Stability-Based Topology Control in Wireless Multihop Networks with Reservation-Based Distributed-Scheduling Policies

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Stability-Based Topology Control in Wireless Multihop Networks with Reservation-Based Distributed-Scheduling Policies
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Vejarano,Gustavo A,Sr
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University of Florida
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
McNair, Janise Y
Committee Members:
Shea, John M
Fang, Yuguang
Chen, Shigang

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Subjects / Keywords:
power -- scheduling -- stability -- topology -- wimax -- wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
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Electrical and Computer Engineering thesis, Ph.D.
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Abstract:
The topology of wireless multihop networks can be controlled by means of transmission power control, and this control can be performed with the objective of adapting the network topology to the data flows established by the end users. When the network topology is fixed, the network is able to support a finite set of flow data rates while guaranteeing the stability of each of the network's link queues (i.e., the link queues, with some probability greater than zero, return to the empty state). By adapting the network topology, the set of flow data rates that the network supports can be increased, or, if the flow data rates are fixed, the end-to-end delays experienced by the flows can be decreased. The research problem studied in this dissertation is the design of topology-control algorithms that maximize the size of the set of supported flow data rates. Therefore, the algorithms control the node transmission powers in order to produce a network topology that is able to support the largest set of flow data rates. The design of the distributed topology-control algorithms is approached in three main steps. In the first step, the supported set of flow data rates is characterized mathematically for fixed network topologies. This characterization describes the dependence on the network topology of the set of flow data rates. This result is obtained by means of a queueing-system stability analysis, in which the queueing system represents the links and link-queues of the network. Given that the link-scheduling policy determines the efficiency with which the links are activated in order to empty the queues with no packet collisions, the policy plays an important role in the stability analysis. The policies considered in this dissertation are reservation-based, i.e., the links coordinate the reservation of future frames for the transmission of packets without collisions. The stability analysis for these policies is a novel technique that is based on the classic stability analysis for non-reservation-based policies available in the literature. A new reservation-based distributed scheduling (RBDS) policy, called greedy-maximal RBDS (GM-RBDS), is proposed and analyzed with the new technique. It is shown that networks that implement this policy are able to support a larger set of flow data rates when compared with the policies currently available in the literature. Therefore, in this first step two main contributions are achieved. These are (1) the novel stability-analysis technique for reservation-based scheduling policies and (2) the GM-RBDS policy that outperforms, in terms of throughput, the current policies available in the literature. In the second step, the mathematical characterization of the set of flow data rates supported by GM-RBDS networks is used for the design of a heuristic and centralized topology-control algorithm. The algorithm uses the dependence on the network topology of the supported set of flow data rates in order to increase the set's size. The result is a network topology that supports higher flow data rates for a given set of flows. This is a novel topology-control approach that outperforms the classic approach based on spatial reuse. Therefore, in this second step, one main contribution is achieved. This is the topology-control algorithm that is based on the stability of the GM-RBDS network and that outperforms the classic spatial-reuse-based algorithms. The third step consists of the design of distributed topology-control algorithms that also use the mathematical characterization of the set of flow data rates. These algorithms are designed using game theory. The players in these games are the flows of the network. They control the transmission power of neighboring nodes for maximizing their utilities. The utilities are given in terms of the supported set of flow data rates, which depends on the network topology. The game-theoretical distributed algorithms are compared with the centralized topology control of the second step. The network scenarios in which each of these two approaches (i.e., centralized and distributed) outperforms the other are identified. Finally, this dissertation also includes the design, implementation, and evaluation of a simulation framework for Institute-of-Electrical-and-Electronics-Engineers (IEEE) 802.16 wireless mesh networks using optimized network engineering tools (OPNET). The GM-RBDS policy and the topology-control algorithms are evaluated in this framework. To the best of our knowledge, this is the first OPNET simulation framework for this type of networks.
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Gustavo A Vejarano.
Thesis:
Thesis (Ph.D.)--University of Florida, 2011.
Local:
Adviser: McNair, Janise Y.

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STABILITY-BASEDTOPOLOGYCONTROLINWIRELESSMULTIHOPNETWORKSWITHRESERVATION-BASEDDISTRIBUTED-SCHEDULINGPOLICIESByGUSTAVOVEJARANOADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011GustavoVejarano 2

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ACKNOWLEDGMENTS IwouldliketoexpressmygratitudetomydoctoraladvisorProf.JaniseMcNair.Hersupportandencouragementwereessentialforthesuccessfulcompletionofmydoctorateanddissertation.Workingunderherdirectionwastrulyinspiringandenjoyable.IthankallthemembersoftheWirelessandMobileSystems(WAMS)Laboratoryfortheirvaluablefeedbackformyresearchandforallthegreattimeswehadinthelaboratory.MygratitudealsogoestoProfessorsYuguangMichaelFang,JohnShea,andShigangChenforservinginmycommittee. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 3 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 10 CHAPTER 1INTRODUCTION ................................... 12 1.1ResearchProblem ............................... 12 1.1.1Assumptions .............................. 12 1.1.2TechnicalRelevance .......................... 12 1.1.3ProposedSolution ........................... 13 1.2DissertationOrganization ........................... 14 2RESERVATION-BASEDDISTRIBUTEDSCHEDULING(RBDS) ........ 16 2.1RelatedWork .................................. 17 2.1.1CentralizedPolicies ........................... 18 2.1.2DistributedPolicies ........................... 19 2.1.3Contributions .............................. 20 2.2NetworkModel ................................. 20 2.3Reservation-BasedDistributedScheduling ................. 23 2.3.1RBDSPolicies ............................. 24 2.3.2StabilityAnalysisofRBDSPolicies .................. 26 2.3.2.1RBDSMarkoviansystemmodel .............. 26 2.3.2.2RBDSMarkoviansystemstateupdate ........... 27 2.3.2.3SchedulinginanRBDSwirelessnetwork ......... 29 2.3.2.4StabilityanalysisofanRBDSwirelessnetwork ...... 30 2.3.3StabilityandComplexityAnalysisoftheGreedyMaximalRBDS(GM-RBDS)Policy ........................... 38 2.3.3.1Complexityanalysis ..................... 38 2.3.3.2Sufcientconditionforthestabilityofoutput-queues ... 40 2.3.3.3Sufcientconditionforthestabilityofinput-queues .... 47 2.3.3.4StableregionandefciencyratioofGM-RBDS ...... 48 2.4SimulationResults ............................... 55 2.4.1GM-RBDSThroughputEvaluation .................. 55 2.4.2OverheadComparisonofGM-RBDSandEnhancedLocalGreedyScheduling ............................... 57 2.5Summary .................................... 61 4

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3HEURISTICCENTRALIZEDTOPOLOGYCONTROL .............. 63 3.1RelatedWork .................................. 64 3.1.1Link-SchedulingPoliciesandtheStabilityRegion .......... 64 3.1.2Stability-RegionExpansionAlgorithms ................ 65 3.1.3Contributions .............................. 68 3.2NetworkModel ................................. 68 3.3Queue-Stability-BasedTransmissionPower(TP)ControlAlgorithm .... 71 3.3.1GM-RBDSanditsStabilityRegion .................. 71 3.3.2TPControlAlgorithm .......................... 73 3.4SimulationResults ............................... 79 3.5Summary .................................... 89 4DISTRIBUTEDTOPOLOGYCONTROLUSINGPOTENTIALGAMES ..... 91 4.1RelatedWork .................................. 93 4.2NetworkModel ................................. 95 4.3Stability-RegionAdaptation .......................... 97 4.3.1AccessSchemesandtheStabilityRegion .............. 98 4.3.2DistributedTP-ControlAlgorithmsusingPotentialGames ..... 102 4.4Stability-RegionAdaptationinInstitute-of-Electrical-and-Electronics-En-gineers(IEEE)802.16WirelessMultihopNetworks(WMN) ........ 106 4.4.1NashEquilibriaandLinearIntegerProgramming .......... 114 4.4.2PerformanceBound .......................... 119 4.5SimulationResults ............................... 121 4.6Summary .................................... 124 5WORLDWIDE-INTEROPERABILITY-FOR-MICROWAVE-ACCESSRBDSSI-MULATOR(WIMAX-RBDS-SIM):ANOPTIMIZED-NETWORK-ENGINEE-RING-TOOLS(OPNET)SIMULATIONFRAMEWORKFORWIRELESSMESHNETWORKS ..................................... 125 5.1RelatedWork .................................. 126 5.1.1Distributed-SchedulingSimulators .................. 127 5.1.2Contributions .............................. 128 5.2IEEE802.16WMNOverview ......................... 129 5.2.1Data-SlotScheduling .......................... 130 5.2.2LinkEstablishment ........................... 131 5.3WiMAX-RBDS-SimArchitecture ....................... 132 5.3.1TheLink-Establishers ......................... 135 5.3.2TheSchedulers ............................. 137 5.4OPNETImplementation ............................ 141 5.5SimulationResults ............................... 143 5.6PerformanceEvaluation ............................ 148 5.7Summary .................................... 149 5

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6SUMMARYOFCOMPLETEDWORK ....................... 150 APPENDIX AINPUTANDOUTPUTDATA-PACKETARRIVALRATEOFSGs .......... 154 BPROOFOFTHEOREM 2.1 ............................. 155 CPROOFOFLEMMA 1 ................................ 159 DPROOFOFTHEOREM 3.1 ............................. 161 EFORMULATIONOFTHESTABILITY-REGION-ADAPTATION-FOR-THROUGH-PUT-MAXIMIZATIONPROBLEMASAMIXEDINTEGERPROGRAMWITHNON-LINEARCONSTRAINTS ........................... 168 FCHAPTER 4 'sNOTATION .............................. 170 GPROOFOFTHEOREM 4.5 ............................. 172 REFERENCES ....................................... 180 BIOGRAPHICALSKETCH ................................ 188 6

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LISTOFTABLES Table page 3-1InstituteofElectricalandElectronicsEngineers(IEEE)808.16meshnetworkconguration ..................................... 80 3-2False-alarmratecomparisonfortheheuristic-stability-region-adaptation(HSRA),MinPower,MaxPower,andOptPowercongurations ............... 81 F-1Chapter 4 'snotation:networkmodel ........................ 170 F-2Chapter 4 'snotation:potentialgame ........................ 171 G-1Objectivefunctionvalues:Transmission-powerconguration1,2-single-ow 174 G-2Objectivefunctionvalues:Transmission-powerconguration2-multiple-ows 174 G-3Objectivefunctionvalues:Transmission-powerconguration3 ......... 174 7

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LISTOFFIGURES Figure page 2-1Framestructure .................................... 21 2-2Datapackettransmissionsbetweennodesiandj ................ 22 2-3Linkdatapacketarrivalanddepartureprocesses ................. 28 2-4Reservation-baseddistributedschedulingintheinterfering-linksetI(i,j) .... 35 2-5QueueQjo(n)updateprocess ............................ 36 2-6Output-queueupdatesduringoneschedulingcycleina2-hopneighborhood 41 2-7Gmax'sspanningtreeTmax .............................. 42 2-8Gmax'scycletypes ................................... 43 2-9Averageinputandoutputqueuelengthsforincreasingtrafcloads ....... 56 2-10Performancecomparisonofthegreedy-maximalreservation-based-distribu-ted-scheduling(GM-RBDS)policy,policyW[ 36 ],andtheenhanced-local-gree-dy-scheduling(ELGS)policy[ 35 ] .......................... 58 2-11EffectoftheoverheadoftheGM-RBDSandELGS[ 35 ]policies ......... 59 2-12ELGSworst-casescenario ............................. 60 3-1Datapackettransmissionsbetweennodesiandj ................ 70 3-2Averageoutput-queuelengthcomparisonfortheheuristic-stability-region-a-daptation(HSRA),MinPower,MaxPower,andOptPowercongurations .... 84 3-3PerformancecomparisonoftheHSRAandMinPoweralgorithms ........ 85 3-4ProbabilitythatHSRAcalculatestheoptimalsolutionasafunctionofM .... 89 4-1Anexampleofpotentialhiddennodes ....................... 118 4-2Hidden-nodeexample:nodehisahiddennodeoflink(i,j),i.e.,jcannotlistentohwhileicanlistentoit .............................. 120 4-3PercentageoftopologieswhoseTiswithin4% .................. 123 5-1FramestructureoftheInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16meshmode .................................. 130 5-2Architectureoftheworldwide-interoperability-for-microwave-accessreserva-tion-based-distributed-schedulingsimulator(WiMAX-RBDS-Sim) ........ 133 5-3Link-establisherprocessmodel ........................... 137 8

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5-4Data-slotreservationintheSliced-GM-RBDSalgorithm ............. 138 5-5Schedulerprocessmodel .............................. 139 5-6ImplementationoftheWiMAX-RBDS-Simarchitecture .............. 142 5-7Control-subframeaccessdelayhistogram ..................... 144 5-8Data-slotreservationoftwo1-hopneighborsinanetworkwithgridtopologyof78nodes .................................... 145 5-9Averageoutput-queuelengthforincreasingnumberofslices(trafcload=64packetspersecond) ................................. 146 5-10Averageoutput-queuelengthforincreasingtrafcloads(numberofslices=16) 146 5-11Input-queueandoutput-queuelengthcomparisonformaximumtrafcload .. 147 5-12Link-establishmentdelayhistogram ........................ 147 5-13Simulationspeed ................................... 148 5-14Memoryusage .................................... 149 D-1Nodequeuingmodelperowitbelongsto ..................... 162 G-1Suboptimaltransmission-power(TP)congurations ............... 173 G-2Nodesoff3withmaximumcontentioninTP-conguration1 ........... 177 G-3Worst-casescenario ................................. 179 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophySTABILITY-BASEDTOPOLOGYCONTROLINWIRELESSMULTIHOPNETWORKSWITHRESERVATION-BASEDDISTRIBUTED-SCHEDULINGPOLICIESByGustavoVejaranoAugust2011Chair:JaniseMcNairMajor:ElectricalandComputerEngineering Thetopologyofwirelessmultihopnetworkscanbecontrolledbymeansoftransmissionpowercontrol,andthiscontrolcanbeperformedwiththeobjectiveofadaptingthenetworktopologytothedataowsestablishedbytheendusers.Byadaptingthenetworktopology,thesetofowdataratesthatthenetworksupportscanbeincreased,or,iftheowdataratesarexed,theend-to-enddelaysexperiencedbytheowscanbedecreased.Theresearchproblemstudiedinthisdissertationisthedesignoftopology-controlalgorithmsthatmaximizethesizeofthesetofsupportedowdatarates. Thedesignofthealgorithmsisapproachedinthreesteps.Intherststep,thesupportedsetofowdataratesischaracterizedmathematicallyforxednetworktopologiesbymeansofaqueueing-systemstabilityanalysis.Twomaincontributionsareachievedinthisstep.Theseare(1)anovelstability-analysistechniqueforreservation-baseddistributedscheduling(RBDS)policiesand(2)thegreedy-maximalRBDS(GM-RBDS)policythatoutperforms,intermsofthroughput,thecurrentpoliciesavailableintheliterature.Inthesecondstep,themathematicalcharacterizationofthesetofowdataratessupportedbyGM-RBDSnetworksisusedforthedesignofaheuristicandcentralizedtopology-controlalgorithmwhichoutperformstheclassicapproachbasedonspatialreuse.Thethirdstepconsistsofthedesignofdistributedtopology-controlalgorithmsthatalsousethemathematicalcharacterizationoftheset 10

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ofowdatarates.Thesealgorithmsaredesignedusinggametheoryandcomparedwiththecentralizedtopologycontrolofthesecondstep.Thenetworkscenariosinwhicheachofthesetwoapproaches(i.e.,centralizedanddistributed)outperformstheotherareidentied. Finally,thisdissertationalsoincludesthedesign,implementation,andevaluationofasimulationframeworkforInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16wirelessmeshnetworksusingoptimizednetworkengineeringtools(OPNET).TheGM-RBDSpolicyandthetopology-controlalgorithmsareevaluatedinthisframework.Tothebestofourknowledge,thisistherstOPNETsimulationframeworkforthistypeofnetworks. 11

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CHAPTER1INTRODUCTION 1.1ResearchProblem Themainproblemtobesolvedistocontrolthetopologyofwirelessmultihopnetworks(WMNs)inordertomaximizetheperformanceofend-to-enddataowsestablishedonthenetworks.Thisperformanceisgivenintermsofthesetofend-to-enddata-packetratesthatthenetworksupportswhileguaranteeingstability(i.e.,thelinkqueuesarepositiverecurrent).Thetopologyiscontrolledbymeansoftransmission-po-wercontrol. 1.1.1Assumptions Itisassumedthatthenodetransmissionsareomnidirectional.Theinterferencemodelissuchthatapacketreceptionissuccessfulonlyifnoothernodethatcoversthenodereceivingthepackettransmitsduringthepacketreception.Also,thefollowingparametersoftheWMNaregiven. Thepathsoftheend-to-endows Thelink-schedulingpolicy Themaximumtransmissionradiusofthenodes 1.1.2TechnicalRelevance ThephilosophybehindtheproposedresearchproblemistoprovideanindependentautonomousWMNcapableoforganizingandmaintainingitself,andadaptingandintegratingitstopologywithitsenvironmentbyconsideringtheclients'dataowsandtheavailablenetworkresources.Inthisway,theWMNwouldprovidenaluserswithaneasydeploymentofthenetworkandlessmaintenanceneeds,andwouldalsoincreasethenumberofscenarioswheretheWMNcanbeused. Weproposethefollowingguidelinesforndingthesolutionofthetopology-controlproblem. 12

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Autonomousformationandmaintenance:ItshouldbepossiblefornaluserswithverybasicknowledgeofWMNstoseamlesslydeployaWMN,andforexperiencedengineerstoachieveagoodperformanceifthenetworkismorecarefullydesigned. Resilience:Thedependenceonanycentralizedcontrollershouldbeavoidedforthesolutiontoberesilient.TheWMNshouldnotdependonaspecicentityforitscorrectoperation.Itshouldrelyonthecapabilitiesenabledindifferentnodesinadistributedway. 1.1.3ProposedSolution Thesolutiontothetopology-controlproblemshouldconsistofdistributedalgorithms,whicharemorefeasibleforimplementationduetotheirlowercomplexitywhencomparedwithcentralizedalgorithms. Inordertosolvethetopology-controlproblem,thefollowingobjectiveshavebeenproposed. 1. Todesigndistributedlink-schedulingpoliciesthataremorethroughputefcientthanthecurrentpoliciesavailableintheliterature 2. Toidentifythedependenceonthenetworktopologyoftheperformanceofthelink-schedulingpoliciesofObjective1 3. Todesigndistributedtopology-controlalgorithmsusingthedependenceofthelink-schedulingpoliciesonthenetworktopology 4. Todevelopasimulationframeworkthatallowstheevaluationofthelink-schedulingpoliciesofObjective1andthetopology-controlalgorithms Objectives1to3providetheroadmapforreachingthesolutiontothetopology-con-trolproblem,andObjective4providestheframeworkfortheperformanceevaluationofthesolution. Thebasicideabehindthetopology-controlapproachofObjectives1to3isasfollows.Themaximumdata-packetratethatagivenowsupportsdependsonthethroughputthateachofthenodesalongtheow'spathsupports,andthemaximumdata-packetratethatanodesupportsdependsonitsschedulingpolicyandthe 13

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conicts1withsurroundingnodeswhichalsoneedtoschedulepackettransmissions.Therefore,bymeansoftransmission-powercontrol,thenodesareabletoreducethenumberofconictseitherbydecreasingtheinterference(i.e.,reducingtransmissionpower)and/orcoordinatingfuturepacket-transmissiontimessuchthatnoconictingtransmissionsareperformedsimultaneously.Inthelatterapproach,i.e.,coordinatingfuturepackettransmissions,thetransmissionpowerneedstobeincreasedinsomecasesinordertoenablethenodestolistentoeachother'sschedules. ThetopologycontrolliesonthestabilityregionoftheWMN.Themaximumthroughputthatnodescansupportischaracterizedbythephysical-linkcapacityandthestabilityregionofthelinkschedulingpolicy.Thisregionisthesetofinput-packetratessupportedbythelinksofthenetworkthatguaranteethattheirqueuesarestable(i.e.,thelinkqueuesarepositiverecurrent).Thephysical-linkcapacitydeterminesthemaximumlengthinbitsofthepackets.Inthetransmission-powercontrolapproach,thenetworktopologyismodiedsuchthatthestabilityregionisadaptedtothegivensetofows.Thegoalofthisadaptationistomaximizethehighestinput-packetratesupportedbytheowsthatguaranteestability. 1.2DissertationOrganization InChapter 2 ,themathematicalframeworkfortheanalysisofreservation-baseddistributedscheduling(RBDS)policieswasdeveloped.Thisframeworkconsistsofaqueuing-systemmodelfortheWMN.Usingthismodel,thesetofend-to-endpacketrates(i.e.,theowpacketrates)thatguaranteethestabilityofthenetworkischaracterizedmathematically.Thisisanovelstability-analysistechniquethatisanadaptationoftheclassicstabilityanalysisdonefornon-reservation-basedscheduling 1Schedulingconictsarisebetweennodeswhentheyattempttotransmitpacketssimultaneouslyandtheinterferencetheycauseoneachotherishighenoughtocausepacketcollisions. 14

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policies.Also,inChapter 2 ,anewRBDSpolicyisproposed.Thispolicyiscalledgreedy-maximalreservation-baseddistributedscheduling(GM-RBDS).ItisshownthattheGM-RBDSpolicyoutperforms,intermsofthroughput,thepoliciescurrentlyavailableintheliterature.ThisresultisachievedbymeansofthemorescalableoverheadoftheGM-RBDSpolicywhencomparedwiththeotherpolicies.InChapter 3 ,therelationbetweenthesetofend-to-endpacketratessupportedbytheGM-RBDSpolicyandthetopologyofthenetworkischaracterizedmathematically.Basedonthisrelation,aheuristicandcentralizedtopology-controlalgorithmisproposed.Thisisanoveltopology-controlapproachthatoutperformstheclassicapproachesbasedonspatialreuseand/orheuristicsbasedonthehidden/exposednodes.Ourapproachisabletoachievehigherlevelsofend-to-endpacketratesthatguaranteethestabilityofthenetwork.Chapter 4 consistsofthedesignandevaluationofdistributedtopology-controlalgorithmsthatadaptthestabilityregionoftheWMNtothepathsoftheowsinthenetwork.Thealgorithmsaredesignedusingagame-theoreticalapproachinwhichtheowsareplayersthatcollaboratetoadaptthestabiliytregion.Thealgorithmsareevaluatedbymeansofsimulation.InChapter 5 ,thesimulationframework,usedinallpreviouschapters,isdevelopedforInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16WMNsusingoptimizednetworkengineeringtools(OPNET).Tothebestofourknowledge,thisistherstOPNETsimulationmodelforthistypeofnetworks.Finally,theresultsofthedissertationaresummarizedinChapter 6 15

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CHAPTER2RESERVATION-BASEDDISTRIBUTEDSCHEDULING(RBDS) Amajorchallengeinwirelessnetworksistheabilitytoachievemaximumthroughputvialinkscheduling.Linkschedulingreferstotheselectionofasubsetoflinksforsimultaneoustransmissionthathavethefollowingcharacteristic:Whenthelinksareactivatedsimultaneously,theinterferencebetweenthemislowenoughtoallowsuccessfulreceptionforeveryactivatedlink.Aschedulingpolicyspecieshowtodeterminethesubsetoflinksthattsthischaracteristicandcalculatesthesubsetoflinksforeachframe.Aschedulingpolicy'sthroughputperformanceisdeterminedfromitsefciencyratio,whichisdenedasthefractionoftheoptimalcapacityregioninwhichthepolicyguaranteesthestabilityofthenetwork,i.e.,thatguaranteesthatthelinks'queuesareallpositiverecurrent[ 69 ].Anoptimalschedulingpolicyhasanefciencyratioofunity.Whentheschedulingpolicyhasanoptimalefciencyratio,thewirelessnetworkisabletosupportthelargestsetofinputrates,andsoitachievesmaximumthroughput. Thechallengeinschedulingisthatthepoliciesarehighlycomplex.Theschedulingproblemingeneralisnondeterministicpolynomialtime(NP)hard[ 66 ].Therefore,theresearchliteraturehasfocusedonpolicieswithlowercomplexitythataremoreamenabletoimplementation[ 49 ]. Mostdistributedschedulingpoliciesthatachieveprovableratioscalculate,attheonsetofeveryframe,asubsetoflinksthatisallowedtotransmitdataintheimmediatelyfollowingframeonly.Inthischapter,weproposeadistributedschedulingpolicythatselectslinkstotransmitdatainanyfutureframesbymeansofframereservations.Also,weproposeanewtheoreticalframeworkforthestabilityanalysisforreservation-baseddistributedscheduling(RBDS)policies.Sinceourframeworkconsidersreservationsofanyfutureframes,thecurrentpoliciescorrespondtoaspecialcasewithinourframework,i.e.,thecasethatlinksareallowedtoreservethenextframeonly. 16

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Therestofthischapterisorganizedasfollows.TherelatedworkandthemajorcontributionsarediscussedinSection 2.1 ,followedbyadescriptionofourwirelessnetworkmodelinSection 2.2 .InSection 2.3 ,wecreateaframeworktoanalyzethestabilityofRBDSsystems.Todemonstrateitsefcacy,weproposeanRBDSpolicy,whichwecallgreedymaximalRBDS(GM-RBDS),forInstitute-of-Electrical-and-E-lectronics-Engineers(IEEE)802.16meshnetworksandusethenewframeworktoevaluatethestabilityofthispolicy.InSection 2.4 ,wevalidateourtheoreticalresultswithsimulationresults,andcomparethecapacityoftheRBDSpolicywithnon-reservation-basedtechniques.Finally,asummaryofthechapterispresentedinSection 2.5 2.1RelatedWork Theconceptofoptimalcapacityregionandacentralizedschedulingpolicywithefciencyratioofunitywereintroducedin[ 69 ].Thecentralizedschedulingpolicyattemptstosolveacomplexglobaloptimizationproblemsothattheentirenetworkisstableforthelargestpossiblesetofinputrates.Stabilityisdenedasthepositiverecurrenceofallofthelinkqueues.Underthe1-hopinterferencemodel,theproblemisshowntocorrespondtoamaximumweightedmatching(MWM),wheretheweightsofthelinksaredeterminedfromthelengthoftheirqueues.ThesolutiontoMWMhascomplexityO(N3)[ 29 59 ],whereNisthenumberofnodes.Underthek-hopinterferencemodel,theproblemhasbeenproventobeNP-Hard[ 66 ].Therefore,theoptimalschedulingpolicyisnotconvenientforimplementationduetoitshighcomplexity.Asaconsequence,lesscomplexschedulingpoliciesthatachieveonlyafractionoftheoptimalcapacityregionforgeneralnetworktopologieshavebeendeveloped[ 12 14 15 17 26 35 37 61 63 65 66 68 83 84 ],andtheirimpactonhigherlayershasbeenstudiedin[ 49 ]. Proposedsuboptimalschedulingpoliciescanbecharacterizedbythetechniquestheyuseforcalculatingthenextschedule.Theschedulecalculationdependsonthe 17

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interferencemodelassumedforthenetworkandthelinks'weightsattheonsetofeveryframe.Theschedulingcanbebasedoncentralizedordistributedapproaches.Thus,ourrelatedworkdiscussionconsidersthesetwoseparatecategories(Unlessotherwisespecied,theschedulingpoliciesreviewedinthissectionconsider1-hoptrafconly.SeeSection 2.2 forthedenitionof1-hoptrafc). 2.1.1CentralizedPolicies In[ 68 ],acentralizedschedulingapproachknownaspick-and-compare[ 65 ]thatachievestheoptimalefciencyratioisdened.Thepick-and-compareschedulingpolicyselectstheoptimalscheduleateveryframewithsomeprobabilitygreaterthanzero.First,theschedulingalgorithmrandomlypicksanewschedulesuchthatthelinkscansatisfytheinterferencemodelconstraints.Then,thenewlypickedscheduleiscomparedtothecurrentschedule.Ifthepickedschedulereducesthetotalweightofthenetwork(i.e.,queuelengths)bymorethanthecurrentschedule,thenthepickedscheduleisselectedasthenextschedule;otherwisethecurrentscheduleisusedagain.Thepick-and-comparepolicyrequiresthecalculationandcomparisonoftheupdatedtotalweightforeveryframe.Therefore,thecomplexityofthistechniquegrowslinearlywithN,whichmakesitdifculttoimplementinnetworkswithahighnumberofnodesorinnetworkswherenodeshavelowprocessingcapabilities. Greedymaximalscheduling(GMS)isasuboptimal,centralizedschedulingpolicy.InGMS,thelinksofthenetworkareorderedaccordingtotheirweights,wherethelinkwithmaximumweightisplacedatthetopofthisgloballyorderedlist.Avalidscheduleisfoundbyselectinglinksfromthelistfromtoptobottomthatdonotinterferewitheachother.ThecomplexityofGMSisO(Llog(N)),whereListhenumberoflinks[ 60 ].GMShasefciencyratioof1 2underthe1-hopinterferencemodel[ 49 ],andunderthek-hopinterferencemodel,GMShasefciencyratioof1,1 6,and1 49fortree,geometric,andgeneralnetworkgraphsrespectively[ 37 66 ]. 18

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2.1.2DistributedPolicies Adistributedversionofthepick-and-compareschedulingpolicywasproposedin[ 61 ].Inthispolicy,anodeisselectedwithsomeprobabilitylessthanonetoinitiatethecalculationofascheduleforthelinksinitsneighborhood.Thenewscheduleisselectedforthenextframeifthenewschedulereducestheneighborhood'sweightbymorethanthecurrentschedule.Thealgorithmhasconstantcomplexity,soitdoesnotdependonthenumberofnodesofthenetwork.Itdoesdepend,however,onthediameteroftheneighborhood.Theefciencyratioincreasesasthediameteroftheneighborhoodincreases.Thealgorithmassumesthe1-hopinterferencemodel,soitcanonlybedirectlyusedonnetworkswithphysicallayerssuchasfrequency-hoppingcode-division-multiple-access(FH-CDMA)thatallowthatassumptiontobemade. Greedyscheduling(GS)policies[ 60 ]havebeendevelopedthatachievethesameefciencyratioofGMS[ 17 35 65 ].IntheGSpolicies,nodescalculatelocallythenextschedulebasedonthelinksthathavethemaximumlocalweights. In[ 14 15 63 65 84 ],amaximalscheduling(MS)approachisdescribed.Inthisapproach,maximumweightisnotrequiredtoschedulealink.Alinkiseligibleforthenextscheduleaslongasithasenoughpacketsinthequeuetotransmitduringtheentiredurationofaframe.TheefciencyratioofMSschedulingpoliciesis1 ,whereisthemaximumnumberofnon-interferinglinksintheinterferencesetofanylinkinthenetwork.MSpolicieshavealsobeenadaptedtomulti-hopowscenariosinwhichasetofowswiththeirrespectiveratesandroutesaregiven[ 14 15 63 83 84 ]. Lastly,distributedschedulingpoliciesofcomplexityO(1)havebeendevelopedin[ 26 36 50 ].Theseareknownasconstanttime(CT)schedulingpolicies[ 65 ].TheCTapproachdiffersfromtheMSapproachinthatwhenalinkdoesnotinterferewiththelinksinaschedule,itisselectedwithprobabilitylessthanone.Therefore,inCTschedulingpolicies,framescanbewastedwithsomeprobabilitygreaterthanzero.In[ 50 ],CTpoliciesareproposedforthe1-hopand2-hopinterferencemodels.The 19

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efciencyratiosofthesepolicieswereimprovedin[ 26 36 ].In[ 35 ],theimprovedefciencyratiosare1 2)]TJ /F6 7.97 Tf 17.99 4.7 Td[(1 p mand2 ^n1 2)]TJ /F6 7.97 Tf 17.98 4.7 Td[(1 p mforthe1-hopand2-hopinterferencemodelsrespectively,where^nisthemaximumnumberof1-hopneighboringlinksforanylinkofthenetwork. 2.1.3Contributions Themajorcontributionsofthischapterare: 1. AnRBDSpolicyisproposedfortheschedulingofIEEE802.16wirelessmeshnetworkstoincreasethroughput. 2. Toevaluatethenewpolicy,aMarkoviansystemmodelisdevelopedforRBDSpoliciesthatenablesthestabilityanalysisofthesepolicies. 3. ThisstabilityanalysisofRBDSpoliciesisamoregeneralframework.Inpreviouspolicies,thelinkscompeteforaccesstothenextframeonly.Sinceourapproachconsidersreservationsforallfutureframes,theprevioustechniquescorrespondtoaspecialcasewithinourframework. 4. ThenewRBDSpolicyisproposedandanalyzedwithintheproposedframework.Specically,sufcientconditionsonthedata-packetarrivalratesthatguaranteethestabilityofthenetworkarefound.Fromtheseconditions,thestabilityregionoftheRBDSpolicyisdescribedandalower-boundforthepolicy'sefciencyratioiscalculated.Itisshownthatthisbounddependsontwocharacteristicsofthenetworktopology. 5. Theresultsarevalidatedthroughtheoreticalandsimulationanalysis,andperformancecomparisonsaremadetoexistingpolicies. Inthenextsection,weintroducetheproposedframeworkbydescribingthenetworkmodel. 2.2NetworkModel WeconsiderawirelessnetworkrepresentedbythegraphG=(N,L),whereNandLarethesetsofnodesandlinksrespectively.Thelinksaredirectional.Thelinkdirectedfromnodeitonodejisdenotedby(i,j).Also,itisassumedthatif(i,j)2L,then(j,i)2L.Theinterferencesetof(i,j)isdenotedbyI(i,j).Atransmissionover(i,j)issuccessfulifandonlyifnoothertransmissionoccurssimultaneouslyoveranyofthe 20

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linksthatbelongtoI(i,j).Ifthisinterferenceconstraintisnotmet,thereisacollision,andtwolinksconictwitheachotheriftheybelongtoeachother'sinterferenceset. Thesetsofnodei's1-hopand2-hopneighborsaredenotedbySi1andSi2respectively.The1-hopneighborhoodand2-hopneighborhoodofnodeiaredenotedbySi1andSi2respectively.Si1isthesetofnodesformedbynodeiandnodei's1-hopneighbors)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Si1,i[Si1.Si2isthesetofnodesformedbynodeiandnodei's1-hopand2-hopneighbors)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Si2=i[Si1[Si2. AsintheIEEE802.16meshmodestandard[ 1 ],timeisdividedintoframes,andeachframeisdividedintoacontrol-subframeandadata-subframe.Control-subframesaredividedintocontrol-time-slotsthatareusedfortheexchangeofschedulingpackets,anddata-subframesaredividedintodata-time-slotsthatareusedforthetransmissionofdatapackets.ThisstructureisshowninFigure 2-1 .Frames,control-time-slots,anddata-time-slotsarenumberedindependentlystartingfrom0.Linkscantransmitonlyoneschedulingpacketpercontrol-time-slotandonlyonedatapacketperdata-time-slot.Therearemcsandmdscontrol-time-slotsanddata-time-slotsperframerespectively. Figure2-1. Framestructure Inthischapter,tointroducethereservation-basedapproach,weconsiderthe1-hoptrafcmodel,asin[ 35 37 61 65 66 ].Infuturework,wewillconsidermulti-hoptrafcmodels.Inthe1-hoptrafcmodel,alldataowsconsistofonlyonehop.Further,weassumethatthereisonlyoneowperlink.Therefore,datapacketsarriveateachlinkaccordingtoarandomarrivalprocessandleavethenetworkoncetheyreachtheirdestinationnodewhichisonehopawayfromthesourcenode.Therandomarrivalprocessfortheowatlink(i,j)isdenotedbyA(i,j)(k),wherekisthecurrentdata-time-slot,andithasmeanarrivalrate(i,j).Itisassumedthatthearrivalprocesses 21

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areindependentandidenticallydistributed(i.i.d.)sequenceswhicharealsoindependentacrosslinks. Intraditionalnetworkmodels,thedatapacketsthatarrivetoalinkarestoredinonequeueuntiltheyaretransmitted.Inournewreservation-basednetworkmodel,weproposethateachlinkhastwoqueues(Figure 2-2 ).Theinput-queuestoresdatapacketsthatarrivetothelinkandarewaitingtobegivenagrant,whichmeanstheyarewaitingtobegrantedadata-time-slot.Theoutput-queuestoresthedatapacketsthathavereceivedgrants,i.e.,havealreadybeenscheduled,andarewaitingtobetransmitted.Whenalinkreceivesagrant,someofitsunscheduleddatapacketsaremovedfromitsinput-queuetoitsoutput-queue.Thelengthsoftheinput-queueandoutput-queueoflink(i,j)atcontrol-time-slotmaredenotedbyQ(i,j)i(m)andQ(i,j)o(m)respectively. Figure2-2. Datapackettransmissionsbetweennodesiandj Wedenestabilityandtheoptimalcapacityregionofawirelessnetworkasin[ 49 69 ].Thenetworkisstablewhenthesystemofqueuesacrossthenetworkispositiverecurrent[ 69 ].Theoptimalcapacityregion,denotedby,istheconvexsetCo(R),whereRisthesetofallfeasibleschedules,andCo(R)isitsconvexhull1.isoptimalinthesensethat,when[(i,j)]isoutside,thereisnoschedulingpolicythatcan 1Notethat=Co(R)istrueonlyforthetrafcmodelconsideredinthischapter.Pleasereferto[ 69 84 ]forthemulti-hoptrafcscenario. 22

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stabilizethesystem.AfeasiblescheduleisabinaryvectorofsizejLjthatspeciesasubsetoflinksthatcanbeactivatedsimultaneouslywithoutconicts. Finally,wedenetheefciencyratioofaschedulingpolicyasin[ 84 ].Theefciencyratioisthelargest2(0,1]suchthattheschedulingpolicystabilizesthesystemifandonlyif[(i,j)]2. 2.3Reservation-BasedDistributedScheduling InanRBDSwirelessnetwork,thenodesnegotiatewiththeirneighborsthereservationoffuturedata-time-slotsfortheirlinks.Thisnegotiationisbasedonathree-wayhandshakethatconsistsofarequest,agrant,andagrantconrmation.Requests,grants,andgrantconrmationsaretransmittedinschedulingpackets.Thenodesaccessthecontrol-time-slotsfortransmittingschedulingpacketsusinganelectionalgorithm.Therefore,inanRBDSwirelessnetwork,thenodesaccessthewirelesschannelusingtwodifferentalgorithms:theelectionalgorithmandtheRBDSalgorithm,whoserolesaretoavoidcollisionsandwaistedtimeslotsinthecontrolanddatasubframesrespectively. Inthischapter,weadopttheelectionalgorithmofIEEE802.16meshnetworkswithcoordinateddistributedscheduling[ 1 ].Also,itisassumedthattheRBDSwirelessnetworkfollowsthe2-hopinterferencemodel,whichisthemodelconsideredintheIEEE802.16meshmodestandard[ 1 ].IntheIEEE-802.16electionalgorithm,thenodesinevery2-hopneighborhoodtaketurnsbycompetingbetweenthemtoaccessthecontrol-time-slotsandtransmitschedulingpackets.Wemodeltheoperationofthiselectionalgorithmasfollows2. Inordertoavoidscheduling-packetcollisions,nomorethanonenodeisselectedineverySi2atanycontrol-time-slot. 2TheoperationoftheelectionalgorithmforIEEE802.16meshnetworkswithcoordinateddistributedschedulingisdescribedindetailin[ 13 ]and[ 78 ]. 23

