Distributed Solutions for Rate Control and Maximum Lifetime in Wireless Networks

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Permanent Link:: https://ufdc.ufl.edu/UFE0024822/00001

Material Information

Title:: Distributed Solutions for Rate Control and Maximum Lifetime in Wireless Networks
Creator:: Zhang, Liang
Place of Publication:: [Gainesville, Fla.]
Publisher:: University of Florida
Publication Date:: 2009
Language:: english
Physical Description:: 1 online resource (106 p.)

Thesis/Dissertation Information

Degree:: Doctorate ( Ph.D.)
Degree Grantor:: University of Florida
Degree Disciplines:: Computer Engineering
Computer and Information Science and Engineering
Committee Chair:: Chen, Shigang
Committee Members:: Liu, Chien-Lian
Sahni, Sartaj
Chow, Yuan-Chieh R.
Wong, Tan F.
Graduation Date:: 8/8/2009

Subjects

Subjects / Keywords:: Algorithms ( jstor )
Bandwidth ( jstor )
Buffer zones ( jstor )
Channel capacity ( jstor )
Flow velocity ( jstor )
Mathematical vectors ( jstor )
Scheduling ( jstor )
Sensors ( jstor )
Simulations ( jstor )
Topology ( jstor )
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
fairness, flow, lifetime, maxmin, mesh, multihop, network, rate, sensor, wireless
Genre:: Electronic Thesis or Dissertation
born-digital ( sobekcm )
Computer Engineering thesis, Ph.D.

Notes

Abstract:: This study focuses on fairness in wireless networks. Two fairness problems are addressed: end-to-end flow rate fairness in multihop wireless networks and lifetime fairness in wireless sensor networks. In recent years, the advent of multihop wireless networks has greatly accelerated the research on bandwidth management in such networks to support new applications. While much research concentrates on the MAC layer, the user?s perception on these networks is however determined mainly based on the networks? end-to-end effectiveness. It is important for us to develop flexible tools for traffic engineering in multihop wireless networks. In this study, two solutions are proposed to achieve end-to-end maxmin flow rate fairness in such networks. A cross-layer design is firstly proposed for achieving end-to-end maxmin fairness in wireless mesh networks. In this approach, a generalized maxmin model is first proposed for multihop wireless networks. At the network layer, our design allocates network capacity to end-to-end flows for maxmin bandwidth allocation. At the MAC layer, our design achieves the allocated bandwidth shares for flows through a two-level weighted fair queuing algorithm. The proposed design is able to equalize the end-to-end bandwidth allocation to competing flows that share common bottlenecks, while fully utilizing the network capacity. Results of simulations are presented to demonstrate the effectiveness of the proposed solution in enhancing end-to-end fairness. We also propose a fully distributed solution that is compatible with IEEE 802.11 DCF for achieving end-to-end maxmin fairness. We transform the global maxmin objective to four local conditions and prove that, if the four local conditions are satisfied in the whole network, then the global maxmin objective must be achieved. We then design a distributed rate adaptation protocol based on the four conditions. Whenever a local condition is tested false at a node, the node informs the sources of certain selected flows to adapt their rates such that the condition can be satisfied. Comparing with previous work, our protocol has a number of advantages. First, it does not modify the backoff scheme of IEEE 802.11. Second, it replaces per-flow queueing with per-destination queueing. Packets from all flows to the same destination is queued together. Third and most important, our protocol achieves far better fairness (or weighted fairness) among end-to-end flows than previous work. Wireless sensor networks have a wide range of applications in habitat observation, seismic monitoring, battlefield sensing, etc. As another type of multihop wireless network, a sensor network consists of battery-powered sensor nodes that are limited in energy supply. An important problem of wireless sensor networks is maximizing the operational lifetime of a sensor network. The lifetime of a sensor network is defined as the lifetimes of all sensors that produce useful data. A centralized solution proposed by previous work requires solving a sequence of linear programming problems. The computation overhead can be prohibitively high for large sensor networks. Collecting the complete information about the network and uploading the complete forwarding policies to all nodes require significant amount of transmissions, particularly for nodes around the sink. We propose a fully distributed progressive algorithm which iteratively produces a series of lifetime vectors, each better than the previous one. Instead of giving the optimal result in one shot after lengthy computation, the proposed distributed algorithm has a result at any time, and the more time spent gives the better result. We show that when the algorithm stabilizes, its result produces the maximum lifetime vector. Furthermore, the algorithm is able to converge rapidly towards the maximum lifetime vector with low overhead. ( en )
General Note:: In the series University of Florida Digital Collections.
General Note:: Includes vita.
Bibliography:: Includes bibliographical references.
Source of Description:: Description based on online resource; title from PDF title page.
Source of Description:: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:: Thesis (Ph.D.)--University of Florida, 2009.
Local:: Adviser: Chen, Shigang.
Statement of Responsibility:: by Liang Zhang.

Record Information

Source Institution:: University of Florida
Holding Location:: University of Florida
Rights Management:: Copyright Zhang, Liang. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:: 489236254 ( OCLC )
Classification:: LD1780 2009 ( lcc )

UFDC Membership

Aggregations:: Institutional Repository at the University of Florida (IR@UF)
University of Florida Theses & Dissertations
University of Florida
INTERNAL UF RETRO ETDS

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Firstofall,Iwouldliketothankmyadvisor,Prof.ShigangChen,forhisconstantguidance,support,andinsightfuladvicethroughoutmygraduatestudy.Heisaterricadvisor,apassionateresearcherandacriticalthinker.Withoutthenumerousdiscussionswithhim,theworkpresentedinthisdissertationwouldneverhaveexisted.IamgratefultoProf.SartajSahni,Prof.RandyChow,Prof.JonathanLiu,Prof.TanWong,andProf.LiuqingYang,fortheirinstructivecommentsandsupportduringmystudy.IwouldalsoliketothankallmycolleaguesinProf.Chen'sresearchgroup,includingZhanZhang,MyungKeunYoon,YingJian,MingZhangandTaoLi,forprovidingvaluablefeedbackandhighlevelofresearchsupport.IwouldalsoliketotakethischancetoexpressmyendlesslovetomywifeXiaojieSun,myparents,andmybrother.Withouttheirlove,understanding,encouragementandsupport,noneofthesewouldhavebeenpossible. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 14 1.1End-to-EndFlowRateFairness ........................ 14 1.2LifetimeFairnessinSensorNetworks ..................... 17 1.3MaximizingLifetimeVectorandMaximizingRateVectorinSensorNetworks 19 1.4RelatedWork .................................. 19 1.4.1FlowRateFairness ........................... 19 1.4.2LifetimeFairnessinSensorNetworks ................. 21 2CROSS-LAYERDESIGNFORACHIEVINGEND-TO-ENDMAXMIN .... 24 2.1NetworkModelandMaxminModel ...................... 24 2.1.1NetworkModel ............................. 24 2.1.2MaxminModel ............................. 25 2.2AGeneralizedMaxminModel ......................... 26 2.2.1ResourcesinWMNs ........................... 26 2.2.2GeneralizedMaxminModel ...................... 27 2.3PacketSchedulingAlgorithm .......................... 29 2.3.1Overview ................................. 29 2.3.2Inter-NodeScheduling ......................... 30 2.4PerformanceEvaluation ............................ 32 2.5Summary .................................... 34 3FULLYDISTRIBUTEDSOLUTIONFORACHIEVINGGLOBALEND-TO-ENDMAXMIN ....................................... 37 3.1Preliminaries .................................. 37 3.1.1NetworkModelandProblemStatement ................ 37 3.1.2CongestionAvoidanceandBuer-BasedBackpressure ........ 39 3.2LinkClassication ............................... 40 3.2.1SaturatedBuer ............................. 40 3.2.2ThreeLinkTypes ............................ 41 3.2.3SaturatedClique ............................ 42 3.3LocalConditionsforGlobalMaxmin:Single-DestinationCase ....... 43 5

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................................ 43 3.3.2NormalizedRate ............................. 44 3.3.3LocalConditionsforGlobalMaxmin ................. 45 3.3.4CorrectnessProof ............................ 47 3.4LocalConditionsforGlobalMaxmin:Multiple-DestinationsCase ..... 52 3.4.1Per-DestinationPacketQueueing ................... 52 3.4.2VirtualNodes,VirtualLinks,andVirtualNetworks ......... 53 3.4.3LocalizedRequirementsforGlobalMaxmin .............. 54 3.5DistributedGlobalMaxminProtocol(GMP) ................. 56 3.5.1Overview ................................. 56 3.5.2MeasurementPeriod .......................... 57 3.5.3AdjustmentPeriod ........................... 59 3.6Simulation .................................... 62 3.6.1EectivenessofGMP .......................... 62 3.6.2PerformanceComparison ........................ 63 3.7Summary .................................... 65 4DISTRIBUTEDPROGRESSIVEALGORITHMFORMAXIMIZINGLIFETIMEVECTORINWIRELESSSENSORNETWORKS ................. 70 4.1NetworkModelandProblemDenition .................... 70 4.1.1SensorNetworkModel ......................... 70 4.1.2VolumeSchedule ............................ 71 4.1.3MaximumLifetimeVectorProblem .................. 72 4.1.4RoutingGraph ............................. 73 4.2NecessaryandSucientConditionsforMaximizingLifetimeVector .... 74 4.3DistributedProgressiveAlgorithm ....................... 77 4.3.1RateSchedule,Volume-BoundDistribution,VolumeSchedule .... 77 4.3.2InitializationPhase ........................... 79 4.3.3IterativePhase|Step1:FromRatestoVolumeBounds ...... 80 4.3.4IterativePhase|Step2:FromVolumeBoundstoVolumesandRates ................................... 82 4.3.5Property ................................. 85 4.3.6TerminationConditions ......................... 89 4.3.7Overhead ................................. 89 4.3.8NetworkDynamics ........................... 90 4.4Simulation .................................... 91 4.4.1ASimpleIllustrativeTestCase .................... 91 4.4.2ConvergenceSpeedofDPA ....................... 92 4.4.3ScalabilityofDPA ............................ 93 4.4.4ComparisonwithHou'sCentralizedAlgorithm ............ 93 4.4.5ComparisonwithOtherCentralizedandDistributedSolutions ... 94 4.5Summary .................................... 95 5CONCLUSION .................................... 100 6

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....................................... 101 BIOGRAPHICALSKETCH ................................ 106 7

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Table page 1-1DualityrelationshipbetweenthetwoproblemsprovedbyHouetal.in[ 1 ] ... 23 2-1SimulationresultsonthetopologyinFig. 2-2 ................... 35 2-2Simulationresultsofthecomplexscenario ..................... 35 3-1SimulationresultsonthetopologyinFig. 3-5 .................... 66 3-2SimulationresultsofweightedmaxmininFig. 3-5 ................. 66 3-3SimulationresultsonthetopologyinFig. 3-6 ................... 66 3-4SimulationresultsonthetopologyinFig. 3-7 .................... 66 3-5SimulationresultsonthetopologyinFig. 3-8 ................... 66 4-1Datasourcelifetimes(indays) ............................ 96 4-2Datasourcevolumes(inthousandsofpackets) ................... 96 4-3SomedatapointsusedtoproduceFig. 4-5 ..................... 96 8

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Figure page 1-1Two-hopowsarestarved. ............................. 23 2-1Asimpleexampleofthegeneralizedmaxminmodel ................ 35 2-2Anexampleofwireless-linkcontentiongraphandcliques ............. 36 2-3Flowsdescribedbythegeneralizedmaxminmodel ................. 36 2-4Schedulingamongcontendingnodes ......................... 36 3-1Amoresophisticatedexampletoillustratethepurposeofthefourlocalconditions 67 3-2Anexampleofrate-limitcondition ......................... 67 3-3Thepathofaowwithanunsaturatedbueratthesource ........... 67 3-4Per-destinationpacketqueueingisnecessarywhentheowspassinganodearedestinedfordierentdestinations. .......................... 68 3-5Networktopologyofasimplescenario ....................... 68 3-6Athree-linkstopology ................................ 68 3-7Networktopology ................................... 68 3-8Networktopology ................................... 69 3-9RatesoftheowsonthetopologyinFig. 3-8 ................... 69 4-1ThereisnoexhaustednodeonP1orP2;nodessandwareunrestrictedfeedingsourcesofi.ThereisanexhaustednodexonP3;nodeuisarestrictedfeedingsourceofi.Thereisnoforwardingpathfromztoi;nodezisapotentialsourceofi. .......................................... 96 4-2IterationsofDPA ................................... 97 4-3Thereisnoexhaustednodefromstoi;nodesisanunrestrictedfeedingsourcesofi.Thereisanexhaustednodexfromutoi;nodeuisarestrictedfeedingsourceofi.Theupstreambottleneckxmaypreventsourceufromfullyutilizingthevolumeboundsetbyionlink(k;i). ...................... 97 4-4Asimpleillustrativetestcase. ............................ 97 4-5MaxdeviationandavgdeviationoflifetimevectorwithrespecttothenumberofiterationsthatDPAhasperformed ........................ 98 4-6DPAscaleswell.Itsoverheadgrowsslowlywiththenetworksize. ........ 98 9

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................ 98 4-8Leftplot:comparisonofnodaloverheaddistributionbetweenLPandDPA.Rightplot:comparisonofmaximumnodaloverheadbetweenLPandDPA ....... 99 4-9NetworklifetimesofDPA,SLPandMPR ..................... 99 4-10AvgandmaxdeviationsofSLPandMPR ..................... 99 10

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Thisstudyfocusesonfairnessinwirelessnetworks.Twofairnessproblemsareaddressed:end-to-endowratefairnessinmultihopwirelessnetworksandlifetimefairnessinwirelesssensornetworks. Inrecentyears,theadventofmultihopwirelessnetworkshasgreatlyacceleratedtheresearchonbandwidthmanagementinsuchnetworkstosupportnewapplications.WhilemuchresearchconcentratesontheMAClayer,theusersperceptiononthesenetworksishoweverdeterminedmainlybasedonthenetworksend-to-endeectiveness.Itisimportantforustodevelopexibletoolsfortracengineeringinmultihopwirelessnetworks.Inthisstudy,twosolutionsareproposedtoachieveend-to-endmaxminowratefairnessinsuchnetworks. Across-layerdesignisrstlyproposedforachievingend-to-endmaxminfairnessinwirelessmeshnetworks.Inthisapproach,ageneralizedmaxminmodelisrstproposedformultihopwirelessnetworks.Atthenetworklayer,ourdesignallocatesnetworkcapacitytoend-to-endowsformaxminbandwidthallocation.AttheMAClayer,ourdesignachievestheallocatedbandwidthsharesforowsthroughatwo-levelweightedfairqueuingalgorithm.Theproposeddesignisabletoequalizetheend-to-endbandwidthallocationtocompetingowsthatsharecommonbottlenecks,whilefullyutilizingthenetworkcapacity.Resultsofsimulationsarepresentedtodemonstratetheeectivenessoftheproposedsolutioninenhancingend-to-endfairness. 11

