0andTthroughT0=T[frg,whereT0V0,r2T0andTV,wegetminT0V0;r2T0T0minTVT=(G): (3) 66

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CombiningEquation 3 andEquation 3 ,wehave(G0)=(G): 3.7ResultsandDiscussionInthischapterweprovideimproveddelayboundsfortheapproximateMWSalgorithminwirelessnetworks.Weillustrateusingourframeworkitispossibletoboundqueue-sumvariance,withinadiminishingsubsetofthecapacityregion.AdditionallyweshowhighermomentsofqueuelengthsumunderLQFcanbeboundedwithin1 withoutathroughputcompromise.Wealsoconsidermiscellaneoustopologicalresultsonlocal-poolingfactorofgraphs.Ourworkcontributestotheeldofsub-optimalschedulingalgorithmsandparticularlytothestudiesontheperformanceofLQF.Ourfutureworkconsistsofextendingthepresentedframework,forhighermoments,tothewholestabilityregionofoftheapproximateMWSpolicy.Itisknownallpolynomialmomentsofthequeuesumdistributionatstationarityarenite[ 30 ].ItmightbeinterestingtondpotentialfunctionstoexploitLemma 14 forderivingboundsonhighermoments.AlsoLQFisknowntobethroughputoptimalwhentheunderlyingconictgraphsatiseslocalpoolingcondition[ 8 ].Anopenquestioniswhether,forlocalpoolinggraphsunderLQF,ourframeworkadmitsboundsforhighermomentswithinthewholestabilityregionofthenetwork.Webelieveifitistrue,thederivedboundscancharacterizeperformanceofLQFmoreprecisely. 67

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CHAPTER4CHARACTERIZINGTHEEXPECTEDQUEUE-SUMBOUNDOFTHEMWSPOLICY 4.1BackgroundThroughputoptimalityhasbeenoneoftheprimaryconcernsofwirelessnetworkre-searchersinthelasttwodecades.Itispossibletoachievethroughputoptimalityviaawiderangeofalgorithms.Recenteffortsinwirelessschedulingresearchconcentrateonndingdelayef-cientthroughputoptimalalgorithms.InthischapterweconcentrateonageneralizationofMWSpolicy,whichscalestheindividualqueuesizeswithconstantstogetqueueweightsthatareusedtocomputeaWMIS.ThisversionoftheMWSpolicyisstudiedin[ 15 ].Authorsshow,underthatgeneralization,anysetofpositiveconstantswillensurestability.Nexttheyinvestigatetheperformanceimprovementondelayusingthealgorithmwithoptimizedparameters.TheprimaryobjectiveoftheauthorsthereinistoprovethegeneralizedMWSpolicyisabletoprovidequeueboundsatsystemstationaritythatisatleastasgoodasanyrandomizedpolicythatassignscon-stantactivationprobabilitiestotheschedules.InotherwordsauthorsshowedthatthegeneralizedMWSpolicyisbetterthananyalgorithmthatusesaprobabilitydistributionoverallpossibleschedules.InthischapterourobjectiveistocharacterizethedelayperformanceofMWSpolicyusingasingleparameter.Towardsthatend,wewillderiveaparameterthatimprovesonthepreviousresultsandistightwhenquadraticpotentialfunctionisbeingutilizedforproducingthedelaybounds.Weprovideacomparisonbetweenthepreviousparameterandthenewoneunderlyingthetopologieswhereaperformanceimprovementisachievedwiththenewchoice.Aneasyestimationofthedelayperformanceisprovidedforthecompleteness.Wealsoextendouranalysistothealgorithmconsideredin[ 15 ],andprovetheweightselectionproposedin[ 15 ]isanoptimalonefromdelayperspective.Intheprocessweidentifytheimportantparametersinthequadraticpotentialfunctionframeworkandprovidetechniquestoassesstheiroptimality. 4.2DelayCharacterizationUsingaSingleParameterSystemmodelinthischapterfollowsdirectlyfromchapter 3 .Wereferreadertosection 3.2 fordetailedsystemmodelandnotations. 68

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InChapter 3 weprovidedthestationaryqueuesumboundf(G) 21)]TJ /F5 7.97 Tf 6.58 0 Td[(f(G;)BwhereB=PjVji=1i+E[A2i(t)])]TJ /F7 11.955 Tf 12.8 0 Td[(22i.Wesignifythedeningfactorinthatinequalitywith()=1)]TJ /F5 7.97 Tf 6.59 0 Td[(f(G;) f(G).Nextwewillshowthatanotherscalar()suchthatf(G;+e())=1,canreplace()providingabetterbound.Firstweshow()isanupper-boundfor(). Lemma31. ()()2RN+. Proof. Westartwithasimpletriangleinequalitydenedwithrespecttothefractionalcoloringnumber:f(G;)+f(G;)f(G;+)Forany;2RN+: (4)UsingEquation 4 anddenitionof()weget:f(G;)+f(G;()e)f(G;+e())f(G;)+()f(G)1()1)]TJ /F3 11.955 Tf 11.96 0 Td[(f(G;) f(G)=(): Weusequadraticpotentialfunctionagain.We'llshowthersttermontherighthandsideofEquation 2 canbeboundedusing().Forsome2Co(ML)and>,usingthedenitionof()wehave:2A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t)TQ(t)2)]TJ /F3 11.955 Tf 11.96 0 Td[(TQ(t)2)]TJ /F7 11.955 Tf 11.96 0 Td[((+()e)TQ(t))]TJ /F3 11.955 Tf 35.87 0 Td[(()eTQ(t): 69