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ThenodesinSi2,whereicanbeanynodeinN,areselectedincycles.Werefertothesecyclesasschedulingcycles. Withinaschedulingcycle,thenodesinSi2areselectedonceandonlyonceeach.Theorderinwhichtheyareselectedisuniformlydistributedamongallthepossibleordersofselection. TheorderthatnodesinSi2areselectedisindependentacrossschedulingcycles. 2.3.1RBDSPolicies Whennodesiandjexchangeschedulingmessagestoscheduledatapacketswaitingtobetransmittedonlink(i,j),theyfollowathree-wayhandshakeinordertomulticastthenegotiatedgranttoalllinksinI(i,j).Thehandshakeconsistsofthefollowingsteps3. 1. Nodeisendsarequesttonodejforacertainnumberofdata-time-slotsalongwithasetofdata-time-slotnumbersthatareavailableforreservationatnodei. 2. Nodejsendsagranttonodeifortherequestednumberofdata-time-slotsaccordingtoitssetofdata-time-slotsavailableforreservationandthoseofnodei. 3. Nodeiconrmsthesuccessfulreceptionofthegrantbyechoingthegrantinitsnextscheduling-packettransmission. Thereservationofthedata-time-slotstakesplaceatsteps2and3.Whennodejtransmitsitsschedulingpacket,j's1-hopneighborsreceivethegrantandmarkthegranteddata-time-slotsasunavailable.Whennodeiconrmsthegrant,i's1-hopneighborsreceivethegrantandmarkthegranteddata-time-slotsasunavailabletoo.Therefore,attheendofstep3,alllinksinI(i,j)havemadethegranteddata-time-slotsunavailable)]TJ /F1 11.955 Tf 5.47 -9.68 Td[(i.e.,thegranthasbeenmulticasttoalllinksinI(i,j). Therequestsandgrantstransmittedbythenodesaredenedasfollows. Denition1. Requestr(i,j)m,(fs,fx,z),where(fs,fx,z)2N3,istherequesttransmittedbynodeiatcontrol-time-slotmthatrequestsforlink(i,j)thedata-time-slotsofz 3Itisassumedthatinthishandshakenodejgrantsnodei'srequestandthatthedata-packet-slotreservationissuccessfulatbothiandj. 24

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consecutivedata-subframesstartingatframefsoranyotherframeafterfs.Requestr(i,j)mexpiresattheonsetofframefx. Denition2. Grantg(i,j)m,(fs,fe),where(fs,fe)2N2,isthegranttransmittedbynodejatcontrol-time-slotmthatassignstolink(i,j)thedata-time-slotsoftheseriesofframesthatstartsandendswithframesfsandferespectively.Grantg(i,j)mexpiresattheendofframefe. Denition3. Thelengthofgrantg(i,j)m,denotedbyjg(i,j)mj,isthenumberofdata-subframesassignedinthegrant.Therefore,jg(i,j)mj,fe)]TJ /F5 11.955 Tf 11.96 0 Td[(fs+1. InordertoimplementRBDSpolicies,eachnodemaintainstwotablesperlinkthatthenodebelongsto.Thesearetheunavailable-data-time-slotstableandtherequested-data-time-slotstable.Thetablesareupdatedwiththegrantsandrequestsexchangedwiththenode's1-hopneighbors.Anunavailable-data-time-slotstablecontainsthesetofunexpiredgrantsthatinterferewiththelinkthatthetablebelongsto.ThissetisdenotedbyT(i,j)u(m)forlink(i,j)andisgivenbyEq. 2 ,where)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(g(x,y)mfeisthefecomponentofg(x,y)m,andfmisthecurrentframenumber(i.e.,theframethatcontrol-time-slotmbelongsto).Therequested-data-time-slotstablecontainsthesetofunexpiredrequestsmadeforthelinkthetablebelongsto.ThissetisdenotedbyT(i,j)r(m)forlink(i,j)andisgivenbyEq. 2 ,where)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(r(i,j)mfxisthefxcomponentofr(i,j)m.T(i,j)u(m)andT(i,j)r(m)arefunctionsofmgiventhatthetablesareupdatedwiththegrantsandrequeststransmittedateverycontrol-time-slot. T(i,j)u(m),g(x,y)l:)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(g(x,y)lfefm,(x,y)2I(i,j),lm(2) T(i,j)r(m),r(i,j)l:)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(r(i,j)lfx>fm,lm(2) 25

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InRBDSpolicies,twograntsoverlapwitheachotheriftheframerangesgivenbytheirrespectivefsandfeframenumbershaveoneormoreframenumbersincommon. 2.3.2StabilityAnalysisofRBDSPolicies 2.3.2.1RBDSMarkoviansystemmodel InanRBDSnetwork,eachlinkhasaninput-queueandanoutput-queueasdenedinSection 2.2 .Thelengthofaninput-queue)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Q(i,j)i(m)isdenedasthenumberofdatapacketsinthequeue.Thelengthofanoutput-queue)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Q(i,j)o(m)correspondstothenumberofdata-subframesinthefollowingframerange:fromthecurrentframetothelastframescheduledforthepacketsintheoutput-queue.Therefore,thelengthofoutput-queuesdoesnotdependonthenumberofscheduledpacketswaitingtobetransmittedbutontheschedulesofsuchpackets.Thelengthofoutput-queuesisformallydenedbyEq. 2 4,whereT(i,j)gisthesetofunexpiredgrantsoflink(i,j))]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,T(i,j)g(m),g(i,j)l:)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(g(i,j)lfefm,lm. Q(i,j)o(m),max\010(g)fe:g2T(i,j)g(m))]TJ /F5 11.955 Tf 11.95 0 Td[(fm+1+(2) Anodetransmitsschedulingpacketsbyaccessingcontrol-time-slotsaccordingtotheelectionalgorithm.Thenextcontrol-time-slotthatnodeiisgoingtoaccessisdeterminedbytheelectionalgorithm.Thiscontrol-time-slotisdenotedbyMi(m).Thatis,atcontrol-time-slotm,thefuturecontrol-time-slotthatnodeiusestotransmitaschedulingpacketiscontrol-time-slotMi(m). Basedonthepreviousdenitions,RBDSwirelessnetworkGcanberepresentedasaMarkoviansystemwhosestateSGisgivenbythelengthsoftheinputandoutputqueuesofallthelinksandtheschedulingcontrol-time-slotsofallthenodes.Thatis, SG,nQ(i,j)i(m),Q(i,j)o(m),Mi(m):(i,j)2L,i2No.(2) 4[]+isthepositive-partoperator. 26

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2.3.2.2RBDSMarkoviansystemstateupdate GiventhatSGisupdatedonlyduringcontrol-subframes(i.e.,atcontrol-time-slotmasgivenbyEq. 2 ),thedatapacketarrivalanddepartureprocessesforlink(i,j)canbemodeledasshowninFigure 2-3 5.Therearedata-packetarrivalsanddeparturesattherstcontrol-time-slotofeverycontrol-subframeonly.Thesecorrespondtothetotalnumberofarrivalsanddeparturesthatoccurredduringthedata-subframeprevioustothecontrol-subframe.InFigure 2-3 ,forcontrol-time-slotm,thelastdata-time-slotofthisdata-subframecorrespondstokm.Thedata-packetarrivals,whicharedenotedbyA0(i,j)(m),correspondtothedatapacketsthatareinputtoQ(i,j)i(i.e.,datapacketsthatneedtobescheduled).Thedata-packetdeparturescorrespondtothedatapacketsthatareoutputfromQ(i,j)o.ThenumberofthesedeparturesdoesnotaffectthelengthofQ(i,j)oaccordingtoEq. 2 .However,thedata-subframeinwhichthesedeparturesoccuraffectsQ(i,j)o.Q(i,j)oisdecreasedby1data-subframeeverytimeadata-subframeisover.ThisdecreaseisdenotedbyD0(m).A0(i,j)(m)andD0(m)aregivenasfollows6. A0(i,j)(m),8>><>>:Pmds)]TJ /F6 7.97 Tf 6.58 0 Td[(1l=0A(i,j)(km)]TJ /F5 11.955 Tf 11.96 0 Td[(l)m=multipleofmcs,0otherwise. (2)D0(m),8>><>>:1m=multipleofmcs,0otherwise. (2) SGisalsoupdatedeverytimeagrantisreceivedbyanyofthelinks.WhenMj(m)=mforsomejinN,nodejtransmitsaschedulingpacket.Theschedulingpacketmay 5NotethatinFigure 2-3 ,itisassumedthatmcs=mds=3.6NotethatinFigure 2-3 ,A0(i,j)(m)andD0(m)areindicatedonlyatthecontrol-time-slotsinwhichtheyaredifferentfromzero. 27

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Figure2-3. Linkdatapacketarrivalanddepartureprocesses carryoneormoregrantsforthenode'sincominglinks(k,j),wherenodekcanbeanyofnodej's1-hopneighbors.Eachgrantcarriestheschedulesforasubsetofdatapacketswaitingintheircorrespondinginput-queueQ(k,j)iatnodek.Thegrantreceivedbylink(i,j)atcontrol-time-slotmanddenotedbyG(i,j)misgivenbyEq. 2 .Theconditionk=iinEq. 2 representsthefactthatthegranttransmittedbynodej)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,g(k,j)misdirectedtonodei.Nodej'sgrantscanbedirectedtoanyofits1-hopneighbors. G(i,j)m,8>><>>:g(k,j)mifMj(m)=mandk=i,;otherwise.(2) GivenA0(i,j)(m),D0(m),andG(i,j)m,stateSGisupdatedaccordingtoEq. 2 ,Eq. 2 ,andEq. 2 .Q(i,j)i(m)isupdatedwhentherearedata-packetarrivals)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,A0(i,j)(m)6=0orwhenpacketsarescheduled)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,G(i,j)m6=;.Q(i,j)o(m)isupdatedwhentherearedata-packetdepartures)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,D0(m)6=0orwhendata-subframesaregranted)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(G(i,j)m6=;.Whennodeitransmitsaschedulingpacketatcontrol-time-slotMi(m)=m,thenextcontrol-time-slot(i.e.,Mi(m+1))usedforthenextscheduling-packettransmissionisMimcontrol-time-slotsaway.Giventhatintheelectionalgorithm(Section 2.3 ),theorderthatthenodesinSi2areselected 28

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ineveryschedulingcycleisindependentacrossschedulingcycles,Mjmisani.i.d.randomsequence7,i.e.,P[Mjm=n]=n8j2N,wherenissomepmfsuchthatn=08n0. Q(i,j)i(m+1)=Q(i,j)i(m)+A0(i,j)(m))]TJ /F5 11.955 Tf 11.96 0 Td[(mdsG(i,j)m+(2) Q(i,j)o(m+1)=max\002Q(i,j)o(m))]TJ /F5 11.955 Tf 11.96 0 Td[(D0(m)+,)]TJ /F5 11.955 Tf 10.46 -9.69 Td[(G(i,j)mfe)]TJ /F5 11.955 Tf 11.95 0 Td[(fm+1+(2) Mi(m+1)=8>><>>:Mi(m)+Mimifm=Mi(m),Mi(m)ifm6=Mi(m).(2) 2.3.2.3SchedulinginanRBDSwirelessnetwork LetQbethesetofqueuesinthenetwork)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Q,nQ(i,j)i(m),Q(i,j)o(m):(i,j)2Lo.AccordingtoEq. 2 andEq. 2 ,theupdatesofthequeuesinQtakeplaceonlywheneitheroftwoeventsoccur.Theseeventsarethebeginningofacontrol-subframeandthetransmissionofaschedulingpacket.AccordingtoEq. 2 andEq. 2 ,whenacontrol-subframestarts(i.e.,whenmismultipleofmcs),thenumberofdata-packetarrivalsanddeparturesisdifferentfromzero.Fortherestofthecontrol-subframe,therearenoarrivalsnordepartures.Therefore,queuesQ(i,j)iandQ(i,j)oareupdatedwithdata-packetarrivalsanddeparturesrespectivelyonlyattherstcontrol-time-slotofeverycontrol-subframe.Whenschedulingpacketsaretransmitted)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,whenMj(m)=m9j2N,queuesQ(i,j)iandQ(i,j)oareupdatedatthenodesthatreceivedgrantscarriedbythescheduling 7NotethatthesequenceMimisdenedonlyforthecontrol-time-slotswhennodeitransmitsascheduling-packet. 29

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packets.TheupdateconsistsofschedulingdatapacketsthatarewaitinginQ(i,j)iandmovingthosepacketstoQ(i,j)o. Let1betheeventthatacontrol-subframestarts,andlet2betheeventthatatleastoneschedulingpacketistransmitted. Theupdatesthattakeplacewhena1-eventoccursatcontrol-time-slotmmodifythequeuesinQasfollows. A0(i,j)(m)datapacketsareinputtoQ(i,j)i. D0(m)data-subframesareremovedfromQ(i,j)o. Theupdatesthattakeplacewhena2-eventoccursatcontrol-time-slotmmodifythequeuesinQasfollows. mdsG(i,j)mdatapacketsareremovedfromQ(i,j)i. Q(i,j)oisincreasedwith)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(G(i,j)mfeaccordingtoEq. 2 AnRBDSpolicyhascontrolonlyoverthe2-typequeueupdates.TheRBDSpolicyisresponsiblefordeterminingthegrantsthataretransmittedinschedulingpackets.Specically,theRBDSpolicydeterminesthegrantsforallnodej'sincominglinks,wherejisanynodeinN,everytimenodejtransmitsaschedulingpacket.Therefore,theRBDSpolicydeterminesthenumberofpacketsthataremovedfromeveryQ(i,j)itoitscorrespondingQ(i,j)o,where(i,j)isanyofnodej'sincominglinks. 2.3.2.4StabilityanalysisofanRBDSwirelessnetwork Denition4. WirelessnetworkGisstableifthequeueprocessQinSGispositiverecurrent. InordertoanalyzethestabilityofnetworkGunderRBDSpolicies,aMarkoviansystemdenotedbySGsandderivedfromEq. 2 andtheupdateequationsEq. 2 ,Eq. 2 ,andEq. 2 isconsidered. 30

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SGsisupdatedonlywhentherearescheduling-packettransmissions(i.e.,thesystem'sstateisupdatedonlywhena2eventoccurs),whicharescheduledbytheelectionalgorithmusedforaccessingthecontrol-time-slotsaccordingtoEq. 2 TheelectionalgorithmandtheRBDSpolicydeterminehowthesystem'sstateisupdated.Specically,theelectionalgorithmdeterminesthesubsetofnodesthattransmitschedulingpacketsatevery2eventandthenumberofdata-time-slotsbetweeneverytwoconsecutive2events.TheRBDSpolicydeterminesthegrantsthatarecarriedbytheschedulingpackets. Inthefollowing,thestabilityanalysisofSGsisperformedinfoursteps.Inthersttwosteps,thesystemupdatesduetotheelectionalgorithmandRBDSpoliciesarecharacterized.Thethirdstepisanexamplethatillustratesasystemupdate.Finally,inthefourthstep,thesufcientconditionsthatguaranteestabilityarecalculated. SGsupdateandtheelectionalgorithm .TwoqueuesaredenedforeachnodejinNbasedonthequeuesofthenode'sincominglinks.Thesearetheinput-queueQji(n)andtheoutput-queueQjo(n).QueueQji(n)representsthetotalnumberofpacketsthatarewaitingatallofj's1-hopneighbors'input-queuestobescheduledfortransmissiontonodej(Eq. 2 ).QueueQjo(n)representsthenumberofdata-subframesintherangeofframesthatstartsatthecurrentframeandendsatthelatestframegrantedbynodej(Eq. 2 ). Qji(n),Xi2Sj1Q(i,j)i(mn)(2) Qjo(n),maxi2Sj1Q(i,j)o(mn)(2) Thesubsetofnodesselectedbytheelectionalgorithmforthenthoccurrenceofevent2isdenotedbyN2(n)andisdenedbyEq. 2 ,wheremnisthecontrol-time-slotwhenthenth2eventoccurs. 31

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N2(n),j:j2N,Mj(mn)=mn(2) Theelectionalgorithmdoesnotguaranteescheduling-packettransmissionsateverycontrol-time-slotm.Theindexmndenotesonlythecontrol-time-slotsinwhichthereisatleastonescheduling-packettransmissioninthenetwork(i.e.,theoccurrenceofa2event).Specically,thenthtimethatatleastoneschedulingpacketistransmittedinthenetworktakesplaceatcontrol-time-slotmn. TheelectionalgorithmcalculatestheseriesofsubsetsofnodesN2(n)basedontheupdateequationgivenbyEq. 2 .Eachofthesubsetsintheseriescorrespondstothenodesthattransmitschedulingpacketssimultaneouslyatacertaincontrol-time-slot.OnlythequeuesQji(n)andQjo(n)ofthenodesinN2(n)areupdatedatcontrol-time-slotmnasfollows:Theinputandoutputqueuesaredecreasedandincreasedrespectivelybythegrantscarriedinthetransmittedpackets.Theinputqueuesareincreasedbythenumberofpacketsthatarrivedsincethepreviousscheduling-packettransmissions.Theoutputqueuesaredecreasedbythenumberofdata-subframesbetweenthepreviousandcurrentscheduling-packettransmissions.Inordertocharacterizethisupdateprocess,thefollowingdenitionsareconsidered. Thenumberofdata-packetarrivalstotheinput-queueoflink(i,j)betweennodej'snthand(n+1)thscheduling-packettransmissionsisdenotedbyA(i,j)s(n)andisgivenbyEq. 2 .Thedata-packetarrivalrateinthisprocessisgivenbyEq. 2 8,whereNj1(n)isthenumberof1eventsbetweennodej'snthand(n+1)thscheduling-packettransmissions9. 8Thedata-packetarrivalrate(i,j)sisderivedinAppendix A .Inthischapter, XstandsfortheexpectedvalueofrandomvariableX.9Nj1(n)isdeterminedbythenumberofcontrol-time-slotsbetweenthesetransmissions,andthisnumberisequaltoMjm(Eq. 2 ). 32

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A(i,j)s(n),mn+1Xm=mn+1)]TJ /F6 7.97 Tf 6.59 0 Td[(MjmnA0(i,j)(m)(2) (i,j)s,EA(i,j)s(n)= Nj1mds(i,j)(2) LetMbethesetofgrantscarriedbytheschedulingpackettransmittedbynodejatcontrol-time-slotmn.Thetotalnumberofdata-subframesthatlink(i,j)isgrantedinMisgivenbyEq. 2 G(i,j)(n),XG(i,j)mn2MG(i,j)mn(2) Thelatestdata-subframethatlink(i,j)isgrantedinMisgivenbyEq. 2 )]TJ /F5 11.955 Tf 5.48 -9.68 Td[(G(i,j)(n)fe,max\010)]TJ /F5 11.955 Tf 17.93 -9.68 Td[(G(i,j)mnfe:G(i,j)mn2M(2) Denition5. ThestateSGsofRBDSwirelessnetworkGisgivenbySGs,Qji(n),Qjo(n),Mj(mn):j2NwhereSGsisupdatedaccordingtoEq. 2 ,Eq. 2 ,andEq. 2 Qji(n+1)=8>><>>:Qji(n)+Pi2Sj1A(i,j)s(n))]TJ /F5 11.955 Tf 11.95 0 Td[(mdsPi2Sj1jG(i,j)(n)j+j2N2(n),Qji(n)j=2N2(n). (2)Qjo(n+1)=8>><>>:max\002Qjo(n))]TJ /F5 11.955 Tf 11.95 0 Td[(Nj1(n)+,maxi2Sj1)]TJ /F5 11.955 Tf 12.46 -9.69 Td[(G(i,j)(n)fe)]TJ /F5 11.955 Tf 11.96 0 Td[(fmn+1+j2N2(n),Qjo(n)j=2N2(n). (2) SGsupdateandtheRBDSpolicy .AccordingtoEq. 2 andEq. 2 ,theupdatesofthequeuesconsiderthegrantscarriedbyeachofthetransmittedscheduling 33

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packets.Eachgrantassignsasetofdata-time-slotstooneoftheincominglinksofthenodethattransmittedthegrant,andtheincominglinkschedulespacketsinitsinput-queueatthegranteddata-time-slots.Thisschedulingcausesthosepacketstobemovedfromthelink'sinput-queuetothelink'soutput-queue. ThegrantsarecalculatedaccordingtoanRBDSpolicy.ThetaskperformedinthiscalculationisillustratedinFigure 2-4 .InFigure 2-4A ,awirelessnetwork)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,G=(N,L)andoneofitsinterferingsets)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,I(i,j)areshown10.GiventhattransmissionsonanyofthelinksinI(i,j)interferewithtransmissionsonlink(i,j),nodejshouldgenerategrantsforlink(i,j)thatconsidertheunexpiredgrantsofallotherlinksinI(i,j).ThisisshowninFigure 2-4B .ItisassumedthatFigure 2-4B isasnapshotofalltheunexpiredgrantsoflinksinI(i,j)rightafternodejtransmitsaschedulingpacketwithonlyonegrantforlink(i,j),whichisdenotedbyg(i,j).Thatis,Figure 2-4B showsthedata-subframesfromthetimeofnodej'sscheduling-packettransmissionuntilthelastgranteddata-subframethatinterfereswithlink(i,j).Thedata-time-slotsineachoftheseframesarealsoincluded11.Assumingthatgrantsgl1,gl2,gl3,gl4weretransmittedbysomeinterferinglinksinI(i,j)beforenodej'sscheduling-packettransmission,nodejneedstoselectasetofavailabledata-time-slotsforgrantg(i,j)thathavenotbeenincludedinanyoftheinterferinggrants.TheselectionmadeinFigure 2-4B showstwoeventsthatnodesneedtoavoidinordertoimprovetheefciencyratio.Theseareblanksandoverlaps.Blankscorrespondtodata-time-slotswhichareavailabletoalink,andthelinkdoesnotincludetheminanyofitsgrants.Overlapscorrespondtodata-time-slotswhichareunavailabletoalink,andthelinkincludestheminoneormore 10InFigure 2-4A ,notallthelinksinGareshown.OnlythelinksinI(i,j)areincluded.Withinthissetoflinks,link(i,j)isshownwithastraightline,andallitsinterferinglinksareshownwithdashedlines.11Itisassumedthatthereare3data-time-slotsperdata-subframe(i.e.,mds=3). 34

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ofitsgrants.Blanksandoverlapsturnintowasteddata-time-slotsanddata-time-slotswithcollisionsrespectivelyoncetheyarethecurrentdata-time-slotintheRBDSwirelessnetwork.Therefore,thetaskofanodeistoselectavailabledata-time-slotsforgrantssothatthetotalnumberofblanksandoverlapsisminimizedacrossallthelinksinthenetwork. AInterfering-linksetI(i,j) BGrantsintheinterferinglinksetI(i,j) Figure2-4. Reservation-baseddistributedschedulingintheinterfering-linksetI(i,j) SGsupdateexample .TheupdateprocessofQjo(n)isillustratedinFigure 2-5 .Afterthenthscheduling-packettransmissionofnodej,Qjo(n)isasshowninFigure 2-5A .Ithasalengthof11data-subframes,inwhichgrantsgl1,gl2,gl3,gl4,andg(i,j)havebeenplaced.Itisassumedthatatthenthscheduling-packettransmissionofnodej,nodejtransmitsaschedulingpacketwithonegrantforlink(i,j)only.Itisalsoassumedthatgrantsgl5,gl6,gl7,andgl8weretransmittedbylinksinI(i,j)atsomecontrol-time-slotsbetweennodej'snthand(n+1)thscheduling-packettransmissions,andthatthesegrantsandg(i,j)wereassigneddata-time-slotsaccordingtoFigure 2-5B .Giventhatnodejtransmitsagrantinits(n+1)thscheduling-packettransmissiontoo,thelengthofQjo(n)isupdated.Thisupdateconsistsofadecreaseandanincreaseofdata-subframes,whichareexplainednext. 35

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AQjo(n) BTransitionfromQjo(n)toQjo(n+1) CQjo(n+1) Figure2-5. QueueQjo(n)updateprocess SufcientconditionsforthestabilityofSGs .Thedecreasecorrespondstothenumberofdata-subframesbetweennodej'snthand(n+1)thscheduling-packettransmissionsNj1(n)(Eq. 2 ).Thequeue-lengthdecreasesizeNj1(n)isgivenbyEq. 2 .InFigure 2-5B ,thesizeofNj1(n)is7data-subframes. Nj1(n),mn+1Xm=mn+1)]TJ /F6 7.97 Tf 6.59 0 Td[(MjmnD0(m)(2) Thequeue-lengthincreasecorrespondstothetotalnumberofdata-subframesincludedinthegrants,blanks,andoverlapsthatwerescheduledbetweenthenthand(n+1)thscheduling-packettransmissions.Thissetofdata-subframes,whichisdenotedbyHj(n),isshowninFigure 2-5B .ThespecicRBDSpolicyimplementedonthenetworkisresponsibleforcalculatingthegrants,blanks,andoverlapsthatHj(n)includes.Therefore,thenumberofframescoveredbyHj(n),whichisdenotedbyjHj(n)j,dependsonthenumberofframescoveredbythegrants,blanks,andoverlapsdeterminedbytheRBDSpolicy.GiventhattheRBDSpolicydeterminesthenumberofframescoveredbythegrants,blanks,andoverlapsfromthenumberofunscheduled 36

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packets(i.e.,theinput-queuelengths),jHj(n)jdependsontheinput-queuesandnotontheoutputqueues12. GivenHj(n),theupdateofQjo(n)denedinEq. 2 canbeexpressedasinEq. 2 Qjo(n+1)=8>><>>:Qjo(n)+Hj(n))]TJ /F5 11.955 Tf 11.96 0 Td[(Nj1(n)+j2N2(n),Qjo(n)j=2N2(n).(2) Denition6. AnRBDSwirelessnetworkisstationaryiftherandomprocessesNj1(n),jHj(n)j,jG(i,j)(n)jarestationaryforalljinNandall(i,j)inL. Theorem2.1. Theoutput-queuesinastationaryRBDSwirelessnetwork)]TJ /F26 11.955 Tf 5.48 -9.69 Td[(i.e.,Q(i,j)o(m):(i,j)2LarestableifEq. 2 holds,andtheinput-queues)]TJ /F26 11.955 Tf 5.47 -9.68 Td[(i.e.,Q(i,j)i(m):(i,j)2LarestableifEq. 2 holds.Therefore,anRBDSwirelessnetworkisstableifbothEq. 2 andEq. 2 hold. maxj2N jHjj)]TJ ET q .478 w 243.99 -336.95 m 261.87 -336.95 l S Q BT /F5 11.955 Tf 243.99 -349.61 Td[(Nj1<0(2) maxj2N0@ Nj1Xi2Sj1(i,j))]TJ /F12 11.955 Tf 11.96 11.35 Td[(Xi2Sj1 jG(i,j)j1A<0(2) SeeAppendix B fortheproofofTheorem 2.1 13. Remark. Intuitively,conditionsgivenbyEq. 2 andEq. 2 guaranteestabilitybecausetheyrequirethatallthequeuesdecreasetheirlengthsataratelowerthantheyincrease.Thatis,theconditionsselectthenodewhosedifferencebetweenincreaseand 12TheindependenceofjHj(n)jfromtheoutput-queueswillbeusedintheproofofTheorem 2.1 (Appendix B ).13TheproofofTheorem 2.1 isbasedontheproofforthestabilityofgreedyschedulingpoliciesunder1-hoptrafcpresentedin[ 84 ]. 37

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decreaseratesisthelargestandmakessurethattheincreaserateislowerthanthedecreaserate. WithintheframeworkforRBDSwirelessnetworkspresentedinthissection(i.e., 2.3.2 ),differentRBDSpoliciescanbeevaluatedintermsofstability.Inthefollowing,weproposetheGreedy-Maximal-RBDS(GM-RBDS)policyandevaluateit. 2.3.3StabilityandComplexityAnalysisoftheGreedyMaximalRBDS(GM-RBDS)Policy TheGM-RBDSpolicyisasfollows.WhenanynodeiinNtransmitsaschedulingpacket, Itgrantsthelongestrequestamongalltheunexpiredrequestsmadebyitsincominglinks,andsetsthegrant'sfscomponentattheframefollowingtheinterferinggrantthatexpiresthelatest. Foreveryofitsoutgoinglinks,itrequestsasmanyconsecutivedata-subframesasunscheduleddatapacketscoverentirely,setseveryrequest'sfscomponentattheframefollowingtheinterferinggrantthatexpiresthelatest,andsetseveryrequest'sfxcomponentattheframescheduledforitsnextscheduling-packettransmission. AndwhenanynodeiinNreceivesaschedulingpacket,itcheckswhetherthereisagrantinthepacketandwhetherthegrantisdirectedtooneofitsoutgoinglinks.Ifthatisthecase,itconrmsthegrantonlyifthegrantdoesnotoverlapwithanyofthegrantsinthelink'sunavailable-data-time-slotstable. TheGM-RBDSpolicyisgreedymaximalinthesensethattherequeststhataregrantedarethelongestrequestsandeachrequestcorrespondstothemaximumintegernumberofdata-subframesthatarecoveredbyalink'sunscheduleddatapackets)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,eachrequestcorrespondstoQ(i,j)i mds,whereQ(i,j)iisthenumberofunscheduleddatapacketstobetransmittedonlink(i,j). 2.3.3.1Complexityanalysis Letthecomplexitybedenedintermsoftheamountofcontrolinformationthatistransmittedduringacontrol-subframe(i.e.,overhead).IntheGM-RBDSpolicy,wheneveranodetransmitsinacontrol-time-slot,itsendsatmost1requestper 38

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outgoinglink,1grant,and1grantconrmationperoutgoinglink.Eachgrantandgrant-conrmationcorrespondstoarangeofframesthatisspeciedwithtwointegers:fsandfe(Denition 2 ).Eachrequestcorrespondstoarangeofframesandanexpirationframe(Denition 1 )whichisatotalofthreeintegers(i.e.,fs,fx,andz).Therefore,intheGM-RBDSpolicy,intheworst-casescenario,anodetransmitsatmost5+2integerspercontrol-time-slot,whereisthemaximumnodedegree14,soatotalofmcs(5+2)integersaretransmittedduringacontrol-subframe.Then,inWMNswithrandomlylocatednodes,theGM-RBDSpolicyhasO(logjNj)complexity15. Remark. IntheGM-RBDSpolicy,anodeneedstoknowtheunavailable-data-time-slottableofthelinksthatinterferewithitsincominglinksinordertocalculatethegrantsitsends.ThisprocessisdescribedinSections 2.3.2.4 and 2.3.2.4 .Notethatthemcs(5+2)integersareenoughforspecifyingtheunavailable-data-time-slotstablesthenodesneedtoknow.Thisistruebecausetheunavailable-data-time-slotstablesareimplicitlyspeciedbytheendframes(i.e.,fe)ofgrantsandgrant-conrmations.Thatis,alltheframesprevioustothoseendframeshavealreadybeenreservedbecauseintheGM-RBDSpolicy,thenodesgranttheclosestframesintimethatareavailable. Thestabilityanalysisisdoneasfollows.Thesufcientconditionsforthestabilityoftheoutput-queuesarefoundrstinthenextsection(i.e.,Section 2.3.3.2 ),thenthesufcientconditionsforthestabilityofinput-queuesarefoundinSection 2.3.3.3 ,andnally,theGM-RBDSstableregionandefciencyratioarecalculatedfromtheconditionsinSection 2.3.3.4 14Thenodedegreeisthenumberofbidirectionallinksconnectedtothenode,whereabidirectionallinkisthepairofoutgoingandincominglinksthatconnecttoandfromthesame1-hopneighborrespectively.15Thenumberofcontrol-time-slotsperframe(i.e.,mcs)isadesignparameterthatisindependentofthenumberofnodes. 39

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2.3.3.2Sufcientconditionforthestabilityofoutput-queues GiventhatMjmisani.i.d.randomsequence(Section 2.3.2.2 )andthattheGM-RBDSpolicyhasnodynamicparameters,GM-RBDSwirelessnetworksarestationary.Therefore,thesufcientconditionsforstabilityofGM-RBDSnetworkscanbefoundusingTheorem 2.1 Everytimeanodeisselectedatacertaincontrol-time-slotbytheelectionalgorithm,thenodetransmitsaschedulingpacket.Thenodewritesonlyonegrantforthelongestofallincominglinks'requestsontheschedulingpacket16.Thetransmissionofthisgrantupdatestheoutput-queueofthenodeandtheunavailable-data-time-slotstablesofits1-hopneighbors.Theoutput-queueofthenodeisincreasedbythelengthofthegrant.Thisisduetothefactthatthegrant'sfscomponentissetattheframefollowingtheinterferinggrantthatexpiresthelatest(GM-RBDSpolicyinSection 2.3.3 .).Theunavailable-data-time-slotstableofevery1-hopneighborisupdatedbyaddingthegranttoit(Eq. 2 ). Forexample,ifthenodesinanypathinGhaveallthesameoutput-queuelengthandtransmitschedulingpacketsconsecutivelysuchthateverynode'stransmissionisfollowedbya1-hopneighbor'stransmission,theoutput-queueofeverynodeisincreasedbythesummationofallthepreviouslytransmittedgrants.Also,iftwopathsconnectatsomenode,theoutput-queueofthisnodeisupdatedwiththemaximumsummationofthetwopaths.ThisprocessisillustratedinFigure 2-6 forthecaseofthreepathswiththreenodeseachandallconnectedatoneoftheirends(i.e.,node7).Nodeiistheithnodeselectedbytheelectionalgorithm,andGiisthegranttransmittedbynodei.Qioistheoutput-queuelengthofnodeiafterthenodehastransmitteditsschedulingpacket. 16Onlyonegrantiswrittenoneveryschedulingpacketduetothefactthatonlythelongestrequestisgranted. 40

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Figure2-6. Output-queueupdatesduringoneschedulingcycleina2-hopneighborhood Therefore,giventhatthepathsthatcanbeestablishedbetweenthenodesinSi2determinethegrantsthataffecttheoutput-queuelengthincreaseofnodei,whereicanbeanynodeinN,thestabilityoftheoutput-queuesofthenodesinNcanbeguaranteedbyconsideringthelengthofsuchpaths. Thefollowingdenitionsareusedforcharacterizingthesufcientconditionsthatguaranteetheoutput-queues'stability. LetPibethepathofmaximumlength17intheundirectedgraphinducedbySi2andthatoriginatesatnodei.PmaxisthelongestPi)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Pmax,argmaxPi:i2NjPij.Let N1betheexpectednumberof1eventsbetweentwoconsecutivescheduling-packettransmissionsofanynode18.Letnmaxbethenodewiththehighestoutput-queueincreaserate)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,nmax,argmaxi2N)]TJ ET q .478 w 203.13 -437.67 m 223.16 -437.67 l S Q BT /F2 11.955 Tf 203.13 -448.33 Td[(jHij)]TJ ET q .478 w 239.37 -438.36 m 257.25 -438.36 l S Q BT /F5 11.955 Tf 239.37 -448.33 Td[(N1,andletGmax,fNmax,LmaxgbetheundirectedgraphinducedbySnmax2)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Nmax,Snmax2andLmax,(i,j):i,j2 17Inthefollowing,thelengthofpathPisdenedasthenumberoflinksinPanddenotedbyjPj.18Notethat,giventheassumptionthattheordernodesinevery2-hopneighborhoodareselectedinaschedulingcycleisuniformlydistributedamongallthepossibleordersofselection(Section 2.3.1 ),thenodeshavethesamepmfforMjm.Therefore,theexpectednumberof1eventsbetweentwoconsecutivescheduling-packettransmissionsisthesameforallthenodes)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e., Ni1= N18i2N. 41

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Snmax2;(i,j)2L.GmaxmisthegranttransmittedinGmaxsuchthat jGmaxmj=max\010 jG(i,j)mj:j2Nmax;(i,j)2L. Theorem2.2. Theoutput-queuesinG)]TJ /F26 11.955 Tf 5.48 -9.68 Td[(i.e.,Q(i,j)o(m):(i,j)2LarestableundertheGM-RBDSpolicyif jGmaxmj< N1 2jPmaxj+1. TheproofofTheorem 2.2 isasfollows.LetTmaxbesomespanningtreeofGmaxrootedatnmax.Therefore,bydenition,theheightofTmaxis2,andithasthestructureshowninFigure 2-7 .LetthegrantstransmittedbythenodesinNmaxduringsomeschedulingcyclebedenotedasshowninFigure 2-7 .GrantG0istransmittedbythenodeinTmaxwhoseheightis0(i.e.,nmax).GrantsG1,G2,...,GjS01jaretransmittedbythenodesinS01,whichisthesetofnodeswhoseheightis1(i.e.,S01=Snmax1).GrantsGi,1,Gi,2,...,Gi,jSi1jaretransmittedbythenodesinSi1,whichisthesetofnodeswhoseheightis2andare1-hopneighborsoftheithnodeinS01. Figure2-7. Gmax'sspanningtreeTmax WhenaddingoneofthelinksinGmaxnTmaxtoTmax,acycleiscreated.ThiscyclecanbeofonlyoneofthetypesshowninFigure 2-8A ,Figure 2-8B ,andFigure 2-8C .IfanotherlinkisaddedtoTmax,theexistingcyclemaybemadelongeroranothercycleofonlyoneofthespeciedtypesiscreated.ThisprocessrepeatsitselfasallthelinksinGmaxnTmaxareaddedtoTmax.Therefore,Gmaxmaycontainoneormorecycleswhichareofonlyonetypeeach. ThecycletypesinGmaxaredenedasfollows.Atype-1cycle(cycleC1inFigure 2-8A )containstherootnodeandtwoormorenodeswhoseheightis1only.Atype-2 42

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AType-1Cycle BType-2Cycle CType-3Cycle Figure2-8. Gmax'scycletypes cycle(cycleC2inFigure 2-8B )containsonenodewhoseheightis1andtwoormorenodeswhoseheightis2only.Atype-3cycle(cycleC3inFigure 2-8C )containstherootnode,twoormorenodeswhoseheightis1,andanevennumberofnodeswhoseheightis2suchthatnoneofthesenodesare1-hopneighborsiftheyareconnectedtothesamenodeofheight1. LetibesomenodeinN.LetPbesomepathintheundirectedgraphinducedbySi2thatoriginatesatnodei.jGjPdenotestheincreaseonQio(n)whenonlythegrantstransmittedbythenodesinpathPduringaschedulingcycleareconsidered.Considerthedifferencebetweentheoutput-queuelengthsofeverynodeofheight2inGmaxandnmaxatthebeginningofaschedulingcycle.LetthisdifferencebedenotedbyQi,jforthenodeofheight2thattransmitsgrantGi,j. 43

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ExpectedjGjPwhennocyclesareconsidered .ConsiderthepathsinTmax.ThemaximumjGjPinTmax,denotedbyjGjTmax,isupperboundedasfollows. jGjTmaxmaxi=1,2,...,jS01jj=1,2,...,jSi1j)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jGij+jGi,jj+[Qi,j]++jG0j Therefore,jGjTmaxisupperboundedbythesummationofthreegrantlengthsandanoutput-queuedifferencethathavemaximumtotallengthamongallthepathsfromtheleafnodestotherootnode.Letthosegrantlengthsandoutput-queuedifferencebeGim,Gim,jm,andQim,jm. Letn2bethenodethattransmitsgrantGim,jm,andletn1bethenodethatconnectsn2andnmax.Whennocyclesareconsidered,Qim,jmcanbeatmostthesummationoftwogrants.Thesetwograntscorrespondtooneofthepathsinn2's2-hopneighborhoodthathasnonodesinnmax's2-hopneighborhood.Thegrantscauseanincreaseontheoutput-queuesofn2thatisnotconsideredinjGjTmaxuntiln1transmitsitsgrant.Therefore,E[Qim,jm]+2E[jGmaxmj],andtheexpectedvalueofjGjTmaxisupperboundedasfollows. EjGjTmaxEjGimj+jGim,jmj+jG0j+[Qim,jm]+5E[jGmaxmj] ExpectedjGjPwhenonlytype-1cyclesareconsidered .ConsiderthepathsinFigure 2-8A .ThemaximumjGjP,denotedbyjGjC1,isupperboundedasfollows. jGjC1maxjGjTmax,maxi=1,2,...,jC1j)]TJ /F6 7.97 Tf 8.94 0 Td[(1j=1,2,...,jSi1jjGi,jj+[Qi,j]++XG2C1jGj Therefore,jGjC1isupperboundedbyeitherjGjTmaxorbythesummationofallthegrantlengthsalongC1,agrantlengthofaleafnode,andanoutput-queuedifference.LetTbetheeventthatjGjC1isupperboundedbyjGjTmax,andletGim,jmandQim,jmbethegrantandoutput-queuedifferencethatsatisfymaxi=1,2,...,jC1j)]TJ /F6 7.97 Tf 8.93 0 Td[(1j=1,2,...,jSi1j)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jGi,jj+[Qi,j]+. 44