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Wirelesssensornetworkshaveawiderangeofapplicationsinhabitatobservation,seismicmonitoring,battleeldsensing,etc.Asanothertypeofmultihopwirelessnetwork,asensornetworkconsistsofbattery-poweredsensornodesthatarelimitedinenergysupply.Animportantproblemofwirelesssensornetworksismaximizingtheoperationallifetimeofasensornetwork.Thelifetimeofasensornetworkisdenedasthelifetimesofallsensorsthatproduceusefuldata.Acentralizedsolutionproposedbypreviousworkrequiressolvingasequenceoflinearprogrammingproblems.Thecomputationoverheadcanbeprohibitivelyhighforlargesensornetworks.Collectingthecompleteinformationaboutthenetworkanduploadingthecompleteforwardingpoliciestoallnodesrequiresignicantamountoftransmissions,particularlyfornodesaroundthesink.Weproposeafullydistributedprogressivealgorithmwhichiterativelyproducesaseriesoflifetimevectors,eachbetterthanthepreviousone.Insteadofgivingtheoptimalresultinoneshotafterlengthycomputation,theproposeddistributedalgorithmhasaresultatanytime,andthemoretimespentgivesthebetterresult.Weshowthatwhenthealgorithm 12

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Thetechnologyofwirelessnetworkinghasbeenwidelyadoptedduetoitsadvantagesonaccessibilityandportability.Inamultihopwirelessnetwork,eachnodeoperatesbothasanendhostandasarouter,forwardingpacketsforothernodesthatcannotcommunicatedirectly.Multihopwirelessnetworksprovidemoreexibilityastheyoperateinadecentralizedandself-organizingmanneranddonotrelyonxednetworkinfrastructure.Inrecentyears,theadventofvariousmultihopwirelessnetworks,includingwirelessmeshnetworksandwirelesssensornetworks,hasgreatlyintensiedresearchonsuchnetworkstoimprovetheirapplicabilityinpractice. 2 3 ].\Fair"isdeneddierentlyinthosealgorithms.Onecommonfeatureofthosealgorithmsisthatatleastcertainamountofbandwidthisguaranteedforeverysingle-hopowinthenetwork. Theuser'sperceptiononmultihopnetworksishoweverdeterminedmainlybasedonthenetworks'end-to-endeectiveness.Forexample,fornewuserstoparticipateinawirelessmeshnetwork,theywanttobesurethattheirend-to-endtracistreatedfairlyaseveryoneelse.Moreover,ifausercontributesmoretothenetwork,shemaydemandthathertracisgivenmoreweightthanothers'trac.Inordertomeetdiverseuserrequirements,itisimportantforustodevelopexibletoolsfortracengineeringinmultihopwirelessnetworks. However,thesolutionsforsingle-hopowfairnesscannotbeextendedtoachieveend-to-endowfairnessbecausetheyignoretherelationshipamongthesubowsfromthesamemultihopow.Ifanupstreamsubowisallocatedmorebandwidththanits 14

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Previouswork[ 4 ]haspointedouttheaboveproblemexistinginmultihopowsifsimplyapplyingtheabovesingle-hopowalgorithmstomultihopowsbybreakingeachmultihopowintomultiplesingle-hopows.In[ 4 ],Lipointedoutarelationshipamongallsubowsofamultihopow,whichisthatallsubowsfromthesamemultihopowareexpectedtohavethesamerateandtoreceiveequalamountofbandwidth.However,thebasicfairnessmodelproposedin[ 4 ]hasseriouslimitationthathindersitsapplicabilityinWMNs.Themodelensuresabasicshareofbandwidthforeachend-to-endowinacontendingowgroupandthentriestomaximizetheoverallnetworkthroughput.Thebasicshareiscalculatedasthechannelcapacitydividedbythetotaleectivelengthoftheroutingpathsoftheowsinthegroup.Theeectivelengthofapathisthesmalleroneofthepathlengthand3.Twoows,f1andfn,belongtothesamecontendingowgroupifthereexistsasequenceofows,f2throughfn1,suchthatficontendswithfi+1,1i
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Westudyafundamentalproblem,howtosupportweightedbandwidthallocationamongallend-to-endowsinamultihopwirelessnetworkbasedonIEEE802.11DCF.Amoreprecisebutlessintuitivedenitionoftheproblemishowtoadapttheowratestoachievetheglobalmaxminobjective[ 5 ]:therateofanyowinthenetworkcannotbeincreasedwithoutdecreasingtherateofanotherowwhichhasanequalorsmallernormalizedrate,wherethenormalizedrateisdenedastheowratedividedbytheowweight.Twosolutionsareproposedtoachievetheglobalmaxminobjective. Therstsolutionisacross-layerdesign.Ageneralizedmaxminmodelisproposedformultihopwirelessnetworks.Atthenetworklayer,itallocatesnetworkcapacitytoend-to-endowsformaxminbandwidthallocation.AttheMAClayer,ourdesignachievestheallocatedbandwidthsharesfortheowsthroughatwo-levelweightedfairqueuingalgorithm.Theproposeddesignisabletoequalizetheend-to-endbandwidthallocationtocompetingowsthatsharecommonbottlenecks,whilefullyutilizingthenetworkcapacity. ThesecondsolutionproposedisafullydistributedsolutionthatiscompatiblewithIEEE802.11DCF.Wetransformtheglobalmaxminobjectivetofourlocalconditionsandprovethat,ifthefourlocalconditionsaresatisedinthewholenetwork,thentheglobalmaxminobjectivemustbeachieved.Wethendesignadistributedrateadaptationprotocolbasedonthefourconditions.Wheneveralocalconditionistestedfalseatanode,thenodeinformsthesourcesofcertainselectedowstoadapttheirratessuchthattheconditioncanbesatised.Comparingwith[ 4 ],whichwebelieveisthemostrelatedwork,ourprotocolhasanumberofadvantages.First,itdoesnotmodifythebackoschemeofIEEE802.11.Second,itreplacesper-owqueueingwithper-destinationqueueing.Packetsfromallowstothesamedestinationisqueuedtogether.Thirdandmostimportant,ourprotocolachievesfarbetterfairness(orweightedfairness)amongend-to-endowsthanthebasicfairschemein[ 4 ]. 16

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Thenetworkcannotcarryoutitstaskafterthenodes'energyisexhausted.Hence,maximizingtheoperationallifetimeofasensornetworkisacriticalproblem. Whatisexactlythelifetimeofasensornetwork?Manypriorworks[ 6 { 15 ]denethenetwork'slifetimeasthetimebeforetherstsensorinthenetworkrunsoutofenergy,orbeforetherstlossofcoverage[ 16 ].ThisdenitionsimpliestheproblemofmaximizinglifetimetoalinearprogrammingproblemoranNP-hardnon-polynomialprogrammingproblemifthesinkisallowedtomove[ 15 ].However,inreality,theoperationallifetimeofthenetworkisnotlimitedtothesmallestlifetimeofallnodes.Whenonesensordies,therestofthenetworkcanstillwork,aslongasusefuldatageneratedbyothersensorscanstillreachthesink.Itisnottruethat,sincesensorsaroundthesinkforwardothers'data,theywillalwaysexhausttheirenergyrstandpreventtherestofthenetworkfromreachingthesink.Onecandeploymoresensorsaroundthesink,uselargerbatteriestoboosttheenergylevelthere,orperformin-networkdataaggregation. Anappropriatedenitionforthelifetimeofasensornetworkshouldincludethelifetimesofallsensorsthatproduceusefuldata.Asensor'slifetimeisthedurationfromthetimewhenitbeginstogeneratetherstdatapackettothetimewhenitgeneratesthelastpacketthatisdeliverabletothesink.Thenetwork'slifetimecanbedenedasthevectorofallsensors'lifetimessortedinascendingorder,whichiscalledthelifetimevector.Thevalueofthelifetimevectorisdeterminedbythenodes'packetforwardingpoliciesthatspecifyhowpacketsareforwardedfromthesensorsthroughthenetworktothesink.Morespecically,foreverynode,itsforwardingpolicyspeciestheproportionofpacketsthatshouldbeforwardedoneachoutgoinglinktowardsthesink. 17

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1 17 ]denetheproblemofmaximizingasensornetwork'slifetimeastondthepacketforwardingpoliciesforallnodesthatcollectivelyproducethelexicographicallylargestlifetimevector,calledthemaximumlifetimevector.Inlesspreciseterms,itrstmaximizesthesmallestlifetimeofallnodes,thenmaximizesthesecondsmallestlifetimeofallnodes,andsoon.Houetal.showthatthisproblemcanbemodeledasaseriesoflinearprogramming(LP)problems.AftersolvingtheLPproblems,thesinkuploadstheoptimalpacketforwardingpoliciestothesensors.Basedonitsforwardingpolicy,eachsensorforwarditspackets.Suchasolutionishoweveracentralizedone.ItrequiressolvingO(jNj)LPproblemsofsizeO(jEj),wherejNjisthenumberofsensorsinthenetwork,jEjisthenumberoflinks,andLPhashigh-orderpolynomialcomplexity.Thecomputationoverheadcanbeprohibitivelyhighforlargesensornetworksthatneedtobeoperationalsoonafterdeployment.Collectingthecompleteinformationaboutthenetworkanduploadingthecompleteforwardingpoliciestoallnodesrequiresignicantamountoftransmissionsinthenetwork,particularlyfornodesaroundthesink.Toavoidtheseproblems,adistributedalgorithmthatspreadstheoverheadevenlyonallnodesbecomesimportant. Weproposetherstdistributedsolutionfortheproblemofmaximizingthelifetimevectorofasensornetwork.Ourstrategyistodesignadistributedprogressivealgorithmthatworksinaseriesofiterations,eachproducingaresult(inourcase,alifetimevectoranditscorrespondingforwardingpolicies)thatisbetterthanthepreviousone.Thesequenceofresultsapproachestotheoptimalsolution.Adistributedprogressivealgorithmispracticallyattractivebecausearesultisavailableatanytimeandisgettingbetterasmoretimeisspent.Weshowthatwhenthealgorithmstabilizes,itsresultproducesthemaximumlifetimevector.Wehaveperformedthousandsofsimulationrunsonrandomnetworksofvarioussizes,andcomparedwithHou'scentralizedalgorithmaswellasotherrelatedalgorithms.Theresultsdemonstratethatouralgorithmrapidlyconvergestothemaximumlifetimevectoranditsoverheadissmall.Fornetworksofthousandsofnodes,it 18

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1 ]thatthereexistsanunderlyingdualityrelationshipbetweentheproblemofmaximizingthelifetimevectorofasensornetworkandtheproblemofmaximizingtheratevectorinasensornetwork.ThedualityrelationshipissummarizedinTable 1-1 ,inwhich,giisthelocaldatarateofnodeiandtiisthelifetimeofnodei. 1.4.1FlowRateFairness 18 { 20 ]notonlyrequireper-owqueueingbutalsoassumeaxedbandwidthcapacityforeachlink,whichmakesthemnotapplicableinrandom-accesswirelessnetworks. ItiswellknownthatTCPdoesnotperformwellinwirelessnetworks[ 21 22 ].MuchresearchhasbeendonetoimproveTCP'sperformance,andarecentsurveycanbefoundin[ 23 ].Mostexistingsolutionsemployheuristicmechanismsforbettercongestion 19

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Utility-basedsolutionsonwirednetworks[ 24 { 27 ]alsorequireeachlinktohaveaxedcapacity.Eortshavebeenmadetoadaptutility-basedsolutionsinwirelessnetworksbyconsideringonlysingle-hopows[ 28 29 ],eliminatingcontentionamongneighboringnodesbyusingseparateCDMA/FDMAchannelsforwirelesslinks[ 30 ],modelingresourcesasmaximalcontentioncliquesinsteadofwirelesslinks[ 31 ],relyingoncross-layerdesigntointegrateend-to-endrateadaptationwithMAC-layerpacketscheduling[ 32 ],assumingallwirelesslinkssharethesamechannel[ 33 ],orassumingaxedbandwidthcapacityforeachwirelessnode[ 34 ].Assigningoneseparatechannelforeachcontendingwirelesslink[ 30 ]requiresalargenumberchannelsforadensenetwork,causinglowcapacityforeachchannel.Thisapproachdoesnotworkwellwithwidely-deployedIEEE802.11b/gthathasonlythreenon-overlappingchannels.Themaximalcliqueapproach[ 31 ]requiresthateachclique'seectivecapacityisknown,butitisnotclearhowtoaccuratelymeasuresuchcapacity,whichisacomplexfunctionofnearbycontentionandenvironmentalnoise.Thecross-layerapproach[ 32 ]requiresthenodestodynamicallyestablishgloballycoordinated(orlocallyapproximated[ 35 ])time-slottedtransmissionschedulesattheMAClayer,whichdoesnottwellwithIEEE802.11'srandomaccessmodel.Moreimportantly,theutilityfunctionthatapproximatesmaxminfairnesscontainsanexponentapproachingtoinnity[ 36 ],whichmakesthesystemhardtostabilize.Insummary,existingutility-basedapproachesdonotprovideamaxminsolutionforIEEE802.11DCF. Thereareotherworksthatarenotutility-based.MostofthemaredesignedtoachieveMAC-layerfairness[ 3 37 38 ]ormaxminfairness[ 2 39 ]amongone-hopows.Whilesomestudymultihopows,eachhasitslimitation.Basicend-to-endfairnessinwirelessad-hocnetworksisachievedin[ 4 ].However,thebasicfairshareguaranteedforeachowishighlyconservative;itcanbefarbelowthemaxminrate.End-to-endmaxminisinvestigatedin[ 40 ],whichassumesaseparateCDMA/FDMAchannelforeach 20

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41 ],whichhoweverdoesnotprovideanalgorithmthatcomputesthetemporally-fairrates.Adistributedalgorithmthatachievesaggregatefairnessinsensornetworksisproposedin[ 42 ],assumingthatallowsaredestinedtothesamebasestation.Tothebestofourknowledge,nodistributedalgorithmhasbeenproposedtoprovideweightedmaxminbandwidthallocationinamultihopwirelessnetworkbasedonIEEE802.11DCF. IEEE802.11e[ 43 ]hasbeenunderdevelopmenttosupportQoS,primarilyforWLAN.ItsEDCAprovidesprioritizedchannelaccessonlyforfouraccesscategories(background,besteort,video,andvoice).Itdoesnotprovidene-levelcontrolforweightedbandwidthallocationamongend-to-endows. 6 { 13 44 ]denethenetwork'slifetimeasthetimebeforetherstsensorinthenetworkrunsoutofenergy,orbeforetherstlossofcoverage[ 16 ]. Houetal.showin[ 1 17 ]thattheproblemofmaximizingthelifetimevectorofasensornetworkcanbemodeledasaseriesofcentralizedlinearprogrammingproblems.Houetal.alsoprovein[ 1 ]thatthereexistsanunderlyingdualityrelationshipbetweentheproblemofmaximizingthelifetimevectorofasensornetworkandtheproblemofmaximizingtheratevectorofasensornetworkwithaglobalnodelifetimerequirement. Someresearchersalsodesignenergy-ecientroutingalgorithmstoachievethegoalofminimizingenergyconsumption[ 45 { 49 ].Thetypicalapproach[ 45 46 ]istouseashortestpathalgorithminwhichtheedgecostisthepowerconsumedtotransmitapacketalongthisedge.Thougheectivelyreducingtheenergyconsumptionrate,thisapproachcancauseunbalancedconsumptiondistribution.Thenodesontheminimum-energypatharequicklydrainedofenergy,causingnetworkpartition. 21