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AtstationaritywehaveE[(Q(t))]=0.andE[(A(t))]TJ /F3 11.955 Tf 11.95 0 Td[(D(t))T(A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))]=B.UsingEquation 2 ,wehave:0=E[2(A(t))]TJ /F3 11.955 Tf 11.96 0 Td[(D(t))TQ(t)]+B0)]TJ /F7 11.955 Tf 21.92 0 Td[(2()E[eTQ(t)]+BE[eTQ(t)]B 2(): (4)Thenewparameter,(),providesabetterboundingmechanismforthestationaryqueuesumvaluewhichinturncanbeusedtobounddelaythroughLittle'sLaw.Nextwedetailontheissuespertainingtothecomputationof().Westartwithaformulationthatequivalentlydenes().Anoptimizationproblemtocalculate()canbeposedasfollows,consideranarrivalratevector2o:max()s.t.ML+()e;NXi=1i=1;i0: 4.2.1ChallengesinComputing()Computing()isNP-hardthroughreductiontothefractionalcoloringproblem.Considerthefollowingreduction:givenagraphG,onecancompute(~e),whichisequalto1)]TJ /F3 11.955 Tf 12.52 0 Td[(f(G).Thusaproceduretocompute()canbeusedtosolvethefractionalchromaticnumberproblemforanygraphG.Alsonotethattheequality(~e)=1)]TJ /F3 11.955 Tf 12.65 0 Td[(f(G)holdsforonlyvector~eandisnotcorrectingeneral.Duetointractabilityoftheproblem;weprovideasimpleframeworktoestimate().Considerthefollowingequationsfor2o(G)thatrelatestoitsschedulingnorm(seeSection 2.7 forthedenition): 70

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minf1;NXiigf(G;)maxii:ItisNP-hardtocomputef(G;)ingeneral.AlsoapolynomialtimealgorithmtoprovideapproximationwithinN 2p logNimpliesP=NP[ 54 ].ConsequentlywewillconcentrateonaspecialcaseofarrivalvectorswherePNii<1.Nowconsiderthefollowingnumber()=1)]TJ /F25 7.97 Tf 6.59 5.98 Td[(PNii N.SinceNf(G)and1)]TJ /F12 11.955 Tf 12.42 8.96 Td[(PNii<1)]TJ /F3 11.955 Tf 12.42 0 Td[(f(G;)weconclude()<()thusB 2()canbeusedtoprovideanupperboundforstationaryqueue-sumunderMWS.Computationof()canbeperformedinlineartime.NotethatB 2()B 2()thustheexpectedqueue-sumboundprovidedinEquation 4 isatleastasgoodasthepreviousboundin[ 4 ].Nextwepresentanexamplewheretheimprovedboundisrealizedi.e.B 2()3thequeueboundisstrictlyimproved. 71

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Nextweinvestigatetheclassofgraphsforwhichthequeueboundsareidenticalusingtheoldparameterandthenewone.We'lldenotethisclasswithsymbolanddemonstratevariousclassesthatarecontainedwithin. Denition12. ThegraphclasscontainsallgraphsGsuchthat82o(G):()=(). Lemma32. AgraphGisinifandonlyifintheLinearProgrammingformulationoff(G)alloptimalsolutionshaveML=e. Proof. SupposeagraphGisin.Then82o(G):()=().AssumethereexitsanindexisuchthatanoptimalsetofdecisionvariablesforLPformulationof(seeDenition 1 )f(G)has(ML)i>1.Leti=(ML)i)]TJ /F6 7.97 Tf 6.58 0 Td[(1 f(G).Nowconsiderthearrivalvectorf:i=iandj=0ifj6=ig.Notethatf(G;+1 f(G)e)=f(G;1 f(G)e)=1whichimplies()=1 f(G).However()=1)]TJ /F5 7.97 Tf 6.59 0 Td[(i f(G)<(),contradictingtheassumptionGisin.Thusalloptimalsetofdecisionvariablesforf(G)satisesML=e.Supposeforalloptimalsolutionsoff(G)foragraphGhasML=e.AssumeG6=thenforsome2o(G)()>()orequivalentlyf(G;)+f(G;()e)>f(G;+e()).Howeverforoptimalsolutionsoff(G;ke)fork>0thereexistsnoslackintheinequalitiesoftheformMLke.Thusforall>0wehavef(G;)+f(G;()e)=f(G;+e())contradictingtheassumptionG6=. Werstneedtointroducesomedenitions(see[ 55 ]foramoredetaileddiscussion).Weconsiderundirectedgraphswithnoloopsandnomorethanoneedgebetweenanytwodifferentnodes,i.e.,thesimplegraphs. Denition13. AnisomorphismfromagraphG=(VG;EG)toagraphH=(VH;EH)isabijectionf:VG!VHsuchthat(u;v)2EGifandonlyif(f(u);f(v))2EH. Denition14. AnautomorphismofagraphGisanisomorphismfromGtoG. Denition15. AgraphG=(V;E)isvertex-transitiveifforeverypairu;v2Vthereisanautomorphismthatmapsutov. Lemma33. Theclassofvertextransitivegraphsiscontainedwithin. 72

