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Distributed Solutions for Rate Control and Maximum Lifetime in Wireless Networks

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

Material Information

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

Subjects

Subjects / Keywords: fairness, flow, lifetime, maxmin, mesh, multihop, network, rate, sensor, wireless
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Liang Zhang.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Chen, Shigang.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024822:00001

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

Material Information

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

Subjects

Subjects / Keywords: fairness, flow, lifetime, maxmin, mesh, multihop, network, rate, sensor, wireless
Computer and Information Science and Engineering -- Dissertations, Academic -- UF
Genre: Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: 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.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Liang Zhang.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Chen, Shigang.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024822:00001


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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