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2 andChapter 3 proposetwosolutionstoachievetheglobalend-to-endowratemaxminobjectiveinmultihopwirelessnetworks:across-layersolutionandafullydistributedsolution.Chapter 4 proposesadistributedprogressivealgorithmformaximizinglifetimevectorinwirelesssensornetworks.Chapter 5 concludesourstudy. 22

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Maximizinglifetimevector Totaldatavolumeatnodei:giT=tiR DualityrelationshipbetweenthetwoproblemsprovedbyHouetal.in[ 1 ] Figure1-1. Two-hopowsarestarved. 23

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Inthischapter,across-layerdesignisproposedforachievingend-to-endmaxmininwirelessmeshnetworks(WMNs).Ageneralizedmaxminmodelisrstproposedformultihopwirelessnetworks.Atthenetworklayer,itallocatesnetworkcapacitytoend-to-endowsformaxminbandwidthallocation.AttheMAClayer,ourdesignachievestheallocatedbandwidthsharesfortheowsthroughatwo-levelweightedfairqueuingalgorithm.Theproposeddesignisabletoequalizetheend-to-endbandwidthallocationtocompetingowsthatsharecommonbottlenecks,whilefullyutilizingthenetworkcapacity. Thischapterisorganizedasfollows.Section 2.1 describesthenetworkmodelandourobjective.Section 2.2 presentsageneralizedmaxminmodel,basedonwhichwedesignthemaxminbandwidthallocationalgorithmsforWMNs.Section 2.3 presentsthetwo-levelweightedfairqueuingschedulingalgorithm.Section 2.4 evaluatestheperformanceofoursolution.Section 2.5 summarizesthechapter. 2.1.1NetworkModel 24

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Westudyend-to-endows,whicharereferredtosimplyasows.Aowconsistsofoneormoresingle-hopows,whicharecalledsubows.Twosubowscontendiftheyarecarriedbythesamelinkortwocontendinglinks.Twoowscontendifanyoftheirsubowscontend. Eachowisentitledtoafairshareofnetworkbandwidthinproportiontoitsweight.Itiswellknownthatmaintainingfairnessandmaximizingnetworkthroughputarecontradictivegoals[ 3 ].Stricterfairnesscanbeachievedoftenattheexpenseoflowernetworkthroughput.Somepreviousstudiesfocusedmoreonthroughputoptimizationundercertainbasic,relaxedfairnesscriteria[ 4 37 ].Weputmorefocusonfairness.Specically,wewanttoachievetheclassicalmaxminfairnessamongend-to-endowsin 25

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Theclassicmaxminmodelforwirednetworksisdescribedasfollows.GivenasetQofresources(i.e.,links),acapacitybqforeachresourceq2Q(i.e.,bandwidth),asetFofows,anominalweightwfforeachowf2F,andaroutingpathpfforeachowf,theproblemistoassignaraterfforeachowfsuchthat 1. 2. foranyowf2F,itsraterfcannotbeincreasedwithoutdecreasingtheraterf0ofanotherowf0,forwhichrf0=wf0rf=wf. ThesetofratesR=frfjf2Fgthatsatisfytheaboveconditionsarecalledthemaxminrates. Theabovemodelassumesthateachresourcehasaxedcapacityandthataresourcecanappearinaow'sroutingpathatmostonce.InordertoapplythismodeltoWMNs,wehavetoidentifywhattheresourcesare.Wirelesslinkscannotbeusedastheresourcesbecausetheydonothaveindividuallyxedcapacities.FollowingHuangandBensaou'swork[ 2 ]whichconsidersonlyone-hopows,weshalluse\cliques"fromthecontentiongraphastheresources,whichwillbeexplainedindetailinSection 2.2 .However,inordertoaccommodatethe\cliqueresources"inthecontextofend-to-endows,wemustgeneralizethemaxminmodelrstinthefollowingsectiontoallowaresourcetoappearinaow'sroutingpathformultipletimes. 2.3 26

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Awireless-linkcontentiongraphcanbeemployedtodescribethespatialcontentionrelationshipamongcontendinglinks.Verticesinawireless-linkcontentiongraphrepresentwirelesslinksinthecorrespondingnetworktopology.Twoverticesareconnectedifthecorrespondinglinkscontendwitheachother.Awirelesslinkisidleifthereisnoowpassingit.Asimpliedwireless-linkcontentiongraphcanbeconstructedfromanetworktopologywithallidlelinksremoved.Anexampleofsimpliedwireless-linkcontentiongraphisgivenbyFig. 2-2 (b). Acliqueisacompletesubgraphwithalinkbetweeneverypairofnodes.Amaximumcliqueisacliquethatisnotcontainedinanotherclique.Inthischapter,werefertomaximumcliquesascliqueshenceforth.Acliqueinawireless-linkcontentiongraphrepresentsagroupofmutuallycontendingwirelesslinksinwhichonlyonelinkcanbeintransmissionatanytime.Thechannelbandwidthissharedbyallwirelesslinksofaclique.Thecliquesfromthewireless-linkcontentiongraphcanbeusedasresources. Followingtheroutingpathofaow,wecanobtainasequenceofcliquesthattheowpasses.Whenaowpassesmultiplelinksofaclique,weconsidertheowpassesthecliquemultipletimes.Forawirelesslinkbelongingtomultiplecliques,ifaowpassesthislink,weconsidertheowpassesthosecliquesinturn. Inthegeneralizedmodel,aresourceisallowedtoappearinaow'sroutingpathformultipletimesindierentpositions.Thenumberofappearancesofresourceq2Qinowf'spathpfisdenotedbynqf.Wehavethefollowingfeasibilityconstraintforasetofow 27

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Asetofratesthatsatisestheaboveconstraintissaidtobefeasible.Itismaxminfairifitisfeasibleand,foreachf2F,rfcannotbeincreasedwhilemaintainingfeasibilitywithoutdecreasingrf0foranotherowf0,forwhichrf0=wf0
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28 ].Tosolvethisproblem,weusetheeectivechannelcapacityofacliqueqasbqinourmodel.Anodemeasurestheeectivebitratesofitsincidentlinks.Theconceptofeectivebitrateissimilartotheonein[ 50 ],whichincorporateslinklayerdetails.Aclique'seectivechannelcapacitycanbeobtainedbysumminguptheeectivebitratesofalllinksofthatclique. IninfrastructureWMNs,meshroutershaverelativelystrongcomputingcapabilityandstablepositionswhichmakethecentralizedimplementationofthealgorithmfeasible.Theimplementationcanalsobedistributed.Nodesonlyworkonlocallinkcontentiongraphwhichismuchsmallerthantheglobalone.Theworkofcliquedecompositionisreducedremarkably.Somedistributedmaxminalgorithmsforwirelinenetworks(e.g.,[ 51 ])couldbecustomizedtocalculateowmaxminfairshares. 29

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52 ])canbeadoptedtoachievethis. Therestofthissectiondescribestheinter-nodeschedulingalgorithmthatisbasedonthe802.11DCFwithRTS-CTS-DATA-ACKhandshake. Fig. 2-4 showsallpossiblecontendingpackettransmissionswithintwohopsawayfromrouterxandroutery.InFig. 2-4 ,circlesrepresentrouters.Alinebetweentworoutersmeanstheyarewithinthetransmissionrangeofeachother.Anarrowfromrouterxtoymeansthenext-to-sendpacketofxneedtobetransmittedtoy.GiventheexampleinFig. 2-4 ,thetransmissionfromxtoyconictswithallothertransmissionsindicatedbythearrowsinFig. 2-4 Ifanodexhasapackettotransmit,thecontendingnodesetofx,denotedbyx,isdenedasthegroupofnodesthatarecompetingforthemediaaccesswithx.InFig. 2-4 ,x=fi;j;m;y;n;v;wg.Let+x=x[fxg.Whenxhasapackettotransmitanditsbackotimerbecomeszero,itshouldcompareitstagwiththoseofthenodesinx.Ideally,thepacketfromxshouldbetransmittedimmediatelyifitstagisthesmallest.Otherwise,x'stransmissionshouldbewithhelduntilallpacketsfromxwithsmallertags 30

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Whenxhasapackettobetransmittedtoy,ifTsx>TnxorTsx>Tny,xcanknowitdoesnothavethesmallesttagin+xandthetransmissionshouldbewithheld.Inordertoobtainmostup-to-dateTnxandTny,RTS,CTS,DATAandACKpacketscanpiggybacknecessarytags.Eachroutermaintainsatabletokeeptrackofitsneighbors'tags. However,itisdiculttoenforcetheabovestrictconditionsforeachtransmission.Thereasonisthatsenderxcannotalwayshavethefreshtagsofitsneighbors,especiallyTnyfromreceivery,whicharebasedonthetagsofnodesasfarasthreehopsawayfromx.Staletagsmaycausedeadlocks.Toavoidpotentialdeadlocks,aheuristicmethodisusedbyxtoestimateTny.ThebasicideaistoestimatetheincrementrateofTny,denotedbyrny.Foreachi2Nx,besidesTni,xalsorecordsrniitestimatesandthetimetiwhenTnigetsupdated.Whenxneedstotransmitapackettoy,xuses^TnyinsteadofTnytocheckthesecondcondition,where ^Tny=Tny+rny(tty)(2{2)tisthecurrenttime.OnceTnygetsupdatedandbecomeslarger,thenewrnyiscomputedas: whereisaparametertocontroltheinuenceofTny'snewincrementrateonrny.Ifaboveapproachisemployed,^Tnywilleventuallybeincreasedlargeenoughsuchthatthesecondwithholdingconditionwillbecomefalse. 31

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1. 2. WecomparetheperformanceofMMFwith(1)802.11DCF(abbreviatedas802.11);and(2)thetwo-phaseprotocol(abbreviatedas2PP)proposedin[ 4 ].Wecomparethealgorithmsfromtwoaspects:end-to-endowfairnessandspatialreuseofspectrum. Toevaluatetheend-to-endfairness,weadoptthemaxminfairnessindex[ 5 ](denotedbyImm)andtheequalityfairnessindex[ 53 ](denotedbyIeq).Imm=minf2Ffrfg

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Tomeasurethespatialreuseofspectrum,weemploytheeectivenetworkthroughputU,whichisdenedasPf2Frflf,wherelfisthenumberofhopsontheroutingpathofowf.Thepacketsdroppedbytheintermediatenodesdonotcounttowardstheeectivenetworkthroughputastheydonotcontributetoend-to-endthroughput.Theeectivenetworkthroughputgivesusameasurementfornetworkbandwidthutilizationandtheeciencyofaprotocol. Wepresentsimulationresultsintwonetworkscenarios:asimplenetworktopologyshowninFig. 2-2 andacomplexnetworktopologythatwillbedescribedlater.Allowsinbothscenarioshavetheequalnominalweights.Intherestofthissection,theunitoftheowornetworkthroughputispacketspersecond(PPS). ThesimulationresultsoftheexampleinFig. 2-2 areshownbyTable 2-1 .Thelengthofaowisthenumberofitssubows.MMFshowsgoodend-to-endfairnessandcomparablebandwidthutilization.In2PP,theobjectiveofthebasicfairnessmodelistomaximizethetotalend-to-endthroughput.Thussinglehopowh9;8ihasmuchhigherratethanotherows. ThecomplexscenariosimulatesthetracinthebackboneofaWMN.27nodesareplacedina900900region,inwhich25arenon-gatewaynodesand2aregatewaynodes.Gatewaynodesareevenlyplacedinthehorizontalmidlineoftheregion.Theregionisdividedinto25grids.Eachnon-gatewaynodeisplacedintoagrid.Thelocationofanon-gatewaynodeinitsgridisrandomlychosen.Anon-gatewaynodeconnectstotheInternetthroughthenearestgatewaynode.Everynon-gatewaynodehasadownloadowfromitsgatewaynode.5non-gatewaynodesarerandomlypickedtohave5uploadowstotheirgatewaynodes.Wealsorandomlycreate5internalowsamongnon-gatewaynodes.ThesimulationresultsareshowninTable 2-2 33

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34

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2PP MMF ow length thro. ee.weight thro. ee.weight thro. 114.44 1.00 115.60 1.00 173.74 198.82 2.50 288.65 1.00 173.74 272.21 1.00 114.36 1.67 291.49 414.32 6.00 682.03 1.67 292.24 ee.networkthro. 1814.13 1950.46 1917.66 0.168 0.595 0.627 0.940 Table2-1. SimulationresultsonthetopologyinFig. 2-2 802.11 2PP MMF ee.networkthro. 1550.15 998.86 1528.45 0.026 0.500 0.453 0.895 Table2-2. Simulationresultsofthecomplexscenario Figure2-1. Asimpleexampleofthegeneralizedmaxminmodel 35

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Anexampleofwireless-linkcontentiongraphandcliques Figure2-3. Flowsdescribedbythegeneralizedmaxminmodel Figure2-4. Schedulingamongcontendingnodes 36

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Inthischapter,wepresentafullydistributedapproachtosupportweightedbandwidthallocationamongallend-to-endowsinamultihopwirelessnetwork.Ourgoalistoenablethenetworktoadapttheowratessuchthatglobalmaxmincanbeachieved.InordertodesignafullydistributedsolutionthatiscompatiblewithIEEE802.11DCF,wetransformtheglobalmaxminobjectivetofourlocalconditionsandprovethat,ifthefourlocalconditionsaresatisedinthewholenetwork,thentheglobalmaxminobjectivemustbeachieved.Wethendesignadistributedrateadaptationprotocolbasedonthefourconditions.Wheneveralocalconditionistestedfalseatanode,thenodeinformsthesourcesofcertainselectedowstoadapttheirratessuchthattheconditioncanbesatised.Comparingwith[ 4 ],whichwebelieveisthemostrelatedwork,ourprotocolhasanumberofadvantages.First,itdoesnotmodifythebackoschemeofIEEE802.11.Second,itreplacesper-owqueueingwithper-destinationqueueing.Packetsfromallowstothesamedestinationisqueuedtogether.Thirdandmostimportant,ourprotocolachievesfarbetterfairness(orweightedfairness)amongend-to-endowsthanthebasicfairschemein[ 4 ]. Therestofthechapterisorganizedasfollows.Section 3.1 denesthenetworkmodel.Section 3.2 classieswirelesslinksintothreecategories.Section 3.3 presentsthelocalconditionsforglobalmaxmininwirelessnetworkswithasingledestination.Section 3.4 presentsthelocalconditionsfornetworkswithmultipledestinations.Section 3.5 designsadistributedglobalmaxminprotocolbasedthelocalconditions.Section 3.6 evaluatestheprotocolbysimulations.Section 3.7 summarizesthechapter. 3.1.1NetworkModelandProblemStatement 37