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Letn2bethenodethattransmitsgrantGim,jm,andletn1bethenodethatconnectsn2andnmax.Whenonlytype-1cyclesareconsidered,Qim,jmcanbeatmostthesummationofthegrantsacrossatype-1cycle.Thesegrantscorrespondtooneofthecyclesinn2's2-hopneighborhoodthathasnonodesinnmax's2-hopneighborhood.LetCauxbethiscycle.Thegrantscauseanincreaseontheoutput-queuesofn2thatisnotconsideredinjGjTmaxuntiln1transmitsitsgrant.Therefore,E[Qim,jm]+jCauxjE[jGmaxmj],andtheexpectedvalueofjGjC1isupperboundedasfollows19. EjGjC1ETjGjTmaxPT+EnotThjGim,jmj+Qim,jm+XG2C1jGji)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F1 11.955 Tf 11.95 0 Td[(PT5E[jGmaxmj]PT+)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(1+jCauxj+jC1jEjGmaxmj)]TJ /F8 11.955 Tf 10.46 -9.68 Td[(1)]TJ /F1 11.955 Tf 11.96 0 Td[(PT)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jC1j+jCauxj+1EjGmaxmj ExpectedjGjPwhenonlytype-2cyclesareconsidered .ConsiderthepathsinFigure 2-8B .ThemaximumjGjP,denotedbyjGjC2,isupperboundedasfollows. jGjC2maxnjGjTmax,maxi=1,2,...,jC2j)]TJ /F6 7.97 Tf 8.94 0 Td[(1[Qi,j]++XG2C2jGj+jG0jo Therefore,jGjC2isupperboundedbyeitherjGjTmaxorbythesummationofanoutput-queuedifference,allthegrantlengthsalongC2,andjG0j.LetTbetheeventthatjGjC2isupperboundedbyjGjTmax,andletQim,jmbetheoutput-queuedifferencethatsatisesmaxi=1,2,...,jC2j)]TJ /F6 7.97 Tf 8.94 0 Td[(1)]TJ /F8 11.955 Tf 5.48 -9.68 Td[([Qi,j]+. Letn2bethenodethattransmitsgrantGim,jm,andletn1bethenodethatconnectsthetype-2cyclen2belongstoandnmax.Whenonlytype-2cyclesareconsidered,Qim,jmcanbeatmostthesummationofthegrantsalongatype-2cycle.Thiscycle,denotedbyCaux,correspondstoatype-2cycleinn2's2-hopneighborhoodthathasnonodesin 19EX[Y]denotestheexpectationofYgivenX. 45

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nmax's2-hopneighborhood.Thegrantscauseanincreaseontheoutput-queueofn2thatisnotconsideredinjGjTmaxuntiln1transmitsitsgrant.Therefore,E[Qim,jm]+jCauxjE[jGmaxmj],andtheexpectedvalueofjGjC2isupperboundedasfollows. EjGjC2ETjGjTmaxPT+EnotThQim,jm+XG2C2jGj+jG0ji)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(1)]TJ /F1 11.955 Tf 11.96 0 Td[(PT5E[jGmaxmj]PT+)]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jCauxj+jC2j+1EjGmaxmj)]TJ /F8 11.955 Tf 10.47 -9.69 Td[(1)]TJ /F1 11.955 Tf 11.95 0 Td[(PT)]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jC2j+jCauxj+1EjGmaxmj ExpectedjGjPwhenonlytype-3cyclesareconsidered .ConsiderthepathsinFigure 2-8C .ThemaximumjGjP,denotedbyjGjC3,isupperboundedasfollows. jGjC3maxnjGjTmax,maxi=1,2j=1,2,...,jSi1jjGi,jj+[Qi,j]++XG2C3jGjo BydoingananalysissimilartotheoneperformedfortheupperboundsofEjGjC1andEjGjC2,itcanbeshownthatEjGjC3isupperboundedasfollows,whereCauxisatype-3cycle. EjGjC3)]TJ /F2 11.955 Tf 5.47 -9.68 Td[(jC3j+jCauxj+1EjGmaxmj Therefore,whentherearenocyclesorthereisonlyonecycleCofanytypeinGmax,theexpectedvalueofthemaximumjGjPisupper-boundedby)]TJ /F8 11.955 Tf 7.47 -9.68 Td[(max)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(5,2jCj+1 jGmaxmj,whereCisthecycleofmaximumlengthamongallthecyclesinthe2-hopneighborhoodsofthenetwork)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,ifCiisthemaximum-lengthcycleinSi2,thenC=argmaxCi:i2NjCij.Thisupper-boundcanalsobeexpressedintermsofPmaxas(2jPmaxj+1) jGmaxmj)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,2jPmaxj+1=max)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(5,2jCj+1. Thisresultcanbedirectlyextendedtothecaseinwhichtherearetwoormorecyclesofanytype. 46

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ExpectedjGjPinthegeneralscenario .ConsiderthesetGPofallpathsinthe2-hopneighborhoodsofthenetworkandwhichstartatthecorrespondingneighborhoods'rootnodeswhenallthecyclesarepresent(i.e.,ifGiisthesetofallpathsintheundirectedgraphinducedbySi2andthatoriginateatnodei,thenGP=Si2NGi).ThejGjPofeachofthesepathsisthesummationofthelengthofthegrantstransmittedbythenodesalongthepathplusanoutput-queuedifferenceiftheheightofthepath'sendnodeis2.Theincreaseontheoutput-queueQnmaxo(n)(i.e.,jHnmax(n)j)isupper-boundedbythemaximumjGjP,whichisdenotedbyjGjmaxP.TheexpectedvalueofjGjmaxPisupperboundedasfollows,whereIPisanindicatorthatPinGPhasthemaximumjGjP. EjGjmaxP=XP2GPEjGjPPIP=1XP2GPEjGmaxmj(2jPj+1)PIP=1=EjGmaxmj)]TJ /F8 11.955 Tf 10.46 -9.69 Td[(2XP2GPjPjPIP=1+1EjGmaxmj)]TJ /F8 11.955 Tf 10.46 -9.69 Td[(2jPmaxj+1 TheprooffollowsfromTheorem 2.1 andbyobservingthatmaxj2N jHjj)]TJ ET q .478 w 74.53 -457.78 m 92.41 -457.78 l S Q BT /F5 11.955 Tf 74.53 -470.43 Td[(Nj1= jHnmaxj)]TJ ET q .478 w 163.62 -459.78 m 189.68 -459.78 l S Q BT /F5 11.955 Tf 163.62 -470.43 Td[(Nnmax1EjGjmaxP)]TJ ET q .478 w 270.15 -460.46 m 288.03 -460.46 l S Q BT /F5 11.955 Tf 270.15 -470.43 Td[(N1)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(2jPmaxj+1EjGmaxmj)]TJ ET q .478 w 437.13 -460.46 m 455.02 -460.46 l S Q BT /F5 11.955 Tf 437.13 -470.43 Td[(N1. 2.3.3.3Sufcientconditionforthestabilityofinput-queues Thestabilityoftheinput-queuesisalwaysguaranteedbytheGM-RBDSpolicyduetothefollowing.Anodealwaysrequestsasmanydata-subframesascanbeentirelycoveredwiththepacketswaitingtobescheduledinitsinput-queues(GM-RBDSpolicyinSection 2.3.3 ),andalltherequestsaresuccessfullygrantedwithsomeprobabilitygreaterthanzeroaccordingtoEq. 2 .Therefore,theinput-queues,withsome 47

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probabilitygreaterthanzero,maybecomeemptyindependentlyoftheircurrentlength,andtheconditiongivenbyEq. 2 alwaysholds. 2.3.3.4StableregionandefciencyratioofGM-RBDS ItremainstondtheexpectedvalueofjGmaxmjintermsofthemeandata-packetarrivalratesf(i,j):(i,j)2Lginordertocalculateastableregionandalower-boundfortheefciencyratiooftheGM-RBDSpolicyusingTheorem 2.2 Foreveryoutgoinglinkofeverynode,thenoderequeststhemaximumnumberofdata-subframesthatcanbeentirelycoveredwiththecurrentnumberofdatapacketsinthelink'sinput-queue.Also,thelengthofthegrantforeveryrequestisalwaysequaltotherequest'slength.Therefore,theexpectedvalueofagrantlengthisupper-boundedbytheexpectedvalueofthelengthofitscorrespondinginput-queueasgivenbyEq. 2 ,wheremiristhecontrol-time-slotwhentherequestforlink(i,j)istransmitted. EjG(i,j)mj=EhjQ(i,j)i(mir) mdskiEhQ(i,j)i(mir) mdsi(2) LetQ(i,j)s(n)beQ(i,j)i(mjn) mds,wheremjnisthecontrol-time-slotusedfornodej'snthscheduling-packettransmission.Q(i,j)s(n)isincreasedbythenumberofdatapacketarrivalstolink(i,j)betweeneverytwoconsecutivenodej'sscheduling-packettransmissions)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,A(i,j)s(n)asgivenbyEq. 2 ,anditisdecreasedbythegrantstransmittedbynodejforlink(i,j).AccordingtotheGM-RBDSpolicy(Section 2.3.3 ),foranyofthesegrantstobeconrmedbynodei,twoindependenteventsmusttakeplace.First,whennodejtransmitsaschedulingpacket,link(i,j)'srequesthasmaximumlengthamongtheunexpiredrequeststransmittedbyallofnodej'sincominglinks.Second,whennodeireceivesnodej'sschedulingpacket,noneoftheunexpiredgrantsreceivedbynodeioverlapwiththegrantforlink(i,j)carriedbytheschedulingpacket.Theoccurrenceofeachofthesetwoeventsatnodej'snthscheduling-packettransmissionisdenotedbyI(i,j)Qmax(n)andI(i,j)s(n)respectively. 48

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Thegrantlengthisequaltothenumberofdata-subframescoveredbytheinput-queuelengthoflink(i,j))]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Q(i,j)iatthecontrol-time-slotthatnodeitransmitslink(i,j)'srequest)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,mir.Therefore,thegrantlengthcanbeexpressedasQ(i,j)s(n)minusthefollowing:thenumberofdata-subframescoveredbythedatapacketarrivalstolink(i,j)sincetherequestistransmittedbynodeiuntilthegrantistransmittedbynodej)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,Q(i,j)i(mjn))]TJ /F7 7.97 Tf 6.59 0 Td[(Q(i,j)i(mir) mds.Letthisnumberofdata-subframesbedenotedbyQ(i,j)s(n)fornodej'snthscheduling-packettransmission.Q(i,j)s(n)isupdatedaccordingtoEq. 2 andEq. 2 asfollows. Q(i,j)s(n+1)=Q(i,j)s(n)+A(i,j)s(n+1) mds)]TJ /F5 11.955 Tf 9.79 0 Td[(I(i,j)Qmax(n+1)I(i,j)s(n+1))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Q(i,j)s(n))]TJ /F8 11.955 Tf 9.8 0 Td[(Q(i,j)s(n)(2) InordertondtheexpectedvalueofQ(i,j)s(n),itisnecessarytocalculatetheprobabilitythatI(i,j)Qmax(n)andI(i,j)s(n)areeachequaltoone. ProbabilitythatI(i,j)s(n)equalsone .LetT(i,j)u(m)andT(i,j)u(m)(Eq. 2 )betheunavailable-data-time-slotstablesforlink(i,j)maintainedbynodesiandjrespectively.Thesetwotablesarenotnecessarilyequalduetothe1-hopneighborsofjhiddenfromiandviceversa.Letthelengthofanunavailable-data-time-slotstablebethenumberofframesintherangethatstartsatthecurrentframeandendsatthelatestunavailableframeinthetable.LetthislengthbedenotedbyjT(i,j)u(m)j.Theeventthatgrantsforlink(i,j)transmittedbynodejdonotoverlapanyoftheunexpiredgrantsinnodeiwhenthisnodereceiveslink(i,j)'sgrants)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,I(i,j)s(n)=1isequivalenttothefollowingevent:jT(i,j)u(mjn)jisgreaterthanorequaltojT(i,j)u(mjn)j.LetTis(mjn)bethesubsetofT(i,j)u(mjn)thatcontainsallthegrantsreceivedbynodeifromthemomentitsendslink(i,j)'srequesttonodej(i.e.,mir)tothemomentnodejreplieswithitsnthtransmittedschedulingpacket(i.e.,mjn),whichcontainslink(i,j)'sgrant.jT(i,j)u(mjn)jisgreaterthanorequaltojT(i,j)u(mjn)jonlyifthemaximumexpirationdata-subframenumberin 49

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Tis(mjn)20islessthanthemaximumexpirationdata-subframenumberinT(i,j)u(mjn).Thisisduetotheminimumdata-subframenumberspeciedbynodeithattherequestcanbegranted(GM-RBDSpolicyinSection 2.3.3 ).Thisminimumisnolongervalidifanyothergrantsthatoccupydata-subframesgreaterthanorequaltoitarereceivedbynodeiandnotbynodejbeforenodejsendsthegrantforlink(i,j).Thatis,ifsuchgrantsarereceivedbynodeiandmakejT(i,j)u(mjn)jbegreaterthanjT(i,j)u(mjn)j,nodejgrantsdata-subframesthatwerealreadyincludedinthegrantsreceivedbynodei,sonodeiisnotabletoconrmnodej'sgrant. T(i,j)u(m)andT(i,j)u(m)areupdatedwiththegrantstransmittedbythenodesinSi1andSj1respectively.ConsiderthesetsSi1nSj1,Si1\Sj1,andSj1nSi1.GiventhatthegrantstransmittedbythenodesinSi1\Sj1arereceivedbybothiandj,theydonotmakeT(i,j)u(m)andT(i,j)u(m)havedifferentlengths.Therefore,I(i,j)s(n)isindependentofsuchgrants.ThegrantstransmittedbythenodesinSi1nSj1affectT(i,j)u(m)anddonotaffectT(i,j)u(m),andthegrantstransmittedbythenodesinSj1nSi1affectT(i,j)u(m)anddonotaffectT(i,j)u(m).Therefore,I(i,j)s(n)dependsonsuchgrantsonly.LetTinjs(mjn)bethesubsetofTis(mjn)thatcontainsallthegrantstransmittedbythenodesinSi1nSj1.Whennodejsendslink(i,j)'sgrant,themaximumexpirationdata-subframenumberinTis(mjn)islessthanthemaximumexpirationdata-subframenumberinT(i,j)u(mjn)ifitisnotanfecomponentofanyofthegrantsinTinjs(mjn).Therefore,theprobabilitythatjT(i,j)u(mjn)jisgreaterthanorequaltojT(i,j)u(mjn)jislower-boundedbytheprobabilitythatTinjs(mjn)isempty.ConsideringthatthenodesinSi2transmitonlyoneschedulingpacketper 20Theexpirationdata-subframenumbersinTis(mjn)correspondtothefecomponentsofthegrantstheycontain(Denition 2 ). 50

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schedulingcycleandthattheorderofthesetransmissionsisindependentacrosstheschedulingcycles,thelower-boundisgivenasfollows21. PI(i,j)s(n)=1PTinjs(mjn)=;=1 1+Si1nSj1(2) Inordertondanupper-boundfortheexpectedvalueofalltheinput-queuelengths,inthefollowing,itisassumedthatPI(i,j)s(n)=1equalsthelower-boundinEq. 2 .Thisistruebecausetheinput-queuesdecreaseatalowerrateunderthisassumption. ProbabilitythatI(i,j)Qmax(n)equalsone .Theeventthatlink(i,j)'srequestisgrantedbynodej)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,I(i,j)Qmax(n)=1dependsonthelengthsoftherequestsinSi2Sj1T(i,j)r(mjn)only.Whennodejtransmitsaschedulingpacket,therecanbeonlyoneunexpiredrequestperincominglink)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,f(i,j):i2Sj1g.Thisisduetothefollowing.EverynodeinSj2transmitsonlyoneschedulingpacketperschedulingcycle,thereisonlyonerequesttonodejineveryschedulingpacket,andthisrequestalwaysexpiresatthenextscheduling-packettransmissionofnodei,whereiisanynodeinSj1(GM-RBDSpolicyinSection 2.3.3 ). Letmjnandmjn)]TJ /F6 7.97 Tf 6.59 0 Td[(1bethecontrol-time-slotsusedbynodejforitscurrentandpreviousscheduling-packettransmissionsrespectively,andletmirbethecontrol-time-slotusedbynodeiinSj1foritsscheduling-packettransmissionprevioustomjn.Atmjn,nodejgrantsthelongestrequestamongthelastrequestssentbyeachofits1-hopneighbors(i.e.,requestssenttonodejbeforemjn).Letisbethe1-hopneighborthatsentthelongestrequest.Nodeisisgivenasfollows. 21Thelower-boundiscalculatedbasedonthefactthattheeventthatTinjs(mjn)isemptyisequivalenttothefollowingevent:withinaschedulingcycle,nodej'sscheduling-packettransmissionprecedesthescheduling-packettransmissionsofthenodesinSi1nSj1. 51

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is=argmaxi2Sj1Q(i,j)s(n))]TJ /F8 11.955 Tf 11.96 0 Td[(Q(i,j)s(n)=argmaxi2Sj1nQ(i,j)i(mir) mdso Therefore,giventhatthemir'sarei.i.d.acrosstheschedulingcyclesandthatgrantsdecreaseinput-queuelengthswithprobabilityPI(i,j)s(n)=1,theprobabilitythatnodei'srequestisgranteddependsontheratethattheinput-queuesofnodej'sincominglinksincreaseasfollows. PI(i,j)Qmax(n)=1=P[i=is]=(i,j))]TJ /F1 11.955 Tf 5.48 -9.69 Td[(PI(i,j)s(n)=1)]TJ /F6 7.97 Tf 6.59 0 Td[(1 Pk2Sj1(k,j))]TJ /F1 11.955 Tf 5.48 -9.69 Td[(PI(k,j)s(n)=1)]TJ /F6 7.97 Tf 6.59 0 Td[(1(2) ExpectedvalueofjGmaxmj .FromEq. 2 andEq. 2 andgiventheindepen-denceofI(i,j)s(n)andI(i,j)Qmax(n)22andtheassumptionthatPI(i,j)s(n)=1equalsthelower-boundinEq. 2 ,theprobabilitythatnodei'srequestforlink(i,j)isgrantedbynodejisgivenbyEq. 2 ,wheresj,max1+Sk1nSj1:k2Sj1,andjisthetotalpacket-arrivalrateatnodej)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,j=Pi2Sj1(i,j). PI(i,j)Qmax(n)=1,I(i,j)s(n)=1=PI(i,j)Qmax(n)=1PI(i,j)s(n)=1(i,j) Pk2Sj1(k,j))]TJ /F8 11.955 Tf 5.48 -9.69 Td[(1+Sk1nSj1(i,j) sjj (2) 22TheindependenceofI(i,j)sandI(i,j)Qmaxcomesfromthefollowing.TheGM-RBDSpolicycalculatesthegrantstransmittedbythenodesinSi1nSj1andSj1nSi1withoutconsideringthenumberofunscheduledpacketsoflink(i,j))]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,Q(i,j)i.Theprobabilitythatagrantforlink(i,j)overlapstheunexpiredgrantsreceivedbynodeidependsonthegrantstransmittedbythenodesinSi1nSj1andSj1nSi1only(Section 2.3.3.4 ).Therefore,thisprobabilitydoesnotdependonthenumberofunscheduledpacketsoflink(i,j),i.e.,I(i,j)sisindependentofQ(i,j)i.Thesameanalysiscanbedoneforanyincominglinkofnodej,soI(i,j)sisindependentofQ(k,j)i,wherenodekisany1-hopneighborofnodej.Therefore,I(i,j)sisindependentoftheeventthatQ(i,j)ihasmaximumlengthamongtheinputqueuesoftheincominglinksofnodej,i.e.,I(i,j)sisindependentofI(i,j)Qmax. 52

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Theexpectedvalueofthelengthofanygrantcanupper-boundedusingEq. 2 ,Eq. 2 ,andEq. 2 asfollows. Lemma1. TheexpectedvalueofgrantG(i,j)misupper-boundedasfollows. G(i,j)m=EQ(i,j)s(n))]TJ /F8 11.955 Tf 11.95 0 Td[(Q(i,j)s(n)sjj N1 SeeAppendix C fortheproofofLemma 1 AsufcientconditionforthestabilityofaGM-RBDSnetworkcanbefoundusingLemma 1 GM-RBDSstableregionandefciencyratio Theorem2.3. Letbegivenby=(2jPmaxj+1)maxsj:j2N.GM-RBDSwirelessnetworkGisstableif Xi2Sj1(i,j)<1 8j2N.(2) TheproofofTheorem 2 isasfollows.ThesufcientconditioninTheorem 2.2 forthestabilityoftheoutput-queuesinGisequivalentto G(i,j)m< N1 2jPmaxj+18(i,j)2L. AccordingtoLemma 1 ,thisconditionissatisedif Xk2Sj1(k,j)<1 sj(2jPmaxj+1)8j2N. ThisconcludestheproofofTheorem 2 Remark. NoticethatthesufcientconditionforstabilitygivenbyEq. 2 isofthesameformoftheconditionforthenon-reservation-basedgreedypoliciesanalyzedin[ 84 ](Eq.(4)in[ 84 ]).Thatis,thetotalpacket-arrivalrateofasetofinterferinglinksneedstobelowerthansomeconstantinordertoguaranteestability,andtheconstantdependson 53

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somecharacteristicofthenetworktopology(i.e.,jPmaxjandsjfortheGM-RBDSpolicy,andforthegreedypoliciesin[ 84 ]). Finally,alower-boundfortheefciencyratiooftheGM-RBDSpolicycanbefoundbasedontheregioninwhichthedata-packetarrivalratessatisfythestabilityconditiongivenbyEq. 2 Theorem2.4. ConsiderRBDSwirelessnetworkG=(N,L)with1-hoptrafc,itsoptimalcapacityregion,anditsmaximumnumberofnon-interferinglinksintheinterferencesetI(i,j)ofanylink(i,j)inL.GisstableundertheGM-RBDSpolicyforanysetofdata-packetarrivalratesf(i,j):(i,j)2Lgthatliesinsidetheregion TheproofofTheorem 2.4 isasfollows.LetC1bethesetofdata-packetarrivalratesthat,forall(i,j)inL,satisfyEq. 2 ,andletC2bethesetofdata-packetarrivalratesthat,forall(i,j)inL,satisfy X(i,j)2I(i,j)(i,j)<1 8(i,j)2L. Therefore,C2C1,andnetworkGisstableforanysetofdata-packetarrivalratesthatliesinC2. In[ 84 ],itisshownthatanecessaryconditionfornetworkstabilityunderanyschedulingpolicyis X(i,j)2I(i,j)(i,j)<8(i,j)2L Therefore,networkGisstablewhenthesetofdata-arrivalratesliesinsidetheregion .ThisconcludestheproofofTheorem 2.4 Therefore,thestabilityofIEEE802.16meshnetworksundertheGM-RBDSpolicy(Theorem 2.3 )dependsononly,whichisalocalcharacteristicofthetopologyofthenetwork.Thatis,dependsonthe2-hopneighborhoodsofthenetworkbutnotonthenetworkasawhole. 54

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2.4SimulationResults ThestabilityofanIEEE802.16meshnetworkundertheGM-RBDSpolicywith1-hoptrafcandPoissondata-packetarrivalsisevaluatedbymeansofsimulationusingthesimulatorproposedinChapter 5 .Thenetworktopologyisagridof16nodes(i.e.,44)and48links.Theframe-structureparametersmcsandmdsaresetat9and256respectively23.Theframelengthis10ms,andatotalof3000framesissimulated. Figure 2-9 showstheaverageinputandoutputqueuelengthsforincreasingtrafcloads.Thetrafcloadsare4 24,8 24,12 24,16 24,17 24,18 24,20 24,and1oftheoptimalcapacity.AccordingtoFigure 2-9A ,theinput-queuesarealwaysstable(i.e.,theydonotgrowindenitely)asexpected(Section 2.3.3.3 ).Ontheotherhand,accordingtoFigure 2-9B ,theoutputqueuesbecomeunstablefortrafcloadsgreaterthan18 24.Therefore,theGM-RBDSpolicyreachesanefciencyratioofapproximately18 24inthesimulation.Thisisinagreementwiththetheoreticallower-bound.Theefciencyratioislower-boundedby1 153accordingtoTheorem 2.4 (i.e.,Pmax=8,maxj2Nsj=3,and=3). 2.4.1GM-RBDSThroughputEvaluation TheperformanceoftheGM-RBDSpolicyiscomparedinFigure 2-10A 24withtheWandEnhanced-Local-Greedy-Scheduling(ELGS)policiesproposedin[ 36 ]and[ 35 ].FromthepoliciesreviewedinSection 2.1 ,WandELGSwereselectedforthecomparisonbecausetheyconsiderthesameinterferencemodelofIEEE802.16meshnetworks(i.e.,2-hopinterferencemodel),whichisthemodelofGM-RBDStoo.Also,theGM-RBDS,W,andELGSpoliciesrequirethesamecontrolinformationformakingschedulingdecisions.Eachnoderequiresthequeuelengthsofthenodesin 23Inthesimulation,theNextXmtMxandXmtHoldoffExponentparametersoftheelectionalgorithm[ 1 ]aresetat3and4respectively.24InFigure 2-10 ,thequeuelengthshavebeennormalizedtotheirrespectivemaximumvaluesinordertomakethecomparison. 55

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AInput-queues BOutput-queues Figure2-9. Averageinputandoutputqueuelengthsforincreasingtrafcloads itstwo-hopneighborhoodformakinganylinkscheduling.Figure 2-10A includesthebestperformancecurvesoftheWandELGSpoliciesreportedin[ 36 ]and[ 35 ]whentheoverhead(i.e.,thefractionofaframethatisdedicatedtothecontrolsubframe)isnotconsidered.ItshowsthatGM-RBDSoutperformsW.Thatis,asthetrafcloadapproachestheoptimalcapacityofthenetwork,thequeuesunderpolicyWgrowfasterthanthequeuesunderpolicyGM-RBDS.Therefore,theGM-RBDSpolicyisabletomaintainthenetworkstableunderhigherthroughputlevelswhencomparedwithW.Ontheotherhand,ELGSoutperformsGM-RBDSinFigure 2-10A .However,thecapacityachievedbytheELGSpolicycanbereducedconsiderablybyitsoverhead 56

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whencomparedwithGM-RBDS.TheGM-RBDSpolicydoesnotrequiretheamountofoverheadrequiredbytheELGSpolicy.Thisisduetoitsabilityforschedulinganyfuturedata-subframes.Figure 2-10B showsthatifthemaximumnodedegreeis8(i.e.,=8)andthecontrolpacketsareformattedaccordingtotheIEEE-802.16-mesh-networksstandard[ 1 ],theGM-RBDSandELGSpoliciesperformequally.Forhighervaluesof(i.e.,8),theGM-RBDSpolicyoutperformstheELGSpolicy25.ThisisshowninFigure 2-11 .Figure 2-11A showsthetrafcloadnecessarytoreachthemaximumaveragequeuelengthreportedinFigure 2-10 asafunctionof,andFigure 2-11B showstheoverheadasafunctionof.Theoverheadisspeciedastheportionofthetotalbandwidthittakes.AccordingtoFigure 2-11 ,theGM-RBDSpolicyachievesgreaterthroughputthantheELGSpolicywhen8.ThereasonisthattheGM-RBDSoverheadgrowsmoreslowlywiththantheELGSoverheaddoes.ThisisshowninFigure 2-11B 2.4.2OverheadComparisonofGM-RBDSandEnhancedLocalGreedySchedul-ing IntheELGSpolicy,thelinksarescheduledonalargest-queue-rstbasis.Thisisdonelocallyinordertomakethepolicydistributed.Therefore,theinformationrequiredbythenodesisthequeuelengthoftheinterferinglinkswhichcorrespondtooneintegereach.Also,thenodesneedtoknowlinkidentications(ID)inordertosolvetheproblemoftwoormorelinkswiththesamequeuelength[ 35 ].ThisIDisanotherintegerperlinkthatthenodesneedtoknow.Intheworst-casescenario(i.e.,whenthehighestnumberoflinksofnodesina2-hopneighborhoodareactivated),thereare()]TJ /F8 11.955 Tf 12.04 0 Td[(1)linkactivations.Thiscaseoccursasfollows.Considerthenodewithmaximumdegreeandactivateoneoftheoutgoinglinksofevery2-hopneighbor.Thisisshown 25ThisalsoimpliesthattheGM-RBDSpolicyoutperformstheWpolicybecausetheoverheadoftheWpolicyisgreaterthantheoverheadoftheELGSpolicy[ 35 ]. 57

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AWithnooverhead BWithoverhead(=8) Figure2-10. Performancecomparisonofthegreedy-maximalreservation-based-distributed-scheduling(GM-RBDS)policy,policyW[ 36 ],andtheenhanced-local-greedy-scheduling(ELGS)policy[ 35 ] 58

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AMaximumtrafcloadsupportedbythepolicies BPolicies'overhead Figure2-11. EffectoftheoverheadoftheGM-RBDSandELGS[ 35 ]policies 59

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inFigure 2-12 ,whereitisassumedthat=3.Therefore,intheworstcasescenario,thetotalamountofcontrolinformation(i.e.,overhead)thatneedstobecommunicatedtocalculateascheduleconsistsof2()]TJ /F8 11.955 Tf 12.7 0 Td[(1)integers.Inadditiontothis,theELGSoverheadalsoincludesaseriesofcontrol-time-slotsforcalculatinglinkeligibilitieswhichareofO(log2jNj)complexity[ 35 ].ThismakestheELGSoverheadhavelengthofatmost2()]TJ /F8 11.955 Tf 12.54 0 Td[(1)integersplustheseriesofcontrol-time-slotsusedforELGSlinkeligibility.Therefore,inWMNswithnodeslocatedrandomly,theELGSoverheadisofO(log2jNj)complexity.ThisresultshowsthattheGM-RBDSoverheadismorescalable.TheGM-RBDSoverheadisofO(logjNj)complexity(thecomplexityanalysisoftheGM-RBDSpolicyisinSection 2.3.3.1 ). Figure2-12. ELGSworst-casescenario Remark. EventhoughtheGM-RBDSpolicyhasgoodperformanceintermsofthroughputaccordingtothepreviousresults,thepolicyhasafairnessproblem.Thelinksareabletoreserveanynumberoffuturedata-subframes,andtherefore,alinkisabletomonopolizethechannelcausingstarvationonotherlinks.Webelievethatfurtherresearchisnecessaryinthisdirectioninordertodesignthroughput-efcientpoliciesthatarefairtoo. Remark. Notethatthepurposeoftheelectionalgorithmistwo-fold.Itallowsthenodestoexchangerequests,grants,andgrant-conrmations,anditallowsthemtopropagatewithinevery2-hopneighborhood,thestateoftheoutputqueues(i.e.,thelistofreservedframesareimplicitlyspeciedingrantsandgrant-conrmations).To 60

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thebestofourknowledge,themechanismsforperformingtheprevioustasks(i.e.,propagatingthecurrentstateofthesystem)inthepoliciesrevisedinSection 2.1 havenotbeenspecied.Intheanalysisofthosepolicies,itisassumedthatthenodesknowthequeuelengthsoftheinterferinglinks.Therefore,giventhatthecounterpartoftheelectionalgorithmforthepoliciesofSection 2.1 hasnotbeenspecied,theoverheadoftheelectionalgorithmwasnotconsideredinthepreviousanalysis.Theoverheadoftheelectionalgorithmaccountedfor0.09%ofthetotalbandwidthinthescenariosimulatedfortheGM-RBDSpolicy26. 2.5Summary Anewframeworkforthestabilityanalysisofschedulingpoliciesforwirelessnetworksthatallowthereservationoffuturedata-subframeshasbeenproposed.Theconceptsofinput-queueandoutput-queuewereintroducedintotheframeworkinordertoaccountforthepacketswaitingtobescheduledandtheschedulesassignedtothesepackets.Basedontheseconcepts,sufcientconditionsforthestabilityofRBDSwirelessnetworkswerefound. Withintheproposedframework,anRBDSpolicywhichusestheconceptofgreedy-maximalschedulingwasanalyzed.ThenodesimplementthispolicybyexchangingschedulingpacketsusingtheIEEE802.16electionalgorithm.Aregioninwhichtheproposedreservation-basedschedulingpolicyisstablewasfoundusingtheframework.Itwasshownthatthesizeofthisregiondependsonthefactorwhichisdeterminedbytwocharacteristicsofthenetworktopologyonly(i.e.,sjandjPmaxj).AnIEEE802.16meshnetworkwiththeproposedschedulingpolicy(i.e.,GM-RBDS) 26TheoverheadoftheelectionalgorithmwascalculatedasthebandwidthconsumedbytheMSH-NCFGcontrolpacketsspeciedintheIEEE-802.16standard[ 1 ].Thesepacketscarrytheinformationnecessaryforthenodestoexecutetheelectionalgorithm.Inthesimulatedscenario,theyaretransmittedonceevery32frames,whichisoneoftheallowedsettingsintheIEEE-802.16standard[ 1 ]. 61

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wassimulated.Itwasshownthatthepolicyalwaysguaranteedthestabilityoftheinput-queuesandthattheoutput-queueswerestablewhentheloadwaswithin18 24oftheoptimalregion.Finally,theperformanceoftheGM-RBDSpolicywascomparedwiththeWandELGSpolicies.ItwasshownthattheGM-RBDSpolicyhasanadvantageovertheWandELGSpoliciesintermsoftherequiredoverhead. 62

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CHAPTER3HEURISTICCENTRALIZEDTOPOLOGYCONTROL Transmissionpower(TP)controlinwirelessmultihopnetworks(WMNs)isanimportantproblemduetotheeffectsithasonthedifferentlayersoftheprotocolstack[ 39 ].Forexample,thenetworkconnectivity,energyconsumption,totalphysical-linkthroughput,spatialreuse,andtotalend-to-endthroughputasafunctionoftheTPhavebeeninvestigatedin[ 19 20 41 62 67 ]respectively.Inthischapter,welookattheproblemofTPcontrolforadaptingthestabilityregion1oftheWMNtoagivensetofowssuchthatthetotalthroughputandend-to-enddelayareimproved.Specically,weaskthequestionofwhatarethenodes'TPsthatadaptthestabilityregiontotheowsinthenetworkwhenasetofsource-destinationpairs,theroutingalgorithm,andthelink-schedulingpolicyaregiven. ByadaptingthestabilityregionoftheWMN,thequeuelengthsacrossthenetworkaredecreasedinaverageforagivensetofinput-packetrates.Inthisway,theowsamongthesource-destinationpairsareabletomaintainhigherlevelsofend-to-endthroughputandlowerlevelsofend-to-enddelaywhileguaranteeingqueuestability.Therefore,theproblemconsideredinthischapterisofparticularinterestforapplicationsthatestablishnon-burstysessionsbetweensource-destinationpairssuchasaudio/videocalls. Inordertoadaptthestabilityregion,weproposeanalgorithmthatisexecutedbytheowsestablishedbetweenthesource-destinationpairs.Theideabehindthealgorithmistoadaptalower-boundregionofthestabilityregion(i.e.,aregioncoveredbythestabilityregion)bymodifyingtheTPs.Thelower-boundregionisawidelyacceptedtheoreticalperformancemetricusedforcomparingdifferentlink-scheduling 1Thestabilityregionisdenedforlink-schedulingpoliciesasthesetofinput-packetratesunderwhichthequeuesinthenetworkarestable(i.e.,positiverecurrent). 63

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policies[ 84 ].Inthealgorithm,oncetheows'pathsaredeterminedbytheroutingalgorithm,theowscalculatethemaximuminput-packetratetheycansupportwithinthelower-boundregion;then,eachowtriestostretchthelower-boundregionbymodifyingtheTPofnodessurroundingit.Theeffectthatthestretchofthelower-boundregionhasonthestabilityregionisanotherstretchonthisregion.Therefore,theresultisastabilityregionadaptedtotheowsthatallowsthemtosupporthigherinput-packetrateswhileguaranteeingthestabilityofthenetwork. Inthischapter,weconsidertheminimumhop(min-hop)routingalgorithmandthegreedy-maximalreservation-based-distributed-scheduling(GM-RBDS)policy(Chapter 2 )inInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16meshnetworks.However,ourresultscanbereadilyextendedtootherWMNs,routingalgorithms,andlink-schedulingpolicies. Therestofthischapterisorganizedasfollows.TherelatedworkandcontributionsarediscussedinSection 3.1 .ThenetworkmodelispresentedinSection 3.2 .InSection 3.3 ,ourTPcontrolalgorithmisexplained.TheperformanceofthealgorithmisevaluatedinSection 3.4 bymeansofsimulation.Finally,asummaryofthechapterispresentedSection 3.5 3.1RelatedWork 3.1.1Link-SchedulingPoliciesandtheStabilityRegion ThestabilityregionforWMNswasrstdenedin[ 69 ]asthesetofinput-packetratesunderwhichthequeuesintheWMNarestable.Thestabilityregionisdenedforlink-schedulingpolicies.Differentlink-schedulingpoliciesachievedifferentstabilityregions,anditissaidthatalink-schedulingpolicyoutperformsanotherpolicyintermsofthroughputwhenithasalargerstabilityregion.Theoptimallink-schedulingpolicyistheonewhosestabilityregionisasupersetofthestabilityregionofanyotherpolicy[ 69 ].Intermsofcomplexity,itisusuallythecasethatthelesscomplexthelink-schedulingpolicyis,thesmalleritsstabilityregionis.Therefore,basedonthetradeoffbetween 64

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thesizeofthestabilityregionandthecomplexity,differentlink-schedulingpolicieshavebeenproposedintheliterature[ 26 35 36 50 60 61 69 71 74 83 84 ].Thesepoliciesarecharacterizedwithaprovableperformanceguaranteewhichisaregionwithinthepolicy'sstabilityregion.Thatis,asetofinput-packetratesiscalculatedforwhichthepolicyisguaranteedtobestable.TheWMNmaybestableunderinput-packetratesoutsidethatset,butthisisnotguaranteed.Therefore,thestabilityregionofthelink-schedulingpolicyisatleastaslargeasthecalculatedsetofinput-packetrates.Wecallthissetthelower-boundregion. Thelower-boundregiondependsoncertaincharacteristicsofthephysicaltopologyofthenetwork.Forexample,thestabilitypropertiesofthegreedymaximalscheduling(GMS)[ 37 ]andthebipartitesimulation(BP-SIM)[ 26 ]policiesdependonthelocal-poolingfactorandthemaximumnodedegreeofthenetworkrespectively.Thelocal-poolingfactorisatopologicalpropertyofthenetworkwhosedenitioncanbeconsultedin[ 37 ],andthenodedegreeisdenedasthenumberoflinksthatthenodebelongsto. 3.1.2Stability-RegionExpansionAlgorithms Themainideapresentedinthischapter(i.e.,adaptingthestabilityregionofagivenlink-schedulingpolicybymeansofTPcontrol)isbasedontheresultsobtainedin[ 11 89 ].In[ 11 ],thenetworkispartitionedbasedonthenotionoflocalpooling,andeachpartitionisassignedtoachannelofthenetwork.Inthisway,theGMSpolicyisguaranteedtoachievetheoptimalstabilityregionineachchannel.In[ 89 ],networktopologiesareidentiedforwhichdistributedlink-schedulingpoliciesachievetheoptimalstabilityregion.However,thesenetworktopologiesarenotsuitableforrealscenarios[ 37 ]becauseoftheirsufcientconditionsthatguaranteetheoptimalstabilityregion.Theseconditionsinclude[ 89 ],1-hopinterference,1-hoptrafc,andatopologythatisagraphthatbelongstooneofthefollowingperfect-graphclasses:chordalgraphs,chordalbipartitegraphs,cographs,andasubgroupofco-comparabilitygraphs.In 65