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Proof. Firstweshowforvertextransitivegraphsinalloptimalsolutionsoff(G),ML=ewhereisthevectorofoptimaldecisionvariables.LPformulationoff(G)isgiveninDenition 1 :ThedualLPwhichisknownasfractionalweightedcliquenumberisthesolutionofthefollowingLP:wf(G;)=maxTs.t.(ML)Te;i0:LetGbeavertextransitivegraph.Forvertextransitivegraphsitisgivenin[ 12 ]thatwf(G)admitsanoptimalsolutionwherealli=1 (G)i=1;:::;N.Bycomplementaryslacknesstheorem,sincei>0wehaveatleastoneoptimalsolutiontotheprimalproblemwhereML=e.Assumethereexistsanotheroptimalsolution,i,forwhich(MLi)i>1.Nowconsidernodevi;vj2V:j6=i.Bythedenitionofvertex-transitivity,thereisanautomorphismfthatmapsvitovjandforthatmappingthereisanewsetofoptimaldecisionvariablesijforwhich(MLij)j>1.Nowifweapplytheautomorphismtogeteachnodekandgetikfork=1;:::;Nwegetacollectionofdecisionvariablesforeach(MLik)k>1.ConsiderthesumMLPNk=11 Nik>ehoweversinceeachikisoptimal;PNi=11 NeTik=f(G).ThusthevectorPNi=11 Nikcansatisfyconstraintsforf(G)andcanbescaleddownwithinconstraintstoachieveanewsolutionf(G)suchthatf(G)
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ACyclegraphwith5nodes BGraphwithanasymmetrictopologyFigure4-2. Vertextransitivitydoesnotrequireorimplymembershipinclass.A)Cyclegraphwith5nodes,C5,isvertex-transitiveandbelongstoclass.B)Thegraphdisplayedisnotvertex-transitivebutbelongstoclass. Figure4-3. IllustrationofK-cliqueandanodelooselyconnectedforK=6. ConsideragraphthatconsistsofaK-clique(forKinpositiveintegers)andasinglenodeconnectedtoonlyoneofthenodesinK-clique.Figure 4-3 illustratesthisgraphwhenK=6.WewillrefertoitasGinthefollowingdiscussion.Nowconsiderarrivalratevector=f1)]TJ /F6 7.97 Tf 15.56 4.7 Td[(2 K;0;:::;0gwherethelooselyconnectednoderepresentsthelinkhavingnon-zeroarrival(indexedwiththerstentry).Firstwenotef(G;)=1)]TJ /F6 7.97 Tf 15.41 4.71 Td[(2 Kandf(G)=K.Nextwehave()=1 K,toseethatobservefor=f1)]TJ /F6 7.97 Tf 15.4 4.71 Td[(2 K;1 K;:::;1 Kgwehavef(G;)=1and=+1 Ke.Thus()=1)]TJ /F6 7.97 Tf 6.59 0 Td[((1)]TJ /F20 5.978 Tf 9.22 3.25 Td[(2 K) K=2 K2where()=1 K.Thuswehave()K 2(),whichshowsthedifferencecanscalewithnodecountlinearlyunderthegivencircumstances. 74

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4.3TightnessofDelayParameters 4.3.1DerivingDelayBoundsforaGeneralizedMWSPolicyUsingaQuadraticPotentialFunctionIn[ 15 ]authorsproposedamodiedMWSpolicywherequeue-lengthsarescaledwithwi2R+thusactivationvectorm(t)attime-slottsatisesm(t)Tz=(G;z)wherezi=wiQi(t).Inthissectionwewilljustifytheparameterchoicemadein[ 15 ].Ouranalysisreveals,usingthepotentialfunctionfrom[ 15 ],theweightfunctionchosenoptimizesthederivedqueue-sumbound.Werevealtheservicevectorthatiscomparedwiththeactivationandthequeueweightsaretwosetsofdecisionvariablesareintegraltotheunderlyingoptimization.Inthissectionweperformoptimizationoverbothsets.HoweverinSection 4.3.2 wewillrestricttheweightstotheform~ewhilederivingasingleparameterperformancecharacterizationofdelayintraditionalMWSpolicy.Considerany2Co(ML)suchthat>andthefollowingLyapunovfunctionV:V(Q(t))=1 2NXi=1wiQ2i(t):Potentialdifferencebetweentwotime-slotscanbecalculatedas:V(Q(t))=1 2NXi=1wiQi(t+1))]TJ /F3 11.955 Tf 11.96 0 Td[(Qi(t)Qi(t+1)+Qi(t)=1 2NXi=1wiAi(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t)2Qi(t)+Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t)=NXi=1wii)]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t)Qi(t)+1 2jVjXi=1wiAi(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t)2NXi=1wi(i)]TJ /F3 11.955 Tf 11.95 0 Td[(i)Qi(t)+1 2jVjXi=1wiAi(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t)2: (4)AcommonwaytoboundE[PjVji=1Qi(t)]istonoteE[V(Q(t))]=0atstationarityandrelatetermPNi=1wi(i)]TJ /F3 11.955 Tf 12.11 0 Td[(i)Qi(t)toPNi=1Qi(t).Inthefollowingderivationtheonlyconstraint 75

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onqueuesizesistheyarerequiredtobenon-negativeintegerswhichindicatesthenalboundshouldbefreefromtermsQi(t).ThesecondterminEquation 4 ,wiAi(t))]TJ /F3 11.955 Tf 12.12 0 Td[(Di(t)2,playsapassiveroleinthederivationoftheboundssinceitisgenerallyupper-boundedeasily.LetBi=1 2i+E[A2i(t)])]TJ /F7 11.955 Tf 11.96 0 Td[(22ithenusingEquation 4 ,wehave:E[1 2jVjXi=1wi(Ai(t))]TJ /F3 11.955 Tf 11.96 0 Td[(Di(t))2]=NXi=1wiBi:Denei=wi(i)]TJ /F3 11.955 Tf 12.36 0 Td[(i)thenouraimistondisuchthatatstationarityforallQi(t)2ZN+,whereZN+isthesetofnon-negativeintegers,wehavePNi=1Qi(t)PNi=1iQi(t)PNi=1wiBi.Thatinturnreducestotheoptimizationproblemofmaximizingthequeue-sumundertheimpliedconstraintswhichcanbeposedas:Q=maxE[NXi=1Qi(t)]s.t.E[NXi=1iQi(t)]NXi=1wiBi:Ifwecanndaconstantk1suchthatkQPNi=1iQi(t)wecanconcludeourexpectedqueue-sumboundwillbeatleastasgoodasPNi=1wiBi k.Toachievethesmallestkwestartwiththefollowingsteps:E[NXi=1iQi(t)]NXi=1wiBi(minii)E[NXi=1Qi(t)]NXi=1wiBiE[NXi=1Qi(t)]PNi=1wiBi minii:Tondthebestpossiblestationaryqueueboundweneedtosolvethefollowingoptimizationproblemwithdecisionvariableswand: 76