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54 ].Mobilead-hocnetworksarebeyondthescopeofthisstudy.TwonodesareneighborsofeachotherifthereareabletoperformRTS/CTS/DATA/ACKexchange.Twonodesthatarenotneighborscommunicateviaamultihopwirelesspath.Timeisnotslotted.Radiointerferenceisresolvedbyrandombacko.Twowirelesslinkscontendiftheycannottransmitsimultaneously.BasedonthemostpopularMACprotocol,thismodelexcludesthemajorityofrelatedworks[ 30 32 34 35 40 41 ]. LetFbeasetofend-to-endowsinthenetwork.Eachowfhasadesirablerated(f)andaweightw(f).Buttheowsourcewillgeneratenewpacketsatasmallerrateifthenetworkcannotdeliveritsdesirablerate.Theactualrateofowfisdenotedasr(f)(d(f)).Thenormalizedrateofowfisdenedas Inthischapter,whenwereferto\owrate"or\normalizedrateofaow",wemean\end-to-endrate".Theglobalmaxminobjectiveisdenedasfollows:Thenormalizedrate(f)ofanyowfcannotbeincreasedwithoutdecreasingthenormalizedrate(f0)ofanotherowf0,forwhich(f0)(f). Inamoreintuitivebutlessprecisedescription,ourgoalistoequalizethenormalizedratesofallowsasmuchaspossible,particularly,raisingthesmallestones.Directlycompetingowstendtoreceivebandwidthinproportionaltotheirweights.Achievingglobalmaxminisafundamentalfunctionofend-to-endtracengineeringinmultihopwirelessnetworks.Itaddsanewentryintheexistingtoolbox(whichincludesprice-basedandothersolutions)fortracdierentiationamongapplications.Forexample,wemayestablishseveralserviceclassesinthenetworkandassignlargerweightstoapplicationsbelongingtohigherclasses.Howtoenforceacertainweightassignmentschemethroughservicecontractorothermeansisbeyondthescopeofthisstudy. Weassumethereexistsaroutingprotocolthatestablishesaroutingtableateachnode.Theroutingtablemaybeimplicitundergeographicrouting[ 55 56 ],orexplicitly 38

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57 ]orlink-stateroutingprotocol.Consideraspecicdestination.Anodemayreceivepacketsfrommultipleupstreamneighborsandforwardthemtoadownstreamneighbortowardsthedestination.Thelinksfromtheupstreamneighborstoanodearecalledupstreamlinksofthenode,andthelinkfromanodetoitsdownstreamneighboriscalledthedownstreamlink. 3.4.1 .Notethatotherworks[ 4 40 ]requireper-owfairqueueing,whichisamorestringentrequirement.Nowconsiderthepacketstoasinglearbitrarydestination.Anodebuerspacketsreceivedfromupstreamlinksbeforeforwardingthemthedownstreamlink.Thebuerspaceforthequeueislimited.Toavoidpacketdropsduetobueroverow,weadoptthecongestionavoidanceschemein[ 54 ],whichallowsanodeitosenditsdownstreamneighborjapacketonlywhenjhasenoughfreebuerspacetoholdthepacket.Supposethebuerspaceisslottedwitheachslotstoringonepacket.Tokeeptheneighborsupdatedwithj'sbuerstate,wheneverjtransmitsapacket(RTS/CTS/DATA/ACK),itpiggybacksitscurrentbuerstate,forexample,usingonebittoindicatewhetherthereisatleastonefreebuerslot.Whenanupstreamneighborioverhearsapacketfromj,itcachesthebuerstateofj.Ifj'sbuerisnotfull,itransmitsitspacket.Ifj'sbuerisfull,iwillholditspacketandwaituntiloverhearingnewbuerstatefromj.Notethattheresidualbueratnodejchangesonlywhenjreceivesorsendsadatapacket.Wheneverthishappens,jwillsendeitherCTS/ACKorRTS/DATA,immediatelyinformingtheneighborsofitsnewbuerstatethroughpiggybacking.Nocyclicwaitingispossibleifroutingisacyclic.Tohandlefailedoverhearing,iwillstopwaitingandattempttransmittingifitdoesnotoverhearj'sbuerstateforcertaintime.Readersarereferredto[ 54 ]fordiscussiononotherissues. 39

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54 ],whenthebueratjbecomesfull,itforcestheupstreamneighborstoslowdowntoacombinedratethatmatchestherateonthedownstreamlink.1Wheneverjsendsoutapacket,itfreessomebuerspacesuchthattheupstreamneighborscancompetefortransmission.Wheneverjreceivesapacket,itsbuermaybecomefullagainandtheupstreamneighborsmayhavetowaitforthenextreleaseofbueratj.Abuerissaturatedifitcontinuouslyswitchesbetweenfullandunfull,whichslowsdowntheratesofupstreamlinksastheupstreamneighborshavetospenttimewaitingforbuerrelease.Abuerisunsaturatedifitstaysunfull(formostofthetime). 40

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Basedontheabovethreecases,wirelesslinksareclassiedintothreetypes:bandwidth-saturatedlinks,buer-saturatedlinks,andunsaturatedlinks. 41

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Abottlenecklinkmustbebandwidth-saturatedasthereissucientdatatouseupallbandwidthavailabletothelink.Anon-bottlenecklinkiseitherbuer-saturatedlinkoranunsaturatedlink.Theavailablebandwidthisnotfullyutilizedbecauseofadownstreambottleneckintheformercaseorshortageofdatasupplyfromupstreaminthelattercase. 2 4 31 ].Apropercliqueisacliquethatisnotcontainedbyalargerclique.Inthefollowing,whenwerefertoacontentionclique,wealreadymeanaproperclique.Alinkmaybelongtomultiplecliques,consistingofnearbycontendinglinks.Packettransmissionsonthelinksofacliquemustbemadeserially.Therefore,thecombinedrateonalllinksofacliqueisboundedbythechannelcapacity.Acliqueissaturatedifthelinkshaveutilizedallavailablebandwidthsuchthatincreasingtherateononelinkwillalwaysleadtodecreasingtherateonanotherlinkintheclique.Becauseabandwidth-saturatedlinkuses 42

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3.5 .Fornow,weassumethatallowsgotothesamedestination.Theassumptionwillberemovedinthenextsection. 43

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Fortheaboveexample,onemayarguethat,althoughIEEE802.11DCFdoesnotprovidefairness,manyMACprotocols[ 2 3 37 { 39 ]havebeenproposedtoachievethat.IEEE802.11ecanalsoprovidecoarse-levelratecontrol.Buttheexampleisasingleonewithonlyone-hopows.TheseMACsolutionscannotprovideend-to-endfairness,letaloneprovableweightedmaxmin. wherep(f)istheroutingpathofowf.Thereisaneasywayforeachlinktoknowitsnormalizedrate.Whenthesourceofaowproducesnewpackets,itletsthepacketscarrytheow'snormalizedrate.Thenodesofalinkinspectthepassingpacketsandtakethelargestnormalizedratecarriedinthepacketsasthelink'snormalizedrate. Thesetofowsthatpass(i;j)consistsofallowspassingtheupstreamlinksandallowsthatbeginfromi.Bythedenitionofnormalizedrate,wehavethelemmabelow.

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Checkingtheaboveconditionsdoesnotrequireglobalstateoftheentirenetwork.AswewillseeinSection 3.5 wherewedesigntheprotocol,thersttwoconditionscanbetestedbyeachnodeindividuallyandthethirdconditiononlyrequiresinformationexchangeamongnearbynodes,whichcanbeecientlydone.Thefourthconditionrequirestheratelimitataowsourcetobeadditivelyincreaseduntilasource,buer-saturatedorbandwidth-saturatedconditionisviolatedinthenetwork.Whenthishappens,thesourcewillbesignaledtotightenitsratelimit.Forexample,ifthebandwidth-saturatedconditionisviolated,alinklthathasthehighestnormalizedrateinthesaturatedcliquewillbeaskedtoreduceitsrateinordertogiveupsomebandwidthforthebandwidth-saturatedlink.Linklwillidentifythepacketscarryingthelargestnormalizedrateandinformthesourcesofthosepacketstoreducetheirrates.Inresponse,thesourceswillself-imposetighterratelimits. Weillustratethepurposeofthefourlocalconditionsbyacoupleofexamples.First,examinethesimplecaseinSection 3.3.1 ,wherethenetworkhasonlytwowirelesslinks,(i;t)and(j;t).Therearethreeows,onefromitotandtwofromjtot.Assumebothiandjhavesaturatedbuer.Satisfyingthesourceconditionensuresthatthetwoowson(j;t)havethesamenormalizedrate.Satisfyingthebandwidth-saturatedcondition 45

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3.5 .) Figure 3-1 givesamoresophisticatedexample.Whitecirclesrepresentowsources.Greycirclesrepresentothernodes.Thickarrowsrepresentbandwidth-saturatedlinks.Thinarrowsrepresentunsaturatedlinks.Thindashedarrowsrepresentbuer-saturatedlinks.InFigure 3-1 (a),aportionofthenetworkisshownwitheacharrowpointingfromanupstreamnodetoitsdownstreamneighbor.InFigure 3-1 (b),therearesixows,f1throughf6,whoseweightsareshownbesidetheirsources.TheactualdataratesofthelinksareshowninFigure 3-1 (c).(i;j)isabandwidth-saturatedlink,whichsendsbuer-basedbackpressureupstream,creatingbuer-saturatedlinksallthewaytotheowsourcesandslowingtheowrates.InFigure 3-1 (d),thenormalizedratesoftheowsareshownbesidethesources.ThenormalizedratesonthelinksareshowninFigure 3-1 (e). Satisfyingthesourceconditionensuresthatthenormalizedrateofowf4isashighasthatofanyotherupstreamow.Thebuer-saturatedconditionrequiresthatowf1hasthesamenormalizedrateasf2,f3andf4.Becausef1'sweightis2,itsactualrateshouldbetwicethatoff2,f3orf4.Tosatisfythiscondition,ratelimitsmustbeappliedatv,wandxtogivemorebandwidthtou.Satisfyingthebandwidth-saturatedrequirementensuresthatthenormalizedratesofows(f1throughf5)passingthebandwidth-saturatedlink(i;j)areaslargeasanycontendingows(f6).Thismayrequire 46

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Itisnotedthat,therateincrementofaowsourcemayleadtotheviolationofoneormorelocalconditionsbuttheowsourceisnotrequiredbyothernodestoreduceitsrate.Inthatcase,therateincrementattheowsourcedoesnotviolatethelocalconditions.AnexampleisgiveninFig. 3-2 .Thethreelinksareinasaturatedclique.Link(k;t)isbandwidth-saturated.InFig. 3-2 (c),thenormalizedrateoff1islowerthanthoseoftheothertwoows.Byrate-limitcondition,iincreasesthenormalizedrateoff1from1to3.Duetothelimitedbandwidthinthesaturatedclique,thenormalizedrateof(k;t)dropsto4,asshowninFig. 3-2 (d).Bandwidth-saturatedconditionisviolatedas(j;t)haslargernormalizedratethan(k;t).Thusjisrequiredtoreducetherateoff2.Inthisexample,althoughtheincrementoff1'srateresultsintheviolationofthebandwidth-saturatedcondition,onlythesourceoff2,whichisj,violatesthelocalcondition.Fig 3-2 (e)showsthenalrateallocationofthethreeowsthatsatisesallfourlocalconditions.Theratelimitoff1orf2cannotbefurtherincreasedwithoutviolatingthebandwidth-saturatedcondition. Theportionofaow'sroutingpathfromtherstnodewhosebuerissaturatedtotherstbandwidth-saturatedlinkiscalledtheprimarysaturatedsubpathoftheow.Itiseasytoseethattheprimarysaturatedsubpathofaowconsistsofachainofbuer-saturatedlinksandabandwidth-saturatedlinkattheend.Thechainofbuer-saturatedlinksintheprimarysubpathistheresultofbuer-basedbackpressureoriginatedfromthebandwidth-saturatedlink,whichisdemonstratedinFigure 3-1 (c),wherethebottlenecklink(i;j)causestheupstreamlinksbuer-saturated.Itispossiblethattheprimarysaturatedsubpathofaowdoesnothaveabuer-saturatedlink.For 47

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Foraowfwithr(f)
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3-3 .InFig. 3-3 ,whitecirclerepresentstheowsource.Greycirclesrepresentothernodes.Thickarrowsrepresentbandwidth-saturatedlinks.Thinarrowsrepresentunsaturatedlinks.Thindashedarrowsrepresentbuer-saturatedlinks.\"ontopofanodeindicatesanunsaturatedbueratthatnode.\+"indicatesasaturatedbuer.Itispossiblethataowdoesnothaveaprimarysaturatedsubpath.Inthatcase,theprimaryunsaturatedsubpathoftheowformstheentireroutingpath. Proof:Bytherate-limitcondition,whenr(f)isincreased,oneormorelocalconditionsareviolatedandthesourceoffisrequiredtoreducetherateoff.Supposetheamountoff'srateincrementisverysmallandthebueratthesourceoffisstillunsaturated.Theviolationwillnotappearatthesourcebecausethesourceconditionandthebuer-saturatedconditionarenotapplicableatanodewithanunsaturatedbuer.Therefore,theviolationmustappearonatleastonelinkontheroutingpathoff.Becauseanysmallamountofrateincrementonfcanintroduceaviolationandtheratereductionrequestwillalwaysbesenttothesourceoff,thenormalizedrateofthelinkwheretheviolationappearsmustbeequalto(f)beforetherateincrementonf.2