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realscenarios,theseconditionslimitthesuitabilityofWMNs.Forexample,onlyafewphysical-layertechnologiessuchascode-division-multiple-access(CDMA)canbeapproximatedwiththe1-hopinterferencemodel,andthetrafcinWMNsismultihopbydenition.Also,makingthetopologyfallwithinthepreviousgraphfamiliesimposesconstraintsonthelocationsandTPsofthenodesandtheavailableroutes.Themultihoptrafccasewasconsideredin[ 89 ],anditwasshownthatonlyasubsetofthepreviousgraphfamiliesguaranteetheoptimalstabilityregioninthemultihop-trafcscenario.Thesewereidentiedasforestofstars,whereeveryconnectedcomponentofthenetworkgraphisastargraph.Also,theresultsin[ 11 89 ]arevalidonlyforGMSpoliciesunder1-hoptrafcorbackpressurerouting-schedulingpoliciesundermultihoptrafc2. Ourapproachisbuiltupontheideaof[ 11 89 ]thatundercertaintopologiesalinkschedulingpolicyperformsbetter.WemodifyrealisticallythenetworktopologyusingTPcontroltoadaptthepolicy'sstabilityregiontotheows.Thealgorithmreceivesanysetofend-to-endpaths,nodelocations,andschedulingpolicy,andadaptsthepolicy'sstabilityregiontothepaths.Suchanapproachisbenecialbecauseitimprovestheend-to-endthroughputanddelaywithouttherestrictionspreviouslydiscussed.Inthischapter,weconsiderthecaseofmin-hoprouting,GM-RBDSscheduling,andrandomlychosensource-destinationpairsofnodesinIEEE802.16meshnetworks. Otherheuristicalgorithmshavebeenproposedintheliteraturethatimprovetheperformanceofthelink-schedulingpolicyintermsofthroughputbymeansofTPcontrol.Thesealgorithmsincludetheonesreportedin[ 54 56 57 ]whosebasicideaistoincreasethetotalthroughputinthenetworkbymeansofspatialreuse.ThespatialreuseisincreasedbyreducingtheTPofthenodes.Thealgorithmsdifferbetweentheminthe 2Itshouldbenotedthattheobjectivein[ 11 89 ]wasmainlytoidentifythetopologiesthatenabletheoptimalityoftheGMSpolicy,andnottodesigntopology-controlalgorithms. 66

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waytheyareadaptedtorequest-to-send(RTS)/clear-to-send(CTS)basedprotocols.In[ 30 33 ],itisshownthatbetterthroughputimprovementscanbeachievednotonlybydecreasingtheTPtoincreasethespatialreusebutalsobyconsideringthehiddenandexposednodes.Thealgorithmsproposedin[ 30 ]performTPcontrolwiththeobjectiveofavoidinghiddennodes.Inthisway,thelinksinthenetworkareabletosustainhigherinput-packetrates.In[ 81 ],aTPcontrolalgorithmforRTS/CTS-basedprotocolsisproposedthatdecreasestheareaoccupiedbylinksduringtheirtransmissions,whichisdenedastheareainwhichothernodesmustremainsilentduringthetimethelinkisactive.Then,itisshownthatwiththisscheme,routingalgorithmsthatfavorshorthopsachievehigherlevelsofthroughput.Thegoalofouralgorithmissimilartothegoalofthepreviousalgorithms[ 30 33 54 56 57 81 ],i.e.,toincreasetheinput-packetratesthatagivenlink-schedulingpolicycansupportbymeansofTPcontrol.However,ourapproachdiffersinthatitisdirectlybasedonaquantitativemetricwhichisthestabilityregion.Itisnotbasedonqualitativeobservationsoftheoperationofthelink-schedulingpolicysuchasthehiddenandexposednodesinRTS/CTS-basedpolicies.Therefore,itcanbereadilyadaptedtoanylink-schedulingpolicywhosestabilityregionhasbeencharacterizedsuchastheonesdiscussedinSection 3.1.1 AdifferenttypeofTPcontrolalgorithms,whicharebasedonoptimizationtechniques,arediscussedin[ 7 38 ].In[ 7 ],theproblemofintegratedlinkschedulingandTPcontrolforthroughputoptimizationisshowntobenondeterministicpolynomialtime(NP)complete.Therefore,aheuristicalgorithmisdeveloped.Thegoalofthealgorithmistominimizetheschedulelengthnecessarytosatisfyallthelinkloadsdeterminedbyagivenroutingalgorithm.Byminimizingtheschedulelength,thetotalthroughputofthenetworkisincreasedbecausemoreschedulingcyclescanbeperformedpertimeunit.In[ 38 ],theproblemofjointlyoptimizingtheowroutes,linkschedules,TP,modulationandcodingschemesisaddressed.Thisisamoregeneralproblemthantheoneconsideredin[ 7 ]giventhatitdoesnotonlyincludethecalculationofTPs 67

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andlinkschedulesbutalsoincludestheroutingandphysicallayers(i.e.,owroutes,modulation,andcodingschemes).Inouralgorithm,weareonlyconcernedintheTPcontrolproblemwhentheowsandlink-schedulingpolicyaregiven.Thatis,foragivensetofows,wedetermineTPsthatimprovetheperformanceofthelink-schedulingpolicyintermsofthroughputandend-to-enddelay. 3.1.3Contributions Thecontributionsofthischapterareasfollows. AnewTPcontrolalgorithmisproposedwhichincreasestheinput-packetratesthatowscansupportanddecreasestheend-to-enddelayswhileguaranteeingqueuestabilityacrossthenetwork.Also,thealgorithmdoesnotmakeanyassumptiononthepathsfollowedbytheowswhichisnotthecaseforthealgorithmsproposedin[ 30 33 54 56 57 81 ],anditisnotlimitedtoRTS/CTS-basedpolicieseither.Ouralgorithmcanbereadilyadaptedtolink-schedulingpolicieswhosestabilityregionhasbeencharacterized. Ouralgorithmisbasedontheadaptationofthestabilityregionofthegivenlink-schedulingpolicywhenonlythelinksthatbelongtothegivenowsareconsidered.Tothebestofourknowledge,thistechniquehasnotbeenusedbefore,anditisinspiredontheresultsreportedin[ 11 89 ]. TheimprovementonthroughputachievedbyouralgorithmisevaluatedbymeansofsimulationusingthesimulatorproposedinChapter 5 forthemin-hoproutingalgorithmandtheGM-RBDSpolicy(Chapter 2 )inIEEE802.16meshnetworks[ 1 ]. 3.2NetworkModel WeconsideraWMNwhosecommunicationgraphisdenotedbyG=(N,L),whereNandLarethesetsofnodesandlinksrespectively.Thelinksaredirectional.Thelinkdirectedfromnodeitonodejisdenotedby(i,j).Thenodetransmissionsareomnidirectional.Link(i,j)belongstoLifandonlyifnodejiswithinnodei'stransmissionrange.The2-hopinterferencemodelisadopted.Therefore,twolinks 68

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interferewitheachotheronlyiftheyareatleasttwohopsawayfromeachother.WeadoptthisinterferencemodelbecauseitisthemodelthatthelinkschedulinginIEEE802.16WMNsisbasedon.Inthistypeofnetworks,anodegrantsaccesstoanincominglinkonlyifnoneoftheincominglinksofthenodesthatareatmost2hopsawayhasalreadybeengrantedaccess. Timeisdividedintoframes,andeachframeisdividedintoacontrol-subframeandadata-subframe.Control-subframesaredividedintocontrol-time-slotsthatareusedfortheexchangeofschedulingpackets,anddata-subframesaredividedintodata-time-slotsthatareusedforthetransmissionofdatapackets3.Linksareallowedtotransmitonlyoneschedulingpacketpercontrol-time-slotandonlyonedatapacketperdata-time-slot.Apackettransmissionoverlink(i,j)issuccessfulifandonlyifnootherlinkthatinterfereswithlink(i,j)isactivatedatanytimeduringthetransmission.Iftwoormoreinterferinglinksareactivatedsimultaneouslyatanytime,thereisacollisionandnoneofthelinks'packettransmissionsaffectedbythecollisionaresuccessful. Thedatatrafcconsistsofasetofows.ThissetisdenotedbyF.TheowsinFareenumerated.Thenthowisdenotedbyfn.Itconsistsofapathandameaninput-packetratewhicharedenotedbypnandnrespectively(i.e.,fn,(pn,n)).Pathpnisthesequenceofnodesonwhichowfnisestablished.Themthnodeinpnisdenotedbypn(m).Itisassumedthatthepathsarecalculatedusingmin-hoprouting.Therstandlastnodesofpnarethesourceanddestinationnodesoffnrespectively.Theintermediatenodesofpnaretheforwardingnodesofowfn.Atthesourcenodeofowfn,thedata-packetarrivalprocessisPoissondistributedwithmeann.Everynodethatisanintermediateordestinationnodeofatleastoneowhasamaximumpacketratethatitcanassigntoitsincomingtrafcwhileguaranteeingthestabilityofitsqueues.Eachofthesenodesequallydividesitsmaximumpacketrateamongalltheowsfor 3ThisistheframestructureadoptedforIEEE802.16meshnetworks[ 1 ]. 69

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whichitisanintermediateordestinationnode.Themaximumpacketratethatnodeicansupportforeachoftheseowsisdenotedbyif. Thenumberofowsthattraversealinkisthedegreeofthelink.Thelinkdegreeoflink(i,j)isdenotedbyd(i,j).Ifaowisforwardedfromnodeitonodej,thensuchaowcountsforthedegreeoflink(i,j)andnotforthedegreeoflink(j,i),i.e.,boththedirectionoftheowanditspathdeterminewhichlinkshavetheirdegreeaffectedbytheow. Wefollowthereservation-basednetworkmodeldescribedinChapter 2 inwhichtherearetwoqueuesinvolvedintheschedulingprocessofeverylink.ThesearetheinputandoutputqueueswhicharedenotedbyQ(i,j)i,fnandQ(i,j)o,fnforowfnonlink(i,j)respectively(Figure 3-1 ).Theinput-queuestoresdatapacketsthatarewaitingtobescheduled,whichmeanstheyarewaitingtobegrantedafuturedata-time-slot.Theoutput-queuestoresthedatapacketsthathavebeenscheduled,i.e.,havealreadybeengrantedafuturedata-time-slot,andarewaitingtobetransmitted.Whenalinkschedulesdatapackets,someofitsunscheduleddatapacketsaremovedfromitsinput-queuetoitsoutput-queue.Thenodesimplementtheconceptofregulators4introducedin[ 84 ].Theroleofregulatorsistoregulatetheburstinessofthenode'sincomingtrafc.Thistaskisperformedforowfnonlink(i,j)usingqueueQ(i,j)r,fn. Figure3-1. Datapackettransmissionsbetweennodesiandj 4Adetaileddescriptionoftheoperationofregulatorsisprovidedin D 70

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Nodejisa1-hopneighborofnodeiifnodeiiswithinnodej'stransmissionrange.The1-hopneighborhoodofanodeisthesetofallthenode's1-hopneighbors.ItisdenotedbySi1fornodei.The2-hopneighborhoodofanodeisdenedasthesetofnodesthatareatleasttwohopsawayfromthenodeandthenodeitself.ItisdenotedbySi2fornodei.Theactive1-hopneighborhoodofanodeisdenedasthesetof1-hopneighborsthatareintermediateordestinationnodesofatleastoneow.ItisdenotedbySiafornodei.Thedirect1-hopneighborhoodofanodeisthesetof1-hopneighborsthatsenddatapacketstothenode.Therefore,thedirect1-hopneighborsofanodealwaysprecedethenodeinatleastonepath.Nodei'sdirect1-hopneighborhoodisdenotedbySid. Thetransmissionrangeofnodeiisdenotedbyri.Themaximumtransmissionrangeofanynodeisdenotedbyrmax,andtheEuclideandistancebetweennodesiandjisdenotedbyjji,jjj. 3.3Queue-Stability-BasedTransmissionPower(TP)ControlAlgorithm 3.3.1GM-RBDSanditsStabilityRegion Thelink-schedulingadoptedforthenetworkistheGM-RBDSpolicydescribedinChapter 2 ,whichisbasedontheschedulingframeworkspeciedforIEEE802.16meshnetworkswithdistributedscheduling.Thisframeworkcanbesummarizedasfollows.Nodestaketurnstotransmitschedulingpacketsoncontrol-time-slotsusingtheelectionalgorithmspeciedforIEEE802.16meshnetworks[ 1 13 78 ].Theobjectiveoftheelectionalgorithmistoavoidscheduling-packetcollisions.Thelinksfollowathree-wayhandshakebymeansofscheduling-packetexchanges.Thegoalofthishandshakeistoscheduledata-packettransmissionsinfuturedata-time-slotsandtoreservethosedata-time-slotsamongtheinterferinglinkssuchthatdata-packetcollisionsareavoided.Thethree-wayhandshakeconsistsof(1)arequestoffuturedata-time-slots,(2)agrantforassigningtherequesteddata-time-slots,and(3)agrantconrmationforacknowledgingthegrant. 71

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InGM-RBDS,wheneveranodeisselectedbytheelectionalgorithm,ittransmitsaschedulingpacketthatcontainsthefollowingrequests,grants,andgrantconrmations.Thenoderequestsforeveryoutgoinglinkasmanydatasubframesascanbecoveredcompletelywithunscheduleddatapacketsi.e.,iftherearemdsdata-time-slotsperdata-subframe,thenumberofdata-subframesrequestedforlink(i,j)isPfn2FQ(i,j)i,fn mds.Thenodegrantsthelongestunexpiredrequestithasreceivedsofarandwhichhasnotalreadybeengranted.Thenodeconrmsanygrantsithasreceivedwhichhavenotbeenconrmedbeforeandthatdonotcollidewitheachother.Inthisway,byfollowingGM-RBDS,thenodesattempttoscheduleasmanydata-packettransmissionsastheyneed,andthesetransmissionsarescheduledonanas-soon-as-possiblemanner. ThesizeofthestabilityregionoftheGM-RBDSpolicydependsontheabilityofthelinkstoperformthethree-wayhandshakessuccessfully(Chapter 2 ).Iftheprobabilitythatalinknishessuccessfullyathree-wayhandshakeislow,thelink'squeuewilldecreaseatalowerrate.Therefore,thelink'sabilitytoforwarddatapacketswithinsometimerangeisgoingtobelower(i.e.,thehighestpacketratesupportedbythelinkislowered),andthisreducesthesizeofthestabilityregion.Theprobabilitythatathree-wayhandshakeoflink(i,j)issuccessfuldependsonthenodesthaticanlistentobutjcannot(i.e.,theactivenodeshiddenfromj,whichareSianSja)andthedegreeoflink(i,j). Basedontheprobabilityofsuccessfulthree-wayhandshakes,sufcientconditionsthatguaranteequeuestabilityundertheGM-RBDSpolicyinthemultihopscenarioaregivenasfollows. Theorem3.1. NetworkGunderGM-RBDS,min-hoprouting,andthe2-hopinterferencemodelisstableifthepacketratejfsupportedbyeverynodejinNsatisfyEq. 3 5. 5Thesetoperatornreferstotherelativecomplement,i.e.,SianSja,nk2Sia:k=2Sjao. 72

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jf<1 5Pi2Sjdd(i,j)SianSja8j2N(3) SeeAppendix D fortheproofofTheorem 3.1 Therefore,inordertoguaranteestabilityinthemultihopscenariowithmin-hoproutingandGM-RBDS,theinput-packetrateofowfn(i.e.,n)mustbelessthanthefollowingrate:theminimumpacketrateamongallthepacketratesthatnodesalongtheow'spathcanassigntotheow.ThisisshowninEq. 3 ,wherejmaxistheupper-boundfornodej'sratejfaccordingtoEq. 3 i.e.,jmax,5Pi2Sjdd(i,j)SianSja)]TJ /F6 7.97 Tf 6.59 0 Td[(1. n
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Giventhattheowsaredeterminedbythemin-hoproutingalgorithm,thefollowingparametersinEq. 3 arexed:pn2F,nSjd:j2No,andnd(i,j):(i,j)2Lo.Therefore,inordertoincreaseT,theonlyparametersthatcanbemodiedaretheactive1-hopneighborhoodsi.e.,nSja:j2No.TheycanbemodiedbymeansofTPcontrolsuchthatTismaximized.Thisoptimizationproblem,whichwecallstabilityregionadaptationforthroughputmaximization(SRA-TM),isgivenasfollows. Denition7. GivenasetofowsFcalculatedbythemin-hoproutingalgorithm,theSRA-TMproblemconsistsofthemaximizationofTbymeansofTPcontrolsuchthatnoneofthenodesexceedthemaximumTPandnoneofthepathsarebroken.Thatis, maximizeXpn2F minj2pn1 5Pi2Sjdd(i,j)SianSja!subjectto0rirmax8i2Nri,rjjji,jjj8i2Sjd,j2N(3) Remark. IntheSRA-TMproblem,theowpathsaregivenandleftunmodied.HighervaluesforTcouldbeachievediftheowpathsweremodiedbyincludingthemasdecisionvariables.Forexample,aroutingschemecanuniformlydistributethetrafcloadsacrossthelinksofthenetworksothatlinkswithhighlevelsofcongestionareavoided.Thisproblemcorrespondstoajointoptimizationofthetopologyandowpathsbasedonthestabilityregion.ThisproblemcanbefurtherstudiedduetoitspotentialbenetsonT.However,thischapterdealsonlywiththestability-region-basedtopologycontrolasarststeptowardstheproblemofstability-region-basedjointtopologyandroutingcontrol. Remark. Ifthedatatrafcinthenetworkchangesdynamically,theowpathsmaychangeaswell.Inthisscenario,theSRA-TMproblemneedstobesolvedforeveryow-pathchange.Therefore,thespeedofconvergenceofalgorithmsthatsolvetheSRA-TMproblemisanimportantmetricforsuchascenario.Thealgorithmsshould 74

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beabletokeepupwiththerateofchangeoftheowpaths.Ontheotherhand,ifthedata-trafclevelsofasetofowschangebuttheowpathsdonotchange,theSRA-TMproblemdoesnotneedtobesolvedagain.ThereasonisthatthesolutionoftheSRA-TMproblemisthetopologythatallowsthoseowstosupportthemaximumlevelofdatatrafcwhileguaranteeingstability.Thismeansthatthedata-trafclevelsintheowsmayvaryaslongastheydonotexceedsuchmaximumlevels(i.e.,minnjmax:j2pno8fn2F),andthiscanbeguaranteedbymeansofcall-admission-controlalgorithms. InordertosolvetheSRA-TMproblem,theTPalgorithmshowninFigure 1 isproposed6.Thisalgorithmiscalledheuristicstabilityregionadaptation(HSRA). ThefollowingdenitionsarenecessaryfortheoperationoftheHSRAalgorithm. Denition8. Thebottlenecknodeofowfnisthenodewiththelowestmaximumrateamongalltheintermediateanddestinationnodesoftheow,i.e.,letjbethebottlenecknodeoffn,thenj=argmini2fpn(m):2mjpnjgif. Denition9. Nodehishiddenfromnodejifandonlyifh2SianSjaforsomei2Sjd. Denition10. TheMinPowersetupisthesetofminimumTPswhosetransmissionrangesguaranteethatnoneofthelinksoftheowsinFisbroken. Theoperationofthealgorithmisasfollows.First,thenodes'TPs)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,fri:i2NgaresetaccordingtotheMinPowersetup(line 2 inAlgorithm 1 ).ByreducingtheTPs(Denition 10 ),thespatialreuseinthenetworkisincreased,andasaconsequence,thetotalthroughputisincreasedaswell7.Then,themaximumthroughputthatintermediateanddestinationnodescansupportfortheowstheybelongtoiscalculated(line 3 6TheSRA-TMproblemisformulatedasamixedintegerprogramwithnon-linearconstraintsin E .ThisformulationisusedinSection 3.4 forcalculatingtheoptimalsolutionofthesimulatedinstancesoftheSRA-TMproblem.7Thisspatial-reuse-basedTPcontrolisthebasisofthealgorithmsproposedin[ 54 56 57 81 ]. 75

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Algorithm1HSRAAlgorithm procedureHSRA(N,F,M) frig MINPOWERSETUP(N) fimaxg NODEMAXRATES(N) T TOTALTHROUGHPUT(F,fimaxg) form 0,mTthen fimaxg fiauxg T Taux else ri raux endif endif endfor endprocedure inAlgorithm 1 ).ThisisdoneusingEq. 3 ,whichdenesthenodes'maximumthroughput.Basedonthesemaximums,thetotalthroughputthenetworkcansupportiscalculated(line 4 inAlgorithm 1 ). OncethetotalthroughputundertheMinPowersetupisknown,owsareselectedrandomlyone-by-oneforanumberofMtimes(line 5 inAlgorithm 1 ).Everytimeaowisselected,themaximumthroughputtheowcansupportisincreasedifthiscausesthatthetotalthroughputbeincreasedaswell.Otherwise,theowisleftunmodied.Thethroughputoftheselectedowisincreasedasfollows. Lettheselectedowbedenotedbyfn(line 6 inAlgorithm 1 ).Thebottlenecknodeoffnisfoundrstbytrackingthenodeoftheowwiththelowestmaximumthroughput 76

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(Eq. 3 ).Letthisnodebedenotedbyj(line 7 inAlgorithm 1 ).Themaximumrateofji.e.,jfisincreasedbyincreasingtheTPofoneofj's2-hopneighbors(lines 8 to 15 inAlgorithm 1 ).However,thisTPincreaseisconrmedonlyifthetotalthroughput(i.e.,T)isincreasedaswell(lines 16 to 20 inAlgorithm 1 ).Otherwise,theTPofj's2-hopneighborisleftunmodied(line 22 inAlgorithm 1 ).ThetotalthroughputmaybedecreasedgiventhattheTPincreaseofj's2-hopneighbormaydecreasethemaximumrateofotherbottlenecknodesinthenetwork,andthismaximum-ratedecreasemaybehigherthantheincreaseonj'smaximumrate. The2-hopneighborofnodejwhoseTPisincreasedisselectedsothatthefactorjSianSjajonthedenominatoroftheupper-boundforjfisdecreased(Eq. 3 ).Qualitatively,thisTPincreasecanbeexplainedasfollows.Nodej(i.e.,thebottlenecknode)hasasetof1-hopneighborsthataresendingdatapacketstoiti.e.,Sjd.Letibeoneofthesenodes,andconsiderthelink(i,j)andtheinputandoutputqueuesQ(i,j)iandQ(i,j)oofnodeiasshowninFigure 3-1 .Inorderforitotransmitpacketstoj,areservationoffuturedata-time-slotsisrequired.Whennodesiandjnishthisreservationsuccessfully,datapacketsinnodei'sinput-queuei.e.,Q(i,j)iaremovedtonodei'soutput-queuei.e.,Q(i,j)o,andthesepacketsarelaterpulledfromQ(i,j)ofortheirtransmission.Therefore,forthequeuesQ(i,j)iandQ(i,j)otohavetheirlengthsdecreased,thereservationperformedbynodesiandjneedstobesuccessful,i.e.,thethree-wayhandshakeforschedulingdata-packettransmissionsonlink(i,j)needstobesuccessful.Theprobabilitythatthehandshakeissuccessfulandthatthequeuesdecreasetheirlengthdependsonthegrantsreceivedbynodeiandnotreceivedbynodej.Inthefollowing,werefertothesegrantsashiddengrants.Ifirequestsfuturedata-time-slotstojandahiddengrantisreceivedbyibeforejtransmitsitsgranttoi,j'sgrantmaynotbeconrmedbyi.Thisisbecausethehiddengrantmayinterferewithj'sgrant.Ontheotherhand,ifjisabletolistentothehiddengrant,jisabletogenerateitsgrantsuchthatitdoesnotinterferewiththehiddengrant,andiwillbeableconrmj's 77

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grant8.Therefore,inordertoincreasetheprobabilityofhandshakesuccessandqueuedecrease,theTPofthenodethattransmitsthehiddengrant(i.e.,thenodehiddenfromj)canbeincreasedsuchthatnodejisabletolistentothehiddennode'stransmissions. Nodejmayhavemorethan1hiddennodesineveryincominglinkfromthenodesinitsdirect1-hopneighborhood.TheHSRAalgorithmchoosesonlyoneofthosehiddennodesforincreasingitsTP.Thenodethatischosenisthenodethatishiddenfromthehighestnumberofnodes(i.e.,nodejandalltheotherintermediateordestinationnodesunabletolistentothehiddennode).Thisisperformedinlines 8 to 12 inAlgorithm 1 .Inthisway,themaximumrateisincreasedforallthosenodessothat,ifoneormoreofthosenodesarebottlenecknodes,higherimprovementsonthetotalthroughputcanbeachieved. TherolethattheobjectivefunctionoftheSRA-TMproblem(Eq. 3 )playsintheHSRAalgorithmisthequanticationofthethroughputimprovementbytheTPincreaseonhiddennodes.ByincreasingtheTPofanodehiddenfromabottlenecknode,thefactorSianSjainthedenominatorofEq. 3 isdecreasedforthebottlenecknode,andasaconsequencethebottlenecknode'smaximumrateisincreased.However,theTPincreaseonthehiddennodemayalsocauseanincreaseontheSianSjafactorofotherbottlenecknodes.Therefore,theobjectivefunctionallowsthealgorithmtotradeoffbetweendecreasingthenumberofhiddennodesbyincreasingTPandmaintainingspatialreusebynotincreasingTP.Inthealgorithm,thistradeoffisachievedbytestingtheimprovementonthetotalthroughput(lines 14 to 23 inAlgorithm 1 ). 8Theproblemofnodejnotbeingabletolistenhiddengrantsisthehidden-nodeproblemversionforreservation-baseddistributedschedulingpolicies.ThisproblemisstudiedindetailinChapter 2 78

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3.4SimulationResults TheperformanceevaluationoftheHSRAalgorithmwasperformedbymeansofsimulationusingthesimulatorproposedinChapter 5 .ThesimulatednetworkisanIEEE802.16meshnetworkwithdistributedschedulingunderthecongurationshowninTable 3-1 .Thenumberofnodeswasspeciedasasimulationparameter.Thenodeswereuniformlydistributedinasquareareasuchthatthenodedensitywasalwayskeptat15nodesperunitarea.Themaximumtransmissionrangeofthenodeswassetat0.3(i.e.,rmax=0.3).Theconnectivityofthenetworkunderbidirectionallinksandwiththenodes'transmissionrangessetatrmaxwasconrmedbeforeexecutingthemin-hoproutingalgorithm.Thenumberofowswasspeciedasasimulationparameter.Thesourceanddestinationnodesofeveryowwereuniformlydistributedamongallthenodesinthenetwork.Themin-hoproutingalgorithmcalculatedtheowpathsundertheMaxPowersetupwhichisthepowerassignmentwhenallthenodes'transmissionrangesaresetatthemaximum(i.e.,rmax).Oncethepathswerecalculated,thetransmissionrangesofthenodeswerefoundusingtheHSRAalgorithm.Also,theoptimaltransmissionranges(i.e.,thesolutiontotheSRA-TMproblem(Eq. 3 )),whichwecallOptPower,werefoundusingtheformulationoftheSRA-TMproblemasamixedintegerprogramwithnon-linearconstraints(MIP-NLC)(Appendix E ).TheMIP-NLCwassolvedusingtheBranchAndReduceOptimizationNavigator(BARON)Solver[ 4 ],whichisasystemforsolvingnon-convexoptimizationproblemstoglobaloptimality.Finally,thenetworkwassimulatedundertheMaxPower,MinPower,OptPower,andHSRAsetups. Figure 3-2 showstheaverageoutput-queuelengthforthreenetworkswith20nodeseachandincreasingnumberofows(i.e.,10owsinFigure 3-2A ,15owsinFigure 3-2B ,and20owsinFigure 3-2C ).Theinput-queueshavebeenomittedbecausetheyareguaranteedtoalwaysbestable(Chapter 2 ).Theowratesineachnetworkwereallsetatthesamevalue.Theseare8,6,and5packetsperframeforthenetworksinFigures 3-2A 3-2B ,and 3-2C respectively.Thesevaluesweresetsothattheir 79

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Table3-1. InstituteofElectricalandElectronicsEngineers(IEEE)808.16meshnetworkconguration ParameterValue Framelength10msControl-time-slotlength63sNumberofcontrol-time-slotsperframe4Numberofdata-time-slotsperframe256NextXmtMxy7XmtHoldoffExponenty6LinkschedulingGM-RBDSRoutingmin-hop y Thisisaparameteroftheelectionalgorithmusedforspecifyingthefrequencythatnodestransmitschedulingpackets. correspondingHSRAnetworkbecameunstableiftheywereincreasedbyatleastonepoint.Inthisway,thenetworkoperatesatapointinsidethestabilityregionandclosetoitsboundary.Therefore,whenanyoftheratesisincreasedbyatleastonepoint,thenetworkoperatesoutsidethestabilityregion,andtherefore,itisunstable. ThestabilityoftheHSRAnetworkwasdeterminedbyavisualtestonFigure 3-2A ,Figure 3-2A ,andFigure 3-2A .However,amorerigorousprocedurefortestingstabilityisbymeansofaconstantfalse-alarmrate(CFAR)test.TheCFARtestproposedin[ 82 ]teststhehypothesisthatanetworkisstablefromsamplesofthequeuelengths.Accordingtothistest,thefalse-alarmrate(i.e.,theprobabilitythatastablenetworkisregardedasunstable)oftheHSRAnetworksofFigure 3-2A ,Figure 3-2A ,andFigure 3-2A are0.17,1.910)]TJ /F6 7.97 Tf 6.59 0 Td[(7,and3.110)]TJ /F6 7.97 Tf 6.58 0 Td[(30respectively9.Thismeansthataccordingtothetest,theHSRAnetworksarelikelytobeunstable.Inotherwords,ifthenetworksareregardedasunstable,theprobabilitythatthisjudgmentisincorrect 9Theseresultswereobtainedbysamplingthequeuelengthsat25pointsequallyspacedduringthelast25%ofthesimulationtime.ThisisdoneinordertoapproximatetheassumptionsoftheCFARtestin[ 82 ].Theassumptionsarethatthesamplesarestatisticallyindependentandthattheinitialtransientisover. 80

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Table3-2. False-alarmratecomparisonfortheheuristic-stability-region-adaptation(HSRA),MinPower,MaxPower,andOptPowercongurations MaxPowerMinPowerHSRAOptPower 10ows1.210)]TJ /F6 7.97 Tf 6.58 0 Td[(1313.210)]TJ /F6 7.97 Tf 6.59 0 Td[(50.170.5715ows00.831.910)]TJ /F6 7.97 Tf 6.58 0 Td[(70.9020ows6.610)]TJ /F6 7.97 Tf 6.58 0 Td[(1360.753.110)]TJ /F6 7.97 Tf 6.59 0 Td[(301.310)]TJ /F6 7.97 Tf 6.59 0 Td[(6 (i.e.,theprobabilitythatthenetworksarestable)is0.17,1.910)]TJ /F6 7.97 Tf 6.58 -.01 Td[(7,and3.110)]TJ /F6 7.97 Tf 6.59 -.01 Td[(30respectively.Forthepurposesoftheanalysisoftheresults,thisfactisignored10.Theaveragequeuelengthsmayincreaseindenetly(i.e.,theyareunstable)accordingtotheCFARtestbutatalowrate.Thus,theHSRAnetworksmaybeoperatingatapointoutsidethestableregionbutclosetoitsboundary,whichistheimportantfactwhencomparingthealgorithms(i.e.,MaxPower,MinPower,OptPower,andHSRA). Table 3-2 showsthefalse-alarmratesofallthecasesinFigure 3-2A ,Figure 3-2A ,andFigure 3-2A .Theseresultsareusefulforthecasesthataredifculttocomparevisually.Forexample,inFigure 3-2A ,theMinPower,HSRA,andOptPowernetworksallhavesimilaraveragequeuelengths.However,Table 3-2 showsthatamongthesethreenetworks,thenetworksthataremostandlesslikelytobeunstableareMinPowerandOptPowerrespectively. Closetotheendofthesimulationtime,whenthetransientbehaviorofthequeuesisover,theaveragequeuelengthsofthedifferentpowersetups(i.e.,MaxPower,MinPower,OptPower,andHSRA)canbecompared.InFigure 3-2A ,theMaxPowersetuphastheworstperformance(i.e.,thelargestaveragequeuelength),andtheMinPower,OptPower,andHSRAhavesimilarperformance.Therefore,whenthenumberofowsislow(i.e.,10ows),theMinPowerandtheHSRAalgorithmsareableto 10Otherwise,thesimulationswouldhavetoberunseveraltimestonetunethepacketratessothatstabilityisguaranteedaccordingtotheCFARtest.Thisnetuningisnotpracticalduetothelengthofthesimulations.Thesearelongduetothefactthatthenetworksoperateneartothestable-regionboundary.Onaverage,onenetworksimulationtakes72hourstocomplete. 81

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achievequeuelengthsthatareclosetothelengthsachievedbytheoptimalsolution(i.e.,OptPower).Ontheotherhand,whenthenumberofowsincreases,theMinPoweralgorithmdoesnotachieveaperformanceclosetotheoptimalonewhiletheHSRAalgorithmdoes.ThisisshowninFigures 3-2B and 3-2C .TheMaxPowerandMinPoweralgorithmshavesimilarperformancewhichisworsewhencomparedwiththeHSRAalgorithm.TheHSRAalgorithmachievesaveragequeuelengthsthatareclosetothelengthsachievedbytheoptimalsolution.Therefore,theHSRAalgorithmenablestheowstocarrymoretrafcwhileguaranteeingstabilitythantheMaxPowerandMinPoweralgorithmsdo.Also,itisconrmedthatthetechniqueofonlymaximizingthespatialreusebyreducingthetransmissionranges(i.e.,MinPower)doesnotperformwellwhentheowdensityincreases(i.e.,whenthenumberofowsincreasesandthenumberofnodesiskeptconstant.).Ontheotherhand,thetechniqueofadaptingthestabilityregiontothegivensetofowsbymeansofTPcontrol(i.e.,HSRA)doesperformwellwhentheowdensityincreases. TheHSRAalgorithmachievesaveragequeuelengthsthatareslightlylowerthanthoseoftheOptPowersetupinsomecases(Figure 3-2A andFigure 3-2C attheendofthesimulationtime).ThisisduetothefactthattheSRA-TMproblemisbasedonthelower-boundregion(Section 3.3.2 ).IntheSRA-TMproblem,thelower-boundregion,andnotthestabilityregionitself,isadaptedtothesetofows,i.e.,thestabilityregionisadaptedthroughthelower-boundregion.Thereasonforthisisthattheexactformulationofthestabilityregionisnotavailable.Thestabilityregionisusuallycharacterizedwiththelower-boundregionbecauseitsexactcharacterizationisnotfeasibleduetoitscomplexity11.Therefore,theaccuracyofthesolutionsbasedonthe 11See[ 26 35 36 50 60 61 69 71 74 83 84 ]fortheliteratureontheproblemofcharacterizingthestabilityregionoflink-schedulingpolicies. 82

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lower-boundregionsuchasHSRAandOptPowerareaffectedbythetightnessofthisregion. Figure 3-3 showsacomparisonoftheperformanceimprovementachievedbytheHSRAandMinPoweralgorithms.Thiscomparisonisbasedontheave-rage-queue-lengthreductionachievedbytheHSRAandMinPoweralgorithmswithrespecttotheaveragequeuelengthoftheMaxPoweralgorithm.Forexample,inFigure 3-3A ,forthecaseof10nodes,theMinPowerandHSRAqueue-length-reductionbarsindicatethattheMinPowerandHSRAalgorithmsachieveaveragequeuelengthsthatare16%and23%smallerthantheMaxPoweraveragequeuelengthrespectively.Figure 3-3A showstheperformanceimprovementwhenthenumberofnodesincreasesandthenumberofowsisxedat10.Figure 3-3B showstheperformanceimprovementwhenthenumberofowsincreasesandthenumberofnodesisxedat20.Theowratesweresetat10,10,8,and10packetsperframeinFigure 3-3A ,andtheyweresetat8,6,5,and3,inFigure 3-3B .Theseframerateswerealsosetsuchthatiftheyareincreasedbyatleastonepoint,theircorrespondingnetworksbecomeunstable. InFigure 3-3A ,whenthenumberofnodesincreases,theHSRAalgorithmalwaysachievesanimprovementhigherthantheoneachievedbytheMinPoweralgorithm.Theaverage-queue-lengthreductionachievedbytheHSRAalgorithmisonaverage6.8pointsabovethereductionachievedbytheMinPoweralgorithm.Thisdifferenceofimprovementsdoesnotincreaseasthenumberofnodesinthenetworkincreases.TheHSRAimprovementisabovetheMinPowerimprovementby7.02,9.68,3.07,and7.37pointsforthecasesof10,15,20,and25nodesrespectively.Therefore,bothalgorithms,MinPowerandHSRA,respondsimilarlyasthenumberofnodesincreases.Thatis,theMinPowerandHSRAalgorithmsareabletomaintainreductionsoftheaveragequeue 83

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A10ows B15ows C20ows Figure3-2. Averageoutput-queuelengthcomparisonfortheheuristic-stability-region-adaptation(HSRA),MinPower,MaxPower,andOptPowercongurations 84

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AIncreasingnumberofnodes BIncreasingnumberofows Figure3-3. PerformancecomparisonoftheHSRAandMinPoweralgorithms lengthsasthelengthofthepathsincrease12.ThisresultsinhighertransportcapacitieswhentheMinPowerandHSRAalgorithmsareused13.Betweenthetwoalgorithms,HSRAachievesthehighesttransportcapacity. 12Notethatgiventhatthenodedensityandnumberofowsarekeptconstantasthenumberofnodesincreases,theaveragelengthofthepathsincreaseswiththenumberofnodes.13Thetransportcapacityofaowistheproductofthemaximumratesupportedbytheowandthelengthofitspath.Thetransportcapacityofthenetworkissummationoftheows'transportcapacities. 85

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InFigure 3-3B ,whenthenumberofowsincreases,theHSRAalgorithmalwaysachievesanimprovementthatishigherthantheoneachievedbytheMinPoweralgorithmtoo.Theaverage-queue-lengthreductionachievedbytheHSRAalgorithmisonaverage27.0pointsabovethereductionachievedbytheMinPoweralgorithm.Thisdifferenceofimprovementsisgreaterthantheoneachievedwhenthenumberofnodesincreases(Figure 3-3A ),whichis6.8points.Also,thedifferenceofimprovementsincreaseswiththenumberofowsinthenetwork,whichwasnotthecasewhenthenumberofnodesincreases.TheHSRAimprovementisabovetheMinPowerimprovementby3.1,19.5,29.3,and56.1pointsforthecasesof10,15,20,and25owsrespectively.Therefore,whentheowdensity,whichisdenedastheratioamongthenumberofowsandnodesinthenetwork,increases,theHSRAalgorithmachievesaverage-queue-lengthreductionsthatincreasetoowhiletheMinPoweralgorithmachievesaverage-queue-lengthreductionsthatdecrease.ThisbehaviorshowsthattheHSRAalgorithmoutperformstheMinPoweralgorithmwhentheowdensityincreases.Forexample,whenthenumberofowsis25,theMinPoweralgorithmhasanaverage-queue-lengthimprovementof)]TJ /F8 11.955 Tf 9.3 0 Td[(36.4%(i.e.,itincreasestheaverage-queue-lengthby36.4%)whiletheHSRAalgorithmhasanaverage-queue-lengthimprovementof19.7%(Figure 3-3B ). AccordingtoFigure 3-3 ,thegeneraltendencyoftheHSRAqueue-length-reductionisthatitincreaseswhentheowdensitydecreases(Figure 3-3A ),anditdecreaseswhentheowdensityincreases(Figure 3-3B ).Therefore,theHSRAalgorithmperformsbetterthanMaxPower,andtheHSRAreacheshigherperformancelevelswhentheowdensityislow.InFigure 3-3A andFigure 3-3B thereisanexceptiontothegeneraltendencywhenthenumberofnodesincreasesfrom10to15andwhenthenumberofowsincreasesfrom15to20respectively.Thereasonisthatfortheparticularsimulatedscenarioswith15nodesand20owsrespectively,thetopologyinducedbytheMaxPowersetuphappenedtobesimilartothetopologyinducedbytheOptPower 86