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min;wPNi=1wiBi miniis.t.>;2Co(ML);w0:Theproblemabovecanberewrittenasatwoleveloptimizationproblem,rstaminimiza-tionoverallweightsandsecondaminimizationoverallservicevectors,asfollows:minminwPNi=1wiBi miniis.t.>;2Co(ML);w0:Wecandothefollowingscalingwithoutaffectingtheoptimalvaluesofthedecisionvariables(thoughtheoptimalvalueoftheoptimizationproblemwillbescaled)minii=miniwi(i)]TJ /F3 11.955 Tf 11.95 0 Td[(i)=1,thenwehave:minminwNXi=1wiBis.t.>;2Co(ML);w0;miniwi(i)]TJ /F3 11.955 Tf 11.95 0 Td[(i)=1:Whenisxedweget:minwNXi=11 2wii+E[A2i(t)])]TJ /F7 11.955 Tf 11.96 0 Td[(22is.t.w0;miniwi(i)]TJ /F3 11.955 Tf 11.95 0 Td[(i)=1:Nowsinceminiwi(i)]TJ /F3 11.955 Tf 12.21 0 Td[(i)=1isaconstraint,ifwesetwi=1 i)]TJ /F5 7.97 Tf 6.59 0 Td[(ithatconstraintwillbesatisedforallindicesianditwouldbeanoptimalsolutiontothegivenoptimizationproblem 77

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sincetheconstraintswillallbesatisedwithequality.NowpluggingthenewdenitionofwtotheoriginalproblemwehaveminNXi=1i+E[A2i(t)])]TJ /F7 11.955 Tf 11.95 0 Td[(22i 2(i)]TJ /F3 11.955 Tf 11.95 0 Td[(i) (4)s.t.>;2Co(ML); (4)asthetightestqueueboundthatisofferedbytheframeworkthatcanbederivedusingthequadraticpotentialfunctionchosen.Notethatthepotentialfunctionisanaturalone,sinceitcorrespondstothesumofintegrationsoftheweightfunctions.Nextwewillstudythisclaimindetail. 4.3.2EstablishinganIntuitiveMeaningfor():AStepTowardsSingleParameterDelayCharacterizationofMWSPolicyThissectionderivesthedelayperformanceofMWSpolicyusingthetechniquesemployedinSection 4.3.1 .MWSpolicyreducestothegeneralizedalgorithmwiththeidenticalqueueweights,wi=wj8i;j.UsingthesamepotentialfunctionfromSection 4.3.1 wewillconductasimilarderivation.Ifwesettheweightstobeidenticali.e.:w=~ewehavethefollowingoptimizationproblem:minPNi=1Bi minii)]TJ /F3 11.955 Tf 11.96 0 Td[(is.t.>;2Co(ML):Byomittingtheconstantfactors,avariationoftheproblemabovecanberewrittenas(argminispreserved): 78

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maxminii)]TJ /F3 11.955 Tf 11.96 0 Td[(is.t.>;2Co(ML):Theoptimizationproblemabovewillbesatisedbyavectorontheboundaryofthecapacityregion.Thuswecanmodifytheconstraintsaccordingly:maxminii)]TJ /F3 11.955 Tf 11.96 0 Td[(is.t.>;ML=;MLXi=1i=1;i0:Let=minii)]TJ /F3 11.955 Tf 11.95 0 Td[(i.Thentheoptimizationproblemcanberewrittenas:maxs.t.ML)]TJ /F3 11.955 Tf 11.95 0 Td[(>e;MLXi=1i=1;i0;whichistheequivalenttotheLPformulationof().Asaresult,whenwerestrictourselvestotheunweightedcaseweobtain()asthetightestsingle-parameterperformancecharacteriza-tion.Anothervariationofthequestionaddressesaboveiswhatifwerestrictthevectorstobeoftheform+efor>0. 79

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min=+e;>0minwPNi=1wiBi miniis.t.>;2Co(ML);w0:min>0minwPNi=1wiBi miniwis.t.+e2Co(ML);w0:ThescalingargumentusedabovefortheminiiThusw=eisoptimalforminwPNi=1wiBi miniwi.Now,againbyomittingtheconstantterm,theoptimizationproblemcanbetranslatedto:min1 s.t.+e2Co(ML);w0;whichwithminorarrangementsisequivalenttooptimizationproblemofnding()fromSection 4.3.1 .Theanalysisconductedaboveestablishesthesignicanceof();withthenaturalchoiceofapotentialfunction,()providesthetightestboundthatcanbeachievedthroughquadraticpotentialfunctionframework.In[ 15 ]authorsprovetheboundsderivedusinggeneralizedMWSpolicywiththeweightselectioninEquation 4 with 4 areatleastasgoodastheboundsthatareachievedbyarandomizedpolicyusingaprobabilitydistributionofactivationoverthesetofschedules.IntuitivelyoptimalinEquation 4 with 4 actsastheaverageservicerateachievedoverthelinks,whentheweightvectorispickedtobe1 )]TJ /F5 7.97 Tf 6.58 0 Td[(.SimilarlyitispossibletoreasonovertheservicerateofMWSpolicywhichis+(). 4.4ResultsandDiscussionInthischapterweattempttoanswerthequestion:WhatisthebestparameterofchoicetocharacterizethedelayperformanceofMWSpolicy?Althoughthereareearlierattemptstobound 80