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Proof:Theremustbearatelimitatthesourceoffbecauser(f)
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,wehave(i1;i2)=(f).Bythebuer-saturatedconditionandLemma 1 ,allotherlinksontheprimarysaturatedsubpathhavethesamenormalizedrateas(i1;i2).Then(ik;ik+1)=(f).Therefore,(ik;ik+1)istheprimarybandwidth-saturatedlinkoff. Iftheviolationhappensonalinkafter(i0;i1)ontheroutingpath,thenormalizedrateof(i1;i2)mustbeequalto(f)beforer(f)isincreased.Thiscanbeprovedbycontradiction.Assume(i1;i2)>(f)beforer(f)isincreased.ByLemma 1 ,alllinksontheroutingpathafter(i1;i2)alsohavenormalizedrateslargerthan(f).Amongalllinksontheroutingpathfrom(i1;i2),thereisalinkonwhichtheviolationofowfoccurs.Thenormalizedrateofthatlinkislargerthan(f)beforer(f)isincreased,whichcontradictswithLemma 3 .Bythebuer-saturatedconditionandLemma 1 ,allotherlinksontheprimarysaturatedsubpathhavethesamenormalizedrateas(i1;i2).Then(ik;ik+1)=(f).Therefore,(ik;ik+1)istheprimarybandwidth-saturatedlinkoff. Bythebandwidth-saturatedcondition,(ik;ik+1)hasthelargestnormalizedrateinatleastonesaturatedclique.2 2 andLemma 4 ,wecangetthelemmabelow. Proof:Supposethelocalrequirementsareachieved.Foranarbitraryowfwithr(f)
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ByLemma 5 ,owfhasaprimarylink(i;j)and(i;j)=(f).Itmeansthatthenormalizedratesofallotherowspassing(i;j)arenotgreaterthan(f).Whenweincrease(f)byincreasingtherateoff,basedontheassumption,thenormalizedratesofallotherowspassing(i;j)willnotbedecreased,whichmeansthat(i;j)'sratewillgoup.ByLemma 5 ,(i;j)hasthelargestnormalizedrateinasaturatedclique.When(i;j)'srategoesup,therateofanotherlink(i0;j0)inthesaturatedcliquewillhavetogodown.Amongallowspassing(i0;j0),atleastoneowf0hastodecreaseitsrate(andthus(f0)).Since(i;j)hasthelargestnormalizedrateintheclique,wehave(f0)(i0;j0)(i;j)=(f),whichcontradictswiththepreviousassumption. 3-4 .Whitecirclesrepresentowsources.Blackcirclesrepresentdestinations.Thickarrowsrepresentbandwidth-saturatedlinks.Thinarrowsrepresentunsaturatedlinks.Thindashedarrowsrepresentbuer-saturatedlinks.Figure 3-4 (a)showsaportionofthenetworkwithtwoowswhoseweightsarebothoneanddesirableratesareboth5.InFigure 3-4 (b),eachnodehasonequeueforalldestinations.First,weshowthatonequeuepernodewillunnecessarilyreducetherateoff2inFigure 3-4 (b),where(z;t)isabandwidth-saturatedlink,causingbuer-basedbackpressuretosaturatethebuersatj,i,xandy.Supposetherateoff1is1duetothebottleneck(z;t).Becausethesourcenodes,xandy,competefairlyfortransmissiontoiwheneveri'sbuer 52

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Anodeissaidtoserveadestinationifitisontheroutingpathofaowwiththatdestination.Anodeshouldmaintainaseparatequeueforeachserveddestination,notforeachpassingow.Itshouldbenotedthat,inameshnetwork,manyowsmaydestineforthesamedestination,i.e.,thegatewaytotheInternet.InFigure 3-4 (c),eachnodehasonequeueperserveddestination.Wheniandjkeepseparatequeuesfordestinationstandv,f2willbeabletosendatitsdesirablerateof5. Separatequeuesachieve\isolation"betweenpacketsfordierentdestinations,whichallowsustomodelthephysicalwirelessnetworkasasetofoverlappingvirtualnetworks,eachforonedestination.Figure 3-4 (d)showsthatf1andf2aredeliveredintwovirtualnetworkswithseparatepacketqueuesbutsharingthesamechannel. 53

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3-4 (d),wherethewirelessnetworkismodeledastwovirtualnetworks,and(i;j)astwovirtuallinks. Eachvirtualnetworkcarriesasubsetofows,whichisdisjointfromthesubsetscarriedbyothervirtualnetworks.Buer-basedbackpressure(Section 3.1.2 )isperformedindependentlywithineachvirtualnetwork.Thenormalizedrateofavirtuallinkisdenedasthelargestnormalizedrateofanyowpassingthelink.Withinavirtualnetwork,weclassifyvirtuallinksasbandwidth-saturated,buer-saturated,orunsaturatedinthesamewayaswedidinSection 3.2.2 .Otherconceptscanalsobetriviallyextendedtovirtualnetworks. 3.3.3 tosuitforawirelessnetworkwhoseowshavedierentdestinations. Lemma 1 -Lemma 5 canbeeasilyextendedtovirtualnetworks.

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Proof:Supposethelocalrequirementsareachieved.Foranarbitraryowfwithr(f)
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3.4.3 ,whichisequivalenttomeetingtheglobalmaxminobjectiveinwirelessnetworkswithmultipledestinations. Evenaftertheconditionsaresatised,thenetwork/tracdynamicsmaycausethemtobeviolatedagain.Theprotocolwillcontinuouslychangetheowratestorestoretheconditionsandachieveglobalmaxmininthecurrentnetwork/tracenvironment. Intheprotocoldescription,werefertoaphysicalnodesimplyas\node",denotedas\i",incontrasttoa\virtualnode",denotedas\it"fordestinationt.Werefertoalinkbetweentwophysicalnodesas\wirelesslink",denotedas\(i;j)",whichmaycontainmultiple\virtuallinks",denotedas\(it;jt)".Werefertotheoriginalnetworkas\wireless 56

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Flowfisalocalowatnodeiifiisthesourceoff.Flowfisalocalowofvirtualnodeitiffisalocalowofianditsdestinationist.Theprimaryowofa(virtual)linkistheowthathasthelargestnormalizedrateamongallowspassingthat(virtual)link.Whenmultipleowshavethelargestnormalizedrate,theyareallprimaryows. Belowweexplaintheoperationsperformedinthemeasurementandadjustmentperiods.Notethattheoperationsbyavirtualnodeitareactuallyperformedbythephysicalnodei. 57

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Attheendofameasurementperiod,foreachvirtuallink(it;jt),theendnodesexchangetheirbuerstate,whichcanbepiggybackedinRTS/CTS/DATA/ACKpacketswithoneextrabit(saturatedornot).Basedontheirbuerstate,bothitandjtcandeterminethetypeof(it;jt),whichisbuer-saturated,bandwidth-saturated,orunsaturated. 58

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3.4.3 ,anodemustalsoknowthenormalizedratesandthechanneloccupanciesofallwirelesslinksthatcontendwithanyofitsadjacentlinks. Werefertothenormalizedrateandthechanneloccupancyofawirelesslinkasthestateofthelink.Wemustdisseminatethestateofeachwirelesslink(i;j)toallnodesthathavealinkcontendingwith(i;j).ForIEEE802.11DCF,itincludesallnodesthatarewithintwohopsfromeitheriandj.Thedisseminationprotocolisdescribedasfollows.Recallthatweonlyconsiderstaticwirelessnetworks.Afterdeployment,weassumeeachnodeidiscoversthewirelesstopologyinitstwo-hopneighborhood,andidentiesaminimumsubsetofone-hopneighbors,calledi'sdominatingset,whoseadjacentlinksreachalltwo-hopneighbors.Nodeiinformsthenodesinitsdominatingsetoftheirmembershipintheset.Attheendofeachmeasurementperiod,ifthestateof(i;j)changesfromthepreviousperiod,bothiandjbroadcastthenewstatetotheirone-hopneighbors.Whenanodeintheirdominatingsetsoverhearsthisinformation,thenodere-broadcaststheinformationtoitsneighbors. Thestateofalinkisverysmall.Insteadofmakingaseparatetransmission,suchinformationcanbedisseminatedbypiggybackinginRTS/CTS/DATA/ACKpackets,whichareoverheardbyallnodesinone-hopneighborhood.Inthisdesign,ipiggybacksthestateof(i;j)initsnormaltransmission,andafteroverhearingtheinformation,anodeini'sdominatingsetdoesthesamething.Toovercomefailedoverhearing,thesameinformationshouldbebepiggybackedinanumberoftransmissions.Westressthatthepiggybackdesigncanbeappliedtodisseminateotherinformationintherestoftheprotocolaswell. 59

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Theoperationsperformedbythenodesinthisperiodareexplainedbelow.RemovingUnnecessaryRateLimits 1. If(jt;it)isequaltoL1,thenaratereductionrequestisissuedfortheprimaryowsonvirtuallink(jt;it).IfL1>3S1,itrequeststheprimaryowstohalvetheirrates;otherwise,itrequeststheprimaryowstoreducetheirratesbypercentage.(Themotivationfortheaboveratereductionschemeisstraightforward.Whilereducingbypercentageisthenorm,anoptimizationisadded|whenthegapbetweenL1andS1istoobig,reducingbyhalfhelpstoclosethegapquickly.Thenumber3isarticiallyset.) 2. If(jt;it)isabuer-saturatedlinkand(jt;it)isequaltoS1,thenarateincreaserequestisissuedfortheprimaryowsonvirtuallink(jt;it).IfL1>3S1,itrequeststheprimaryowstodoubletheirrates;otherwise,itrequeststheprimaryowstoincreasetheirratesbypercentage. 60

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Atthebeginningofeachadjustmentperiod,icomputesthechanneloccupancyofeachclique,whichisequaltothesumofthechanneloccupanciesofthewirelesslinksintheclique.Forawirelesslink(i;j)thathasatleastonebandwidth-saturatedvirtuallink,wedothefollowing:First,amongitsbandwidth-saturatedvirtuallinks,weidentifytheone(it;jt)withthesmallestnormalizedrate.Amongallcliquesthat(i;j)belongsto,wetreatthosethathavethelargestchanneloccupancyasbeingsaturated.Second,wecheckwhether(it;jt)satisesthebandwidth-saturatedcondition.If(it;jt)isnotthelargestnormalizedrateinanyofitssaturatedcliques,wemustincrease(it;jt)byissuingrateadjustmentrequests.LetL2bethelargestnormalizedrateonwirelesslinksinallsaturatedcliquesthat(i;j)belongsto.NodeidisseminatesL2,(it;jt),andtheidentiersofsaturatedcliquesviaitsdominatingsettoallnodesintwo-hopneighborhood.Whenanodekreceivesthisinformation,ifawirelesslink(k;m)belongstooneofthosesaturatedcliques,kcallsAdjust(kv;mv;L2;(it;jt))foreachofitsvirtuallinks(kv;mv),with 61

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3-5 .First,weassignallowsthesameweight1,sothataow'snormalizedrateisthesameastheow 62

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3-1 areconsistentwiththeaboveanalysis.Inthesimulation,aftertheowratesarestabilized,(0;1)and(1;2)arebandwidth-saturatedlinks,while(3;4)and(4;5)areunsaturatedlinks.Thebandwidth-saturatedconditionensuresthat,inthesaturatedclique1,thenormalizedrateoff2isnolessthanthoseoff3andf4.Therate-limitconditionensuresthatf1willsendatthehighest-possiblerateaslongasitdoesnotdrivetherateoff2toolowthatviolatesthebandwidth-saturatedconditionoff2inclique1. Nextwetestweightedmaxminonthesamenetworktopologybyassigningdierentweightstoows.ThesimulationresultsaregiveninTable 3-2 .Theratesofthethreeowsinclique1areapproximatelyproportionaltotheirpre-assignedweights.Flowf1hasahigherratethanowf2eventhoughitsweightissmaller.Thatisbecauseitopportunisticallyutilizesallremainingbandwidthinclique0thatcannotbeusedbyf2. 4 ].Thesethreeprotocolsusedierentbuermanagementstrategiestoaccommodatetheirpacketqueuingalgorithms.In802.11,allowspassinganodesharethesamebuerspace.Whenapacketarrivesatanodewhosebuerisfull,itwilloverwritethepacketatthetailofthequeue.In2PP,eachowisallocatedaseparatedqueuethatcanhold10packets.InGMP,allowstothesamedestinationshareacommonqueuethatcanhold10packets. 63

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AsinSection 2.4 ,weadoptthemaxminfairnessindexImmandtheequalityfairnessindexIeqtoevaluatetheend-to-endfairnessandeectivenetworkthroughputUtomeasurethespatialreuseofspectrum. FirstwesimulatethescenarioinFig. 3-6 .ThesimulationresultsareshowninTable 3-3 .GMPismuchfairerthan2PP,whichisinturnmuchfairerthan802.11.Duetothehiddenterminalproblemunder802.11,asevereunfairnessinmediaaccessexistsbetweenlink(0;1)and(2;3)[ 2 ].Node0hasmuchlesschancetograbthechannelwhenithaspacketstobetransmittedtonode1.Thisexplainswhytheowfromnode0tonode3,whichpasses(0;1),hasthelowestrateunder802.11.Theeectivenetworkthroughputsof2PPandGMParecomparable,andtheyarehigherthanthatof802.11,whichdropsmorepacketsduetobueroverow. Thedesignof2PPistoensureabasicfairshareofbandwidthforallowsandthenfavorshortowsinallocatingtheremainingbandwidth.Thebasicfairsharecanbeverysmall,andtherearecasesinwhichitisoutperformedby802.11.WeperformsimulationsonthetopologyinFig. 3-7 ,andtheresultsareshowninTable 3-4 .Withthistopology,thebasicfairsharecalculatedbasedontheformulain[ 4 ]issmall,andtheremainingbandwidthisdistributedheavilybiasedtowardsf2andf8basedonthelinearprogrammingapproachinthesamepaper.Under802.11,theowsinthemiddle(f3,f4,f5andf6)havelowerratesthantheowsonthesides(f1,f2,f7andf8).Thereasonisthataowinthemiddleneedcompeteforbandwidthwithmoreowsthanaowontheside.WithGMP,allowshaveapproximatelyequalratesregardlessoftheirlocationsandlengths.Theowsinthemiddlehaveslightlylowerratesfortwopossiblereasons.First,underGMP,twoowratesareconsideredtobe\equal"iftheirdierenceisbelow,whichis10%inoursimulations.Second,themaximumcombinedrateofthefourlinks 64

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Finally,weperformsimulationsonamorecomplexnetworktopologyshowninFig. 3-8 .Thenetworkconsistsof25nodesthataredeployedina900900m2region.Wecreate25multihopowsinthenetwork,wherethesourceandthedestinationofeachowarerandomlychosen.ThedestinationsoftheowsstartingfromanodearelistedinsquarebracketsafterthesourcenodeID.Thewirelesslinksareshownasedgesinthegraph.Asolidlinemeansthatthelinkisontheroutingpathofatleastoneow.Thetotalnumberofowspassingalinksisshowninparenthesesbesidethatlink.Adottedlinerepresentsanunusedlink.ThesimulationresultsareshowninFig. 3-9 andTable 3-5 .InFig. 3-9 ,theowratesthatareunder100pps(packetspersecond)usethenumbersontheleftverticalaxis;theowratesabove100ppsusethenumbersontherightverticalaxis. Under802.11,halfofallowshaveratesunder10pps.Severalows(e.g.f7andf13)arealmoststarved.Under2PP,threeone-hopows,f0(fromnode6tonode5inFig. 3-8 ),f3(fromnode6tonode1),andf5(fromnode12tonode7),haveveryhighratesandcontributemorethan50%ofthetotalend-to-endthroughput.Thethreeowswhoseratesarearound40pps(f11,f14andf24)arealsoshortowsthatareonlyone-hoportwo-hopslong.GMPachievesfarbetterfairnessasshowninFig. 3-9C 65