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setupduetotherandomnodelocationsandows'sourcesanddestinations.Therefore,inthesescenarios,theHSRAalgorithmimprovedtheinitialMaxPowertopologybysmalleramounts. Figure 3-3 alsoshowstheincreasethattheHSRAalgorithmcausesonthetotalthroughput(i.e.,theobjectivefunctionToftheSRA-TMproblem(Eq. 3 )).Thisincreasecorrespondstothestretchthatthealgorithmdoestothelower-boundregionofthelink-schedulingpolicy.Initially,thenetworkstabilityregionisdeterminedbytheMaxPowersetup.Oncethealgorithmisexecuted,thestabilityregionisdeterminedbytheHSRApowersetup.Therefore,theincreasethattheHSRAalgorithmcausesonthetotalthroughputcorrespondstotheincreaseachievedbymodifyingtheMaxPowerstabilityregionaccordingtotheHSRAalgorithm.Forexample,inFigure 3-3A ,forthecaseof10nodes,theHSRAalgorithmincreasesthetotalthroughputby24.4%.Thetotal-throughputincreasedoesnotshowadirect-proportionalnoraninverse-proportionalrelationwiththenumbernodesorows.InbothFigures 3-3A and 3-3B ,thetotal-throughputimprovementincreasesanddecreaseswithboththenumberofnodesandows.Also,whenthetotal-throughputimprovementsachievedwhenthenumberofnodesincreases(Figure 3-3A )arecomparedwiththeonesachievedwhenthenumberofowsincreases(Figure 3-3B ),theyaresimilaronaverage.Theaveragetotal-throughputimprovementsinFigures 3-3A and 3-3B are40.5%and40.9%respectively.Thetotal-throughputimprovementsinFigure 3-3A are24.4%,76.5%,36.7%,and24.4%forthecasesof10,15,20,and25nodesrespectively,andthetotal-throughputimprovementsinFigure 3-3B are36.7%,34.3%,46.4%,and46.1%forthecasesof10,15,20,and25owsrespectively.Therefore,theHSRAalgorithmperformssimilarlywheneithertheaverageowlengthortheaverageowdensityincreases.Thisresultextendstotheaverage-queue-lengthreductionsachievedbytheHSRAalgorithmwhenthenumberofnodesandowsincrease.Theseaveragesaresimilartoeachother;theyare35.2%and32.5%respectively.However,thereis 87

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adifferencebetweentheaverage-queue-lengthandtotal-throughputimprovementsachievedbytheHSRAalgorithm.Forexample,inFigure 3-3A ,forthecaseof25nodes,theaverage-queue-lengthimprovementis30.4pointsabovethetotal-throughputimprovement,andinFigure 3-3B ,forthecaseof25ows,theaverage-queue-lengthimprovementis26.4pointsbelowthetotal-throughputimprovement.Therefore,althoughtheimprovementonthemaximumtotal-throughputachievedbytheHSRAalgorithmimprovestheaveragequeuelengthinthenetwork,theamountofthetotal-throughputimprovementdiffersfromtheamountofaverage-queue-lengthimprovement.ThisdifferenceisduetothefactthattheHSRAalgorithmisbasedonthelower-boundregion(Eq. 3 )andnotonthestabilityregionitself.Therefore,thetightnessofthelower-boundregion(i.e.,howcloseitistothestabilityregion)affectstheperformanceofthetechniquetheHSRAalgorithmisbasedon.Thealgorithmcanpredictmoreaccuratelythesizeoftheexpansionitperformsonthestabilityregionwithatighterlower-boundregion. Figure 3-4 showstheperformanceoftheHSRAAlgorithmasafunctionofM.ThisperformanceisevaluatedintermsoftheprobabilitythattheHSRAAlgorithmndstheoptimalsolutionoftheSRA-TMproblemwhenMincreases.Figure 3-4A showstheperformancewhenthenumberofowsisxedat10andthenumberofnodesis10,15,20,and25.Figure 3-4B showstheperformancewhenthenumberofnodesisxedat20andthenumberofowsis10,15,20,and25.AccordingtoFigure 3-4A ,theperformanceissimilarwhenthereare10and15nodes,anditisalsosimilarwhenthereare20and25nodes.However,thegeneraltrendisthattheperformanceimprovesasthenumberofnodesincreases(i.e.,astheowdensitydecreases).Whentheowdensitydecreases,theprobabilitythataowinterfereswithanotherowdecreases.Therefore,thenumberofbottlenecknodeswhosemaximumthroughputcanbeincreasedbyincreasingtheTPofnodesinadjacentows(i.e.,hiddennodes)islower,sotheHSRAalgorithmisabletondthosehiddennodeswithalowerM. 88

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AccordingtoFigure 3-4B ,theperformancedecreaseswiththenumberofows.ThisresultisalsoduetotherelationbetweentheHSRAAlgorithmandtheowdensity.AsthenumberofowsincreasesinFigure 3-4B ,theowdensityincreases,andtherefore,theHSRAneedsahigherMtondthehiddennodeswhoseTPneedstobeincreased. AIncreasingnumberofnodes BIncreasingnumberofows Figure3-4. ProbabilitythatHSRAcalculatestheoptimalsolutionasafunctionofM 3.5Summary TheHeuristic-Stability-Region-Adaptationalgorithmhasbeenproposedfortransmissionpowercontrol.Thisalgorithmincreasestheinput-packetratesthatowscansupportanddecreasestheend-to-enddelays.Itisisbasedontheadaptationofthestabilityregionofagivenlink-schedulingpolicywhenonlythelinksthatbelong 89

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toagivensetofowsareconsidered.Thealgorithmcanbereadilyadaptedtoanylink-schedulingpolicywhosestabilityregionhasbeencharacterized,soitisnotlimitedtoanyspecicschedulingapproachsuchasRTS/CTS-basedpolicies.Theimprovementonthroughputachievedbyouralgorithmwasevaluatedbymeansofsimulationforthemin-hoproutingalgorithmandtheGM-RBDSlink-schedulingpolicyinIEEE802.16meshnetworks.Itwasshownthatitoutperformstheclassicalsolutionofreducingtransmissionpowerstoincreasespatialreuse.Also,itsperformancewasevaluated.Itwasfoundthatitdependsontheowdensity.Theperformanceincreasesastheowdensitydecreases. 90

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CHAPTER4DISTRIBUTEDTOPOLOGYCONTROLUSINGPOTENTIALGAMES Inthischapter,theproblemofmaximizingthetotalend-to-endthroughputwhenasetofowsisgivenisaddressed.Thesetofowsrepresentsthetrafcbetweenendusersthatisgeneratedbytheirapplicationssuchasend-to-endvideo/audiosessions.Themaximizationofthetotalend-to-endthroughputisperformedbymeansoftransmissionpower(TP)controlthattheowsperformcollaboratively. Intuitively,ourTPcontrolcanbedescribedasfollows.Themaximumthroughputthatagivenowsupportsdependsonthethroughputthateachofthenodesalongtheow'spathsupports,andthemaximumthroughputthatanodesupportsdependsonitsschedulingpolicyandtheconicts1withsurroundingnodeswhichalsoneedtoschedulepackettransmissions[ 49 ].Therefore,bymeansofTPcontrol,thenodesareabletoreducethenumberofconictseitherbydecreasingtheinterference(i.e.,reducingTP)and/orcoordinatingfuturepacket-transmissiontimessuchthatnoconictingtransmissionsareperformedsimultaneously.Thelatterapproachisusedinthischapter,i.e.,theowscontroltheTPssothatnodesareabletolistentoeachother'sschedulesinordertocoordinatefuturepackettransmissionsmoresuccessfully. Mathematically,ourTPcontrolcanbecharacterizedusingthestabilityregionoftheWMN.Themaximumthroughputthatnodessupportischaracterizedbythephysical-linkcapacityandthestabilityregionofthelinkschedulingpolicy.Thisregionisthesetofinput-packetratessupportedbythelinksofthenetworkthatguaranteethattheirqueuesarestable(i.e.,thelinkqueuesarepositiverecurrent)[ 69 ].Thephysical-linkcapacitydeterminesthemaximumnumberofbitsthatthepacketscancarry.IntheTPcontrolapproach,thenetworktopologyismodiedsuchthatthestabilityregionis 1Schedulingconictsarisebetweennodeswhentheyattempttotransmitpacketssimultaneouslyandtheinterferencetheycauseoneachotherishighenoughtocausepacketcollisions. 91

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adaptedtothegivensetofows.Thegoalofthisadaptationistomaximizethehighestinput-packetratesupportedbytheowsthatguaranteestability.ThisapproachwasoriginallyproposedinChapter 3 ,inwhichtheTP-controlalgorithmwasheuristicandcentralized.Inthischapter,weproposeandanalyzeadistributedTP-controlalgorithmbymeansofpotentialgames[ 53 ].Specically,weformulatetheTP-controloptimizationproblemasapotentialgameinwhichtheows(i.e.,players)collaboratetomaximizethepacketratestheycansupportbyadaptingthestabilityregion.Thegameisformulatedsuchthattheowscompeteforhelpingeachother.Theyhelpeachotherbymakingsurethatnoneoftheirindividualactionsaffectotherowsnegatively.Therefore,inthegame,theowsarerewardedwhentheybenetnotonlythemselvesbutotherowsaswell,i.e.,theyarerewardedwhentheycollaborate. TheTP-controloptimizationproblemisamixedintegerprogramwithnon-linearconstraints(Chapter 3 andAppendix E ).Therefore,inordertoanalyzethepotentialgame,theproblemismodiedheuristicallysuchthatitcanbeformulatedasanintegerlinearprogram.Basedonthismodication,theNashequilibriumofthegameischaracterized.However,duetothemodication,theNashequilibriumissuboptimalwithrespecttotheobjectivefunctionoftheoriginalproblem(i.e.,theproblemwithnomodications).Therefore,aperformanceboundfortheequilibriumisfound.Inthisway,theequilibriumcanbecomparedwiththeoptimalvalueoftheobjectivefunctionoftheoriginalproblem. Thechapterisorganizedasfollows.TherelatedworkandcontributionsarediscussedinSection 4.1 .Section 4.2 describesthenetworkmodel.InSection 4.3 ,thepotentialgameformaximizingthetotalend-to-endthroughputisproposedandanalyzed.Inordertodemonstratethehowthepotentialgamecanbeusedforaspecicnetwork,adistributedtopology-controlalgorithmforInstitute-of-Electrical-and-Elec-tronics-Engineers(IEEE)802.16WMNsisdevelopedinSection 4.4 basedonthegame.Aperformanceboundforthealgorithmisalsocalculated.InSection 4.5 ,the 92

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boundiscomparedwithsimulationresults.Also,theproposedalgorithmiscomparedwiththecentralizedandheuristicalgorithmofChapter 3 andthealgorithmbasedonspatial-reusemaximization(i.e.,nodestransmittingatminimumpowerlevelsthatguaranteeconnectivity).Finally,asummaryofthechapterispresentedinSection 4.6 4.1RelatedWork Ourstability-basedtopology-controlapproachisprimarilybasedontheideaspresentedin[ 10 11 89 ]andChapter 3 .In[ 11 ],thenetworkispartitionedbasedonthenotionoflocalpooling2,andeachpartitionisassignedtoachannelofthenetwork.Inthisway,greedymaximalscheduling(GMS)isguaranteedtoachievetheoptimalstabilityregionineachchannel.In[ 10 89 ],networktopologiesareidentiedforwhichGMSpoliciesachievetheoptimalstabilityregion.Although[ 10 89 ]provideinsightfulresultsfortheunderstandingofGMSpolicies,thenetworktopologiestheyidentifyarenotsuitableforrealscenarios[ 37 ].Thisisduetotheconditionsthatguaranteeoptimalityforsuchnetworktopologies.Theseconditionsinclude1-hopinterference,1-hoptrafc,andatopologythatcorrespondstoanF-freegraph3.RealWMNshardlymeettheserestrictiveconditions.Inouralgorithm,theowsmodifyrealisticallythenetworktopologyusingTPcontroltoadaptthestabilityregion.Thealgorithmisdistributed,anditisbasedontheheuristicandcentralizedalgorithmproposedinChapter 3 In[ 58 ],thedynamicroutingandpowercontrol(DRPC)algorithmwasproposed.DRPCndstheoptimalsolution.Ateverytimeslot,DRPCisgiventhesetofend-to-endnodeswiththeircorrespondingend-to-enddatarates,thechannelstates,andthequeuelengths,andndstheoptimalsetofroutesandTPsthatmaximizethroughput.However,thisisaconstrainedoptimizationproblemthatrequiresglobalinformationofthenetwork 2SeeSection 4.3.1 forthedenitionoflocalpooling.3See[ 10 ]forthedenitionofF-freegraphs. 93

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(i.e.,channelstatesandqueuelenghts)ateverytimeslot,so,in[ 58 ],asuboptimaldistributedalgorithmwasalsoproposedforrandom-accessschedulingpolicies.In[ 25 ],anotherrandom-power-selectionalgorithmforrandom-accessschedulingpolicieswasproposed.Itisshownthatitachievesmaximalthroughputinthefollowingsense:thethroughputachievedbyanyxedpowerselectionisatmostequaltothethroughputachievedbytherandom-power-selectionalgorithm. Othertopology-controlalgorithmsforthroughputmaximizationwereproposedin[ 7 30 33 54 56 57 81 ].In[ 25 54 56 57 ],thetotalthroughputisincreasedbyincreasingthespatialreuse.Thisisachievedbyreducingthenodes'TPs.In[ 30 33 ],thetotalthroughputisincreasedfurtherbynotonlyconsideringthespatialreusebutalsotheexposedandhiddennodes.In[ 81 ],aTPcontrolalgorithmforrequest-to-send(RTS)/clear-to-send(CTS)basedprotocolsisproposedthatdecreasestheareaoccupiedbylinksduringtheirtransmissions,anditisshownthatwiththisscheme,routingalgorithmsthatfavorshorthopsachievehigherlevelsofthroughput.In[ 7 ],theproblemofintegratedlinkschedulingandTPcontrolforthroughputoptimizationisshowntobenondeterministicpolynomialtime(NP)complete.Therefore,aheuristicalgorithmisdeveloped.Itsgoalistominimizetheschedulelengthnecessarytosatisfyallthelinkloadsdeterminedbyagivenroutingalgorithm.Inthisway,thetotalthroughputofthenetworkisincreasedbecausemoreschedulingcyclescanbeperformedpertimeunit. Ourapproachdiffersfromthepreviousalgorithmsinthatitisbasedonthestabilityregionofthelink-schedulingpolicy.Therefore,ourapproachcanbeusedforanypolicywhosestabilityregionhasbeencharacterized.Also,ourapproachdiffersinthatitisbasedonadistributedcollaborationoftheowstoincreasetheircorrespondingthroughput. 94

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Contributions Thecontributionsofthischapterareasfollows. Anewframeworkforthedevelopmentofdistributedalgorithmsthatmaximizethetotalend-to-endthroughputinWMNsisproposed.ThisframeworkisbasedonthestabilityregionoftheWMN'slink-schedulingpolicy.Itconsistsofapotentialgameinwhichagivensetofowsactasplayersthatcollaboratetomaximizethepacketratestheycansupportwhileguaranteeingstability. Basedontheproposedframework,anewdistributedTP-controlalgorithmisdevelopedforIEEE802.16WMNs.TheNashequilibriumofthisgameischaracterizedbymeansofinteger-linear-programmingtechniques. AperformanceboundforthenewTP-controlalgorithmisfoundandcomparedwithsimulationresults.Also,itisshownthatinmorethan30%ofthesimulatednetworks,ouralgorithmndsatopologythatreachesathroughputthatis96%to100%ofthemaximumthroughput.ThisperformanceissuperiorwhencomparedwiththeoneachievedbytheclassicTP-controlapproachbasedspatial-reusemaximization. 4.2NetworkModel AWMNwhosecommunicationgraphisdenotedbyG=(N,L)isconsidered.Nisthesetofnodesinthenetwork,andListhesetoflinks.Linksaredirectional.Thelinkdirectedfromnodeitonodejisdenotedby(i,j).Thenodes'transmissionsareomnidirectional,andalinkbelongstoLifandonlyifthetransmissionrangeofthesourcenodecoversthedestinationnode.Theinterferencesetoflink(i,j)isdenotedbyE(i,j)andcontainsallthelinksinLthatinterferewithlink(i,j).Thetransmissionpacketrateoflink(i,j)isdenotedby(i,j),andthevectoroflinkpacketratesisdenotedby=[(i,j)](i,j)2L. Timeisdividedintoframes,andeachframeisdividedintoacontrol-subframeandadata-subframe.Control-subframesaredividedintocontrol-time-slotsthatareusedforthetransmissionofschedulingpackets,anddata-subframesaredividedintodata-time-slotsthatareusedforthetransmissionofdatapackets4.Apacketreception 4ThisistheframestructurefortheWMNsin[ 1 71 74 84 ]andreferencestherein. 95

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overlink(i,j)issuccessfulifandonlyifnootherlinkinE(i,j)isactivatedwhilethepacketisbeingtransmitted.Ifsuchlinkisactivated,thereisacollisionatnodej(i.e,thedestinationnodeoflink(i,j))andthepacketreceptionisunsuccessful. AfeasibleschedulehonH,whereHissomesubsetofL,isalink-activationvector[0,1]jHjthatwhenallthelinksinitareactivatedsimultaneously,allthepacketreceptionsaresuccessful.AfeasibleschedulehonHismaximalif,whenallthelinksinhareactivated,nomorelinkscanbeactivatedwithoutviolatingtheinterferenceconstraints.ThesetofallpossiblemaximalschedulesonHisdenotedbyMH,anditsconvexhullisdenotedbyCo(MH). ThedatatrafcconsistsofasetofowsdenotedbyF.TheowsinFareenumerated.Then-thowisdenotedbyfn.Itconsistsofapathandameanin-put-packetratewhicharedenotedbyPfnandfnrespectively(i.e.,fn,(Pfn,fn)).PathPfnisthesetofnodesonwhichowfnisestablished.Thedatatrafconowfnisgeneratedatitssourcenodebyadata-packet-arrivalprocessthatisPoissondistributedwithmeanfn.Thepacketsleavethenetworkoncetheyarriveattheow'sdestinationnode.Thenodesinthepaththatforwardthedatapacketsfromsourcetodestinationaretheintermediatenodes.Everynodethatisanintermediateordestinationnodeofatleastoneowhasamaximumpacketratethatitcanassigntoitsincomingtrafcwhileguaranteeingthestabilityofitsincominglinks'queues.Eachofthesenodesequallydividesitsmaximumpacketrateamongalltheowsforwhichitisanintermediateordestinationnode.Themaximumpacketratethatnodejsupportsforeachoftheseowswhileguaranteeingstabilityisdenotedbyjmax. Thedegreeofalinkisdenedasthenumberofowsitbelongsto.Itisdenotedbyd(i,j)forlink(i,j).ThesetoflinksthatbelongtoatleastonowisdenotedbyL.ThevectoroflinkpacketratesofthelinksinLisdenotedby=[(i,j)](i,j)2L.ThesetofnodesthatbelongtoatleastoneowisdenotedbyN. 96

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Nodejisa1-hopneighborofnodeiifnodeiiswithinnodej'stransmissionrange.Nodejisa2-hopneighborofnodeiifnodejisa1-hopneighborofanyofnodei's1-hopneighbors.Theactive1-hopneighborhoodofanodeisdenedasthesetof1-hopneighborsthatareintermediateordestinationnodesofatleastoneow.ItisdenotedbySiafornodei.Thedirect1-hopneighborhoodofanodeisthesetof1-hopneighborsthatsenddatapacketstothenode.Therefore,thedirect1-hopneighborsofanodealwaysprecedethenodeinatleastoneow'spath.Nodei'sdirect1-hopneighborhoodisdenotedbySid. Thetransmissionrangeofnodeiisdenotedbyri.ThevectoroftransmissionpowersofthenodesinNisdenotedbyr,andthevectoroftheirmaximumtransmissionrangesisdenotedbyrmax.TheEuclideandistancebetweennodesiandjisdenotedbyjji,jjj. Table F-1 inAPPENDIX F summarizesthepreviousnotationofthenetworkmodel. 4.3Stability-RegionAdaptation WedenethenormalizedtransportcapacityofaWMNbasedonthetransportcapacitydenedin[ 28 ]andthequeueing-systemstabilityregiondenedin[ 69 ]. Denition11. Foragivensetofows,thenormalizedtransportcapacity(NTC)isthemaximumtotalnumberofpacketstransportedintheWMNfromtheows'sourcestodestinationsperdistanceunitpertimeunitthatguaranteesthestabilityofalllinkqueues. Forexample,ifthereare2owsinthenetwork,thedistancebetweentheows'sourceanddestinationis5meters,thelinkscantransmitupto103bitsperpacket,andeachowcantransportupto104bitspersecondwhileguaranteeingstability,theNTCis25104 103=102packets-meterpersecond. NotethattheNTCdenitionrequiresapre-speciedsetofows.Therefore,theNTCisnotanabsoluteperformancemetricoftheWMN.Itisaperformancemetric 97

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oftheWMNforthegivensetofows,soaWMNmayhavedifferentNTCvaluesfordifferentsetsofows. Thehighestpacketratethataowcantransportwhileguaranteeingstabilityisdeterminedbythepacketratesthattheintermediateanddestinationnodescanforwardandreceiverespectively.LetPfnintbethesetofintermediateanddestinationnodesoffn(i.e.,fn'spathwithoutthesourcenode),andletfjmax:j2Pfnintforsomefn2Fgbethesetofupper-boundsforthepacketratesthenodescanforward/receivethatguaranteestability.Theseupper-boundsaredeterminedfromthelink-schedulingpolicy'sstabilityregionasexplainedinSection 4.3.1 .InordertoguaranteestabilityoftheWMN,thesourcenodeofeveryowcannotgeneratedatapacketsatanyratehigherthantheminimumupper-boundamongtheupper-boundsoftheintermediateanddestinationnodesoftheow.Thatis,theWMNisstableif fn
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by=f:forsome2Co(ML)g[ 69 ].Intermsofcomplexity,itisusuallythecasethatthelink-schedulingpolicieswithlargerstabilityregionsaremorecomplex.Forexample,theoptimalschedulingpolicy[ 69 ]requiresthesolutionofanNP-hardproblem(i.e.,maximalweightedmatching)[ 66 ]. Thestabilityregionofdifferentlink-schedulingpoliciesischaracterizedbyasetofconditionsthatifsatised,theWMNisguaranteedtobestable5.ThebasicideabehindourNTC-adaptationapproachisbasedonthesesufcientconditionsforstability.Weclaimthatthesecanbecontrolledbymanipulatingthenetworktopologysuchthatthehighestpacketratestheowscansupportwhileguaranteeingstabilityareincreased. Inordertoillustratethisidea,weclassifythesufcientconditionsthatguaranteestabilityintotwomaincategoriesasfollows6. Thelink-schedulingpoliciesproposedin[ 26 44 51 84 ]andChapter 2 (i.e.,greedyandconstant-timepolicies)belongtotherstcategory.ThesepoliciesarecharacterizedbythesetsofinterferinglinksfE(i,j):(i,j)2Lgandanincreasingfunctionf(i,j):E(i,j)!R+,whereE(i,j)isthesetoftransmissionpacketratesofthelinksinE(i,j)(i.e.,E(i,j),f(k,l):(k,l)2E(i,j)g).Underthesepolicies,theWMNsarestableiff(i,j)<0forevery(i,j)inL.Theintuitionbehindthisisthatthecombination(i.e.,f(i,j))ofthetransmissionpacketratesofinterferinglinks(i.e.,E(i,j))cannotexceedcertainlimitinordertoguaranteethestabilityoftheWMN.Otherwise,thequeuesoftheinterferinglinksmaybuildupandneverreturntotheemptystatemakingthenetworkunstable.Forexample,greedyschedulingpolicies[ 84 ]requirethattheconditionsinEq. 4 bemet, 5See[ 27 65 ]forareviewandcomparisonofthedifferentlink-schedulingpolicies,andsee[ 34 ]andChapter 2 forlink-schedulingpoliciesforIEEE802.11and802.16WMNs.6Thelink-schedulingpoliciesproposedin[ 65 68 ](i.e.,pick-and-comparepolicies[ 65 ]),arenotdiscussedheresincetheyalreadyreachtheoptimalstabilityregion,butatahighcomplexitycost[ 65 ]. 99

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wherec(i,j)isthecapacityoflink(i,j)inunitsofpackets-per-slot,andisthemaximumnumberofnon-interferinglinksintheinterferencesetofanylinkinthenetwork. X(k,l)2E(i,j)(k,l) c(k,l)<8(i,j)2L(4) ByconsideringonlythelinksthatbelongtoatleastoneowfninF(i.e.,L),thesetsofinterferinglinksfE(i,j):(i,j)2Lgcanbemodiedbymeansoftopologycontrolinordertominimizetherateatwhichthefunctionsf(i,j)increasewiththetransmissionpacketratesinE(i,j).Inthisway,theNTCofthenetworkcanbeincreasedbymeansoftopologycontrol. Thelink-schedulingpoliciesdiscussedin[ 35 37 45 46 ](i.e.,greedymaximalscheduling)belongtothesecondcategory.Thesepoliciesarecharacterizedbytheconceptoflocal-poolingfactor,whichisanindicatorofhowdifferenttheeffectivenessofthedifferentmaximallinkschedulesarefromeachother[ 45 ].Thiseffectivenessmakesreferencetotheirabilitytoreducethequeuelengths.Thehigherthelocal-poolingfactor,theclosertheeffectivenessofthedifferentmaximalschedules.Whenthedifferentmaximallinkschedulesaresimilarlyeffective,GMSpoliciesareabletosupportpacketratesthatareclosertotheboundariesoftheoptimalstabilityregion.Thelocal-poolingfactorisdenedasfollows[ 37 ].LetHbesomesubsetoflinks(i.e.,HL).Thelocal-poolingfactorofaWMNisthengivenby =supf:8,2Co(MH),8H2Lg. Therefore,dependsonthesetsofmaximalschedulesonH.ThesetsofmaximalschedulesaredeterminedbytheinterferencesetsfE(i,j):(i,j)2Lg,socanbemodiedbyconsideringonlytheinterferencesetsoflinksthatbelongtoatleastoneowfninF(i.e.,L)andcontrollingthetopologyofthenetwork.Inthisway,theNTCoftheWMNcanbeincreasedbymeansoftopologycontrol. 100

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Allthepreviouslink-schedulingpolicesdependonthelinksintheinterferingsetsfE(i,j):(i,j)2Lg,andthesesetsdependonthenodestransmissionrangesr.Therefore,thestabilityregionscanbecontrolledbymeansofrinordertoadaptthemtothegivensetofows.ThegoalofthisadaptationcanbeformulatedastheimprovementoftheNTCbysolvingtheNTC-adapationproblem,whichisdenednext. Thevectoroftransmissionrangesofthenodesthatbelongtoatleastoneow,i.e.,r,isfeasibleifthenodesdonotexceedtheirmaximumTP(i.e.,rrmax)andnoneofthelinksinLisbroken(i.e.,noowisbroken).LetrminbethevectorofminimumtransmissionrangesthatguaranteethatnoneofthelinksinLisbroken(i.e.,rimin,rjminjji,jjj8i2Sjd,j2N).Then,risfeasibleifrminrrmax. Letfnmaxbethehighestpacketratethatowfnsupportswhileguaranteeingstability.AccordingtoEq. 4 ,fnmaxisgivenbytheminimumofthehighestpacketratessupportedbythenodesinPfnint(i.e.,minfjmax:j2Pfnintg).Accordingtothepreviousdiscussiononstabilityregions,thesepacketratesareafunctionofthelink-interferencesetsfE(i,j):(i,j)2Lg,whichinturn,areafunctionofthetransmissionrangesr.ThisdependenceofE(i,j)onrisdenotedbyE(i,j)r.Therefore,thehighestpacketratessupportedbythenodesareafunctionofthetransmissionranges(i.e.,jmax(Ejr):RjLj+!R+,whereEjr,fE(i,j)r:i2Sjdg),andfnmaxdependsonrasshowninEq. 4 fnmax(r),minfjmax(Ejr):j2Pfnintg.(4) Denition12. TheNTC-adaptationproblem(NTC-AP)istheproblemofndingafeasiblevectoroftransmissionrangesrsuchthatthereisnofeasiblerthatmeetsthefollowingcondition:fnmax(r)>fnmax(r)foratleastonefninFandfnmax(r)fnmax(r)forallotherfninF,i.e.,thereisnovector[fnmax(r)]fn2Fthatdominatesvector[fnmax(r)]fn2F. Therefore,thegoaloftheNTC-APistondavectorroffeasiblenode-transmissionrangesthatguaranteesParetooptimality.Thiscanbedone,accordingtoEq. 4 ,by 101

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controllingthevaluesofthehighestpacketratessupportedbythenodesfjmax(Ejr):j2Pfnint,fn2Fgwhicharedenedbytheparticularlink-schedulingpolicyadoptedinthenetwork. Inthefollowingsection,theframeworkforsolvingtheNTC-APisproposed.Itisbasedonpotential-gametheory.Inthisframework,theowsaretheplayersofthepotentialgame,andtheyadaptthenetworktopologybycontrollingtheTPsofthenodesthataffectthehighestpacketratesthatguaranteestability.Usingtheframework,adistributedalgorithmforsolvingtheNTC-APinIEEE802.16WMNsunderthegreedypolicyproposedinChapter 2 isproposedandanalyzedinSection 4.4 4.3.2DistributedTP-ControlAlgorithmsusingPotentialGames ThefollowingdenitionsarenecessarytoformulatetheNTC-APasanormal-formgame.Table F-2 inAppendix F summarizesthenotationintroducedinthisformulation. Theinterferencesetoflink(i,j)underthenodes'transmissionrangesgivenbyrisdenotedbyE(i,j)r.WhenallthenodestransmitattheirmaximumTP,thissetisdenotedbyE(i,j)max. NotethattheinterferencesetE(i,j)rdependsontheTPofthenodesthatbelongtoatleastoneow(i.e.,r)only.ItdoesnotdependontheTPsofnodesthatdonotbelongtoanyowbecausethesenodesdonottransmitnorreceiveanypackets. TheNTC-APisformulatedasanormal-formgameasfollows.IndividualowsformtheplayersetF,ff1,f2,...,fn,...,fNg.EachowfncanautonomouslysettheTPofthenodesinthesetSfnwhichisdenedasfollows.Sfnisthesetofnodesthatareabletoaffectfn'shighestpacketratethatguaranteesstabilitywhenalltheallowedTPlevelsareconsidered.Therefore,thesearethenodesthatarewithinsomemaximumdistancefromtheintermediateanddestinationnodesoftheow(i.e.,Pfnint).Thisdistancedependsonthemaximumtransmissionrangeofthenodesandtheinterferencemodel.Specically,SfnisdenedintermsofthesetsSj.Sjisthesetofnodesthatareabletoaffectthehighestpacketratethatnodejsupportsforanyofits 102

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incomingows(i.e.,jmax)whenalltheallowedTPlevelsareconsidered.Therefore,SjisdeterminedfromthesetsofinterferinglinksE(i,j)max,i.e.,link(i,j)'sinterferinglinkswhenallthenodestransmitatthemaximumTP.SjisgivenbyEq. 4 7.AccordingtoEq. 4 ,inSj,onlytheincominglinksofnodejformedwithitsdirect1-hopneighbors(i.e.,Sjd)areconsidered.TheinterferencesetsoftheselinksarecalculatedwhenallthenodestransmitatthemaximumTPsothatallthenodesthatareabletoaffectjmaxareconsidered. Sj,(k:k2[i2SjdE(i,j)max)(4) SfnisdenedbyEq. 4 .ItistheunionoftheSjofthenodesalongfn'spathwiththeexceptionofthesourcenode(i.e.,j2Pfnint).Thesourcenodeisexcludedbecausethisnodedoesnotlimittheow'shighestpacketratethatguaranteesstability.Itistheabilityoftheintermediateanddestinationnodestoforwardandreceivethedatapacketsgeneratedbythesourcenodewhatlimitsthehighestpacketratesupportedbytheow. Sfn,[j2PfnintSj(4) TheactionspaceofowfnisthesetoffeasibletransmissionrangesofthenodesinSfn.ThissetisdenotedbyRn.Anactionoffnisdenotedbythevectorrn.Therefore,actionrnbelongstoRn,anditspeciesthetransmissionrangescontrolledbyfn(i.e.,thetransmissionrangesofthenodesinSfn).Thegame'sactionspaceRisthesetoffeasibletransmissionrangesofthenodescontrolledbytheows(i.e.,thenodesinSfn2FSfn).Actionr)]TJ /F7 7.97 Tf 6.58 0 Td[(nspeciesthetransmissionrangesnotcontrolledbyfn(i.e., 7InEq. 4 andfortherestoftheanalysis,itissaidthatanodekbelongstoasetoflinksHifkispresentinanyofthelinksinH. 103

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thetransmissionrangesofthenodesin(Sfi2FSfi)nSfn).Therefore,r)]TJ /F7 7.97 Tf 6.59 0 Td[(nbelongstoR)]TJ /F7 7.97 Tf 6.59 0 Td[(n=RnRn. Whenowfnmakesamove(i.e.,updatesitsactionvectorrn),thehighestpacketratesthatitandotherowssupportmaybeaffected.Thisrelationbetweenfnandotherowsisreectedonitsutilityfunctionn:R!Rinordertoenablecollaborationamongows.LetFnbethesetofowswhosehighestsupportedpacketratesareaffectedbyanyofthemovesoffn(Eq. 4 ).Flowfn'sutilityfunctionnisdenedfromthehighestpacketratessupportedbytheowsinFnasgivenbyEq. 4 .Therefore,accordingtoEq. 4 ,owfn'sutilityincreasesnotonlywhenitshighestsupportedpacketratefnmaxincreases,butalsowhenthehighestsupportedpacketratesoftheowsinFnincreaseaswell.Inthisway,fnisencouragedtocollaboratewiththeowsthatareaffectedorpotentiallyaffectedbyitsmoves,i.e.,whenanyowmakesamove,theowtendstobenetotherowsaswellbecauseinthiswayitsutilityfunctioncanbeincreasedmoreeffectively. Fn,ffi:Sfn\Sfi6=;,fi2Fg(4) n(r),Xfi2Fnfimax(r)(4) Thevectorofutilityfunctionsisu=[1,2,...,N]:R!RN. Denition13. Thegame8NTC-AP=hF,R,u(r)iisanordinalpotentialgame(OPG)ifthereexistsafunctionV:R!Rsuchthat8fn2F,8r)]TJ /F7 7.97 Tf 6.59 0 Td[(n2R)]TJ /F7 7.97 Tf 6.59 0 Td[(n,andxn,yn2Rn V(xn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n))]TJ /F5 11.955 Tf 11.96 0 Td[(V(yn,r)]TJ /F7 7.97 Tf 6.58 0 Td[(n)>0,n(xn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n))]TJ /F9 11.955 Tf 11.95 0 Td[(n(yn,r)]TJ /F7 7.97 Tf 6.58 0 Td[(n)>0 Viscalledtheordinalpotentialfunction(OPF)oftheNTC-APgame. 8See[ 53 ]forthetheoryofpotentialgames. 104

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Theorem4.1. ThegameNTC-AP=hF,R,u(r)iisanOPG.AnOPFisgivenbyT(r),whichisthetotalhighestpacketratesupportedbythenetworkforthegivensetofowsF(Eq. 4 )9. T(r),Xfn2Fminfjmax(r):j2Pfnintg(4) TheproofofTheorem 4.1 isasfollows.FromEq. 4 ,Eq. 4 ,andEq. 4 ,T(r)canberewrittenasfollows.T(r)=Xfi2Ffimax(r)=Xfi2Fnfimax(r)+Xfi2FnFnfimax(r)=n(r)+Xfi2FnFnfimax(r) Therefore, T(xn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n))]TJ /F9 11.955 Tf 11.96 0 Td[(T(yn,r)]TJ /F7 7.97 Tf 6.58 0 Td[(n)=n(xn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n))]TJ /F9 11.955 Tf 11.95 0 Td[(n(yn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n)+Xfi2FnFnfimax(xn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n))]TJ /F12 11.955 Tf 19.67 11.35 Td[(Xfi2FnFnfimax(yn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n) ThehighestpacketratessupportedbytheowsinFnFn(i.e.,ffimax:fi2FnFng)areindependentoftheactionsoffn(i.e.,fimax(xn,r)]TJ /F7 7.97 Tf 6.59 0 Td[(n)=fimax(yn,r)]TJ /F7 7.97 Tf 6.58 0 Td[(n)8fi2FnFn,8r)]TJ /F7 7.97 Tf 6.59 0 Td[(n2R)]TJ /F7 7.97 Tf 6.59 0 Td[(n,8xn,yn2Rn)because,bydenition(Eq. 4 ),Fnisthesetofowswhosehighestpacketratesareaffectedbyfn'sactionsrn2Rn.Therefore, Xfi2FnFnfimax(xn,x)]TJ /F7 7.97 Tf 6.58 0 Td[(n)=Xfi2FnFnfimax(yn,x)]TJ /F7 7.97 Tf 6.59 0 Td[(n) ThisconcludestheproofofTheorem 4.1 Inthefollowingsection,theNTC-APgameisanalyzedforIEEE802.16WMNs.Adistributedalgorithmisproposedforowstodeterminetheirindividualactions. 9ThedirectdependenceofjmaxonE(i,j)r,i.e.,jmax(E(i,j)r)(Eq. 4 ),hasbeenomittedinEq. 4 ,i.e.,jmax(r),tosimplifythenotation. 105

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TheconvergenceandperformanceofthealgorithmarecharacterizedwiththeNashequilibriaofthegame. 4.4Stability-RegionAdaptationinInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16WirelessMultihopNetworks(WMN) InIEEE802.16WMNs[ 1 ],timeisdividedintoframesaccordingtothedescriptiongiveninSection 4.2 .Inthecontrol-time-slots,thenodesareselectedbyanelectionalgorithm[ 1 13 78 ]suchthatwhenanodeisselected,noneofits1-hopand2-hopneighborsareselected.Theelectionalgorithmselectsnodesineverycontrol-time-slot,andanodetransmitsaschedulingpacketeverytimeitisselected.Aschedulingpacketcarriesthreetypesofmessages.Thesearerequest,grant,andconrmation.Therecanbemorethanonemessagepertypeinaschedulingpacket(e.g.,aschedulingpacketmaycarry3requests,1grant,and2conrmations).Thenodesmakereservationsoffuturedata-time-slotsfortransmittingdatapacketsstoredintheirlinkqueues.Thereservationsaredoneonaper-linkbasisbymeansofathree-wayhandshake.First,whenthelink'ssourcenodeisselectedbytheelectionalgorithm,itsendsarequesttothelink'sdestinationnodeandwaitsforareply.Then,whenthelink'sdestinationnodeisselected,itrepliesbysendingagranttothelink'ssourcenode.Finally,whenthelink'ssourcenodeisselectedagain,itsendsaconrmation,whichisacopyofthegrant.Thenodesthathaveeitherthelink'ssourceordestinationnodeas1-hopneighbors(i.e.,thenodeswithinthetransmissionrangeofeitherthelink'ssourceordestinationnode)listentothetransmittedgrantandconrmation.EverynodeintheWMNkeepstrackofthereservedfuturedata-time-slots.Basedonthisinformation,thenodesdeterminethefuturedata-time-slotstheyincludeintheirrequestsandgrantssuchthatcollisions 106