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delaythroughstationaryqueue-sumsuchasin[ 48 ],[ 26 ],[ 15 ],andourattemptsinChapter 3 ,ouremphasisinthischapterisnotonlyontheboundbutalsoonthetightnessoftheboundsprovided.WiththatendinviewwerstintroduceanewparametertobounddelayofMWSpolicy,thatoutperformspreviousresults.Nextweprovethecomputationofthenewparameterisintractableandprovideasimplemethodtoestimateit.Tocomparetheoldandnewparametersonthedelayboundweprovideanexamplethatdemonstratesanimprovementtoberealized.Theperformancecomparisonbetweentheoldandnewparameterisalsoconductedanditisshownthattheboundswiththenewparametercanbesuperiortoalinearscale,wherelinearityiswithrespecttothelinkcountinthewirelessnetwork.Thedistinctnessoftheoldandthenewparameter,however,isnotrealizedonallgraphclasses.Weidentifynecessaryandsufcientconditionsforagraphtohaveidenticalvaluesforoldandnewparameters.Asaresultweprovefortheclassofvertex-transitivegraphsbothparametersassumethesamevalue.Thesecondpartofthechapterdealswithndingtightdelayboundsusingquadraticpotentialfunctions.Weshowtheweightschosenin[ 15 ]areoptimalwhenthequadraticpotentialfunctionisoptimizedoverallpossibleservicevectors,andallpossibleweightselections.Nextweutilizethesameframeworktoshow,thenewparameterintroducedisalsooptimalthroughthesamemethodology.AsaresultofourstudywebuiltanintuitionontheservicevectorthatisachievedbybothgeneralizedandnativeversionsofMWSpolicy. 81

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CHAPTER5GENERALIZINGMWSPOLICY 5.1BackgroundInthischapterwestudyageneralizationofMWSpolicy.ThetraditionalMWSpolicy[ 52 ]usesthequeuelengthvectortocomputeWMISforactivation.Althoughthatapproachensuresstability,itisnotimmediateifotherperformancemeasures,e.g.delayorfairnessamonglinks,areoptimized.In[ 15 ]authorsproposeageneralizationofMWSpolicyusingpositiveconstantstoscalequeuesizes.Asaresult,delayboundsareimproved.Thuswhenasecondarysetofobjectivesareinquestionageneralizationmightoutperformthenativealgorithm.Thischapterfurtherextendsthegeneralizationin[ 15 ]byreplacingtheconstantweightswithgeneralfunctions.Inotherwordsateachtimeslotthequeueweightiscomputedusingafunction,whichmightbedistinctfordifferentqueues.Ourmethodologyextendstheframeworkpresentedin[ 27 ],whichstudiesthesufcientconditionsonthosefunctions.Thevalueofthisgeneralizationistwofold.FirsttheforemostdifcultyinMWSpolicyliesinsolvingtheWMISproblemandcomputingweightswitharbitraryfunctionsdonotfurthercomplicatethistaskunlessthefunctionsitselfarehardtocompute.Secondanoptimizationforasecondaryparametersetisdoneoverallthroughputoptimalfunctions.Weneedtoidentifythefeasiblesetoftheoptimizationtostartsuchaprocedure.Alsowithageneralenoughframeworkwecandirectlyconcludethealgorithmpresentedin[ 15 ]isimmediatelythroughputoptimal.Althoughthedelayboundspresentedin[ 15 ]isatleastasgoodasanyrandomizedalgorithmthatusesaprobabilitydistributionoverthesetofpossibleschedules,itrequirestomeasurethearrivalrateandsolveanoptimizationproblemusingtheestimatedarrival.Theproblemwiththisapproachisthatmeasuringthearrivalwithprecisionrequiresalongperiodofsamplecollection(thesamplesizecanbeboundedusingChebyshev'sinequality),whichmightevenbelongerthanthetimenetworkreachesstationarityfromtheinitialstate. 82

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5.2ChoosingWeightFunctioninMWSforStabilitySystemmodelinthischapterfollowscloselyfromchapter 3 .Wereferreadertosection 3.2 fordetailedsystemmodelandnotations.Notethatthereareminordifferencesintroducedbuttheyareunderlinedinthediscussions.In[ 52 ]authorsproposedtheMWSpolicyforthroughoutoptimality.Inthissectionwewillstudyageneralization.Wedeneaf()-MWSscheme,foravectorfunctionf:RN!RN,tobeapolicywhoseactivationvectors,m(t),foralltimeslotsttosatisfy:NXi=1fi(Qi(t))mi(t)maxm2MLNXi=1fi(Qi(t))miAlthoughMWSpolicyisthroughputoptimal,itisnotimmediatelyclearifitispossibleoptimizeotherparametersofinterestusingtheMWSschemewithdifferentweightfunctions.In[ 27 ]authorsprovidedasufcientconditionforweightfunctionstobethroughputoptimalunderMWSschemes.Wewillreneandextendtheirstatementinthefollowingdiscussion.In[ 50 ]authorscommentedthat-MWSschemes(i.e.:8ifi(x)=x)provideoptimalsteadystatedelayboundswhen!0.Laterin[ 15 ]authorsprovedthatthefunction8ifi(x)=wixcanprovidestationaryexpectedqueuesumperformancethatsurpassesanyrandomizedpolicyassigningconstantactivationprobabilitiestotheindividualschedules.Inwhatfollowswewillgeneralizetheresultin[ 27 ]toageneralizedinterferencemodelusinggraphs.Alsotheresultin[ 27 ]assumesBernoulliarrivalprocessforwhichA(t)2f0;1gandwegeneralizedtheschemeforA(t)2f0;g.ConsiderthepotentialfunctionF(Q(t)),PNi=1Fi(Qi(t))whereFi(x)=Fi(0)+Zx0fi(u)du:Let(t),A(t))]TJ /F3 11.955 Tf 11.97 0 Td[(D(t),thus)]TJ /F7 11.955 Tf 9.3 0 Td[(1(t).WeassumethesecondmomentofthearrivalprocessisboundedbyCasinChapter 3 .Nowtheexpectedpotentialdifferencebetweentwoconsecutivetimeslotscanbecomputedas: 83