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h1;2i h3;5i h4;5i 563.96 196.96 217.57 221.41 Table3-1. SimulationresultsonthetopologyinFig. 3-5 ow h1;2i h3;5i h4;5i 1 2 1 3 rate 527.58 225.40 121.90 377.20 Table3-2. SimulationresultsofweightedmaxmininFig. 3-5 ow 802.11 2PP GMP 131.86 164.75 188.76 176.04 240.85 179.21 1013.96 1025.54 0.547 0.919 0.946 0.999 Table3-3. SimulationresultsonthetopologyinFig. 3-6 ow 802.11 2PP GMP 43.31 145.46 347.81 145.94 43.33 134.26 86.67 132.38 43.39 135.44 86.70 133.04 43.36 141.69 346.96 149.07 1214.93 1674.13 0.125 0.888 0.514 0.998 Table3-4. SimulationresultsonthetopologyinFig. 3-7 802.11 2PP GMP 1672.65 2632.74 0.017 0.206 0.298 0.835 Table3-5. SimulationresultsonthetopologyinFig. 3-8 66

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Amoresophisticatedexampletoillustratethepurposeofthefourlocalconditions Figure3-2. Anexampleofrate-limitcondition Figure3-3. Thepathofaowwithanunsaturatedbueratthesource 67

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Per-destinationpacketqueueingisnecessarywhentheowspassinganodearedestinedfordierentdestinations. Figure3-5. Networktopologyofasimplescenario Figure3-6. Athree-linkstopology Figure3-7. Networktopology 68

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Networktopology B2PP CGMP RatesoftheowsonthetopologyinFig. 3-8 69

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Maximizingtheoperationallifetimeofasensornetworkisacriticalprobleminpractice.Manypriorworksdenethenetwork'slifetimeasthetimebeforetherstsensorinthenetworkrunsoutofenergy.However,whenonesensordies,therestofthenetworkcanstillwork,aslongasusefuldatageneratedbyothersensorscanreachthesink.Moreappropriately,weshouldmaximizethelifetimevectorofthenetwork,consistingofthelifetimesofallsensors,sortedinascendingorder.Forthisproblem,thereexistsonlyacentralizedalgorithmthatsolvesaseriesoflinearprogrammingproblemswithhigh-ordercomplexities.Thischapterproposesafullydistributedprogressivealgorithmwhichiterativelyproducesaseriesoflifetimevectors,eachbetterthanthepreviousone.Insteadofgivingtheoptimalresultinoneshotafterlengthycomputation,theproposeddistributedalgorithmhasaresultatanytime,andthemoretimespentgivesthebetterresult.Weshowthatwhenthealgorithmstabilizes,itsresultproducesthemaximumlifetimevector.Furthermore,simulationsdemonstratethatthealgorithmisabletoconvergerapidlytowardsthemaximumlifetimevectorwithlowoverhead. Therestofthischapterisorganizedasfollows.Section 4.1 givesthenetworkmodelandtheproblemstatement.Section 4.2 laysdownthetheoreticalfoundationforouralgorithm.Section 4.3 proposesourdistributedprogressivealgorithmformaximizingthelifetimevector.Section 4.4 presentsthesimulationresults.Section 4.5 summarizesthechapter. 4.1.1SensorNetworkModel 70

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LetNbethesetofsensornodes,amongwhichthesubsetSthatgeneratenewdataarecalleddatasources,whichmaybetheaggregationnodesrepresentinglocalclusters[ 1 17 ].Letgi;i2N,bethesourcerateatwhichnodeigeneratesnewdatapackets.gi>0ifi2S;gi=0ifi62S.Weassumethatthesourceratesaresetlowenoughtonotcausecongestioninthenetwork.Thesinkmayconsistofmultiplegeographicallydispersedbasestations.Assumethebasestationsareexternallyconnectedtoadatacollector.Itmakesnodierencewhichbasestationadatapacketisroutedto. Twonodesareneighborsiftheycanreceivepacketsfromeachother(tosupportDATA/ACKexchange).Theremaybemultipleroutingpathsfromeachnodetothesink.LetDibethesetofneighborsthatnodeiuseasthenexthopstothesink.Theyarecalleddownstreamneighborsofnodei.8j2Di,(i;j)iscalledanoutgoinglinkofi.LetUibethesetofupstreamneighbors,whichuseiasthenexthopontheirroutingpathstothesink.8k2Ui,(k;i)iscalledanincominglinkofi.Ifiisadownstreamneighborofk,thenkmustbeanupstreamneighborofi.LetE=f(i;j)j8i2N;j2Dig.Wecallthegraphconsistingofalltheselinksastheroutinggraphofthesensornetwork,whichcontainsallroutingpathsfromdatasourcestothesink. 71

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Leteibetheenergyavailableatnodei.Letbetheamountofenergythatanodespendsonreceivingadatapacketfromanupstreamneighbor,ibetheamountofenergythatnodeispendsonproducinganewdatapacket,ibetheamountofenergythatnodeispendsonsendingapacket.Theenergyconstraintisgivenbelow. WesayanodeiisexhaustedifXk2Uiv(k;i)+iv(i)+Xj2Diiv(i;j)=ei: Ifitrequiresperiodicmeasurementofmin/max/avgamongreadingsfromsourcesthathavenotexhaustedyetandremainreachabletothesink,thenanodewillsendapacketforeachsetofpacketsreceivedfromitsupstreamneighborsorgeneratedlocally.Theconstraintbecomes 72

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Thelifetimevectorofthesensornetworkisdenedas(ts;s2S)sortedinascendingorder. Eachfeasiblevolumescheduleproducesafeasiblelifetimevector.Allfeasiblelifetimevectorsformthelifetimespace.OnelifetimevectorTisgreaterthananotherT0ifTislexicographicallylarger|forsomex2[1::jSj],TandT0sharethecommon(x1)smallestelementsbutthexthsmallestelementinTisgreaterthanthexthsmallestelementinT0. Themaximumlifetimevectorproblemistondafeasiblevolumeschedulethatproducesthelargest(orsay,maximum)lifetimevector.Intuitively,itsgoalistorstmaximizethesmallestlifetimeofallsources,thenthesecondsmallest,andsoon. Oncewendthevolumescheduleforthemaximumlifetimevector,thenodesmustknowtheirpacketforwardingpoliciesthatwillrealizethevolumeschedule.Toimplementavolumeschedule,eachnodeisimplydoesthefollowing:1)itgeneratesnewpacketsatitssourcerategiforv(i)packets,and2)itforwardsthereceivedpacketstodownstreamneighborsinweightedroundrobin,usingthevolumesontheoutgoinglinksastheweights.Therefore,thepacketratesontheoutgoinglinksareproportionaltothevolumesonthelinks.Thisiscalledthevolume-rateproperty. wherer(i;j)isthepacketrateonlink(i;j). 73

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4{2 ):Identifyaroutingloopandndthelink(i;j)withthesmallestvolumev(i;j).Deductthelinkvolumesalongtheloopbyv(i;j)andremove(i;j).Repeattheaboveprocedureuntilallloopsareremoved.Theresultingvolumeschedulestillsatisesthevolumeconservationconstraintandtheenergyconstraint.Nosourcevolumehasbeenchanged,andthusthelifetimevectorproducedbythenewvolumescheduleontheacyclicroutinggraphremainsthesame.Furthermore,sincesomelinkvolumeshavebeenreduced,whichmayleaveroomforincreasingsomesourcevolumestoproducealargerlifetimevector.Thesamereasoningcanbeappliedtomodel( 4{3 )withaddeddetailsonhowtoreducevolumesalongacycleandonotherlinks.Butthekeypointisthesame|whenremovingacycle,linkvolumesonlyneedtobereduced. Theacyclicroutinggraphcanbeeasilyconstructedwhenpacketsareforwardedbasedonhopcountsorthenodes'geographiclocationstothesink.Forexample,Dimayconsistofalloraselectedsubsetofneighborsthatareclosertothesink(basedonthehopcountorEuclideandistancetotheclosestbasestation),andUimayconsistofalloraselectedsubsetofneighborsthatarefurtherawayfromthesink. Thevolumeofa(directed)pathisdenedastheminimumvolumeofthelinksonthepath.Apathintheroutinggraphiscalledaforwardingpathifitsvolumeisgreaterthanzero.Otherwise,itiscalledanon-forwardingpath. Nodes2Sisafeedingsourceofnodei2Nifthereisaforwardingpathfromstoi.Furthermore,nodesisarestrictedfeedingsourceofnodeiifthereisanexhaustednodeoneveryforwardingpathfromstoi.Nodesisanunrestrictedfeedingsourceofnodeiifthereisnoexhaustednodeonatleastoneforwardingpathfromstoi,wherethepath

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Wewillestablishthenecessaryandsucientconditionsformaximizingthelifetimevectorinatheorembelow.Belowweexplainabasictechniqueusedintheproof,calledvolumeshift.Understandingthistechniquewillalsohelponetounderstandthedesignofthealgorithm. ConsidertheroutinggraphinFig. 4-1 .Supposesandwaretwounrestrictedfeedingsourcesofnodei.LetP1andP2betwoforwardingpathsthatdonothaveanyexhaustednode.Weshowthatthelifetimeofanunrestrictedfeedingsourcecanbeincreasedattheexpenseofthelifetimeofanother.Todoso,wesimplydecreasethesourcevolumeofs,thendecreasethevolumesonthelinksofP1,increasethesourcevolumeofw,andnallyincreasethevolumesonthelinksofP2,allbythesametinyamount,whichshouldbesmallenoughsuchthatitsadditiononP2doesnotviolatetheenergyconstraint.Theaboveoperationiscalledavolumeshiftfromstowwithrespecttoi.Itiseasytoseethat,aftervolumeshift,thevolumescheduleremainsfeasibleandthelifetimeofsisdecreased,thelifetimeofwisincreased,whilethelifetimesofallothersourcesremainunchanged.Itisobviousthat,toimprovethelifetimevector,weshallalwaysperformavolumeshiftfromanodewithalargerlifetimetoanodewithasmallerlifetime. Notonlycanavolumeshiftbeperformedbetweentwounrestrictedfeedingsources,butalsoitcanbeperformedfromarestrictedfeedingsourceutoanunrestrictedfeedingsources,orfromanunrestrictedfeedingsourcestoapotentialsourcez,butnottheotherwayaround|morespecically,i)avolumeshiftcannotbeperformedfromanunrestrictedfeedingsourcestoarestrictedfeedingsourceubecausewecannotaddanyadditionalvolumetoP3thathasanexhaustednodex;ii)avolumeshiftcannotbeperformedfromapotentialsourceztoanunrestrictedfeedsourcesbecausethevolumeofanypathfromztoiiszeroandthusnothingcanbeshiftedout. 75

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1. Thereisanexhaustednodeoneverypathfromasourcetothesink. 2. Allunrestrictedfeedingsourcesofanodemusthavethesamelifetime,whichshouldbenolessthanthelifetimesoftherestrictedfeedingsourcesofthesamenode,andnogreaterthanthelifetimesofthepotentialsourcesofthesamenode. Proof:First,weprovethattheconditionsarenecessary.Ifafeasiblevolumescheduledoesnotmeeteithercondition,weshowthat,bymodifyingthevolumeschedule,wecanproducealargerlifetimevector.IftherstconditionisnottrueonapathPfromasourcestothesink,wecanimprovethelifetimeofsbyincreasingitssourcevolumeaswellasthevolumeofPbyatinyamount,whichresultsinalargerlifetimevector.Nextconsiderthesecondcondition. 4-1 )fromstowsuchthatthelifetimeofwisslightlyincreased(butstillbelowthatofs),whichresultsinalargerlifetimevector.Notethatthevolumeshiftonlychangesthelifetimesoftwonodes,sandw. Second,weprovethattheconditionsaresucient.Thelifetimespace,consistingofallfeasiblelifetimevectors,isconvexandcompact,whichcanbeseenfromthelinear(ormax)natureoftheenergyconstraint( 4{1 )andthevolumeconservationconstraint( 4{2 )or( 4{3 ),aswellasthelifetimedenition( 4{4 ).RadunovicandLeBoudecshowedthat,inaconvex,compactspace,amax-minvectorexists,andmoreoveritisuniqueandmustbelexicographicallylargestinthespace[ 58 ].Hence,weonlyneedtoshowthatafeasiblevolumeschedulethatmeetsthetwoconditionsproducesthemax-minvector,satisfyingthefollowingrequirement:Thelifetimetsofonesourcescannotbeincreased 76

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Basedontherstcondition,datasourcesshouldaggressivelysettheirsourcevolumestothehighestvaluesthattheirpathstothesinkallow. Thelifetimeofasources,whichisv(s)=gs,canbeinterpretedastheaveragevolumeassignedtoeachunitofrate.Thesecondconditionrequiresthateachunitofratereceivedbyanodeifromanunrestrictedfeedingsourcedeservesthesameamountofvolumeallocation.Inotherwords,forunrestrictedfeedingsources,nodeishouldallocatevolumesinproportiontotheirrates(thatireceivesandforwards).However,eachunitofratefromarestrictedfeedingsource(whichencountersanexhaustednodeonitsforwardingpath)mayreceivelessvolumeallocationatnodei.Moreover,asourceshouldalwaysdirectitspacketstopathsthathavehighestvolumeallocationperunitofrate. 77

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DPAbeginswithaninitialrateschedulethatcanbearbitrarilyset.Fromtheratescheduleandenergyavailabilityatthenodes,itcomputesavolume-bounddistributionbasedonthesecondconditioninTheorem 3 .Fromthevolume-bounddistribution,itsetsavolumeschedule,basedonwhichitwillinturnderiveanewrateschedule.Thiscompletestherstiterationofthealgorithm.AsshowninFig. 4-2 ,ineachsubsequentiteration,DPArepeatstheabovecomputationofanewvolume-bounddistribution(basedontherateschedulefromthepreviousiteration),thenanewvolumeschedule,andnallyanewrateschedule.Eachiterationproducesabettervolumeschedulewhoselifetimevectorislargerthanthepreviousone. Therateschedule,volume-bounddistribution,andvolumeschedulearestoredandcomputedinafully-distributedway.Eachnodeonlymaintainstherates,volumebounds,andvolumesofitsadjacentlinkswithaspacecomplexityofO(jDij+jUij).Becauseeachdirectedlinkissharedbyapairofupstream-downstreamnodes.Somepropertiesofthelinkwillbesetbytheupstreamnodeandthensenttothedownstreamnode,whileotherpropertieswillbesetbythedownstreamnodeandthensenttotheupstreamnode.Detailsaregivenbelow. Nodeiwillsetitsoutgoingrates,r(i;j);j2Di,bydistributingthetotalincomingrateamongtheoutgoinglinks.Itwilllearntheincomingrates,r(k;i);k2Ui,fromupstreamneighborskwhosetthoserates.(Wewanttostressthatthelinkrateshereareauxiliaryvariablesusedtofacilitatethecomputationofvolumes.Theyhavenothingtodowiththeactualdata-packetratesonthelinksatthetimewhenDPAisexecuted.In 78