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areavoided.Thenodesareabletouseanylink-schedulingpolicy10thatadaptstotheIEEE-802.16standard[ 1 ]fordeterminingsuchsetsofdata-time-slots(i.e.,requestedandgranteddata-time-slots).Inthefollowing,the2-hopinterferencemodelandthelink-schedulingpolicyproposedinChapter 2 willbeadopted.WegiveabriefdescriptionofthispolicyanditsstabilityregioninordertoillustratehowtheNTC-APgamecanbeusedinIEEE802.16WMNs. Theadoptedpolicyisthegreedy-maximalreservation-based-distributed-scheduling(GM-RBDS)policy(Chapter 2 ),whichissummarizedasfollows. Wheneveranodeisselected, foreveryoutgoinglink,conrmanynon-interferinggrantsreceivedsincetheprevioustimethenodewasselectedandscheduledatapacketsfortransmissionaccordingtothesegrants foreveryincominglink,grantasmanydata-time-slotsasrequestedatthefuturedata-time-slotsthathavenotbeenreservedyetandthataretheclosestintime foreveryoutgoinglink,requestasmanydata-time-slotsasunscheduleddatapacketsstoredinthelink'squeue ThesizeofthestabilityregionoftheGM-RBDSpolicydependsontheabilityofthelinkstoperformthethree-wayhandshakessuccessfully11.Iftheprobabilitythatalinknishessuccessfullyathree-wayhandshakeislow,thelink'squeuewilldecreaseatalowerrate.Therefore,thelink'sabilitytoforwarddatapacketswithinsometimerangeisgoingtobelower(i.e.,thehighestpacketratesupportedbythelinkislowered),andthisreducesthesizeofthestabilityregion.Theprobabilitythatathree-wayhandshakeissuccessfuldependsonthegrantsreceivedbythelink'ssourcenodefromthetimeinstantitsendstherequestuntilthetimeinstantitreceivesthegrantfromthelink's 10TheIEEE802.16standard[ 1 ]doesnotspecifyanyparticularlinkschedulingpolicy.Itonlyprovidestheframeworkforimplementingthem(i.e.,electionalgorithm,controlanddatasubframes,andcontrolmessages).11SeeChapter 2 foradetailedstabilityanalysisoftheGM-RBDSpolicy. 107

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destinationnode.Ifanyofthesegrantsisnotheardbythelink'sdestinationnodeandthelink'sdestinationnode'sgrantoverlapsthem,thethree-wayhandshakeisunsuccessful.Thatis,thelink'ssourcenodewillnotbeabletoconrmthegrantsentbythelink'sdestinationnodebecauseothergrantspreviouslyreceivedalreadyreservedthefuturedata-time-slotsgrantedbythelink'sdestinationnode.Therefore,thehighestpacketratethatanodesupportsfortheowsitforwardsorservesasasinknodedependsonthefollowingnodes:thenodesthattransmitgrantsthatarenotreceivedbyitandarereceivedbyits1-hopneighbors(i.e.,hiddennodes).Forexample,considersomelink(i,j).Thenodesthattransmitthegrantsreceivedbyiandjaretheactive1-hopneighborsSiaandSjaofiandj.ThenodesinSianSjatransmitgrantsthatarereceivedbyibutnotbyj.Theyarehiddenfromj.Therefore,ifthegrantstransmittedbythenodesinSianSjaoverlapthegrantsforlink(i,j),whicharetransmittedbyj,link(i,j)'sthree-wayhandshakesareunsuccessful.Whenalltheincominglinksofnodejthatbelongtoatleastoneowareconsidered,themaximumpacketratejmaxthatnodejisabletograntforeachofitsincomingowsisgivenbyEq. 4 12. Theorem4.2. NetworkGundertheGM-RBDSpolicyandthe2-hopinterferencemodelisstableifthepacketrateofeveryincomingowofnodejisnotgreaterthanjmaxforeveryjinN,wherejmaxisgivenbyEq. 4 jmax=1 5Pi2Sjdd(i,j)jSianSjaj(4) Corollary1. NetworkGundertheGM-RBDSandthe2-hopinterferencemodelisstableifthepacketratefnofeveryowfninFsatisesEq. 4 fn
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TheonlyfactorinEq. 4 thatdependsonthetransmissionrangesrisjSianSjaj,whichwillbedenotedbya(i,j)r.Theexpressionforfnmax(r)thatdenesu(r)(Eq. 4 )andT(Eq. 4 )fortheNTC-APgameinIEEE802.16WMNsundertheGM-RBDSpolicyisthengivenasfollows. fnmax(r)=min(1 5Pi2Sjdd(i,j)a(i,j)r:j2Pfnint)(4) Denition14. Thebottlenecknodeofowfnisthenodeamongtheow'sintermediateanddestinationnodesthathastheminimumhighestpacketratethesenodessupport,i.e.,letjbethebottlenecknodeoffn,thenj=argmini2Pfnintimax. Inthegame,thegoaloftheowsistoincreasethevalueoftheirutilityfunctions(i.e.,u(r))byadjustingthevaluesofthefactorsfa(i,j)r:(i,j)2Lg.However,giventhatfnmax(r)isanonlinearfunctionofa(i,j)r,theutilitiesu(r)andtheOPFT(r)arealsononlinearfunctionsofa(i,j)r. Inthefollowing,theNTC-APgameinIEEE802.16WMNsisapproximatedwithagamewhoseutilitiesandOPFarelinearfunctionsofa(i,j)r.ThislinearversionoftheNTC-APgamewillbecalledLin-NTC-AP.Basedonthelinearity,theNashequilibriaoftheLin-NTC-APgameisanalyzedinSection 4.4.1 ,andthedifferencebetweenthesolutionsreachedbytheNTC-APandLin-NTC-APgamesisupper-boundedinSection 4.4.2 Proposition4.1. ThesolutionsetoftheoptimizationproblemgivenbyEq. 4 isalsoasolutiontotheNTC-AP(Dention 12 ). minimizeXfn2F1 5fnmax(r)subjecttorminrrmax.(4) Theproofisasfollows.First,itisnotedthatasolutiontoEq. 4 isalsoasolutiontotheNTC-AP,andsecond,itisprovedthatEq. 4 andEq. 4 areequivalent. 109

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ndsomersuchthati)fnmax(r)fnmax(r)8fn2Fii)rminr,rrmax(4) AsolutiontoEq. 4 isalsoasolutiontotheNTC-APduetoitsrstconstraint,i.e.,theconstraintthatfnmax(r)fnmax(r)8fn2Fguaranteesthatvector[fnmax(r)]fn2Fisnotdominatedbyanyothervector[fnmax(r)]fn2F. TheequivalenceofEq. 4 andEq. 4 isduetothefactthatrsolvesEq. 4 iffrminimizesEq. 4 Letr1beasolutiontoEq. 4 .Therefore,fnmax(r1)fnmax(r)forallr:rminrrmaxandforallfn2F.Assumethatr1isnotasolutiontoEq. 4 .Therefore,theremustbesomer2:rminr2rmaxandsomefn2Fsuchthatfnmax(r2)>fnmax(r1).However,thisisimpossiblesincer1isanoptimalsolutiontoEq. 4 Thesameargumentcanbemadeintheoppositedirection.Thatis,ifr1isasolutiontoEq. 4 ,thenitisalsoasolutiontoEq. 4 .Thisconcludestheproofoftheproposition. Inthefollowing,theconceptofcontentionlevelisintroducedinordertodenetheLin-NTC-APgameinEq. 4 .ThecontentionlevelofnodejisdenedaccordingtoEq. 4 ,i.e.,thesummationofthenumberofhiddenactivenodes(i.e.,a(i,j)r)ineveryincominglink(i.e.,f(i,j):i2Sjdg)weightedbythelinkdegree(i.e.,d(i,j)).Therefore,whenthecontentionexperiencedbyanodeishigh,theamountoftrafcitcanforwardand/orreceiveislowbecauseitsgrantsmayoverlapwithhigherprobabilityanyofthegrantstransmittedbythehiddenactivenodes. cj(r),Xi2Sjdd(i,j)a(i,j)r(4) 110

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FromEq. 4 ,Eq. 4 ,andEq. 4 ,theproblemgivenbyEq. 4 (i.e.,NTC-AP)canberewrittenintermsofthecontentionlevelasfollows. minimizeXfn2Fmaxfcj(r):j2Pfnintgsubjecttorminrrmax(4) Therefore,theNTC-APreducestondingasetoffeasibletransmissionrangesthatminimizesthehighestcontentionlevelexperiencedbyeveryow. ThesolutiontoEq. 4 (i.e.,NTC-AP)isapproximatedwiththesolutiontotheLin-NTC-AP.TheLin-NTC-APisformulatedasgivenbyEq. 4 .ThetermscfnT(r)andcfnV(r)inEq. 4 aredenedasfollows.ThetotalcontentionexperiencedbyfniscfnT(r),Pj2Pfnintcj(r).Themeancontentionexperiencedbyfniscfn(r),jPfnintj)]TJ /F6 7.97 Tf 6.59 0 Td[(1Pj2Pfnintcj(r).ThecontentionvariationexperiencedbyfniscfnV(r),Pj2Pfnintjcj(r))]TJ /F8 11.955 Tf -457.49 -23.91 Td[(cfn(r)j. minimizeXfn2F(cfnT(r)+cfnV(r))subjecttorminrrmax.(4) Therefore,intheLin-NTC-AP,thegoalistominimizethetotalcontentionandthecontentionvariationexperiencedbytheows.Intuitively,theLin-NTC-APapproximatestheNTC-APbasedonthefollowingtwoobservations.First,bymakingsurethatthemaximumcontentionexperiencedbyaowatacertainnodeisnottoodifferentfromthecontentionattheothernodesoftheow,thecontentionalongtheow'spathismoreuniform.Thisisachievedwhenthecontentionvariationisreduced.Second,giventhatthecontentionismoreuniform,reducingthetotalcontentionexperiencedbytheowalsoreducesthemaximumcontentionalongtheow'spath,whichisthegoalintheNTC-AP(Eq. 4 ). 111

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IntheLin-NTC-APgame,whichisdenednext,eachowcompetesforminimizingthetotalcontentionandcontentionvariationofitselfandofanyotherowsitaffectswithitsactions.Therefore,theplayersetF,theplayers'actionsetsfRn:fn2Fg,andthegame'sactionspaceRarethesameforboththeNTC-APandtheLin-NTC-APgames13.Theonlydifferenceintheformulationofthesetwogamesistheutilityfunctions.IntheLin-NTC-APgame,theutilityfunctionn(r):R!Roffnisdenedasfollows. n(r),Xfi2Fn(cfiT(r)+cfiV(r)) NotethattheLin-NTC-APgameisalsoanOPG14.AnOPFforthisgameisgivenbyEq. 4 .ThisOPFrepresentsthetotalcontentionandcontentionvariationexperiencedbyalltheows. cLin(r),Xfn2F(cfnT(r)+cfnV(r))(4) Algorithm 2 ,calledWiMAX-Mesh-NTC,isproposedfortheowstodecreasethevalueoftheirutilityfunctions15.Thealgorithmrequiresthatthetransmissionrangesbeinitializedwiththeirminimumvaluesthatdonotdisconnectanyoftheows.Thealgorithmalsorequiresthefollowinginformationwhichisconstantforthewholedurationofthegame:thesetPnofpathsoftheowsinFn(i.e,Pn,fPfn:fn2Fng),thedegreesDnoftheincominglinksofthenodesthatbelongtothepathsinPnandthat 13SeeSection 4.3.2 fortheirdenitions.14TheproofthattheLin-NTC-APgameisanOPGfollowsthesameargumentoftheproofthattheNTC-APgameisanOPG(i.e.,proofofTheorem 4.1 ).Ithasbeenomittedforthesakeofbrevity.15TheWiMAX-Mesh-NTCalgorithmisbasedontheHSRAalgorithmproposedinChapter 3 112

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havedegreegreaterthanzero(i.e.,Dn,fd(i,j):j2Pn,i2Sjdg).Finally,thealgorithmrequiresthefollowinginformationthatchangeswiththeactionstakenbytheows:thesetHnrofhiddenactivenodesofthelinkswhosedegreeswereincludedinDn(i.e.,Hnr,fSia(r)nSja(r):j2Pn,i2Sjdg). Algorithm2Playerfn'sAlgorithm Require: TPsaresettotheirminimumvaluesthatdonotdisconnectanyoftheows procedureWIMAX-MESH-NTC(fn,Pn,Dn,Hnr) n CALCULATEMYUTILITY(fn,Pn,Dn,Hnr) b GETMYBOTTLENECKNODE(fn,Pn,Dn,Hnr) C GETHIDDENACTIVENODES(b,Hnr) c SELECTTHEBESTHIDDENNODE(n,C,Pn) ifc6=;then r(c) jjc,bjj endif endprocedure TheWiMAX-Mesh-NTCalgorithmworksasfollows.Inthegame,theowstaketurnstotakeaction(e.g.,bypassingatokenfromowtoow).Foreveryofitsturns,aowcalculatesitsactionaccordingtoWiMAX-Mesh-NTCasfollows.Theowrstcalculatesitsutilityfunctionn(r)forthecurrentsetofTPsr(line 2 inAlgorithm 2 ).Then,itndsitsbottlenecknode(line 3 inAlgorithm 2 )andtheactivenodeshiddenfromthebottlenecknode(line 4 inAlgorithm 2 ).Ifbdenotesthebottlenecknode(line 3 inAlgorithm 2 ),thesetofactivenodeshiddenfromthebottlenecknodeisC=fSia(r)nSba(r):i2Sbdg(line 4 inAlgorithm 2 ).Theowthenselectsoneofthenodesinthisset(line 5 inAlgorithm 2 )asfollows.Foreverynodeintheset,theowcheckswhetherthenode'sTPcanbeincreasedbytheminimumamountofpowernecessarytocoverthebottlenecknode.IfsuchaTPincreaseispossible,theowcalculatesitsutilityfunctionwiththenewTP.Thenodethatwasabletodecreasetheow'sutilityfunctionthemostisselected.Whennoneofthenodesisabletodecreasetheutilityfunction,noneofthethemisselected.Therefore,attheend(lines 6 to 8 ),theowcheckswhetheranyoftheactivenodeshiddenfromthebottlenecknodewasselected, 113

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andifitwas,itsetstheTPoftheselectednodetotheminimumTPthatcoversthebottlenecknode(line 7 inAlgorithm 2 ). Therefore,thegoaloftheWiMAX-Mesh-NTCalgorithmistodecreasetheow'sutilityfunctionbyincreasingtheTPofanodesothattheow'sbottlenecknodeisabletolistentoit.Thisnodeistheactivenodehiddenfromthebottlenecknodethatdecreasestheow'sutilityfunctionthemost. 4.4.1NashEquilibriaandLinearIntegerProgramming ThereasonforapproximatingtheNTC-APgamewiththeLin-NTC-APgameisthatduetothelinear-integer-programmingnatureofLin-NTC-AP(Theorem 4.3 ),itssetofoptimalsolutions(i.e.,argminrminrrmaxPfn2F(cfnT(r)+cfnV(r)))canbecharacterizedwiththeNash-equilibria(Theorem 4.4 ).Therefore,giventhattheLin-NTC-APgameispotential,itisguaranteedtoconvergetotheoptimalsolutionoftheLin-NTC-AP(Corollary 2 ). Theorem4.3. TheLin-NTC-APisalinearintegerprogram. TheproofofTheorem 4.3 isasfollows.TheLin-NTC-APcanbeformulatedintermsofthevectora,[a(i,j)r](i,j)2L,i.e.,thevariablesaretheelementsofvectoraandnottheelementsofvectorr.AnysetofTPsrthatsatisesthesolutionaachievesthesameobjective-functionvalue.TheobjectivefunctioncLinisformulatedintermsofaasfollows16. 16Giventhatthevariablesare[a(i,j)r](i,j)2L,thersubscriptisdropped. 114

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cLin(a)=Xfn2F(cfnT(a)+cfnV(a))=Xfn2F Xj2Pfnintcj(a)+Xj2Pfnintjcj(a))]TJ /F8 11.955 Tf 12.14 0 Td[(cfn(a)j!=Xfn2FXj2Pfnintcj(a)+Xfn2FXj2Pfnintcj(a))]TJ /F8 11.955 Tf 21.95 8.09 Td[(1 jPfnintjXk2Pfnintck(a)=Xfn2FXj2PfnintXi2Sjdd(i,j)a(i,j)+Xfn2FXj2PfnintXi2Sjdd(i,j)a(i,j))]TJ /F8 11.955 Tf 21.95 8.09 Td[(1 jPfnintjXk2PfnintXi2Skdd(i,k)a(i,k) Letdjbethenumberofowsforwhichnodejisanintermediateordestinationnode.Letm(i,j)kbeanindicatorthatlink(i,j)pointstonodek(i.e.,m(i,j)k=1ifj=k,otherwisem(i,j)k=0).Letm(i,j)fnbeanindicatorthatlink(i,j)pointstoanintermediateordestinationnodeofowfn(i.e.,m(i,j)fn=1ifj2Pfnint,otherwisem(i,j)fn=0).TheobjectivefunctioncLincanbeformulatedintermsofdj,m(i,j)k,andm(i,j)fnasfollows. cLin(a)=X(i,j)2Ldjd(i,j)a(i,j)+Xfn2FXk2fnX(i,j)2Lm(i,j)kd(i,j)a(i,j))]TJ /F8 11.955 Tf 21.95 8.09 Td[(1 jPfnintjX(i,j)2Lm(i,j)fnd(i,j)a(i,j) TheobjectivefunctioncLincanbeformulatedasalinearfunctionofausingthevectorsd,[djd(i,j)](i,j)2L,mk,[m(i,j)k](i,j)2L,mfn,[m(i,j)fn](i,j)2L,andf,d+Pfn2FPk2Pfnintjmk)]TJ /F14 11.955 Tf 11.95 0 Td[(mfnjasfollows. cLin(a)=da+Xfn2FXk2Pfnintjmka)]TJ /F14 11.955 Tf 11.95 0 Td[(mfnaj=da+Xfn2FXk2Pfnintjmk)]TJ /F14 11.955 Tf 11.95 0 Td[(mfnja=fa (4) 115

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Inthefollowing,thefeasibleregionofaischaracterizedbasedonthefeasibleregionofr,i.e.,rminrrmax. LetSinjabethesetofnodesinSiathatareclosertonodeithantonodej.NotethatthevalueofSinjavarieswiththenetworktopology.Therefore,Sinjadependsonr.ThisisdenotedbySinja(r).ThenodesinSinja(rmax)arethenodesthatforsomefeasibler(i.e.,rminrrmax)arehiddennodesoflink(i,j),i.e.,thenodesthatdonotbelongtoSinja(rmax)arenothiddennodesoflink(i,j)foranyfeasiblevalueofr.WerefertothenodesinSinja(rmax)asthepotential-hiddennodesoflink(i,j). ThefeasibleregionoftheLin-NTC-APcanbeformulatedintermsofthefollowingthreeconstraints,whicharedenedintermsofjSinjajr,jSinja(r)j. Constraint1. a(i,j)Sinjarmin8(i,j)2L Constraint2. a(i,j)Sinjarmax8(i,j)2L Constraint3. a(i,j)>a(h,k))]TJ /F12 11.955 Tf 11.96 10.16 Td[(ShnkanSinjarmaxifSinha\Sinja\Shnkarmax>Shnia\Sinja\Shnkarmax8(i,j),(h,k)2L,(i,j)6=(h,k) Constraints 1 and 2 guaranteethata(i,j)isnotlowerandgreaterthanitsminimumandmaximumpossiblevaluesrespectively.Thevalueofa(i,j)representsthenumberofactivenodesthatcoverianddonotcoverj(i.e.,activenodesthatiisabletolistentoandthatarehiddenfromj).Thisnumbercannotbelower/greaterthanthenumberofactivenodesthatareclosertoithantoj,andthatcoveriatminimum/maximumTP(i.e.,jSinjajrminandjSinjajrmaxrespectively). 116

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Constraint 3 guaranteesthatwhentwodifferentlinks(e.g.,(i,j)and(h,k))sharepotential-hiddennodes(i.e.,Sinja\Shnkarmax6=;),thelinkthathas,atmaximumTP,thehighestnumberofsharedpotential-hiddennodesclosesttoitssourcenodealwayshasahighernumberofhiddennodes17.WhentheTPsofthesharedpotential-hiddennodesarebeingincreased,theycoverrstthelink'ssourcenodethatisclosesttothem.Therefore,thelinkwiththehighestnumberofsharedpotential-hiddennodesclosertoitssourcenodealwayshashighernumberofhiddennodes.ThisisshownintheexampleofFigure 4-1 .InFigure 4-1A ,twolinks(i.e.,(i,j)and(h,k))andtheirpotential-hiddennodesareshownwiththeircorrespondingmaximumTPs18.Thepotential-hiddennodesoflink(i,j)arenodes1,2,...,5(i.e.,Sinja(rmax)=f1,2,...,5g).Thepotential-hiddennodesof(h,k)arenodes1,2,...,7(i.e.,Shnka(rmax)=f1,2,...,7g).Thepotential-hiddennodessharedby(i,j)and(h,k)are1,2,...,5(i.e.,Sinja\Shnkarmax=f1,2,...,5g).WhenevertheTPofanyofthesenodesisbeingincreased,thesourcenodeoflink(h,k)(i.e.,nodeh)isnevercoveredafterthesourcenodeoflink(i,j)(i.e.,nodei)hasbeencovered.Therefore,link(i,j)alwayshasasmanyhiddennodesaslink(h,k)atleast19.Thepotential-hiddennodesoflink(h,k)whicharenotpotential-hiddennodesoflink(i,j)(i.e.,thenodesinShnka(rmax)nSinja(rmax)=f6,7g)areabletoincreasethenumberofhiddennodesof(h,k)(i.e.,ShanSka)withoutincreasingthenumberofhiddennodesof(i,j)(i.e.,SianSja).Therefore,inordertoaccountforthesenodes,thefactorShnkanSinjarmaxisintroducedinConstraint 3 bysubtractingitfromthenumberofhiddennodesof(h,k).Forexample,nodes6and7inFigure 4-1 arepotential-hiddennodesof 17ItisbeingassumedthatShnkanSinjarmax=0.ThegeneralcaseofShnkanSinjarmax0isconsiderednext.18Itisassumedthatthenodes1,2,...,7inFigure 4-1 areactive.19Nodes6and7havenotbeenconsideredyet,i.e.,itisbeingassumedthatShnkanSinjarmax=0. 117

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(h,k)butnotof(i,j)(i.e.,Shnka(rmax)nSinja(rmax)=f6,7g),sotheyareabletoincreasethenumberofhiddennodesof(h,k)withoutincreasingthenumberofhiddennodesof(i,j). AMaximumTPs(i.e.,rmax) BPotential-hiddennodesoflinks(i,j)and(h,k)atr=rmax Figure4-1. Anexampleofpotentialhiddennodes AccordingtoEq. 4 andConstraints 1 2 ,and 3 ,theLin-NTC-APcanbeformulatedasalinearintegerprogramasfollows,whereAisthefeasibleregiondeterminedbyConstraints 1 2 ,and 3 minimizefasubjecttoa2A(4) ThisconcludestheproofofTheorem 4.3 Theorem4.4. TheoptimalsolutionsetoftheLin-NTC-APandtheNashequilibriasetoftheLin-NTC-APgameareequivalent. TheproofofTheorem 4.4 isasfollows.Letaoptbeanoptimalsolution.AssumethataoptisnotaNashequilibrium.Therefore,thereexistssomeplayerfnandastrategyansuchthatn(an,a)]TJ /F7 7.97 Tf 6.58 0 Td[(n)
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andcLin(a)isanOPF,thepreviousresultimpliesthatcLin(an,a)]TJ /F7 7.97 Tf 6.58 0 Td[(n)
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abletoreachthemaximumNTCwithinaboundederror.Theerror'sboundcanbecharacterizedbasedonthefollowingobservation. Anactivenodethatisahiddennodeofoneormorelinksofaowcontributestothecontentionexperiencedbytheow.WhentheTPofanactivenodepartiallycoversaow,thenodebecomesanactivehiddennodeoftheow'slinksthatoriginatewithintheTPandterminateoutsidetheTP.ThisisshowninFigure 4-2 inwhichlink(i,j)originateswithinh'sTP(i.e.,iiscoveredbyh'sTP)andterminatesoutsideh'sTP(i.e.,jisnotcoveredbyh'sTP).Giventhathisahiddennodeofonefn'slinks(i.e.,(i,j)),hcontributestothecontentionexperiencedbythelink,andasconsequence,itcontributestothecontentionexperiencedbyfn.AccordingtoEq. 4 andEq. 4 ,hcontributesanadditivefactorofd(i,j)andofd(i,j) jPfnintjtofn'stotalcontentionandmeancontentionrespectively.Thecontributionofhtothecontentionvariationinf1,denotedbyfnV,canbepositiveornegativedependingonthenetworktopology.Forexample,inFigure 4-2 ,hincreasesfn'scontentionvariationbecausewithouth,allthenodesinPfnintexperiencethesamecontention,whilewithh,nodej'scontentionisincreasedbyd(i,j)whiletheothernodes'contentionsremainthesame.Therefore,withouth,fn'scontentionvariationiszero(i.e.,fnV=0),whilewithh,fn'scontentionisgreaterthanzero(i.e.,fnV>0). Figure4-2. Hidden-nodeexample:nodehisahiddennodeoflink(i,j),i.e.,jcannotlistentohwhileicanlistentoit LetaoptLinandaoptNTCbeoptimalsolutionstotheLin-NTC-APandNTC-APrespectively.LetcNTC(a)betheobjectivefunctionoftheNTC-APasformulatedbyEq. 4 (i.e., 120

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cNTC(a),Pfn2Fmaxfcj(r):j2Pfnintg).Letdmaxbethemaximumlinkdegree(i.e.,dmax,maxfd(i,j):(i,j)2Lg). Theorem4.5. ThedifferencebetweenthevaluesoftheNTC-APobjectivefunctionevaluatedataoptLinandaoptNTCisupper-boundedasfollows. cNTC(aoptLin))]TJ /F5 11.955 Tf 11.96 0 Td[(cNTC(aoptNTC)dmaxjFj)]TJ /F8 11.955 Tf 31.03 0 Td[(2 2 SeeAPPENDIX G fortheproofofTheorem 4.5 4.5SimulationResults TheperformanceevaluationoftheWiMAX-Mesh-NTCalgorithmwasperformedbymeansofsimulationusingthesimulatorproposedinChapter 5 .ThisevaluationisgivenintermsofthethroughputerrordenotedbyTanddenedasfollows.LetT(r)bethehighesttotalthroughputsupportedbytheowsinFunderpowersettingr(i.e.,T(r),Pfn2Ffnmax(r)).LetroptbethefeasiblesetoftransmissionrangesthatmaximizesT.Thethroughputerrorofthetopologyinducedbyr,wherercanbeanyfeasiblesetoftransmissionranges,isgivenbyEq. 4 T(r)=T(ropt))]TJ /F9 11.955 Tf 11.96 0 Td[(T(r) T(ropt)(4) TheWiMAX-Mesh-NTCalgorithmwascomparedwiththeHSRA,MinPower,andMaxPoweralgorithms.TheHSRAalgorithm(Chapter 3 )isaheuristicandcentralizedalgorithmthataimstondasetoftransmissionrangesthatmaximizesT.TheMinPoweralgorithmaimstomaximizethespatialreusebysettingthenodes'transmissionrangesattheminimumvaluesthatdonotdisconnectanyoftheowsinF.TheMaxPoweralgorithmsetsallthenodes'transmissionrangesattheirmaximumvalues. 121

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Thesimulationwasconguredasfollows20.ThelinkschedulingpolicywasGM-RBDS(Chapter 2 ).Theowpathswereobtainedusingthemin-hoproutingalgorithm.Thenodeswereuniformlydistributedinasquaresuchthatthenodedensitywasalwayskeptat15nodesperareaunit.Therewereatotalof20nodes.Themaximumtransmissionrangeforallthenodeswassetat0.3(i.e.,rmax=[0.3]i2N).Theconnectivityoftheowswiththenodes'transmissionrangessetatrmaxwasveriedbeforeexecutingthemin-hoprouting,WiMAX-Mesh-NTC,HSRA,andMinPoweralgorithms.Thesourceanddestinationofeveryowwereuniformlydistributedacrossallthenodesinthenetwork.Themin-hopalgorithmcalculatedtheowpathswhenthetransmissionrangesweresetatrmax. Eachofthetopology-controlalgorithms(i.e.,WiMAX-Mesh-NTC,HSRA,MinPower,andMaxPower)calculatedthesetoftransmissionrangesfor300differentnetworksconguredasexplainedpreviously.Thesenetworksweredividedintothreegroupsof100networkseachdependingonthenumberofowsinthenetworks.Intherst,second,andthirdgroupsthenetworkshad10,15,and20owsrespectively.Thealgorithmscalculatedasetoftransmissionrangesforeachofthe300networks,andthethroughputerrorTwascalculatedforeachofthesetoftransmissionranges21.Figure 4-3 showstheratiooftopologiescalculatedbythealgorithmsthathadaTlessthanorequalto4%.Therefore,Figure 4-3 showstheabilityofthealgorithmstoreachthemaximumpossibletotalthroughput(i.e.,T(ropt)). AccordingtoFigure 4-3 ,theWiMAX-Mesh-NTCalgorithmalwaysoutperformstheMinPowerandMaxPoweralgorithms.Whenthereare10ows(Figure 4-3A ),44.3%of 20ThisisthesamecongurationusedinChapter 3 .21Tondropt,whichisnecessaryforcalculatingT(Eq. 4 ),theBranchAndReduceOptimization(BARON)Solver[ 4 ]wasused.BARONisasystemforsolvingnon-convexoptimizationproblemstoglobaloptimality. 122

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A10ows B15ows C20ows Figure4-3. PercentageoftopologieswhoseTiswithin4% thetopologiescalculatedbyWiMAX-Mesh-NTChaveaTofatmost4%while37.9%and32.8%ofthetopologiescalculatedbyMinPowerandMaxPowerhavesuchT.Whenthereare15ows(Figure 4-3B ),thesepercentagesare48.5,39.6,and43.1respectively,andwhenthereare20ows(Figure 4-3C ),theyare44.6,39.3,and43.3.Therefore,inallthecasestheWiMAX-Mesh-NTCalgorithmwasabletondmoreeffectivelyatopologythatmaximizesthetotalthroughputT. WhentheWiMAX-Mesh-NTCandHSRAalgorithmsarecompared,theowdensityneedstobeconsidered.Lettheowdensitybetheratioamongthenumberofowsandthenumberofnodes.GiventhatinFigure 4-3 ,thenumberofnodesdoesnotchangewhilethenumberowsincreasesfrom10to20,theowdensityincreases.Figure 4-3A showsthatthealgorithmshavethesameperformancewhentheowdensityislow.Figure 4-3B /Figure 4-3C showsthatWiMAX-Mesh-NTChasworseperformancethanHSRAwhentheowdensityismedium/high.TheintuitionbehindthisbehavioristhattheprobabilitythatanyofthecasesshowninFigure G-1 takesplaceincreases 123

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withtheowdensity,andsuchcasesaffecttheperformanceofWiMAX-Mesh-NTC(Theorem 4.5 ).ThisisnotthecaseforHSRAbecauseHSRAisacentralizedalgorithm,soithasaglobalviewofthenetwork.However,theadvantageofWiMAX-Mesh-NTCisthatitisdistributedwhileHSRAisnot.WiMAX-Mesh-NTCrequiresknowledgeabouttheowsinFnwhileHSRArequiresknowledgeaboutalltheowsinF.Therefore,WiMAX-Mesh-NTCismoreamenableforimplementation. 4.6Summary Anewframeworkforthedevelopmentofdistributedalgorithmsthatmaximizethetotalend-to-endthroughputinWMNswasproposed.Itisbasedonthelink-schedulingpolicy'sstabilityregion,anditconsistsofapotentialgameinwhichagivensetofowsactasplayersthatcollaboratetomaximizethepacketratestheycansupportwhileguaranteeingstability.Basedontheproposedframework,theWiMAX-Mesh-NTCalgorithmwasdeveloped.WiMAX-Mesh-NTCaimstoreducethemaximumcontentionexperiencedbyeveryow.TheconvergenceofWiMAX-Mesh-NTCwascharacterizedbymeansoftheNashequilibrium,andaperformanceboundwascalculatedbyconsideringallthepossibleworst-casescenarios.Finally,theWiMAX-Mesh-NTCperformancewascomparedbymeansofsimulationwiththeperformanceofothertopology-controlalgorithms(i.e.,HSRA,MinPower,MaxPower).ItwasshownthatWiMAX-Mesh-NTCalwaysoutperformsMinPowerandMaxPower,andthatitoutper-formsHSRAwhentheowdensityismedium. 124

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CHAPTER5WORLDWIDE-INTEROPERABILITY-FOR-MICROWAVE-ACCESSRBDSSIMULATOR(WIMAX-RBDS-SIM):ANOPTIMIZED-NETWORK-ENGINEERING-TOOLS(OPNET)SIMULATIONFRAMEWORKFORWIRELESSMESHNETWORKS TheInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16standard[ 1 ]denesthephysicalandmedium-access-control(MAC)layersforwirelessmeshnetworks(WMN).Althoughthestandardprovidesthearchitecturenecessaryfortheimplementationofanyschedulingpolicy,nopolicyisspeciedbythestandard.TheMAClayerisbasedontime-divisionmultipleaccess(TDMA),wheretimeisdividedintoframesthataresimultaneouslyusedbynon-interferinglinks.Eachframeisdividedintoacontrolsubframeandadatasubframe.Schedulingmessagesareexchangedincontrolsubframescontainingschedulinginformationsuchasrequestsforuseofdatasubframesandgrantsofdatasubframes.Anelectionalgorithmdeterminesthesubsetofnodesthataccessthecontrolsubframesuchthatthetransmissionsofschedulingmessagesdonotinterferewitheachother.Theschedulingmessages,controlsubframes,andtheelectionalgorithmenabletheimplementationofanygivenschedulingpolicy. ThestandardalsodenestheproceduresthatnodesneedtofollowforjoininganIEEE802.16WMNandforestablishingphysicallinkswithothernodes.Theseproceduresneedtobenishedbeforeanodebeginsschedulingdatapackets. Inthispaper,anoptimized-network-engineering-tools(OPNET)[ 3 ]simulationframeworkisproposed.Itiscalledworldwide-interoperability-for-microwave-ac-cessreservation-based-distributed-schedulingsimulator(WiMAX-RBDS-Sim).Itspurposeistoenabletheevaluationofdistributedschedulingpoliciesandthelink-establishmentprocessinIEEE802.16WMNs.Itimplementstheschedulingmessages,electionalgorithm,andlink-establishmentandprovidesaninterfacefortheintegrationofdifferentdistributedschedulingalgorithms.Tothebestofourknowledge,WiMAX-RBDS-SimistherstOPNET-basedsimulatorforIEEE802.16 125

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WMNswithdistributedscheduling.InSection 5.1 ,therelatedworkandcontributionsarediscussed.AnoverviewofIEEE802.16WMNsispresentedinSection 5.2 .InSection 5.3 ,thearchitectureofWiMAX-RBDS-SimisexplainedindetailusingtheproposedalgorithmSliced-GM-RBDSasanexample.InSection 5.4 ,theOPNETimplementationoftheWiMAX-RBDS-Simarchitectureisdescribed.InSection 5.5 ,thesimulationresultsobtainedwithWiMAX-RBDS-Simfortheproposedalgorithmareprovidedanddiscussed.TheperformanceofWiMAX-RBDS-SimisevaluatedinSection 5.6 .Finally,asummaryofthechapterispresentedinSection 5.7 5.1RelatedWork Theresearchcommunityhasrecentlyinvestigatedthechallengesofcentralizedanddistributedschedulingin802.16WMNs.Inthispaperwefocusonthedistributedschedulingproblem1. Distributed-schedulingresearchcanbeclassiedintotwogroups:election-algo-rithm-basedpoliciesandRBDSpolicies.In[ 13 23 ],theperformanceoftheelectionalgorithmisevaluatedtheoreticallyandbymeansofsimulation.In[ 6 40 78 87 ],theperformanceisimprovedintermsofnumberofcollisionsandcontrol-subframeutilizationbydynamicallyadjustingtheparametersofthealgorithm.In[ 31 85 86 ],aquality-of-service(QoS)differentiationschemeisproposedbasedontheadjustmentoftheelection-algorithmparameterssothatanodetransmitsschedulingmessagesmoreoftenwhenitsdatatransmissionshavehigherpriority.In[ 47 ],acongestioncontrolmechanismisproposedinwhichoneoftheelection-algorithmparametersisadjustedbyeachnodeaccordingtothecongestionmeasuredlocally.Theeffectofthisadjustmentisthatanodewaitslongerforsendingschedulingmessageswhenthelocalcongestionishigh. 1Formoreinformationoncentralizedscheduling,pleasereferto[ 16 52 ]. 126

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Thesecondgroupfordistributed-schedulingresearchconsidersRBDSpolicies.Thesepoliciesarebasedonthereservationoffuturedatasubframessothatnotwointerferinglinksareassignedthesamedatasubframes.In[ 21 ],thenodesprioritizethetrafcowsgoingthroughthemaccordingtotheweightsofeachoftheows,andthenumberofdatasubframesreservedforeachowisproportionaltoitsweight.Thedatasubframeseligibleforreservationareniteinthesensethatonlytheframesthatareacertainnumberofframesawayfromthecurrentonecanbeincludedinthereservations.Thissetofdatasubframesisknownastheschedulehorizon.In[ 79 ],data-subframeutilizationisimprovedbyallowingeveryreservationtocontainnon-contiguousdatasubframesthatareallassignedtothesamelink.Inthisway,thenumberofwasteddatasubframes(i.e.,datasubframesthatareneverassignedtoanylink)isreduced.In[ 43 ],thereservationsforeachlinkarecalculatedbasedonthestatisticalcharacteristicsofthedatatrafcgeneratedatthesourcenodeinordertoreducetheschedulingoverhead.In[ 24 ],anend-to-endreservationschemeisproposedforconstant-bit-rateowssuchasvoice-over-IP.Othersimplerreservation-baseddistributedschedulingschemesareproposedin[ 18 42 48 ],andabroadcastingalgorithmthataimstominimizethenumberofreservationsperbroadcastisproposedin[ 88 ]. 5.1.1Distributed-SchedulingSimulators TheIEEE802.16standarddenestwooperatingmodesformetropolitanaccessnetworks.Thesearethepoint-to-multipoint(PMP)andmeshmodes.Intheliterature,researchhasbeenmostlyfocusedonsimulatorsforthePMPmode[ 8 55 64 ].Tothebestofourknowledge,therearetwosimulatorsforIEEE802.16WMNswithdistributedschedulingthathavebeendiscussedintheliterature.TheseareWiMsh[ 22 ],whichisapatchfortheNetworkSimulator2(ns-2)[ 2 ],andtheNationalChiaoTungUniversitynetworksimulator(NCTUns)[ 32 ],whichisanetworksimulatorandemulatorthatincludesasimulationtoolforIEEE802.16WMNswithdistributedscheduling. 127

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WiMAX-RBDS-Simisapacket-levelsimulationmodelfortheOPNETdiscreteeventsimulator[ 3 ].ItisanimplementationoftheMACCommonPartSublayergiveninthestandard[ 1 ].ThearchitectureofWiMAX-RBDS-SimwasconceivedforthesimulationoftheRBDSpoliciesdenedinChapter 2 whicharebasedontheinput/outputqueueconcepts. Although,WiMAX-RBDS-SimandWiMshhavesimilarfunctionalities,therearedifferencesbetweenthesetwosimulatorsintermsofarchitectureandnetworktopology.Thesedifferencesareasfollows. ThearchitectureofWiMAX-RBDS-Simisbasedonatheoreticalframeworkthatfocusesonastabilityanalysisforwirelessmulti-hopnetworksunderanyRBDSpolicy.WiMAX-RBDS-Simprovidesthemeansforobtainingsimulationresultsforsuchpoliciesinordertocomparethemwiththetheoreticalresultsobtainedforthesamepoliciesunderthetheoreticalframework. WiMAX-RBDS-Simimplementsthelinkestablishmentalgorithm,whichisnotimplementedinWiMsh,andprovidestheframeworkfortheimplementationandevaluationofimprovementstothelinkestablishmentprocess. InWiMAX-RBDS-Sim,thelinksestablishedbetweenthenodesdependonthelocationofthenodes,thephysicalchannelmodelselectedforthesimulation,andthenodes'antennas.Therefore,thetopologyofthenetworkiscalculatedbasedonthoseparameters.InWiMsh,thelinksaregivenasasimulationparameterbyspecifyingthenetworktopologyfromasetofpredenedtopologies. 5.1.2Contributions Inthispaper,thedesignandimplementationofWiMAX-RBDS-Simispresented.WiMAX-RBDS-Simhasthefollowingcharacteristics: Itprovidesaframeworkfortheimplementationandevaluationofalgorithmsthatadjusttheparametersoftheelectionalgorithmforeachnodedynamically.WiMAX-RBDS-Simallowstheevaluationofthesealgorithmsintermsofschedulingdelay,control-subframeutilization,andnumbercollisionsincontrolsubframes. 128