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E[F(Q(t))jQ(t)]=E[F(Q(t+1)))]TJ /F3 11.955 Tf 11.96 0 Td[(F(Q(t))jQ(t)]=E[F(Q(t)+(t)))]TJ /F3 11.955 Tf 11.95 0 Td[(F(Q(t))jQ(t)]=NXi=1E[Fi(Qi(t)+i(t)))]TJ /F3 11.955 Tf 11.95 0 Td[(Fi(Q(t))jQ(t)]=NXi=1E"Zi(t)0fi(Qi(t)+u)dujQ(t)#: (5)GivenRa0(a)]TJ /F3 11.955 Tf 11.96 0 Td[(x)f0(b+x)dxwecandointegrationbyparts:Zbaudv=[uv]ba)]TJ /F12 11.955 Tf 11.96 16.27 Td[(Zbavdu: (5)Leta)]TJ /F3 11.955 Tf 11.95 0 Td[(x=uandf0(b+x)dx=dvinEquation 5 ,thenwehave;Za0(a)]TJ /F3 11.955 Tf 11.95 0 Td[(x)f0(b+x)dx=[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(x)f(b+x)]a0+Za0f(b+x)dx:=[(a)]TJ /F3 11.955 Tf 11.95 0 Td[(x)f(b+x)]a0+Za0f(b+x)dx:=)]TJ /F3 11.955 Tf 11.95 0 Td[(af(b)+Za0f(b+x)dx: (5)UsingEquation 5 inEquation 5 theexpectedpotentialdifferencebecomes:E[F(Q(t))jQ(t)]=NXi=1E"i(t)fi(Qi(t))+Zi(t)0(i(t))]TJ /F3 11.955 Tf 11.96 0 Td[(u)f0i(Qi(t)+u)dujQ(t)#:Letf(Qi):=maxu2fQi)]TJ /F6 7.97 Tf 6.59 0 Td[(1;Qi+gjf0(u)jthen: 84

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E[F(Q(t))jQ(t)]Ef(Q(t))T(t)jQ(t)+NXi=1E2i(t) 2fi(Q(t))jQ(t) (5)Foreachqueuei,functionDi(t)indicateswhetherornotlinkireceivedserviceattimeslott.Notethat:Di(t)=8>><>>:1;ifQi(t)1andiisscheduled.0;otherwise.Althoughthediscrepancybetweenactivationandbeingabletoserveapacketatatimeslotisseeminglyaminorone,nextwe'llshowduetothisdiscrepancythesufcientconditionin[ 27 ]needsamodication.Thediscrepancybetweenactivationanddeparturecanbeboundedasfollows:f(Q(t))TD(t)+(G;f(Q(t)));where=maxm2MLNXi=1fi(0)mi:ConsideringtherstterminEquation 5 :Ef(Q(t))T(t)jQ(t)=Ef(Q(t))TA(t))]TJ /F3 11.955 Tf 11.96 0 Td[(f(Q(t))TD(t)jQ(t))]TJ /F3 11.955 Tf 21.91 0 Td[(()NXi=1fi(Qi(t))+:Notethat: 85

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2i(t)=(Ai(t))]TJ /F3 11.955 Tf 11.95 0 Td[(Di(t))2=A2i(t))]TJ /F7 11.955 Tf 11.96 0 Td[(2Ai(t)Di(t)+D2i(t)C+1:Returningtodrift:E[F(Q(t))jQ(t)])]TJ /F3 11.955 Tf 21.92 0 Td[(()NXi=1fi(Qi(t))++(C+1) 2NXi=1fi(Qi(t)): (5)NowwewillstateFoster'scriteriawhichprovidesaneasyguidancetoestablishpositiverecurrenceoftheunderlyingMarkovchain.NotethatstabilityandpositiverecurrenceoftheunderlyingMarkovchainareequivalentconditions. Theorem5.1. (Foster'scriteria).SupposeMarkovchainfXngisirreducible.Supposethereexistsalower-boundedfunctionV:E!R,some>0,andanitesubsetE0ofEsuchthatE[V(Xn+1))]TJ /F3 11.955 Tf 11.95 0 Td[(V(Xn)jXn=i])]TJ /F3 11.955 Tf 21.92 0 Td[(;ifi=2E0E[V(Xn+1)jXn=i]<1;ifi2E0:Then,theMarkovchainispositivepositiverecurrent.Letkxk1:=Pni=1jxij.Assumef0i()iscontinuousforalli,x2RN+andthefollowingconditionholds:limkxk1!1PNi=1fi(xi)+maxm2MLPNi=1fi(0)mi PNi=1fi(xi)=0: (5)ThentheresultingMarkovchainispositiverecurrentunderpoliciessatisfyingthef()-MWSscheme.TheoutlineoftheproofincludescombiningTheorem 5.1 andEquation 5 86