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Nodeiwillsetitsoutgoingvolumesv(i;j)bydistributingthetotalincomingvolumeamongtheoutgoinglinks.Itwilllearntheincomingvolumesv(k;i)fromupstreamneighborskwhosetthosevolumes. Nodeiwillsetitsincomingvolumeboundsb(k;i)bydistributingitsforwardingcapacityamongtheincominglinks.Itwilllearntheoutgoingvolumeboundsb(i;j)fromdownstreamneighborsjwhosetthosebounds. Intherestofthesection,wewilldescribethedetailsofDPA,whichconsistsofInitializationphaseanditerativephasewitheachiterationhavingtwosteps.Therststepcomputesvolumeboundsbasedonlinkrates.Thesecondstepdetermineslinkvolumesfromvolumeboundsandthencomputesnewlinksrates,whichsetsthestageforthenextiteration. 79

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Intotal,atmostjNjINITpacketsandjNjRATEpacketsaretransmitted.EachnodeisendsoneINITofsizeO(1)andoneRATEpacketofsizeO(jDij).Theinitializationphasecompleteswithinthemaximumroundtriptimebetweenthesinkandanysourceinthenetwork. First,anodeishouldnotreceiveandforwardmorepacketsthanthedownstreamneighborscanhandle.Iftheapplicationmodelischaracterizedby( 4{2 ),thenthecombinedincomingvolumebound(setbyi)shouldnotexceedthecombinedoutgoingvolumebound(setbydownstreamneighbors). whereb(i;j)islearnedbyifromj.Iftheapplicationmodelischaracterizedby( 4{3 ),thentheconstraintbecomes maxfmaxk2Uifb(k;i)g;b(i)gXj2Dib(i;j)(4{7) Second,nodeishouldnotreceiveandforwardmorepacketsthanitsenergyallows. 80

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4{2 );maxfmaxk2Uifb(k;i)g;b(i)gb(i;j) 4{3 ): 4{6 )-( 4{7 ),b0(i;j)b(i;j),andthereforetheaboveconstraintismorerelaxthanonethatreplacesb0(i;j)withb(i;j). Aswehaveexplainedintheprevioussection,thesecondconditionofTheorem 3 requiresthatvolumeallocationshouldbemadeinproportiontotheincomingrates(whichmustbeadjustedforrestrictedfeedingsources,aswillbediscussedshortlyinStep2).Hence,wehavethefollowingrate-boundproperty. Ifr(k;i)=0,thenb(k;i)=0.Ifgi=0,thenb(i)=0. ThedistributedcomputationofStep1isdescribedasfollows:Thesinkbeginstheprocessofsettingvolumeboundsaftertherateinitializationphaseterminates(atthetimewhenthesinkreceivesRATEsfromallupstreamneighbors),orafterStep2completes(atthetimewhenthesinkreceivesVOL RATEpacketsfromallupstreamneighbors|tobedescribedinSection 4.3.4 ).ThesinksetsitsincomingvolumeboundstobeinniteandsendsaBOUNDpacketstoupstreamneighbors,carryingthevolumeboundsofitsincominglinks.AfteranodeireceivesBOUNDsfromalldownstreamneighborsj2Diandlearnsalloutgoingvolumeboundsb(i;j),itsetstheincomingvolumebounds,b(k;i);k2Ui,anditssourcevolumeboundv(i)aslargeaspossible,basedon( 4{9 )subjecttotheconstraintsof( 4{6 )-( 4{7 )and( 4{8 ).NodeithensendsitsincomingvolumeboundstotheupstreamneighborsinaBOUNDpacket. Intotal,jNjBOUNDpacketsaretransmitted.EachnodeionlytransmitsonepacketofsizeO(jUij). 81

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TherstconditionofTheorem 3 requiresustosetthesourcevolumeashighaspossible.Hence,weassign Inadditionto( 4{10 ),outgoinglinkvolumesarealsosubjecttothevolumeconservationconstraintin( 4{2 )or( 4{3 ).Anodecannotsendmorepacketsthanitreceives.Ifitdoesnotreceiveenoughincomingvolumes,itsoutgoingvolumesmayhavetobesetlowerthanwhatthevolumeboundsallow.Ifthevolumeconservationconstraintis( 4{2 ),tosatisfythisconstraint,nodeiassignsitsoutgoingvolumesasfollows. wherev(k;i)issetbyupstreamneighborkandlearnedbyifromk.Ifthevolumeconservationconstraintis( 4{3 ),nodeiassignstheoutgoingvolumestobe 4{10 ). First,weprovebyinductionthatusing( 4{12 )willsatisfytheboundconstraint( 4{10 ).ConsiderthebasecasewithUi=;.By( 4{12 ),( 4{6 )andthefactthatUi=;,we 82

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Nextwemaketheinductiveassumptionthatv(k;i)b(k;i);8k2Ui,andprovethecasewhenUi6=;.(ThisisavalidinductiveassumptionforaDAGroutinggraph,whichhasnoloopforcircularreasoning.)Togetherwith( 4{6 )and( 4{11 ),wehavev(i;j)=(Xk2Uiv(k;i)+v(i))b(i;j) Theinductionproofforthecaseof( 4{13 )issimilar. Thisresult,togetherwith( 4{8 ),ensuresthattheassignedvolumessatisfytheenergyconstraintrequiredin( 4{1 )|toseethis,onehastousethefactthatv(i;j)b0(i;j)dueto( 4{12 )-( 4{13 )and( 4{10 ),whereb0(i;j)isdenedin( 4{8 ).Consequently,theresultingvolumescheduleisfeasible. Afterwesetthelinkvolumes,weassignnewlinkratesbelowbasedontherate-volumepropertyin( 4{5 ),settingthestageforthenextiteration.Forapplicationmodel( 4{2 ), Forapplicationmodel( 4{3 ), 83

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4-3 ,thevolumeboundassignedbyionlink(w;i)foranunrestrictedfeedingsourceswillbefullyutilized.However,thevolumeboundassignedbyionlink(k;i)forarestrictedsourceumaynotbefullyutilizedduetoanupstreambottleneckxthatmaysetatighterboundonthesourcevolumeofu.Inthiscase,thevolumev(k;i),whichissetbykandconstrainedbythelimitedupstreamenergyatx,issmallerthanthevolumeboundb(k;i).Whenthishappens,weshallreduceb(k;i)tomatchv(k;i),andallowb(w;i)tobelarger,whichwillinturnallowstohavealargersourcevolumeandthusalargerlifetime.SincevolumeboundsaresetatStep1basedonlinkrates,wecanachievethereductionofb(k;i)byarticiallyreducingtherater(k;i). Morespecically,afterthelinkratesarecalculatedbasedon( 4{14 )-( 4{15 ),theymaybereducedbymultiplyingareductionfactorf(i)(2(0;1]),whichhasaninitialvalueof1andisupdatedateachiterationasfollows.Supposenodeiisnotadirectneighborofthesink.Ifiisexhausted,i.e.,Pk2Uiv(k;i)+iv(i)+Pj2Diiv(i;j)=ei,oritwasexhaustedinoneofthepreviousiterations,thenitupdatesf(i): whereB(i)andV(i)arethewould-bevolumeboundandvolumeonalloutgoinglinks,respectively,iftheratereductionhadnotbeenpreformedtoreducetheoutgoingvolumeboundinpreviousiterations.Clearly,thevalueoff(i)willstabilizeatanexhaustednodeionlywhenthevolumePj2Div(i;j)matchestheboundPj2Dib(i;j).Afterupdatingf(i),nodeireducestheoutgoingratesasfollows. 84

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4-3 ,bothiandxwillperformtheaboveoperation.Whenxdoesso,itsratereductionwillpropagatedownstream,causingthereductionofr(k;i),whichinturncausesthereductionofb(k;i)andtheincreaseofb(w;i). ThedistributedcomputationofStep2forsettingvolumes/ratesisanaturalcontinuationofStep1.AfteranodewithnoupstreamneighborreceivesBOUND(denedStep1)fromalldownstreamneighbors,itisabletoassignitssourcevolumeby( 4{11 )andoutgoingvolumesby( 4{12 )-( 4{13 ).Itthenupdatesthelinkratesby( 4{14 )-( 4{15 ),( 4{16 ),and( 4{17 ).Afterthat,itsendstheoutgoingvolumes/ratestothedownstreamneighborsbyaVOL RATEpacket.AfteranodeireceivesVOL RATEpacketsfromallupstreamneighborskandlearnsv(k;i),itisabletoassignitssourcevolumeby( 4{11 ),theoutgoingvolumesby( 4{12 )-( 4{13 ),andthenewoutgoingratesby( 4{14 )-( 4{15 ),( 4{16 ),and( 4{17 ).Itsendstheoutgoingvolumes/ratestodownstreamneighborsinVOL RATE.WhenthesinkreceivesVOL RATEfromallupstreamneighbors,itknowsthatStep2iscompleted. Step2transmitsjNjpackets.EachnodeisendsonlyonepacketofsizeO(jDij).Eachiteration,includingStep1andStep2,completeswithinthemaximumroundtriptimebetweenthesinkandanysourceinthenetwork. First,considerthecomputationofvolumebounds.Thetotalforwardingcapabilityofanode,whichisdeterminedby( 4{6 )-( 4{7 )and( 4{8 ),isdistributedasvolumeboundsbasedontherate-boundpropertyin( 4{9 ),whichessentiallyperformsvolumeshiftfromfeedingsourceswithlargervolumeperunitofrate(i.e.,largerlifetime)tothosewithsmallervolumeperunitofrate.Suchvolumeshiftincreasesthelifetimevector.Theonlyproblemisthatavolumeboundmaynotbefullyturnedintovolumeifthereisanupstreamexhaustednodewhichsetsatightervolumebound.Thisproblemissolvedby 85

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4{16 )untilthevolumematchesthebound. Second,thevolumeassignmentsin( 4{10 )and( 4{12 )-( 4{13 )areaggressiveinthesensethattheytrytofullyutilizeallvolumebounds,bysettingthesourcevolumesashighaspossibleandbyforwardingallincomingvolumesateachnode. Third,theratereductionin( 4{14 )-( 4{15 ),( 4{16 )and( 4{17 )articiallydecreasesthelinkratesifthevolumeboundsarenotfullyturnedintothevolumes.Insubsequentiterations,dueto( 4{9 ),decreasedratesleadtodecreasedvolumeboundsonthoselinks,allowingotherlinksthatcanfullyutilizetheirboundstohavehighervolumebounds. Insummary,thevolumeboundcomputationperformsvolumeshiftfromlarge-lifetimesourcestosmall-lifetimesources;thevolumecomputationandtheratereductiontechniqueensurethatthevolumeboundsarefullyutilized.Together,theyimprovethelifetimevectorasDPAexecutesthroughitsiterations.Asthelifetimevectormovesincreasinglyclosertoitsmaximumvalue,theroomforimprovementbecomessmallerandsmaller.OursimulationswillshowthatDPAconvergesrapidly. Proof:LetGbethesubgraphconsistingofallpathsfromsourcestotherstencounteredexhaustednodesortothesinkifnoexhaustednodesareencountered.RatereductionhasnoimpactonthelinkratesinsideG.WhenlinkvolumesarestabilizedinG,linkratesandvolumeboundsmustalsobestabilizedbecausetheirlinearinter-dependencyin( 4{5 ),( 4{9 ),( 4{12 )-( 4{13 )and( 4{14 )-( 4{15 ).Weprovebyinductionthat ConsiderthebasecasewithUi=;.Nodeiisnotexhaustedandhenceiv(i)+Xj2Diiv(i;j)
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4{11 )and( 4{12 )-( 4{13 ),itcanberewrittenasib(i)+Xj2Diib0(i;j)
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4{14 )-( 4{15 ),itsratewillbecontiguouslyshiftedawayfromfeedingi,andeventuallyturnsitselfintoapotentialsourceofi. WehaveprovedearliertherstconditionofTheorem 3 thatthereisanexhaustednodeoneverypathfromasourcetothesink.LetG0bethesubgraphconsistingofallpathsfromsourcestothelastencounteredexhaustednodes,andC0bethesetofthoselastencounteredexhaustednodes,whichformsacutofthenetworkthatseparatesthesinkfromallsources.WhenlinkvolumesarestabilizedinG0,linkratesandvolumeboundsmustalsobestabilizedbecausetheirlinearinter-dependencyin( 4{5 ),( 4{9 ),( 4{12 )-( 4{13 )and( 4{14 )-( 4{15 ).Weprovebycontradictionthat Suppose,9(k;i)2G0,v(k;i)
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By( 4{5 ),( 4{9 ),( 4{14 )-( 4{15 ),( 4{2 )-( 4{3 )and( 4{19 ),theratioofvolumetorateiskeptconstantonanypathsegmentinG0thatdoesnotcontainanexhaustednode(whichperformsratereduction).Itiseasytoseethatthelifetimeofarestrictedfeedingsourceuofanodeicanonlybeequaltoorsmallerthanthatofanunrestrictedsourcebecause,duetoratereduction,theradioofvolumetoratewilldecreasewhenwetraverseapathbackwardfromitouandcrossanexhaustednode. 4 ,weshallterminateDPAwhenithasstabilizedthelinkvolumes,whichcanbedetectedbyaddingaagthatistransitivelycarriedbythecontrolmessages.Theagisinitiallyunset.Anodesetstheagifitchangesalinkvolumebyanamountthatisnotnegligiblysmall.Itisuptotheapplicationrequirementtodecideonhowsmallisnegligible.Thesinkwillstopifitdoesnotreceiveaagthatisset.Alternatively,DPAmayalsobeterminatedarticiallyafteracertainnumberofiterations,orwhentheresultinglifetimevectormeetstheapplicationrequirement. Whiletheoodingdesignitselfmayappearnon-innovative,thenoveltyofDPAisinthedetailsthatestablishestheconstraintsandformulasfornodestoperformlocalizedoperations|iterativelycomputingtheirindividualvolumeboundsfromrates,volumesfromvolumesbounds,andratesfromvolumeswithreduction|yetglobally,asanetoutcome,produceaprogressivelybetterlifetimevector,approachingtotheoptimalresult. 89