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Itprovidesaframeworkinwhichschedulingpoliciescanbeimplementedandevaluatedintermsofcapacity(i.e.,thesetofmaximumdatainputratesthatthepolicycanhandlewhileguaranteeingthestabilityofallthequeuesinthenetwork),throughput,anddelay. Itprovidesaframeworkfortheimplementationandevaluationoflink-establishmentalgorithms.Thisevaluationisperformedintermsoflink-establishmentdelay(i.e.,thetimerequiredbytwonodestoestablishlinksbetweenthem). 5.2IEEE802.16WMNOverview Inan802.16WMNtherearetwotypesofnodes:basestations(BS)andsubscriberstations(SS).The802.16WMNisconnectedtoexternalnetworkssuchastheInternetthroughBSs.Unidirectionallinkscanbeestablishedbetweenanyofthesenodes,andtheinformationistransmittedonahop-by-hopbasis.Eachlinkisuniquelyidentiedinthenetworkwithboththenodeidentication(ID)andthelinkID.ThelinkIDidentiesthelinkamongthesetofoutgoinglinksofanode,andthenodeIDidentiesthenodefromwhichthelinkoriginates.Thesystemaccessfollowsaframe-basedapproachwhichisshowninFigure 5-1 .Eachchannelisdividedintimeintoseriesofframes.InFigure 5-1 ,eachoftheseseriesconsistsofeightframes(i.e.,fromframentoframen+7).Aframeisfurtherdividedintotwosubframes:acontrolsubframeandadatasubframe.Thecontrolsubframesareusedforestablishinglinksandschedulingdata.Thedatasubframesareusedfordatatransmission.Controlsubframesaredividedintoslots,whichareusedfortransmittingschedulingpacketsandlink-establishmentpackets.Thecontrolsubframeoftherstframeofaseries(e.g.,frameninFigure 5-1 )isusedfortransmittinglink-establishmentpackets2.Thecontrolsubframesofthefollowingframesoftheseries(e.g.,framesn+1,n+2,...,n+7inFigure 5-1 )areusedfortransmittingschedulingpacketsonly.Datasubframesaredividedintoslotswhichareusedfortransmittingdatapackets. 2Thesepacketscarrynetwork-congurationinformationtoo,butthisisoutofthescopeofthispaper. 129

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Figure5-1. FramestructureoftheInstitute-of-Electrical-and-Electronics-Engineers(IEEE)802.16meshmode Nodesaccessthecontrolslots(i.e.,theslotsincontrolsubframes)usingtheIEEE802.16electionalgorithm[ 13 ].Theelectionalgorithmguaranteesthatwheneveranodetransmits,alltheothernodesinits2-hopneighborhoodstayquiet,wherethe2-hopneighborhoodofanodeconsistsofallthenodesthatareatmost2hopsawayfromit.Inthisway,collisionsincontrolslotsareavoidedandlink-establishmentandschedulingpacketsaresuccessfullyreceived. Nodesaccessthedataslots(i.e.,slotsindatasubframes)usingaschedulingpolicythatisimplementedbyexchangingschedulingpacketsinthecontrolsubframe.Thegoaloftheschedulingpolicyistoavoidunuseddataslotsandcollisionsofdatapacketswhileoptimizingoneormoremetricsofthenetworksuchasitscapacity. 5.2.1Data-SlotScheduling TheschedulingpolicyforaccessingdataslotsisnotspeciedintheIEEE802.16standard.Thestandardonlydenesthemesh-distributed-scheduling(MSH-DSCH)message,whichcontainsschedulinginformationandiscarriedbyschedulingpackets,sothatdifferentdistributedschedulingpoliciescanbeimplemented.TheinformationcontainedinanMSH-DSCHmessageisorganizedininformationelements(IE)asfollows. 130

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RequestIE:Anodecanmakeseveralrequestssimultaneouslyonaone-re-quest-per-linkbasis.TheinformationincludedinarequestisthelinkID,numberofrequesteddataslotsperdatasubframe,andnumberofrequesteddatasubframes.Thenumberofdatasubframesmaybeinnitesothatstreamsofinformationcanbetransmittedinthelink. AvailabilityIE:Anodenotiesits1-hopneighbors3ofthedataslotsithasavailableforreservation.ThisIEspeciesasetofavailabledataslotswithastartframenumber,numberofframes,startdataslotnumber,numberofdataslots,direction(i.e.,theminislotsmaybeavailablefortransmission,reception,orbothofthem),andthechannelstheavailabledataslotsbelongto. GrantIE:ThisIEincludesthesameparametersspeciedfortheavailabilityIE.However,theseareusedforspecifyingasetofdataslotsthathavebeenassignedtoalink. Theschedulingprocedurefollowsathree-wayhandshake.First,anodesendsanMSH-DSCHmessagetooneofits1-hopneighborsrequestingasetofdataslots.Inthemessage,thenodealsoincludesthesetofdataslotsthatithasavailableforreservation.The1-hopneighborgrantstherequestbyreplyingwithanotherMSH-DSCHmessagethatspeciesasetofdataslotsthatsatisestheavailabilityofdataslotsatbothnodes.Finally,therstnodeconrmsthereservationofsuchsetofdataslotsbyechoingthegrantinanotherMSH-DSCHmessage.Byfollowingthisthree-wayhandshake,the1-hopneighborsofthetwonodesbecomeawareofthedataslotreservationsothatthedataslotsinthegrantbecomeunavailableforthem. 5.2.2LinkEstablishment Inordertoexchangedatapacketsinbothdirections,two1-hopneighborsneedtoestablishtwolinks(i.e.,onelinkineachdirection).Thisisachievedbymeansofathree-wayhandshakeperformedwithmeshnetworkconguration(MSH-NCFG)messageswhicharetransmittedinlink-establishmentpackets.ThehandshakeisinitiatedbythenodewithlowestID.First,thelowest-IDnodesendsachallengetoits 3Twonodesare1-hopneighborsifandonlyiftheybothcanreceivepacketsfromeachotherdirectly. 131

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new1-hopneighbor.Second,the1-hopneighborreplieswithachallenge-responseandtheIDforitsoutgoinglink(i.e.,theincominglinkofthelowest-IDnode).Finally,thelowest-IDnodereplieswithanacceptandtheIDforitsoutgoinglink(i.e.,theincominglinkofthe1-hopneighbor).Thelink-establishmentinformationthatthenodesexchangeduringthehandshake(i.e.,challenge,challenge-response,accept,andlinkIDs)iswrittenonlink-establishmentIEs,andtheseareattachedtoMSH-NCFGmessages. 5.3WiMAX-RBDS-SimArchitecture TheWiMAX-RBDS-SimarchitectureisshowninFigure 5-2 .Thearchitectureconsistsofthephysical,MAC,andlogical-linklayers.Thenodesallsharethesamearchitecture.Thereare10radiochannelsinthephysicallayer.TheMAClayerimplementstheoptionfordistributedschedulinginwirelessmeshnetworksoftheIEEE802.16standard[ 1 ]4.Itconsistsoffourmainfunctionalmodules.Thesearethemain-processingmodule,thepacket-classiermodule,theoutput-queueandchannel-classiermodule,andthetime-synchronizationmodule.Thelogical-linklayeristhenode'ssourceandsinkofdatapackets.Itdoesnotimplementanylogical-linkprotocol. Eachoftheradiochannelscanbesetatthecongurationmodesspeciedinthestandard[ 1 ].Thesemodesdeterminetheframelength,channelbandwidths,andforward-error-correctionandmodulationschemes.Theyarespeciedassimulationparameters. Inordertomakecomparisonsbetweentheoreticalandsimulationresults,theWiMAX-RBDS-Simarchitecturefollowsthe1-hoptrafcmodel(i.e.,thedestinationnodeofeverypacketisalways1hopeawayfromthesourcenode).Thelogical-linklayer 4Specically,theMAClayerimplementstheMACCommonPartSublayerforIEEE802.16wirelessmeshnetworkswithdistributedscheduling. 132

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Figure5-2. Architectureoftheworldwide-interoperability-for-microwave-accessreservation-based-distributed-schedulingsimulator(WiMAX-RBDS-Sim) canbesetatsomedata-packetgenerationrate,andthedata-packetsourcegeneratesdata-packetsatsuchrateforeachofthenode's1-hopneighbors. IntheMAClayer,thepacket-classiermoduleclassiesallthereceivedpacketsintoscheduling,link-establishment,anddatapacketsandforwardsthemtothemain-processingmodule.Inthismodule,thereisoneclassierperradiochannel.Theoutput-queueandchannel-classiermodulequeuepacketsthathavealreadybeenscheduledfortransmissionandforwardthemtotherightradiochannelatthetransmissiontimeindicatedbytheirrespectiveschedules.Inthismodule,thereisoneoutput-queueperradiochannel. Themain-processingmoduleperformsthefollowingtasks: establishesincomingandoutgoinglinkswitheachofthenode's1-hopneighbors(i.e.,thelinksintheWiMAX-RBDS-Simarchitectureareunidirectional) 133

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queuesunscheduleddatapacketsandforwardsthemtotheoutput-queueandchannel-classiermoduleoncetheyarescheduled schedulesfortransmissiondata-packetsgeneratedatthelogical-linklayer forwardstothelogical-linklayerthereceiveddata-packetsthataredestinedtothenode Themain-processingmoduleconsistsofseveralprocessorsthatarededicatedtospecictasksandasetofqueueswhereunscheduleddatapacketsarestoredtemporarily.Therearethreetypesofprocessors:mainprocessor,scheduler,andlinkestablisher.Thereisonlyonemainprocessorinthemodule.Themainprocessorcontrolstheowofpacketswithinthemoduleandcreatesanddestroysschedulerandlink-establisherprocesses.Theschedulersarecreatedanddestroyedattheonsetandattheendofthesimulationrespectively.Thereisoneschedulerperchannel.Aschedulergeneratesandprocessesforitsassignedchanneltherequest,availability,andgrantIEstransmittedandreceivedbythenoderespectively.Thelink-establisherprocessesarecreatedanddestroyeddynamicallybythemainprocessorduringthesimulation.Alinkestablisheriscreatedwhena1-hopneighborisdetectedforthersttime.Thelink-establishergeneratesanddestroysthelink-establishmentIEstransmittedandreceivedbythenodeforestablishingtheincomingandoutgoinglinkswiththedetected1-hopneighbor.Themainprocessordestroysthelink-scheduleroncetheselinkshavebeenestablished. Theowofpacketswithinthemain-processingmoduleconsistsoftwodirections.Thesecorrespondtothetransmissionandreceptionofpackets.Inthedirectionoftransmission,datapacketsaregeneratedbythelogical-linklayeranddeliveredtothemainprocessor.Themainprocessorstorestheminthemodule'squeues.Thesearetheinputqueues.Thereisoneinputqueueperoutgoinglink.Dependingonthelengthsoftheinput-queues,themainprocessorinvokestheschedulerswhichgeneratetheschedulingIEs(i.e.,requestIE,availabilityIE,andgrantIE).WiththeseIEsandbasedontheelectionalgorithm,themainprocessorgeneratesschedulingpacketsand 134

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calculatestheirschedules.Theschedulingpacketsareforwardedtotheoutputqueuesalongwiththeirschedules.Whentheschedulersarenishedschedulinganypacketsintheinputqueues,themainprocessorremovesthosepacketsfromtheinputqueuesandforwardsthemalongwiththeirschedulestotheircorrespondingoutputqueues.Also,themainprocessorgenerateslink-establishmentpacketswiththelink-establishmentIEsgeneratedbythelink-establishersandcalculatestheirschedulesbasedontheelectionalgorithm.Thesepacketsarealsoforwardedalongwiththeirschedulestotheircorrespondingoutputqueues.Inthedirectionofreception,themainprocessorrstcheckswhetherthereceivedpacketsaredirectedtothenodeitbelongsto.Ifthatisthecase,themainprocessorforwardsdatapacketstothelogical-linklayer,readsIEsfromschedulingandlink-establishmentpackets,anddestroysthesetypesofpackets(i.e.,schedulingandlink-establishmentpackets).SchedulingIEsarepassedtotheircorrespondingschedulers,andlink-establishmentIEsarepassedtotheircorrespondinglink-establishers. 5.3.1TheLink-Establishers Thelink-establisherprocessmodelisshowninFigure 5-3 .Themodelisastatetransitiondiagraminwhichactionsaretakenateverystate,andstatetransitionstakeplacewhenthemainprocessorinvokesthelink-establisher.Thelink-establishercommunicateswiththemainprocessorbywritingonandreadingfromabufferthatcanbeaccessedbybothprocessesonly. Thelink-establisherisabletoperformtwodifferentrolespresentinthethree-wayhandshake.Thesearethechallengerandthereplier.Thechallengercorrespondstothelink-establisheratthe1-hopneighborwithlowerIDwhichinitiatesthehandshakebysendingachallenge.Therepliercorrespondstothelink-establisheratthe1-hopneighborwithhigherIDwhichrepliestothechallengebysendingachallenge-response.Intheinitialstate,namedinit(Figure 5-3 ),theprocessdecidesitsrole.Thechallengerjumpsfrominittoprep-challenge,andthereplierjumpsfrominittoprep-response. 135

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Uponarrivaltoprep-challenge,thelink-establisherpreparesthelink-establishmentIEwiththechallengethatinitiatesthehandshakeandwaitsatstatewaituntilthemainprocessorisreadytoprocessthelink-establishmentIE.Whenthemainprocessorisready,thelink-establisherjumpstotx-challenge,writesthelink-establishmentIEonthebuffer,andsetsatimerfortheretransmissionofthelink-establishmentIEincaseitreceivesnoresponsefromthe1-hopneighbor.Then,thelink-establisherjumpstowait-responsewhereitwaitsforaresponse.Iftheresponseisnottheexpectedchallenge-response,itjumpstoabortwhereitabortsthehandshakeanddestroysitself.Inthiscase,themainprocessorwilllatercreateanotherlink-establisherforestablishingthelinkswiththe1-hopneighbor.Iftheresponseistheexpectedchallenge-response,thelink-establisher,atstateprep-accept,readstheincoming-linkIDassignedbythe1-hopneighbor,andcreatesanotherlink-establishmentIEwithanacceptandoutgoing-linkID.WhentheLink-Establishmentiswrittenonthebufferatstatetx-accept,thelink-establisherjumpstonish,notiesthelogical-linklayerthatanewoutgoinglinkhasbeenestablishedanddestroysitself. Uponarrivaltoprep-response(i.e.,whenthelink-establishertakesthereplierrole),thelink-establishercreatesalink-establishmentIEwithachallenge-responseandoutgoing-linkIDandwaitsatstatewaituntilthemainprocessorisreadytoprocessthelink-establishmentIE.Whenthemainprocessorisready,thelink-establisherjumpstotx-response,writesthelink-establishmentIEonthebuffer,andsetsatimerforaretransmissionincasenoresponseisreceivedfromthe1-hopneighbor.Whentheresponseisreceived,thelink-establishercheckswhetheritisanotherchallenge,theexpectedaccept,oranunexpectedresponse(i.e.,reject,challenge-response).Ifitisanotherchallenge,thelink-establisherassumesthatthepreviouslytransmittedchallenge-responsewasnotreceivedandthelink-establishmentIEisretransmitted.Ifitistheexpectedaccept,thelink-establisherjumpstoget-LinkIDwhereitreadstheincoming-linkIDandjumpstonish.Atstatenish,itnotiesthelogical-linklayerthata 136

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newoutgoinglinkhasbeenestablishedanddestroysitself.Ifitanunexpectedresponse,thelink-establisherjumpstoabort. Figure5-3. Link-establisherprocessmodel 5.3.2TheSchedulers Theschedulersimplementthedistributed-schedulingalgorithmusedforschedulingthedatapacketsstoredinalltheinputqueues.Toillustratetheschedulingprocess,weproposetheSliced-Greedy-Maximal-RBDS(Sliced-GM-RBDS)algorithm,basedontheGM-RBDSpolicyproposedinChapter 2 .TheSliced-GM-RBDSalgorithmusessetsofdataslotsliketheonesshowninFigure 5-4 .Inordertospecifythesesets,thedatasubframesarealldividedorslicedintoagivennumberofdata-slotgroupsofthesamesize.Forexample,inFigure 5-4 ,thereare12dataslotsperframenumberedfrom4to15whicharedividedinto4data-slotgroups.Thesegroupscorrespondtothedataslotsnumbered4-6,7-9,10-12,and13-15.Asetofdataslotsisspeciedwithadata-slotgroupandaframerange.Forexample,ifthegroupcorrespondstodataslotsnumberedfrom7to9andtheframerangeconsistsofframesnumberedfrom11to20,thesetofdataslotsconsistsof30dataslots(i.e.,10framesand3dataslotsperframe).ThissetisshowninFigure 5-4 alongwithotherthreedifferentsets.InWiMAX-RBDS-Sim,there 137

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are256minislotsperframeasspeciedinthestandard[ 1 ],andthesizeofthedata-slotgroupsisasimulationparameter. Figure5-4. Data-slotreservationintheSliced-GM-RBDSalgorithm TheSliced-GM-RBDSalgorithmisbasedonthefollowingschedulingpolicy.Wheneveranodetransmitsaschedulingpacket, foreveryoutgoinglink,requestasetofdataslotsthatcoversthelongestframerangethatcanbeentirelycoveredwithunscheduleddatapackets grantthelongestrequeststhatdonotoverlapwitheachother foreveryrequestandgrantmade,setitsstartframe(i.e.,therstframeoftheframerangecoveredbytherequest/grant)attheearliestpossibleframesuchthatoverlapsareavoided Inthealgorithm,arequestIE(denedinSection 5.2.1 )isalwaysgeneratedalongwithanavailabilityIE.TherequestIEspeciesthesizeofasetofdataslots(i.e.,thenumberofrequesteddataslots).TheavailabilityIEspeciesasetofdataslotswhosesizeisatleastequalthesizeofthesetofdataslotsspeciedintherequestIE.TheavailabilityIE'ssetindicatesallthedataslotsthatthenode's1-hopneighborisallowedtoincludeinthegrantfortherequest.Therefore,thissetshouldnotincludeanyofthedataslotsthatbelongtoanyofthegrantsheardbythenodesendingtherequest, 138

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otherwisethegrantgeneratedatthenode's1-hopneighbormayoverlaponeormoreofsuchgrants,andthenodewillnotbeabletoconrmthegrant.Ontheotherhand,the1-hopneighborisabletogranttherequestifitndsasetofdataslotswithintheavailabilityIE'ssetthatdoesnotoverlapanyofthegrantsthatithasheardandthathasthesizespeciedintherequestIE. TheschedulerprocessmodelisshowninFigure 5-5 .Itconsistsoftwostates.Theseareinitandschedule.Instateinit,theschedulerinitializesitselfaccordingtosimulationparameterssuchasthenumberofdata-slotgroups.Thisinitializationtakesplaceonlyoncewhenthescheduleriscreatedbythemainprocessorattheonsetofthesimulation.Thetasksperformedinstatescheduletakeplaceeverytimetheschedulerisinvokedbythemainprocessor.ThemainprocessorinvokestheschedulereverytimeaschedulingpacketistransmittedorreceivedinordertogenerateandprocessMSH-DSCHmessagesrespectively.ThecontentsofthemessagesaregeneratedandprocessedusingtheSliced-GM-RBDSalgorithm. Figure5-5. Schedulerprocessmodel WhentheschedulerisinvokedforgeneratingthecontentsofanMSH-DSCHthatisgoingtobetransmitted,itgeneratesgrant,request,andavailabilityIEs5. Grant-IEgeneration:Foreverypendingrequest,thesetofoverlappingrequestsisfound.Iftherequestisthelongestinthisset,itisgrantedandalltheotherrequestsinthesetarediscarded(i.e.,theyarenotlongereligibleforanygrant).Thegrantforeachofthegrantedrequestsisassignedasetofdataslotsthatisdeterminedasfollows. 5TheSliced-GM-RBDSalgorithmusestimersfordiscardingrequeststhatexpire. 139

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Theset'sdata-slotgroupistheonespeciedintherequest'savailabilityIE.Thesizeoftheset'sframerangeistheonespeciedintherequestIE,andthenumberoftheset'sstartframeisthemaximumamongtheframenumbersofthefollowingframes:theavailabilityIE'sstartframe,theframefollowingthelastframegrantedinthedata-slotgroupindicatedbytheavailabilityIE,andtheframethatisapre-speciednumberofframesaheadofthecurrentone.Inthisway,itisassuredthatthegrantdoesnotoverlapanyofthegrantsheardbythenodeandthe1-hopneighborthatmadetherequest.Also,itisassuredthatthe1-hopneighborisabletoconrmthegrantbeforethegrant'sstartframebecomesthecurrentframe. RequestandavailabilityIEsgeneration:Foreveryoutgoinglink,ifthereisapre-speciedminimumofunscheduleddatapacketsinitsinput-queue,arequestismadeforasetofdataslotswhoselengthisthelongestthatcanbeentirelycoveredwithunscheduleddatapackets.Theavailabilityofeachrequestisgeneratedaccordingtothefollowingfourconditions. Itdoesnotoverlapanyofthegrantsheardbythenode. Ifthereisadata-slotgroupwhichhasnotbeenincludedinanyoftheavailabilitiesheardbythenodeoranyoftheavailabilitiesthatwerepreviouslycalculatedforotheroutgoinglinks,thatdata-slotgroupisassignedtotheavailabilitybeingcalculated,andtheavailability'sstart-framenumberisthemaximumamongtheframenumbersofthefollowingframes:theframefollowingthelastframegrantedinthedata-slotgroup,theframewhenthenextMSH-DSCHpacketistransmittedbythisnode. Ifthereisnotsuchdata-slotgroup,thedata-slotgroupthathasbeenassignedtothelowestnumberofavailabilitiesamongalltheavailabilitiesheardbythenodeandtheavailabilitiespreviouslycalculatedforotheroutgoinglinksisassignedtotheavailability,andtheavailability'sstart-framenumberisthemaximumamongtheframenumbersofthefollowingframes:theframefollowingthelastframegrantedacrossallthedata-slotgroups,theframewhenthenextMSH-DSCHpacketistransmitted. 140

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Thenumberofthelastframeoftheavailability'sframerangeisalwaysmadeequaltoinnity.Inthisway,thenodethatgrantstherequestisabletouseforthegrantanyoftheframesfollowingtheavailability'sstartframe.Thisispossiblebecausenoneofthedataslotsthatbelongtotheavailability'sdata-slotgroupandtotheframesthatfollowtheavailability'sstartframehasbeengrantedaccordingtotheprevioustwoitems. WhentheschedulerisinvokedforprocessingthegrantIEsofanMSH-DSCHmessagethathasbeenreceived,itperformsthefollowingactions.ForeverygrantIEintheMSH-DSCHmessage,theschedulercheckswhetherthegrantisdirectedtothenode.Ifitis,theschedulerlooksfortheIDoftheoutgoinglinkthatconnectstothe1-hopneighborthatsentthegrant.Then,itschedulesdatapacketsthatareintheinputqueueassignedtotheoutgoinglink.ThesepacketsarescheduledinthesetofdataslotsspeciedinthegrantIE.Finally,itgeneratesagrantIEforconrmingthereceivedgrant.ThisgrantconrmationisacopyofthereceivedgrantIE. 5.4OPNETImplementation TheOPNETimplementationoftheWiMAX-RBDS-SimarchitectureisshowninFigure 5-6 .ThisimplementationmatchestheWiMAX-RBDS-SimarchitectureshowninFigure 5-2 .Thephysicallayerconsistsofaradio-receivermoduleandaradio-transmittermodule.Eachmoduleisconguredwith10radiochannelsandanyofthecongurationmodes(i.e.,bandwidth,forwarderrorcorrection,andmodulationschemes)speciedinthestandard[ 1 ].Also,eachmoduleisconnectedto10packetstreams,whichareOPNETelementsusedforcommunicatingpacketsacrossmodules.Thereis1packetstreamperchannel.Theradio-receivermoduleconnectsto10packet-classiermodulesthroughthepacketstreams,andeachpacket-classiermoduleconnectstothemain-processingmodulethrough3packetstreams.Eachofthesepacketstreamscarriespacketsofonlyonetype(i.e.,link-establishment,scheduling,ordatapacket).Themain-processingmoduleconnectstothedata-pa-cket-source-sinkmoduleofthelogical-linklayerthrough2packetstreams.The 141

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outgoingpacketstreamcarriesreceiveddatapacketsandnotications,intheformofpackets,ofnewoutgoing-linkestablishments.Theincomingpacketstreamcarriesdatapacketsgeneratedbythedata-packet-source-sinkmodule.Themain-processingmoduleconnectsto10output-queue-channel-classiermoduleswith1packetstreampermodule.Thereis1ofeachofthesemodulesperchannel,andtheyareusedfortemporarilystoringanytypeofscheduledpackets.Finally,eachoftheout-put-queue-channel-classiermodulesconnectstotheradio-transmittermodulethrough1packetstreamthatcarriesscheduledpacketsofanytypewhentheyneedtobetransmitted. Figure5-6. ImplementationoftheWiMAX-RBDS-Simarchitecture Themain-processingmoduleimplementsthemainprocessor,link-establishers,andschedulersasshowninFigure 5-2 .Also,itimplementstheinput-queuesinwhichdata-packetsgeneratedatthelogical-linklayerarestoredtemporarilywhiletheyarescheduled. 142

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Thetime-syncmodulesynchronizesthemain-processingandoutput-queue-cha-nnel-classiermoduleswiththeframestructureshowninFigure 5-1 .Itinterruptsthesemodulesattheonsetofeverycontrolanddataslotandinformsthemofthecurrent-timeinformationsuchasthecurrentframenumber,currenttypeofsubframe(i.e.,controlordatasubframe),andthecurrentcontrolordataslotnumber. 5.5SimulationResults AnIEEE802.16meshnetworkwithagridtopologyof78nodeswassimulated6.Thenumberofslotsinthecontrolsubframewassetto9.Theinterferencemodelconsideredwastheprotocolmodel[ 28 ]inwhichareceptionissuccessfulifthetransmitterisclosertothereceiverthananyothernodethattransmitssimultaneously.Thetrafcloadwasvariedfrom0to256pk/swhileaccountingforthenumberofinterferinglinksofeverylink.Forexample,ifalinkhad1or3interferinglinksandthetrafcloadwassetat256pk/s,thetrafcgeneratedforthatlinkwas128pk/sor64pk/srespectively(i.e.,256 1+1pk/sand256 1+3pk/s).Atotalof3000framesweresimulated. TheresultsobtainedwithWiMAX-RBDS-Simareshown7inFigures 5-7 5-8 5-9 5-10 5-11 ,and 5-12 Figure 5-7 showsthehistogramofthecontrol-subframeaccessdelay.Thisisthedelayanodeexperiencestogainaccesstothecontrolsubframe.WhenanIEEE802.16meshnodeaccessesthecontrolsubframe(i.e.,ittransmitseitheralink-establishmentorschedulingpacket),itcompetesforafuturecontrolslottotransmititsnextlink-establishmentorschedulingpacket.Thetimebetweentwoconsecutiveaccessestothecontrolsubframeistheaccessdelay,anditdirectly 6ThesizeofthisnetworkisrepresentativeofrealWMNs[ 9 ].7TheWiMAX-RBDS-Simframeworkalsoallowstheanalysisofthedata-packet-deliverydelay,throughput,andcollisionprobabilityinthecontrolanddatasubframes,buttheseresultsarenotdiscussedhereforthesakeofbrevity. 143

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affectsthetimerequiredtonishanyhandshakeincontrolsubframes.Therefore,thescheduling-handshakedelay,whichisthetimerequiredtoperformthethree-wayhandshakespeciedinSection 5.2.1 fornegotiatingadata-slotreservation,isaffectedbytheshapeofsuchhistogram(i.e.,thedistributionofthecontrol-subframeaccessdelay).Thisshapecanbemodiedbydynamicallycontrollingtheparametersoftheelectionalgorithm,whichisthetechniqueusedin[ 6 31 40 47 78 85 87 ]fortheimplementationofdifferentQoS,collisionavoidance,andcongestioncontrolschemes.Therefore,WiMAX-RBDS-Simisatoolthatcanbeusedfortheevaluationofsuchschemesbyfacilitatingasimulationframeworkinwhichtheycanbereadilyintegrated. Figure5-7. Control-subframeaccessdelayhistogram Figure 5-8 showsthedata-slotreservationsoftwo1-hopneighborsduringanintervalof7seconds.ThesereservationsarecalculatedbytheSliced-GM-RBDSAlgorithmsothatoverlapsbetweenthemareavoided.Thealgorithmwasconguredwith16slices8.Theblankspacescorrespondtosetsofdataslotsthatwerereservedforothernodesinthenetwork.Ideally,thereshouldnotbeanyblankspacesnor 8Thedata-slotreservationexampleshowninFigure 5-4 considersonly4slices. 144

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overlapswhenallthenodesthatinterferewiththetwo1-hopneighborsareconsidered.Figure 5-8 isanexampleofthetypeofresultsthatcanbeobtainedforRBDSalgorithms.Inthisexample,theSliced-GM-RBDSalgorithmisbeingconsidered.However,otherRBDSalgorithms,suchastheonespresentedin[ 18 21 24 42 43 48 79 88 ],canbeintegratedasschedulerprocesses(Figure 5-2 )andevaluated. Figure5-8. Data-slotreservationoftwo1-hopneighborsinanetworkwithgridtopologyof78nodes Figure 5-9 showstheaveragelengthacrossthenetworkoftheoutputqueuesforatrafcloadof64pk/swhenthenumberofslicesincreases.Theaverageoutput-queuelengthsclearlydependonthenumberofslices.Therefore,thecapacityofthenetworkcanbeincreasedbyconguringitwiththeoptimal(i.e.,minimumaverageoutput-queuelength)numberofslices.Whenthedatasubframesarenotpartitioned(i.e.,whenthereisonly1slice),theaverageoutput-queuelengthisnotminimized.ThisisthecongurationusedintheGM-RBDSalgorithmpresentedinChapter 2 .Therefore,theSliced-GM-RBDSalgorithmisanimprovedversionoftheGM-RBDSalgorithmintermsofnetworkcapacity.TheSliced-GM-RBDSalgorithmisabletomaintainlowerqueuelengths,andasaconsequencelowerdata-packet-deliverydelays,thantheGM-RBDSalgorithmdoes.Specically,theSliced-GM-RBDSminimizestheaverageoutput-queuelengthwhenthenumberofslicesis16.Figure 5-10 showstheaverage 145

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output-queuelengthfordifferenttrafcloadswhenthenumberofslicesisxedat16.Theoutput-queuelengthsareniteaslongasthetrafcloadisbelow128pk/s.Forhighertrafcloads,theoutput-queuelengthsincreaseindenitelywithtime. Figure5-9. Averageoutput-queuelengthforincreasingnumberofslices(trafcload=64packetspersecond) Figure5-10. Averageoutput-queuelengthforincreasingtrafcloads(numberofslices=16) Figure 5-11 showstheinstantaneouslengthoftheinputandoutputqueuesinthenetworkwhenthetrafcloadismaximum(i.e.,256pk/s).AsexpectedfromtheresultsinFigure 5-10 ,theoutputqueuesareunstable(i.e.,theyincreaseindenitelywithtime).However,theinputqueuesarestable(i.e.,theyalwaysreturntotheemptystateatsomepointsintime).Therefore,giventhattheinputqueuesarealwaysstableandtheoutput 146

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queuesarestablefortrafcloadslowerthan128pk/s,theSliced-GM-RBDSalgorithmguaranteesthestabilityofthenetwork(i.e.,allthequeuesinthenetworkarestable)forthesimulationscenarioconsideredwhenthetrafcloaddoesnotexceed128pk/s. Figure5-11. Input-queueandoutput-queuelengthcomparisonformaximumtrafcload Figure5-12. Link-establishmentdelayhistogram AsanalexampleoftheresultsandanalysisthatcanbeperformedwithWi-MAX-RBDS-Sim,thehistogramofthelink-establishmentdelayisshowninFigure 5-12 .Thisdelayisthetimerequiredforestablishingtheincomingandoutgoinglinksbetween 147

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1-hopneighborswiththethree-wayhandshakespeciedinSection 5.2.2 .Thereisatotalof97incoming-outgoinglinkpairsinthe78gridtopology.Theresultsshowthatthelink-establishmentalgorithmestablishesabidirectionallinkinnomorethan2.6s.Thisresultisalsoaffectedbythedistributionofthecontrol-subframeaccessdelay(Figure 5-7 )giventhatthelink-establishmenthandshakeisperformedwiththeexchangeofpacketsinthecontrolsubframe.Aschemeforreducingthelink-establishmentdelaywasproposedin[ 80 ]. 5.6PerformanceEvaluation TheperformanceofWiMAX-RBDS-Simintermsofsimulationspeedandmemoryusagewasevaluatedfordifferentnetworksetups.Thesimulatednetworksetupsincluded4differentgridtopologieswithincreasingnumberofnodes(i.e.,48,58,68,and78)withthesamecongurationusedinSection 5.5 .Thesimulationswereperformedinthesequentialmodeinwhichthereisonlyonethreadofexecution9.Figures 5-13 and 5-14 showthatthesimulationspeeddecreasesandthememoryusageincreasesapproximatelylinearlywhenthenumberofnodesincreases. Figure5-13. Simulationspeed 9TheplatformusedinthesimulationincludedaprocessorIntelCoreDuoat3GHzand3GBofRAMat3GHz. 148

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Figure5-14. Memoryusage 5.7Summary AnewsimulationmodelforIEEE802.16meshnetworkswithdistributedscheduling(i.e.,WiMAX-RBDS-Sim)wasdevelopedinOPNET.Itprovidesinterfacesfortheimplementationandevaluationofdistributedschedulingpoliciesandlink-establishmentalgorithms.Thepoliciesmaybebasedonthedynamicvariationoftheelection-algorithmparametersoronthereservationoffuturedataslots. AsanexampleoftheuseofWiMAX-RBDS-Sim,anewRBDSpolicy(i.e.,Sliced-GM-RBDS)wasimplementedandevaluatedwithit.Thispolicyisbasedontheconceptsofinputandoutputqueues.TheresultsshowedthattheSliced-GM-RBDSpolicyoutperformedtheGM-RBDSpolicyintermsofcapacitysincethesystemofqueueswasstableforhighertrafcloadswhentheSliced-GM-RBDSwasused.Also,alink-establishmentalgorithmwasimplementedandintegratedwithWiMAX-RBDS-Sim,anditsperformancewasevaluatedintermsofdelay. Finally,theperformanceofWiMAX-RBDS-Simwasevaluatedintermsofsimulationspeedandmemoryusageforanincreasingnumberofnodes. 149

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CHAPTER6SUMMARYOFCOMPLETEDWORK InChapter 1 ,theresearchproblemwasstated.Themainproblemtobesolvedistocontrolthetopologyofwirelessmultihopnetworks(WMN)inordertomaximizetheperformanceofend-to-enddataowsestablishedonthenetworks.Thisperformanceisgivenintermsofthesetofend-to-enddata-packetratesthatthenetworksupportswhileguaranteeingstability.Thetopologyiscontrolledbymeansoftransmission-powercontrol. Inordertosolvetheresearchproblem,theworkdiscussedinChapters 2 3 ,and 5 hasbeennished. InChapter 2 ,anewframeworkforthestabilityanalysisofschedulingpoliciesforwirelessnetworksthatallowthereservationoffuturedata-subframeswasproposed.Inthesepolicies,thenodescoordinatetheassignmentofdata-subframesbyexchangingschedulingpackets.Theconceptsofinput-queueandoutput-queuewereintroducedintotheframeworkinordertoaccountforthepacketswaitingtobescheduledandtheschedulesassignedtothesepackets.Basedontheseconcepts,sufcientconditionsforthestabilityofreservation-based-distributed-scheduling(RBDS)wirelessnetworkswerefound. Withintheproposedframework,thegreedy-maximalreservation-based-dis-tributed-scheduling(GM-RBDS)policywasanalyzed.ThenodesimplementthispolicybyexchangingschedulingpacketsusingtheInstitute-of-Electrical-and-Elec-tronics-Engineers(IEEE)802.16electionalgorithm.Aregioninwhichtheproposedreservation-basedschedulingpolicyisstablewasfoundusingtheframework.Itwasshownthatthesizeofthisregiondependsonthefactorwhichisdeterminedbytwocharacteristicsofthenetworktopologyonly(i.e.,sjandjPmaxj).AnIEEE802.16meshnetworkwiththeproposedschedulingpolicy(i.e.,GM-RBDS)wassimulated.Itwasshownthatthepolicyalwaysguaranteedthestabilityoftheinput-queuesandthatthe 150

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output-queueswerestablewhentheloadwaswithin18 24oftheoptimalregion.Finally,theperformanceoftheGM-RBDSpolicywascomparedwiththeWandenhanced-lo-cal-greedy-scheduling(ELGS)policies.ItwasshownthattheGM-RBDSpolicyhasanadvantageovertheWandELGSpoliciesintermsoftherequiredoverheadanditsabilitytoreserveanyfuturedata-subframes. InChapter 3 ,theheuristic-stability-region-adaptationalgorithmwasproposedfortransmissionpowercontrol.Thisisacentralizedalgorithmthatincreasestheinput-packetratesthatowscansupportanddecreasestheend-to-enddelays.Itisisbasedontheadaptationofthestabilityregionofagivenlink-schedulingpolicywhenonlythelinksthatbelongtoagivensetofowsareconsidered.Thealgorithmcanbereadilyadaptedtoanylink-schedulingpolicywhosestabilityregionhasbeencharacterized,soitisnotlimitedtoanyspecicschedulingapproachsuchasre-quest-to-send/clear-to-send-basedpolicies.Theimprovementonthroughputachievedbyouralgorithmwasevaluatedbymeansofsimulationforthemin-hoproutingalgorithmandtheGM-RBDSlink-schedulingpolicyinIEEE802.16meshnetworks.Itwasshownthatitoutperformstheclassicalsolutionofreducingtransmissionpowerstoincreasespatialreuse.Also,itsperformancewasevaluated.Itwasfoundthatitdependsontheowdensity.Theperformanceincreasesastheowdensitydecreases. InChapter 4 ,distributedtopology-controlalgorithmsforWMNsunderGM-RBDSweredesignedconsideringtheresultsin[ 73 77 ].Theproblemwassolvedinthreesteps.First,thetopology-controloptimizationproblemwasformulatedasapotentialgameinwhichtheows(i.e.,players)collaborateformaximizingthepacketratestheycansupportwhileguaranteeingstability.Second,inordertoanalyzethepotentialgame,theobjectivefunctions(i.e.,ows'utilities)ofthetopology-controloptimizationproblemwereheuristicallymodiedsuchthattheproblemcouldbeformulatedasanintegerlinearprogram.Thisproblemcorrespondedtotheoptimizationofalinearpotentialfunctionofthemodiedgame.Thislinearpotentialfunctionapproximated 151