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Itispossibletomeettheconditionbyrestrictingtheindividualfunctionstothefollowingclass.If8ilimx!1fi(x) fi(x)=0;fi(0)=0;thenconditioninEquation 5 holds.Alargeclassoffunctionssatisestheconditionabove.fi(x)=log(x);fi(x)=xkk2R+arefewexamples.Howevernotallincreasingfunctionsareinthatclass.Forexamplealthoughexponentialfunction,f(x)=exp(x)isincreasing,itdoesnotsatisfytheconditionabove.Alsotheslightdifferencewiththestatementin[ 27 ]rulesoutsomefunctionsthatwerenotruledoutpreviously,e.g.:fi(x)=xkk2fR)]TJ /F13 11.955 Tf 11.96 0 Td[(R+g.Nextwewilldiscussthemaintheoremin[ 27 ]andtheeffectsoftheoverlookedterm: Lemma34. (Theorem1in[ 27 ])Assumethatf:R!Risnon-negativeandcontinuouslydifferentiableonR+,andthatlimx!1f(x) f(x)=0.Then,foranyBernoullii.i.d.admissibletrafcforqueuesi,f(x)-MWSsatises:E[Qi(t)]<1Notethatfunctionssuchasf(x)=1 x;f(x)=)]TJ /F3 11.955 Tf 9.3 0 Td[(xsatisfytheconditionprovidedinLemma 34 wherethef()-MWSschemefailstoprovidethroughputoptimalitywhenthosefunctionsareused.Theformerfunctionfailstoprovidedriftduetotermbeingindenitewhenx=0andlatterfunctionhasacorrespondingpotentialfunction,F(x)=)]TJ /F6 7.97 Tf 10.5 4.71 Td[(1 2x2,whichisnotlowerboundedarequirementimposedbytheFoster'sCriteria.Choosingaconstantfunctionfortheweightsi.e.:fi(x)=kk2R+willnotestablishstability,sinceitignoresarrivalpatterninmakingdecisions.Aslightmodicationwhere:f(x)=8>><>>:0;ifx=0k;ifx0; 87

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willalsofailtoprovidestabilityduetothethedecisionsbeingmadewithouttakingthequeuesizesintoaccountwhentheygrowlarge.ConsiderthefollowingnetworkconsistingofthreelinkshavingBernoulliarrivalpatternswiththeexpectedarrivalratesof=f1 2;1 3;1 2g. Figure5-1. Conictgraphofathreelinknetworkillustratingthelinetopology. Whentherearepacketstoserve,theMWSalgorithmwiththeabovementionedfunctionactivatesthemaximumcardinalityindependentsetonthegraphthatisinducedbynon-emptyqueues.ForndingthemaximumindependentsetMWSalgorithmisfreetobreaktiesarbitrar-ily,andstillensurestability.Thusthefollowingchoiceonfavoringlink1and3over2isduetothesamefreedom.Giventhatiftherearearrivalstoeithernode1or3(orboth);thealgorithmwillactivatetheschedulef1;3g.Iftherearenoarrivalsinthecorrespondingtime-slotto1and3,thenlink2willbescheduledfortransmission.SincearrivalprocessisBernoulli,theprobabilityofactivationforlink2is1 4.Howeverlink2hasanaveragearrivalrateof1 3thusitsqueuewillgrowindenitely.Anupperboundedincreasingfunctionsatisfyingf(0)=0ontheotherhandwillensurestability.Intuitivelythereisadifferencebetweentreatingqueuesizesequivalentlyafterathresholdandfavoringthelongerones,eventhoughthefavoringprocesslosesitsstrengthasthequeuesizesincrease. 5.3ResultsandDiscussionInthischapterweconsiderageneralizationofMWSpolicywherefunctionsofqueuelengthsareusedtoobtainweights,beforeWMISiscomputed.Weavoidmakingconstrainingassumptionsonthenatureoffunctions,onlyrequirethemtobeintegrableanddifferentiable.Werenethesufcientconditionsprovidedin[ 27 ]andshowthatpreviouslyoverlookedconditionssuchaspotentialfunctionbeinglower-boundedmightplayanimportantroleinfunctionprovidingstability.Aconnectionbetweenthefunctionvaluewhenqueuesareemptyandstabilityisalsoestablished.Accordingly,forachoiceoffunctiontoprovidestability,not 88

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onlyitisrequiredtodominateitsderivativeasymptoticallybutalsothefunctionvalueatpoint0.Alsothestatementshouldbetrueforsumsofentities,whenanon-uniformfunctionsetisbeinganalyzed.Therearestillopenquestionsfollowingourdiscussion.Thewiderangeoffunctionssatisfyingthroughputoptimalitybringsupthequestion:Isispossibletoachieveboundsin[ 15 ]usingareactivealgorithm(respondingtoonlycurrentqueuesizes)asopposedtomeasuringthearrivalpatterntoobtaininformationabouttheaveragerates?Thatresultwouldprovidesatisfactorydelaycharacteristicswhileeliminatingtherequirementtomeasureandestimatearrivalpatternaveragewhichisadifculttasktoaccomplishwithprecisionwithinoperationalconstraintsofawirelesscommunicationnetwork. 89

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CHAPTER6CONCLUSIONS 6.1SummaryofResultsWehaveprovidedanegativityresultonthecomputationalcomplexityofthroughputoptimalalgorithmsthathaspolynomiallyboundedtransientqueuesizeswithpertime-slotpolynomialcomputations.Ourresultsarederivedwithinadeterministicsettingwheretherandomizednatureofthearrivalsarenotfullycaptured.Howevercombiningwiththeresultsprovidedin[ 49 ],ourresultscanalsobeusedtoconcludeaboutsimplerandomizedcaseswithminormodication.Wefoundtheframeworkprovidedin[ 49 ]andourframeworktoposesdifferentstrengthsandeachcapturingdifferentsetofparametersbetter.Ourframeworkincorporatesthearrivalvector'sdistancetotheboundaryandaccordinglyshowsthereexistsapositivecounterparttoourargument:ifthereisWMISsolveravailable,itispossibletoachieveefcientqueuesizeswithpertime-slotpolynomialcomputations.However[ 49 ]eliminatedthedistancetotheboundarybyembeddingittotheapproximationratioandnallyachievingatightboundthatisderivedoutofinapproximabilityofmaximumindependentsetproblem.Asofnowwearenotawareofanymethodstoextendeitherframeworktocontaintheotheronewithin.WealsoprovidedboundsofapproximateMWSschedulingatsystemstationarity.Ourresultprimarilybuildsonagraphtheoreticalinequality,whichcapturesthedriftconditionsonthequadratic,andhigherorder,potentialfunctions.Inashrinkingportionofthecapacityregionitispossibletoderiveboundsforhigherordermomentsaswell.Weprovidedtheformulafortheshrinkingfactorintermsoftopologicalcharacteristicsoftheinterferencegraphandtheapproximationratiok.WehavedisplayedtheefciencyofLQFpolicywhenitiscomparedwithagenericapproximateMWSpolicy.OurresultfurtherjustiesusingLQFinpracticalsettings.AlthoughMWSpolicyprovidesthroughputoptimalityitsdelayperformanceisnotoptimal.In[ 15 ]ageneralizationofMWSisproposedtoachievetighterdelaybounds.WestartwithimprovingdelayboundsforMWSpolicyandprovideresultspertainingtoanewparameterthatdenestheimprovement.Theweightparameterchoicein[ 15 ]providesattractivequeue-sum 90