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RATEofsizeO(jDij).Upstream/downstreamneighborsrepresentasubsetofallnodeswithinthecommunicationrangeofi.Thepacketsizeislimitedwhenwechooseasmallnumberofupstream/downstreamneighborsforroutingpurpose.WeperformedmanysimulationsinSection 4.4 ,whichshowsthatDPAconvergesquicklytowardstheoptimallifetimevector.Toachievenomorethan5%deviationfromtheoptimal,fornetworksof1,000nodes,lessthan25iterationsareneeded.Inaddition,theoverhead(i.e.numberofiterations)increasesslowlywithnetworksize. Ifthenetworkisdesignedtocollecttensofthousandsofdatapacketsfromeachsource,thesmalloverheadofDPA(intensofcontrolpacketspernode)isnegligible.Ifthenumberofiterationsispre-determined,wecantakethesmallenergyconsumptionofDPAintoaccountbyreducingthenodes'energy(ei)foranappropriateamount. Tokeepupwithchanges,DPAmaybere-executedtocomputeanewvolumeschedule.Thereisatradeobetweenoverheadandbetterlifetimevector.ThefrequencyofexecutingDPAisdependentontheamountofoverheadallowed.Forexample,supposethesinkcollectsaggregateinformationfromthenetworkperiodicallybasedontheapplicationmodelcharacterizedby( 4{3 ),anddatapacketsarelongerthancontrolpackets(INIT/RATE/BOUND/VOL RATE).IfDPAisallowedtoconsumenomorethan0.5%of 90

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Theonlyalternativesolution[ 1 17 ]intheliteratureiscentralized.Itismuchharderforacentralizedalgorithmtohandlenetworkdynamicsbecausethatrequiresthesinktocollectthecompletenetworkinformationbeforeeachexecution. 4-4 ,whereacirclerepresentsasourcenode,asquarerepresentsanon-sourcenode,andthetwonumbersbesideanodearetheinitialenergy(inJoules)andthesourcerate(inpackets/min),respectively.DPAitselfdoesnotdictatehowtheroutinggraphshouldbeconstructed.Instead,itcanworkwithanyroutinggraphthatcontainthepotentialroutingpathsitcanchoosefrom(seeSection 4.1 ).DPAworksattheapplicationlevel;itisindependentofwhichMACprotocolisused.Suppose==0:000012Joule/packetand=0:0000432Joule/packet,whicharechosenbasedontheparametersin[ 59 ]andwillbeusedinalloursimulations. Tables 4-1 showsthelifetimevectorsaftertherst,second,10th,and20thiterationsofDPA,aswellasthemaximumlifetimevector(MLV)inthelastcolumn,whichiscomputednumericallybasedonHou'scentralizedalgorithm[ 1 ].TheresultdemonstratesthatthesequenceoflifetimevectorsproducedbyDPAconvergesrapidlytowardsMLV.Table 4-2 showsthesourcevolumesthatareassignedbyDPAtothesourcenodesaftertherst,second,10th,and20thiterations,aswellastheoptimalsourcevolumesthatproduceMLV.Recallthatthesourcevolumeisthenumberofpacketsthatasourcecan 91

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ConsiderthelifetimevectorVxproducedbyDPAafterthexthiteration.WemeasurehowmuchVxdeviatesfromMLVbythefollowingtwometrics.Lettx(s)bethelifetimeofsourcesinVx.Lett(s)bethelifetimeofsinMLV.ThemaxdeviationofVxisdenedasmaxs2Sfjtx(s)t(s)j 4-5 showstheavg/maxdeviationsoflifetimevectorsproducedbyDPAon500-nodesensornetworks.Thedeviationsdropquicklytoaninsignicantlevelafterasmallnumberofiterations.Eachofthedatapointsusedtoproducetheguresinthissectionistheaverageof100simulationrunsondierentrandomnetworks.Table 4-3 presentssomedatapointsforFig. 4-5 .Forexample,theavg/maxdeviationsaremerely0.066and0.013respectivelyafter20iterations|thatmeans,intheworsecase,thelifetimeofany 92

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4-6 .ItshowsthattheoverheadforDPAtosatisfyatargetdeviation,whichismeasuredbythenumberofiterations,growsslowlywiththenetworksize.Recallthatanodesendsatmost2smallcontrolpacketsineachiteration.Evenforanetworkof3,000nodes,only12iterationsareneededtoachieveanavgdeviationof5%,and32iterationsareneededforamaxdeviationof5%. 1 ]basedoniterativelinearprogramming.OurDPAcanalsobeusedasacentralizedalgorithmwhentheinformationaboutthenetworkisavailableatthesink.ThetargetmaxdeviationforDPAissettobe0.025.Fig. 4-7 comparestherunningtimesofthetwoalgorithms.ItshowsthatDPAareordersofmagnitudefasterthanLP,andthegapwidenswhenthenetworksizeincreases. NextwecomparethecommunicationoverheadofthetwoalgorithmswhenLPisusedasacentralizedalgorithmwhileDPAisusedasadistributedalgorithm.ForLP,thesinkhastocollectnetworkinformation,including,foreachnode,sourcerate(4bytes),nodeenergy(4bytes),transmissionpower(4bytes),nodeID,andIDsofitsdownstreamneighbors(2byteseach).Thesinkhasalsotodownloadtheresultingvolumescheduletothenetwork,whichincludes,foreachnode,itssourcevolumeandthevolumesofitsoutgoinglinks(4byteseach).ForDPA,ineveryiteration,anodesendsoutthevolumes/ratesofitsoutgoinglinksandthevolumeboundsofitsincominglinks(4byteseach). 93

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4-8 showsthenodalcommunicationoverheadinascendingorder.Theoverheadismeasuredbythenumberofbytesthatanodehastotransmit.Clearly,somenodesinLP(attherightendofthegure)bearahugeburdenofcommunicationoverhead. TherightplotinFig. 4-8 showsthemaximumnodaloverheadwithrespecttonetworksize.ThemaximumnodaloverheadofLPincreasesmuchfasterthanthatofDPA. 17 ])thatisalinearprogrammingsolutionformaximizingtheminimumlifetimeofallsources,andMPR(Minimum-PowerRouting[ 45 60 ])thatisadistributedalgorithmforenergy-ecientrouting. FirstwerunDPA,SLP,andMPRon100-noderandomnetworks(withallnodesbeingsources).Fig. 4-9 comparesthelifetimevectorsproducedbythealgorithms.Eachcurverepresentsthelifetimevectorinascendingordergeneratedfromoneofthethreealgorithms.ThesmallestlifetimeinthevectorproducedbyDPAismorethan100%largerthanthatbyMPR.ForSLP,theresultshowsthatmaximizingtheminimumlifetimeofsourcesdoesnotmaximizethelifetimevectorofthenetwork.DPAproducesfarbettersourcelifetimesinthelowerthreequartersofthevector.Second,wecomparethealgorithmsonlargernetworks.Fig. 4-10 showstheavg/maxdeviationsofthelifetimevectorsproducedbySLPandMPRonnetworksof500to3,000nodes(with20%beingsources).ThedeviationsarelargewhencomparingwiththoseofDPA,whichcanbemadearbitrarilysmall. 94

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95

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1stiter. 2nditer. 10thiter. 20thiter. MLV k 41.9 41.9 41.3 41.9 41.9 v 41.9 41.9 41.3 41.9 41.9 u 36.9 63.2 131.2 129.4 125.8 x 33.6 60.7 109.0 121.6 125.8 m 129.9 146.2 158.9 157.3 157.3 j 108.8 154.2 160.2 157.3 157.3 w 151.0 140.5 156.7 157.3 157.3 q 335.5 200.5 239.5 251.7 251.6 Table4-1. Datasourcelifetimes(indays) sources 1stiter. 2nditer. 10thiter. 20thiter. MLV k 60.4 60.4 59.4 60.4 60.4 v 120.8 120.8 118.8 120.8 120.8 u 53.1 90.9 188.9 186.3 181.2 x 48.3 87.5 156.9 175.1 181.2 m 187.0 210.6 228.8 226.5 226.4 j 156.6 222.0 230.6 226.5 226.4 w 434.8 404.6 451.2 453.0 452.9 q 483.1 288.7 344.8 362.5 362.3 Table4-2. Datasourcevolumes(inthousandsofpackets) 10thiter. 20thiter. 30thiter. maxdev. 2.28 0.25 0.066 0.031 avgdev. 0.39 0.045 0.013 0.007 SomedatapointsusedtoproduceFig. 4-5 Figure4-1. ThereisnoexhaustednodeonP1orP2;nodessandwareunrestrictedfeedingsourcesofi.ThereisanexhaustednodexonP3;nodeuisarestrictedfeedingsourceofi.Thereisnoforwardingpathfromztoi;nodezisapotentialsourceofi. 96

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IterationsofDPA Figure4-3. Thereisnoexhaustednodefromstoi;nodesisanunrestrictedfeedingsourcesofi.Thereisanexhaustednodexfromutoi;nodeuisarestrictedfeedingsourceofi.Theupstreambottleneckxmaypreventsourceufromfullyutilizingthevolumeboundsetbyionlink(k;i). Figure4-4. Asimpleillustrativetestcase. 97

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MaxdeviationandavgdeviationoflifetimevectorwithrespecttothenumberofiterationsthatDPAhasperformed Figure4-6. DPAscaleswell.Itsoverheadgrowsslowlywiththenetworksize. Figure4-7. ComparisonofrunningtimebetweenLPandDPA 98

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Figure4-9. NetworklifetimesofDPA,SLPandMPR Figure4-10. AvgandmaxdeviationsofSLPandMPR 99

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Twoimportantproblemsinmultihopwirelessnetworksarestudied.Theyareend-to-endowratefairnessandlifetimefairness. Weproposetwoapproachestoachieveglobalend-to-endowratemaxmininmultihopwirelessnetworks.Therstapproachisacross-layerdesign.Ageneralizedmaxminmodelisrstproposedformultihopwirelessnetworks.Atthenetworklayer,ourdesignallocatesnetworkcapacitytoend-to-endowsformaxminbandwidthallocation.AttheMAClayer,itachievestheallocatedbandwidthsharesfortheowsthroughatwo-levelweightedfairqueuingalgorithm.Wedemonstratetheeectivenessoftheproposedsolutioninenhancingend-to-endfairness.Thesecondapproachproposedisafullydistributedapproach.Wetransformtheglobalmaxminobjectivetofourlocalconditionsandprovethat,ifthefourlocalconditionsaresatisedinthewholenetwork,thentheglobalmaxminobjectivemustbeachieved.Wethendesignadistributedrateadaptationprotocolbasedonthefourconditions.OurapproachdoesnotmodifythebackoschemeofIEEE802.11.Itreplacesper-owqueueingwithper-destinationqueueing.Mostimportant,itachievesfarbetterfairness(orweightedfairness)amongend-to-endowsthanexistingapproaches. Weproposeadistributedprogressivealgorithmformaximizingthelifetimevectorinawirelesssensornetwork,therstalgorithmofitskindforthisproblem.Thedesignofthealgorithmisbasedonthenecessaryandsucientconditionsthatwehaveprovedforproducingthemaximumlifetimevector.Withourprogressivealgorithm,aresultisavailableatanytimeandisgettingbetterasmoretimeisspent.Wedemonstratethatthealgorithmisabletoconvergerapidlytowardsthemaximumlifetimevectorwithlowoverhead. 100

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[1] Y.T.Hou,Y.Shi,andH.D.Sherali,\RateAllocationinWirelessSensorNetworkswithNetworkLifetimeRequirement,"Proc.ofACMMobiHoc'04,pp.67{77,2004. [2] X.L.HuangandB.Bensaou,\OnMax-MinFairnessandSchedulinginWirelessAd-hocNetworks:AnalyticalFrameworkandImplementation,"Proc.ofMobiHoc'01,LongBeach,California,October2001. [3] H.Luo,S.Lu,andV.Bharghavan,\ANewModelforPacketSchedulinginMultihopWirelessNetworks,"Proc.ofMobiCom'00,August2000. [4] B.Li,\End-to-EndFairBandwidthAllocationinMulti-hopWirelessAdHocNetworks,"Proc.ofIEEEICDCS'05,June2005. [5] D.BertsekasandR.Gallager,Datanetworks,2nded.Prentice-HallInc,1992. [6] J.ChangandL.Tassiulas,\Energyconservingroutinginwirelessad-hocnetworks,"Proc.ofIEEEINFOCOM'00,2000. [7] Q.Li,J.Aslam,andD.Rus,\Onlinepower-awareroutinginwirelessAd-hocnetworks,"Proc.ofACMMobiCom'01,pp.97{107,2001. [8] M.BhardwajandA.Chandrakasan,\Boundingthelifetimeofsensornetworksviaoptimalroleassignments,"Proc.ofIEEEINFOCOM'02,vol.3,p.1587C1596,2002. [9] K.Kalpakis,K.Dasgupta,andP.Namjoshi,\Maximumlifetimedatagatheringandaggregationinwirelesssensornetworks,"Proc.ofIEEEICN'02,2002. [10] G.ZussmanandA.Segall,\EnergyEcientRoutinginAdHocDisasterRecoveryNetworks,"Proc.ofIEEEINFOCOM'03,2003. [11] A.SankarandZ.Liu,\MaximumLifetimeRoutinginWirelessAd-hocNetworks,"Proc.ofIEEEINFOCOM'04,2004. [12] R.Madan,Z.Q.Luo,andS.Lall,\Adistributedalgorithmwithlinearconvergenceformaximumlifetimeroutinginwirelesssensornetworks,"Proc.oftheAllertonConferenceonCommunication,ControlandComputing,2005. [13] J.Zhu,S.Chen,B.Bensaou,andK.-L.Hung,\TradeobetweenLifetimeandRateAllocationinWirelessSensorNetworks:ACrossLayerApproach,"Proc.ofIEEEINFOCOM'07,2007. [14] Y.Wu,S.Fahmy,andN.B.Shro,\OntheConstructionofaMaximum-LifetimeDataGatheringTreeinSensorNetworks:NP-CompletenessandApproximationAlgorithms,"Proc.ofIEEEINFOCOM'08,2008. [15] Y.ShiandT.Hou,\TheoreticalResultsonBaseStationMovementProblemforSensorNetworks,"Proc.ofIEEEINFOCOM,April2008. 101

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LiangZhangwasborninBeijing,China.HereceivedhisBachelorofEngineeringandMasterofEngineeringdegreesincomputerscienceandtechnologyfromTsinghuaUniversity,China,in1999and2002,respectively.Afterthat,hehadworkedinOracleR&DCenterinChinaforoneyear.In2003,hejoinedtheDepartmentofComputerandInformationScienceandEngineeringattheUniversityofFlorida,topursuehisPh.D.degree.HisadvisorisDr.ShigangChen.Hisresearchfocusedonfairnessinmultihopwirelessnetworks. 106