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bounds,weprovetheweightselectionisdoneoptimallyandusingthesameframeworkweshowournewboundistight.FinallywestudyageneralizationofMWSpolicywhereafunctionofqueuesizereplacesthequeuesizeincalculationofthequeueweights.Werenedpreviouslyprovidedresultsin[ 27 ],andexplainwhyaclassoffunctionssatisfyingtheconditionsthereinfailtoprovidestability.Wealsoconcentrateonthemaximumcardinalityschedulingalgorithmandprovideanexamplewhereisfailstostabilizeanetwork.Thatunderlineswithinthespectrumofaconstantfunctiontoanupper-boundedincreasingfunctionstabilityisachievedsomewhereinthetransition. 6.2FutureResearchGiventherevealedlimitationsofschedulingpolicieswebelievefutureresearchshoulddirecttowardsunderstandingthewirelessinterferencebettersothateitherthegraphbasedK-hopinterferencemodelwillbereplacedwithanewmodelthatadmitseasierper-time-slotsub-problemsortherealistictopologycharacteristicswillbeunderstoodtothepointwherethehardnessresultswillnolongerbeapplicable.Forexampleactionaldiscoveryoninterferencegraphsrevealingtheircontainmentwithinthefamilyofperfectgraphswillrulethehardnessresultsoutimmediatelyduetofractionalcoloringandindependentsetproblemsbeingpolynomiallysolvableontheperfectgraphs. 91

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[42] EytanModiano,DevavratShah,andGilZussman.Maximizingthroughputinwirelessnetworksviagossiping.SIGMETRICSPerform.Eval.Rev.,34(1):27,2006. [43] MichaelJ.Neely,EytanModiano,andCharlesE.Rohrs.Tradeoffsindelayguaranteesandcomputationcomplexityfornxnpacketswitches.InProc.ofCISS,2002. [44] M.J.Neely.Delayanalysisformaximalschedulinginwirelessnetworkswithburstytrafc.InProc.ofIEEEINFOCOM,pages6,April2008. [45] S.RajagopalanandD.Shah.Distributedalgorithmandreversiblenetwork.InProc.ofCISS,pages498,March2008. [46] SujaySanghavi,LocBui,andR.Srikant.Distributedlinkschedulingwithconstantoverhead.SIGMETRICSPerform.Eval.Rev.,35(1):313,2007. [47] S.SarkarandS.Ray.Arbitrarythroughputversuscomplexitytradeoffsinwirelessnetworksusinggraphpartitioning.IEEETransactionsonAutomaticControl,53(10):2307,November2008. [48] DevavratShahandMilindKopikare.Delayboundsforapproximatemaximumweightmatchingalgorithmsforinputqueuedswitches.InProc.ofIEEEINFOCOM,pages1024,2002. [49] DevavratShah,DavidN.C.Tse,andJohnN.Tsitsiklis.Hardnessoflowdelaynetworkscheduling,2009. [50] DevavratShahandDamonWischik.Optimalschedulingalgorithmsforinput-queuedswitches.InProc.ofIEEEINFOCOM,pages1,2006. [51] G.Sharma,R.R.Mazumdar,andN.B.Shroff.Onthecomplexityofschedulinginwirelessnetworks.InProc.ofACMMOBICOM,pages227,2006. [52] L.TassiulasandA.Ephremides.Stabilitypropertiesofconstrainedqueueingsystemsandschedulingpoliciesformaximumthroughputinmultihopradionetworks.IEEETransactionsonAutomaticControl,37(12):1936,Dec1992. [53] LeandrosTassiulas.Linearcomplexityalgorithmsformaximumthroughputinradionetworksandinputqueuedswitches.InProc.ofIEEEINFOCOM,pages533,1998. [54] LucaTrevisan.Non-approximabilityresultsforoptimizationproblemsonboundeddegreeinstances.InProceedingsofthethirty-thirdannualACMsymposiumonTheoryofcomputing,STOC'01,pages453,NewYork,NY,USA,2001.ACM. [55] DouglasB.West.IntroductiontoGraphTheory(2ndEdition).PrenticeHall,2000. [56] XinzhouWu,R.Srikant,andJamesR.Perkings.Schedulingefciencyofdistributedgreedyschedulingalgorithmsinwirelessnetworks.InProceedingsofIEEEINFOCOM,2006. 95

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[57] YungYi,AlexandreProutiere,andMungChiang.Complexityinwirelessscheduling:impactandtradeoffs.InProc.ofACMMOBIHOC,pages33,2008. [58] G.Zussman,A.Brzezinski,andE.Modiano.Multihoplocalpoolingfordistributedthroughputmaximizationinwirelessnetworks.InProc.ofIEEEINFOCOM,pages1139,April2008. 96

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BIOGRAPHICALSKETCH CemBoyacireceivedhisB.S.incomputerengineeringfromBilkentUniversity,Ankara,Turkeyin2005,andhisM.S.incomputerengineeringfromUniversityofFlorida,Gainesville,Floridain2011.Hisresearchinterestsareinperformanceandcomplexitycharacterizationofdelayinwirelesscommunicationnetworksviagraphtheoreticalmodels. 97