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theobjectiveoftheoriginalproblem(i.e.,withoutthemodications).ThereasonforthelinearformulationisthatitguaranteedthatthepotentialgameconvergedtoaNashequilibriumthatisalsoaglobaloptimumofthemodiedobjectivefunction.However,giventhatadifferentobjectivefunctionwasoptimizedinthegameduetothemodications,thetransmissionpowers(TP)calculatedbytheowswereinsomecasesdifferenttotheoptimalTPs.Third,thedifferenceofthemaximumtotalthroughputachievedbymeansofthemodiedpotentialgameandtheoptimalmaximumthroughputwascharacterized.Thischaracterizationwasbasedontheworstcasescenariosinwhichthemodiedutilitiesmisleadtheowstodecidestrategiesthatdidnotoptimizethemaximumtotalthroughput.Thesimulationresultsshowedthatforneworkswithlowowdensity,thegame-theoreticalalgorithmsperformedsimalarlytothecentralizedalgorithmofChapter 3 ,fornetworkswithmediumowdensity,theyoutperformedthecentralizedalgorithm,andfornetworkswithhighowdensity,theywereoutperformedbythecentralizedalgorithm.Thereasonforthisbehaviorwastheviewofthenetworkthatthealgorithmshave.Thegame-theoreticalalgorithmshavelimitedviewofthenetworkduetotheirdistributednature,andthecentralizedalgorithmhasaglobalviewofthenetwork. Finally,inChapter 5 ,anewsimulationmodelforIEEE802.16meshnetworkswithdistributedschedulingwasdevelopedusingoptimizednetworkengineeringtools(OPNET).Itwascalledworldwide-interoperability-for-microwave-accessRBDSsimulator(WiMAX-RBDS-Sim).Itprovidesinterfacesfortheimplementationandevaluationofdistributedschedulingpoliciesandlink-establishmentalgorithms.Thepoliciesmaybebasedonthedynamicvariationoftheelection-algorithmparametersoronthereservationoffuturedataslots. AsanexampleoftheuseofWiMAX-RBDS-Sim,anewRBDSpolicy(i.e.,Sliced-GM-RBDS)wasimplementedandevaluatedwithit.Thispolicyisbasedontheconceptsofinputandoutputqueues.TheresultsshowedthattheSliced-GM-RBDS 152

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policyoutperformedtheGM-RBDSpolicyintermsofcapacitysincethesystemofqueueswasstableforhighertrafcloadswhentheSliced-GM-RBDSwasused.Also,alink-establishmentalgorithmwasimplementedandintegratedwithWiMAX-RBDS-Sim,anditsperformancewasevaluatedintermsofdelay. TheresultsobtainedinChapters 1 to 5 havebeenpublishedasfollows.TheresearchproblemdiscussedinChapter 1 wasoriginallyformulatedin[ 70 ].TheresultsdiscussedinChapter 2 havebeenpublishedin[ 74 ]and[ 71 ].TheresultsdiscussedinChapter 3 havebeenpublishedin[ 73 ]and[ 77 ].TheresultsobtainedinChapter 4 havebeensubmittedforpublicationin[ 76 ].TheresultsdiscussedinChapter 5 havebeenpublishedin[ 75 ]and[ 72 ]. 153

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APPENDIXAINPUTANDOUTPUTDATA-PACKETARRIVALRATEOFSGS Theexpectednumberofdata-packetarrivalstonodej'sinput-queueQji(n)isderivedasfollows. (i,j)s,EA(i,j)s=E24mn+1Xm=mn+1)]TJ /F6 7.97 Tf 6.59 0 Td[(MjmnA0(i,j)(m)35=E24Nj1(mn))]TJ /F6 7.97 Tf 6.58 0 Td[(1Xl=0A0(i,j)jmn+1 mcsk)]TJ /F5 11.955 Tf 11.95 0 Td[(lmcs35=E24Nj1(mn))]TJ /F6 7.97 Tf 6.58 0 Td[(1Xl=0mds)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=0A(i,j)k((bmn+1 mcsc)]TJ /F7 7.97 Tf 6.59 0 Td[(l)mcs))]TJ /F5 11.955 Tf 11.95 0 Td[(k35=1Xr=1PNj1(mn)=rE"r)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xl=0mds)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=0A(i,j)k((bmn+1 mcsc)]TJ /F7 7.97 Tf 6.58 0 Td[(l)mcs))]TJ /F5 11.955 Tf 11.96 0 Td[(k#=1Xr=1Pj1[r]r)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xl=0mds)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=0EhA(i,j)k((bmn+1 mcsc)]TJ /F7 7.97 Tf 6.59 0 Td[(l)mcs))]TJ /F5 11.955 Tf 11.95 0 Td[(ki=1Xr=1Pj1[r]r)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xl=0mds)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xk=0(i,j)=mds(i,j)1Xr=1Pj1[r]r)]TJ /F6 7.97 Tf 6.59 0 Td[(1Xl=01=mds(i,j)1Xr=1rPj1[r]= Nj1mds(i,j) 154

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APPENDIXBPROOFOFTHEOREM 2.1 ConsidertheLyapunovfunctionsVis(n)andVos(n)denednextforthequeueprocessesQji(n)andQjo(n)inSGs. Vi,os(n),Xj2N)]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Qji,o(n)2 ThefollowingLyapunovfunctionsVi(mn)andVo(mn)forqueueprocessesQ(i,j)i(m)andQ(i,j)o(m)inSGarefoundtobeequivalenttoVis(n)andVos(n)respectivelyasfollows. Vi(mn),X(i,j)2LQ(i,j)i(mn)Xk2Sj1Q(k,j)i(mn)=X(i,j)2LXk2Sj1Q(i,j)i(mn)Q(k,j)i(mn)=Xj2NXi2Sj1Xk2Sj1Q(i,j)i(mn)Q(k,j)i(mn)=Xj2NXi2Sj1Q(i,j)i(mn)Xk2Sj1Q(k,j)i(mn)=Xj2N)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qji(n)2=Vis(n)Vo(mn),Xj2Nmaxi2Sj1Q(i,j)o(mn)2=Xj2N)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)2=Vos(n) Therefore,Vi(mn)hasnegativedriftifandonlyifVis(n)hasnegativedrift.ThesamerelationholdsforVo(mn)andVos(n). ThechangeinthevalueofVos(n)isfoundasfollows. 155

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Vos(n+1))]TJ /F5 11.955 Tf 11.96 0 Td[(Vos(n)=Xj2N)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n+1)2)]TJ /F12 11.955 Tf 11.96 11.36 Td[(Xj2N)]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)2=Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n+1)2+Xj=2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n+1)2)]TJ /F12 11.955 Tf 21.2 11.36 Td[(Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)2)]TJ /F12 11.955 Tf 21.19 11.36 Td[(Xj=2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)2=Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)+jHj(n)j)]TJ /F5 11.955 Tf 17.93 0 Td[(Nj1(n)2+Xj=2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)2)]TJ /F12 11.955 Tf 21.19 11.36 Td[(Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Qjo(n)2)]TJ /F12 11.955 Tf 21.2 11.36 Td[(Xj=2N2(n))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Qjo(n)2=Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Qjo(n)+jHj(n)j)]TJ /F5 11.955 Tf 17.93 0 Td[(Nj1(n)2)]TJ /F12 11.955 Tf 21.2 11.36 Td[(Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Qjo(n)2=Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Qjo(n)2+Xj2N2(n)jHj(n)j2+Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.68 Td[(Nj1(n)2)]TJ /F8 11.955 Tf 11.95 0 Td[(2Xj2N2(n)Qjo(n)Nj1(n)+2Xj2N2(n)Qjo(n)jHj(n)j)]TJ /F8 11.955 Tf 11.95 0 Td[(2Xj2N2(n)Nj1(n)jHj(n)j)]TJ /F12 11.955 Tf 27.18 11.36 Td[(Xj2N2(n))]TJ /F5 11.955 Tf 5.48 -9.69 Td[(Qjo(n)2=2Xj2N2(n)Qjo(n))]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jHj(n)j)]TJ /F5 11.955 Tf 17.93 0 Td[(Nj1(n)+Xj2N2(n))]TJ /F2 11.955 Tf 5.48 -9.69 Td[(jHj(n)j)]TJ /F5 11.955 Tf 17.94 0 Td[(Nj1(n)2 ThedriftofVos(n)isfoundasfollows,whereCissomeconstant. EQjo(n)Vos(n+1))]TJ /F5 11.955 Tf 11.95 0 Td[(Vos(n)=2EQjo(n)Xj2N2(n)Qjo(n))]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jHj(n)j)]TJ /F5 11.955 Tf 17.94 0 Td[(Nj1(n)+C=2XN0NEQjo(n),N2(n)Xj2N0Qjo(n))]TJ /F2 11.955 Tf 5.48 -9.68 Td[(jHj(n)j)]TJ /F5 11.955 Tf 17.93 0 Td[(Nj1(n)PN2(n)=N0+C=2XN0NXj2N0Qjo(n))]TJ /F1 11.955 Tf 5.48 -9.68 Td[(EjHj(n)j)]TJ /F1 11.955 Tf 11.95 0 Td[(ENj1(n)PN2(n)=N0+C 156

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Inthepreviousequalities,thefactsthatEQjo(n),N2(n)jHj(n)j=EjHj(n)jandEQjo(n),N2(n)Nj1(n)=ENj1(n)havebeenused.ThesefactsaretruebecausejHj(n)jandNj1(n)areindependentofQjo(n)andN2(n).jHj(n)jistheoutput-queue-lengthincrease,whichcorrespondstothetotalnumberofdata-subframesincludedinthegrants,blanks,andoverlapsthatarescheduledbetweenthenthand(n+1)thscheduling-packettransmissionsofj(Section 2.3.2.4 ).Thissetofdata-subframesisshowninFigure 2-5B .Thedistributionofgrants,blanks,andoverlapsdependsonhowthespecicreservation-based-distributed-scheduling(RBDS)policyimplementedonthenetworkusesthelengthofinput-queuesforcalculatingrequests.Nj1(n)isdeterminedbythenumberofcontrol-time-slotsbetweenthesetransmissions,andthisnumberisequaltoMjm(Eq. 2 ). Giventhatthenetworkisstationary,thedriftofVos(n)isupper-boundedasfollows. EQjo(n)Vos(n+1))]TJ /F5 11.955 Tf 11.96 0 Td[(Vos(n)=2XN0NXj2N0Qjo(n))]TJ ET q .478 w 186.84 -350.95 m 207.1 -350.95 l S Q BT /F2 11.955 Tf 186.84 -361.61 Td[(jHjj)]TJ ET q .478 w 221.72 -348.95 m 239.6 -348.95 l S Q BT /F5 11.955 Tf 221.72 -361.61 Td[(Nj1PN2(n)=N0+C2XN0NXj2N0Qjo(n))]TJ /F8 11.955 Tf 7.47 -9.69 Td[(maxj2N0)]TJ ET q .478 w 217.32 -388.36 m 237.59 -388.36 l S Q BT /F2 11.955 Tf 217.32 -399.03 Td[(jHjj)]TJ ET q .478 w 252.2 -386.37 m 270.09 -386.37 l S Q BT /F5 11.955 Tf 252.2 -399.03 Td[(Nj1PN2(n)=N0+C=2XN0NPN2(n)=N0)]TJ /F8 11.955 Tf 12.45 -9.68 Td[(maxj2N0)]TJ ET q .478 w 248.58 -425.78 m 268.85 -425.78 l S Q BT /F2 11.955 Tf 248.58 -436.44 Td[(jHjj)]TJ ET q .478 w 283.46 -423.79 m 301.34 -423.79 l S Q BT /F5 11.955 Tf 283.46 -436.44 Td[(Nj1Xj2N0Qjo(n)+C2)]TJ /F8 11.955 Tf 7.47 -9.68 Td[(maxj2N)]TJ ET q .478 w 135.39 -463.2 m 155.65 -463.2 l S Q BT /F2 11.955 Tf 135.39 -473.86 Td[(jHjj)]TJ ET q .478 w 170.26 -461.21 m 188.15 -461.21 l S Q BT /F5 11.955 Tf 170.26 -473.86 Td[(Nj1XN0NPN2(n)=N0Xj2N0Qjo(n)+C Byasimilaranalysis,itisfoundthatthedriftofVis(n)isupper-boundedasfollows. EQjo(n)Vis(n+1))]TJ /F5 11.955 Tf 11.95 0 Td[(Vis(n)20@maxj2N0@Xi2Sj1(i,j)s)]TJ /F5 11.955 Tf 11.96 0 Td[(mdsXi2Sj1 G(i,j)1A1AXN0NPN2(n)=N0Xj2N0Qjo(n)+C 157

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Therefore,thedriftsofbothVis(n)andVos(n)arenegativeifmaxj2N)]TJ ET q .478 w 206.37 -37.16 m 226.64 -37.16 l S Q BT /F2 11.955 Tf 206.37 -47.82 Td[(jHjj)]TJ ET q .478 w 241.25 -35.16 m 259.13 -35.16 l S Q BT /F5 11.955 Tf 241.25 -47.82 Td[(Nj1<0,maxj2N0@Xi2Sj1(i,j)s)]TJ /F5 11.955 Tf 11.95 0 Td[(mdsXi2Sj1 jG(i,j)j1A<0, andtheprooffollowsfromthisconditionandEq. 2 158

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APPENDIXCPROOFOFLEMMA 1 Considertheinput-queuegivenbytheMarkovprocessinEq. 2 ,assumethattheprobabilitythattherequestforthisqueueissuccessfullygranted)]TJ /F1 11.955 Tf 5.48 -9.69 Td[(i.e.,PI(i,j)Qmax(n)=1,I(i,j)s(n)=1isalwaysequaltoitslower-boundgivenbyEq. 2 .Letsdenotethislower-bound.TheexpectedvalueofQ(i,j)s(n)isupper-boundedbytheexpectedvalueofQ(i,j)s(n)foundunderthisassumption. Letaxandxbedenedasfollows. ax,P[A(i,j)s(n)=x]x,P[Q(i,j)s(n)=x] TheprobabilitythatQ(i,j)s(n)movesfromstatex1tostatex2,wherex1x2iss(ann)[x2]+(1)]TJ /F5 11.955 Tf 12.75 0 Td[(s)ax2)]TJ /F7 7.97 Tf 6.59 0 Td[(x1.Thatis,theprobabilitythattherequestissuccessfullygrantedandthenumberofpacketarrivalsA(i,j)s(n)plustheremainingpacketsinthequeueQ(i,j)s(n)isx2orthattherequestisnotsuccessfullygrantedandtherearex2)]TJ /F5 11.955 Tf 11.27 0 Td[(x1packetarrivals. Therefore,theequilibriumequationsaregivenasfollowsforallPx,wherePxistheprobabilitythatQ(i,j)s(n)isinstatex. Px=(s(ann)[x]+(1)]TJ /F5 11.955 Tf 11.95 0 Td[(s)ax)P0+(s(ann)[x]+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(s)ax)]TJ /F6 7.97 Tf 6.58 0 Td[(1)P1+...=s(ann)[x]1Xn=0Pn+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(s)1Xn=0ax)]TJ /F7 7.97 Tf 6.59 0 Td[(nPn=s(ann)[x]+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(s)(anPn)[x] TheexpectedvalueofQ(i,j)s(n)isfoundbyevaluatingdP(z) dzz=1,whereP(z)isdenedasP1x=0Pxzx. P(z)anddP(z) dzz=1aregivenasfollows. 159

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P(z)=sA(z)(z)+(1)]TJ /F5 11.955 Tf 11.96 0 Td[(s)A(z)P(z)dP(z) dzz=1=1 sdA(z) dzz=1+d(z) dzz=1=1 sE[A(i,j)s(n)] mds+E[Q(i,j)s(n)] FromEq. 2 ,Eq. 2 ,andtheassumptionthatPI(i,j)Qmax(n)=1,I(i,j)s(n)=1=s,theexpectedvalueofQ(i,j)s(n)canbeupper-boundedasfollows. EQ(i,j)s(n)sjj N1+EQ(i,j)s(n) 160

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APPENDIXDPROOFOFTHEOREM 3.1 Letthefollowingnotationbedenedforthepresentationoftheproof.Link(i,j)belongstoowfn2Fifnodejisprecededbynodeiatanypointintheow'spathpn)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,(i,j)2fnifpn(m)=iandpn(m+1)=jforsomem=1,2,...,jpnj)]TJ /F8 11.955 Tf 18.14 0 Td[(1.Thesetofowsthatlink(i,j)belongstoisdenotedbyF(i,j))]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,F(i,j),fn:fn2F,(i,j)2fn.AnodehasthreequeuesforeveryowthatitbelongstoasshowninFigure D-1 .Forowfnonlink(i,j)atnodei,Q(i,j)r,fn(n),Q(i,j)i,fn(n),andQ(i,j)o,fn(n)aretheregulator,input,andoutputqueuesrespectively.A-regulatorisalogicaldevicewithmaximumservicerate,i.e.,ateachtime-slot,a-regulatorchecksitsqueueand,ifthereareanypackets,ittransfersapacketwithprobability.Thefollowingrule1isusedforeachowfninF:forthersthopalongtheow'spath(i.e.,pn(1)),thereisan-regulatoratitsinputqueue;forthemthhop(i.e.,pn(m),m2),thereisa(+(m)]TJ /F8 11.955 Tf 12.47 0 Td[(1))-regulator,where>0.ThepacketdeparturesofqueuesQ(i,j)r,fn,Q(i,j)i,fn,andQ(i,j)o,fnaredenotedbyR(i,j)fn(n),G(i,j)fn(n),andDi(n)respectively.Thequeuelengthsanddeparturesareindexedbynwhichdenotesthenthtimethatoneormorethantwonodesinthenetworktransmitaschedulingpacket.Thelinkinowfnthatfollowslink(i,j)isdenotedby(i,j)0.Forexample,Q(i,j)0r,fnistheregulatorqueueforthelinkfollowinglink(i,j)inowfn,whichislocatedatthedestinationnodeoflink(i,j)(i.e.,nodej)asshowninFigure D-1 .Thetotaltrafcrateonlink(i,j)isdenotedby(i,j),i.e.,(i,j)=Pn2F(i,j)n. LetthefollowingLyapunovfunctionV)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qi(n),Qo(n)=Vr)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qo(n)+Vi)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qi(n)+Vo)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qo(n)bedenedforthesystemofqueuesgivenbyQr,i,o(n),Q(i,j)r,i,o,fn(n)fn2F,(i,j)2Lasfollows. 1Thisrulewasoriginallyusedin[ 84 ]fortheproofofthestabilityregionofdistributedgreedyschedulingpoliciesunderregulatedmultihoptrafc. 161

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FigureD-1. Nodequeuingmodelperowitbelongsto Vr)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qo(n),Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n)2,Xj2NXi2Sj1Xfn2F(i,j)Q(i,j)o,fn(n)+Q(i,j)0r,fn(n)2 Vi)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qi(n),X(i,j)2L0@0@Xfn2F(i,j)Q(i,j)i,fn(n)1A0@Xk2Sj1Xfn2F(k,j)Q(k,j)i,fn(n)1A1A Vo)]TJ /F14 11.955 Tf 5.47 -9.68 Td[(Qo(n),Xj2N0@maxi2Sj10@Xfn2F(i,j)Q(i,j)i,fn(n)1A1A2 ThedriftofV)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qi(n),Qo(n),isdenedasfollows2. EQr(n),Qi(n),Qo(n)V=EQr(n),Qi(n),Qo(n)V)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n+1),Qi(n+1),Qo(n+1))]TJ /F5 11.955 Tf 11.96 0 Td[(V)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qi(n),Qo(n) TheproofisbasedonthesufcientconditionswhichguaranteethatthedriftofV)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qi(n),Qo(n)isnegative3.Itproceedsasfollows.First,thedriftsofVi)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qi(n)andVo)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qo(n)areobtainedfromtheresultsinChapter 2 .Then,thedrift 2EY[X]denotestheexpectedvalueofXgivenY(i.e.,EY[X],E[XjY]).3AccordingtoFoster'stheorem[ 5 ],aMarkovianqueueingsystemisstableifthedriftoftheLyapunovfunctionisnegative. 162

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ofVr)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qr(n),Qo(n)iscalculated.Finally,thedriftofV)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qr(n),Qi(n),Qo(n)iscalculatedfromthedriftsofVr)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qo(n),Vi)]TJ /F14 11.955 Tf 5.47 -9.68 Td[(Qi(n),andVo)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qo(n),andthesufcientconditioninwhichthedriftofV)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qi(n),Qo(n)isnegativeisfound. AccordingtoChapter 2 ,underthegreedy-maximalreservation-based-distribu-ted-scheduling(GM-RBDS)policy,thedriftofVi)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qi(n)isupper-boundedasfollows,whereC1andC2arepositiveconstantsindependentofQr(n),Qi(n),andQo(n). EQi(n)Vi=EQi(n)Vi)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qi(n+1))]TJ /F5 11.955 Tf 11.95 0 Td[(Vi)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qi(n)<)]TJ /F5 11.955 Tf 9.3 0 Td[(C1Xj,i,fnQ(i,j)i,fn(n)+C2(D) Therefore,undertheGM-RBDSpolicy,theinput-queuesareguaranteedtobealwaysstable.Qualitatively,thisresultisaconsequenceofthefactthatundertheGM-RBDSpolicy,everytimeanodetransmitsaschedulingpacket,itrequestsforeverylinkasmanyfuturedata-subframesascanbecoveredcompletelywithunscheduleddatapackets.Giventhattheserequestsarealwaysgrantedwithsomeprobabilitygreaterthanzero(Chapter 2 ),theprobabilitythattheinput-queuesreturntotheemptystateisalwaysgreaterthanzero. AccordingtoChapter 2 ,undertheGM-RBDSpolicy,thedriftofVo)]TJ /F14 11.955 Tf 5.47 -9.68 Td[(Qo(n)isupper-boundedasfollows,whereC3andareconstantsindependentofQr(n),Qi(n),andQo(n). EQo(n)Vo=EQo(n)Vo)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qo(n+1))]TJ /F5 11.955 Tf 11.95 0 Td[(Vo)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qo(n)<)]TJ /F9 11.955 Tf 9.3 0 Td[(Xj,i,fnQ(i,j)o,fn(n)+C3(D) TheconstantC3ispositive.TheconstantispositiveiftheconditiongivenbyEq. D issatised(Chapter 2 ),wherePmaxisthelongestpathamongallthepathsinthegraphsinducedbyeverynode's2-hopneighborhoodandthatoriginateatthenode)]TJ /F1 11.955 Tf 5.48 -9.68 Td[(i.e.,ifPiisthelongestpaththatoriginatesatnodeiinthegraphinducedbySi2,then 163

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Pmax=argmaxfPi:i2NgjPij.ThelengthofPmaxisdenedasthenumberoflinksinitanddenotedbyjPmaxj. Xi2Sj1(i,j)1+Si1nSj1[j<1 2jPmaxj+18j2N(D) Therefore,undertheGM-RBDSpolicy,theoutput-queuesareguaranteedtobestableifEq. D issatised.Qualitatively,theconditiongivenbyEq. D guaranteesthestabilityoftheoutput-queuesbecauseitlimitstheamountofdatatrafctheeverynodereceives.Thatis,whenthetotaltrafcthatanodereceivesislimited,thegrantsgeneratedbythenodeareguaranteednottoreservefuturedata-time-slotsthatareinnitelyfarawayfromthecurrentdata-time-slot.Inthisway,theprobabilitythatthesystemtimereachesthescheduleofthedatapacketscheduledthefarthestintimeisgreaterthanzero.Thisistheprobabilitythatoutput-queuesreturntotheemptystate. ThedriftofVr)]TJ /F14 11.955 Tf 5.48 -9.68 Td[(Qr(n),Qo(n)iscalculatedasfollows4. Vr=Vr)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qr(n+1),Qo(n+1))]TJ /F5 11.955 Tf 11.95 0 Td[(Vr)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qr(n),Qo(n)=Xj,i,fnQ(i,j)o,fn(n+1)+Q(i,j)0r,fn(n+1)2)]TJ /F12 11.955 Tf 11.95 11.35 Td[(Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n)2=Xj,i,fnQ(i,j)o,fn(n+1)+Q(i,j)0r,fn(n+1))]TJ /F5 11.955 Tf 11.95 0 Td[(Q(i,j)o,fn(n))]TJ /F5 11.955 Tf 11.96 0 Td[(Q(i,j)0r,fn(n)2+2Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n)Q(i,j)o,fn(n+1)+Q(i,j)0r,fn(n+1))]TJ /F5 11.955 Tf 11.96 0 Td[(Q(i,j)o,fn(n))]TJ /F5 11.955 Tf 11.95 0 Td[(Q(i,j)0r,fn(n) GiventhatQ(i,j)o,fn(n+1)+Q(i,j)0r,fn(n+1)=Q(i,j)o,fn(n)+G(i,j)fn(n)+Q(i,j)0r,fn(n))]TJ /F5 11.955 Tf 12.13 0 Td[(R(i,j)0fn(n),Vrisgivenasfollows. 4Thiscalculationisbasedontheproofforthestabilityregionofdistributedgreedyschedulingpoliciesundermultihoptrafcpresentedin[ 84 ]. 164

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Vr=2Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n)G(i,j)fn(n))]TJ /F5 11.955 Tf 11.96 0 Td[(R(i,j)0fn(n)+Xj,i,fnG(i,j)fn(n))]TJ /F5 11.955 Tf 11.96 0 Td[(R(i,j)0fn(n)2 ThedriftofVr)]TJ /F14 11.955 Tf 5.48 -9.69 Td[(Qr(n),Qo(n)isgivenasfollows. EQr(n),Qo(n)Vr=2Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n)EQr(n),Qo(n)hG(i,j)fn(n))]TJ /F5 11.955 Tf 11.96 0 Td[(R(i,j)0fn(n)i+Xj,i,fnEQr(n),Qo(n)hG(i,j)fn(n))]TJ /F5 11.955 Tf 11.96 0 Td[(R(i,j)0fn(n)2i(D) Giventhattheinput-queuesareguaranteedtobealwaysstableaccordingtoEq. D andthatQ(i,j)r,fn'soutputisconnectedtoQ(i,j)i,fn'sinputonly, EQr(n),Qo(n)hG(i,j)fn(n)i=EQr(n),Qo(n)hR(i,j)fn(n)iEQr(n),Qo(n)hR(i,j)0fn(n)i. Therefore,thersttermintheright-handsideofEq. D canbeupper-boundedasfollows,where R(i,j)fnisthedeparturerateofregulatorQ(i,j)r,fnifQ(i,j)r,fnisgreaterthanzero,andC4isapositiveconstantindependentofQr(n),Qi(n),andQo(n)suchthatC4> R(i,j)0fnIQ(i,j)0r,fn>0)]TJ ET q .478 w 113.89 -402.08 m 138.33 -402.08 l S Q BT /F5 11.955 Tf 113.89 -415.62 Td[(R(i,j)fn>0. 2Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n)EQr(n),Qo(n)hR(i,j)fn(n))]TJ /F5 11.955 Tf 11.96 0 Td[(R(i,j)0fn(n)i=2Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n) R(i,j)fnIQ(i,j)r,fn>0)]TJ ET q .478 w 217.34 -502.19 m 244.46 -502.19 l S Q BT /F5 11.955 Tf 217.34 -515.73 Td[(R(i,j)0fnIQ(i,j)0r,fn>0<)]TJ /F8 11.955 Tf 9.3 0 Td[(2C4Xj,i,fnQ(i,j)o,fn(n)+Q(i,j)0r,fn(n) Thesecondtermintheright-handsideofEq. D canbeupper-boundedasfollows,whereC5isapositiveconstantindependentofQr(n),Qi(n),andQo(n). 165

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Xj,i,fnEQr(n),Qo(n)hG(i,j)fn(n))]TJ /F5 11.955 Tf 11.95 0 Td[(R(i,j)0fn(n)2iXj,i,fnEQr(n),Qo(n)hG(i,j)fn(n)2+R(i,j)0fn(n)2i
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Xi2Sj1(i,j)1+Si1nSj1[j=Xi2Sjd(i,j)1+SianSja[j=Xi2Sjd(i,j)SianSja=jfXi2Sjdd(i,j)SianSja Also,giventhatmin-hoproutingisassumed,jPmaxjcanbeatmost2becausetherearetwoandonlytwolinksintheshortestpathfromanynodetoanyofits2-hopneighbors.Thesearethelinkfromthenodeitselftoa1-hopneighborthatconnectstothe2-hopneighborandthelinkthatconnectsthe1-hopneighbortothe2-hopneighbor.Therefore,theboundgivenbyEq. D isequivalentto jfXi2Sjdd(i,j)SianSja<1 58j2N. 167

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APPENDIXEFORMULATIONOFTHESTABILITY-REGION-ADAPTATION-FOR-THROUGHPUT-MAXIMIZATIONPROBLEMASAMIXEDINTEGERPROGRAMWITHNON-LINEARCONSTRAINTS LetK(i,j)betheactive1-hopneighborhoodofnodeiwhenallthenodesinNhavetheirtransmissionrangessetatthemaximumrmax(i.e.,K(i,j),Siawhenrk=rmax8k2N). Let[xij]i,j2Nbetheintegerdecisionvariablesthatindicatewhethernodejiswithinnodei'stransmissionrange(i.e.,xij=1ifrijji,jjj,xij=0otherwise). Theminimumtransmissionrangeofnodeithatguaranteesthatnoneofthelinksusedbytheowsisbrokenisdenotedbyrimin. Theproblemofstabilityregionadaptationforthroughputmaximization(SRA-TM)canbeformulatedasamixedintegerprogramwithnon-linearconstraintsasfollows1. maximizeXn2Fnsubjecttoxii=08i2Nxij=1ificoversjintheMinPowersetupxij=0ifidoesnotcoverjintheMaxPowersetupxijxikifjji,jjj
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ofallthelinkspresentintheMinPowersetup.Thethirdsetofconstraintsguaranteesthatthemaximumtransmissionrangeofthenodesisnotexceededbyanynode.Thefourthsetofconstraintsguaranteesthatthecoverageofeverynodeisomnidirectional.Thefthsetofconstraintsguaranteesthattheowratesarewithinthelower-boundregiongivenbyTheorem 3.1 Thesolutionofthepreviousproblemisgivenintermsofthedecisionvariables[xij]i,j2N,andthesolutionoftheSRA-TMproblemisgivenintermsofthetransmissionranges[ri]i2N.Anysetoftransmissionrangesthatsatisesthesolution[xij]i,j2Nachievesthesameobjective-functionvalueintheSRA-TMproblemanditsmixed-inte-ger-programformulation. 169

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APPENDIXFCHAPTER 4 'SNOTATION Tables F-1 and F-2 summarizethenotationusedthroughoutChapter 4 TableF-1. Chapter 4 'snotation:networkmodel SymbolMeaning NThesetofnodesthatbelongtoatleastoneowLThesetoflinksthatbelongtoatleastoneowHSomesubsetoflinks,i.e.,HLMHThesetofallmaximalschedulesonHCo(MH)TheconvexhullofMHFThesetofowsfnThenthowinFPfnThepathoffnPfnintThepathoffnwithoutthesourcenoderiThetransmissionrangeofnodeiriminTheminimumrithatdoesnotbreakanylinkinLrThevectoroftransmissionrangesofthenodesinNrminThevectorofminimumtransmissionrangesofthenodesinNrmaxThevectorofmaximumtransmissionrangesofthenodesinNSiaTheactive1-hopneighborhoodofi,i.e.,the1-hopneighborsofithatareintermediateordestinationnodesofatleastoneowSidThedirect1-hopneighborhoodofi,i.e.,the1-hopneighborsofithatsenddatapacketstoijji,jjjTheEuclideandistancebetweennodesiandj(i,j)Thelinkdirectedfromnodeitonodejd(i,j)Thedegreeoflink(i,j),i.e.,thenumberofows(i,j)belongstoE(i,j)Thesetoflinksthatinterferewithlink(i,j)E(i,j)rE(i,j)asafunctionofrEjrThesetoflinksthatinterferewithatleastoneincominglinkofnodejasafunctionofrE(i,j)maxE(i,j)revaluatedatrmaxE(i,j)ThesetofpacketratesofthelinksinE(i,j)ThevectorofpacketratesofthelinksinLThevectorofpacketratesofthelinksinLjmaxThehighestpacketratethatnodejsupportsforeachofitsincomingowsthatguaranteesstabilityjmax(Ejr)jmaxasafunctionofEjrfnThepacketrateofowfnfnmaxThehighestpacketratethatowfnsupportswhileguaranteeingstabilityfnmax(r)fnmaxasafunctionofr 170

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TableF-2. Chapter 4 'snotation:potentialgame SymbolMeaning SjThesetofnodesthatareabletoaffectjmaxwiththeirtransmissionranges(Eq. 4 )SfnThesetofnodesthatareabletoaffectfnmaxwiththeirtransmissionranges(Eq. 4 )FnThesetofowswhosehighestsupportedpacketratesareaffectedbyanyofthemovesthatfncanmake(Eq. 4 )RThegame'sactionspace:thesetoffeasibletransmissionrangesofthenodescontrolledbytheows,i.e.,thenodesinSfn2FSfnRnTheactionspaceoffn:thesetoffeasibletransmissionrangesofthenodescontrolledbyfn,i.e.,thenodesinSfnR)]TJ /F7 7.97 Tf 6.59 0 Td[(nThesetoffeasibletransmissionrangesofthenodesnotcontrolledbyfn,i.e.,thenodesin)]TJ 7.47 -.72 Td[(Sfi2FSfinSfnr)]TJ /F7 7.97 Tf 6.58 0 Td[(nThevectoroftransmissionrangesofthenodesnotcontrolledbyfn,i.e.,thenodesin)]TJ 7.47 -.72 Td[(Sfi2FSfinSfnn(r)Theutilityfunctionofowfnasafunctionofrinthenormalized-transport-capacityadaptation-problem(NTC-AP)game(Eq. 4 )(r)Thevectorofutilityfunctions[1(r),2(r),...,N(r)]intheNTC-APgamen(r)TheutilityfunctionofowfnasafunctionofrinthelinearNTC-AP(Lin-NTC-AP)game(Eq. 4 )T(r)Anordinalpotentialfunction(OPF)oftheNTC-APgame(Theorem 4.1 ),andalsothetotalthroughput(Eq. 4 )a(i,j)rThenumberofactivehiddennodesoflink(i,j),i.e.,thenumberofnodesinSianSja,asafunctionofraThevectora(i,j)r(i,j)2Lcj(r)Thecontentionlevelofnodej(Eq. 4 )cfnT(r)Thetotalcontentionexperiencedbyfn(Eq. 4 )cfnV(r)Thecontentionvariationexperiencedbyfn(Eq. 4 )cLin(r)AnOPFoftheLin-NTC-APgame,andalsothetotalcontentionandcontentionvariationexperiencedbyalltheows(Eq. 4 )cNTC(a)TheobjectivefunctionoftheNTC-APasformulatedinEq. 4 171

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APPENDIXGPROOFOFTHEOREM 4.5 Inthenormalized-transport-capacityadaptation-problem(NTC-AP),thegoalistominimizethemaximumcontentionexperiencedbyeachoftheows(Eq. 4 ).Figure G-1 showsthecasesinwhichtransmissionpowers(TP)ofactivenodesdeterminedfromaoptLindonotminimizethemaximumcontentionofaow.ThenotationinFigure G-1 isasfollows.TheowwhosemaximumcontentionisnotminimizedbyaoptLinisf1.TheonlyTPsthatareshownaretheTPsoftheactivenodesthatcanbemodiedinordertominimizethemaximumcontentionoff1.TheTPsshownwithdashedlinesaretheoptimalTPsofthelinearNTC-AP(Lin-NTC-AP)(i.e.,TPsdeterminedfromaoptLin),andtheTPsshownwithsolidlinesaretheoptimalTPsoftheNTC-AP(i.e.,TPsdeterminedfromaoptNTC).InthecaseofonesingleowwhosemaximumcontentionisnotminimizedbyaoptLin,therearethreepossibleTP-congurationsinwhichtheoptimalLin-NTC-APTPsdifferfromtheoptimalNTC-APTPs.Inthegeneralcase,i.e.,whenthemaximumcontentionexperiencedbytwoormoreowsarenotminimizedbyaoptLin,theTP-congurationforeachoftheseowscorrespondstooneofthethreepossibleTP-congurations.Therefore,thethreepossibleTP-congurationsdescribeallthepossiblewaysinwhichoptimalLin-NTC-APTPsdonotminimizethemaximumcontentionexperiencedbyoneormoreowsinthenetwork.TherstoftheseTP-congurationsisshowninFigure G-1A ,thesecondisshowninFigure G-1B andFigure G-1C ,andthethirdisshowninFigure G-1D andFigure G-1E ThefollowinganalysisisbasedonthefollowinghiddennodesinFigure G-1 .InFigure G-1A ,Figure G-1B ,Figure G-1C ,onlyonehiddennodeisconsideredpergure.Thisnodeisthesinkoff2.InFigure G-1D andFigure G-1E ,severalhiddennodesareconsideredpergure.Thesenodesarethesinksoff5,f6,...,f5+jPf1intj. ThecontributionsthatthehiddennodesinFigure G-1A ,Figure G-1B ,Figure G-1C ,Figure G-1D ,andFigure G-1E maketothecontentionvariationoff1andf3aredenoted 172

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ATP-conguration1:sin-gleactivenodeisahiddennodeofasinglelink BTP-conguration2-single-ow:singleactivenodeisahiddennodeofmultiplelinks CTP-conguration2-multiple-ow:singleactivenodeisahiddennodeofmultiplelinks DTP-conguration3(Lin-NTC-APTPs):multipleactivenodesarehiddennodesofmultiplelinks ETP-conguration3(NTC-APTPs):multipleactivenodesarehiddennodesofmultiplelinks FigureG-1. Suboptimaltransmission-power(TP)congurations byf1Vandf3Vrespectively.ThecontributionthatthehiddennodeinFigure G-1C makestothecontentionvariationoff4isdenotedbyf4V.f1VisdifferentfromzeroonlywhentheTPsaredeterminedfromaoptLin(i.e.,dashed-lineTPs).WhentheTPsaredeterminedfromaoptNTC(i.e.,solid-lineTPs),thenodesarenolongerhiddennodesofanyoff1'slinks,sotheydonotcontributetof1'scontentionvariation(i.e.,f1V=0).Ontheotherhand,f3Vandf4VaredifferentfromzeroonlywhentheTPsaredeterminedfromaoptNTC(i.e.,solid-lineTPs).WhentheTPsaredeterminedfromaoptLin(i.e.,dashed-lineTPs),thenodesarenolongerhiddennodesofanyofthelinksoff3andf4,sotheydonotcontributetothecontentionvariationoff3andf4(i.e.,f3V=f4V=0). Itisassumedthatthevaluestakenbyf1V,f3V,andf4Vthataregreaterthanzeromeetthefollowinginequalities.Otherwise,ifthisassumptionisnotmade,the 173

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TableG-1. Objectivefunctionvalues:Transmission-powerconguration1,2-single-ow aoptLinaoptNTC Lin-NTC-APcLin(aoptLin)cLin(aoptLin)+f3V)]TJ /F8 11.955 Tf 11.95 0 Td[(f1VNTC-APcNTC(aoptLin)cNTC(aoptLin))]TJ /F5 11.955 Tf 11.96 0 Td[(dmax TableG-2. Objectivefunctionvalues:Transmission-powerconguration2-multiple-ows aoptLinaoptNTC Lin-NTC-APcLin(aoptLin)cLin(aoptLin)+f3V+f4V)]TJ /F8 11.955 Tf 11.96 0 Td[(f1VNTC-APcNTC(aoptLin)cNTC(aoptLin))]TJ /F5 11.955 Tf 11.96 0 Td[(dmax solutionaoptLinbecomesoptimal.InTP-conguration1(Figure G-1A ),TP-conguration2-single-ow(Figure G-1B ),andTP-conguration3(Figure G-1D andFigure G-1E ),f3V>f1V,andinTP-conguration2-multiple-ows(Figure G-1C ),f3V+f4V>f1V. Tables G-1 to G-3 showtheobjective-functionvaluesoftheLin-NTC-APandtheNTC-APevaluatedataoptLinandaoptNTCforthethreeTP-congurations.ThevaluesoftheobjectivefunctionsevaluatedataoptNTC(i.e.,cLin(aoptNTC)andcNTC(aoptNTC))aregivenintermsofthevaluesoftheobjectivefunctionsevaluatedataoptLin(i.e.,cLin(aoptLin)andcNTC(aoptLin)).Inthisway,thefactorsthatcausethedifferencebetweentheobjective-functionvaluescanbeidentied.Forexample,Table G-1 statesthat cLin(aoptLin))]TJ /F5 11.955 Tf 11.96 0 Td[(cLin(aoptNTC)=f3V)]TJ /F8 11.955 Tf 11.95 0 Td[(f1V,cNTC(aoptLin))]TJ /F5 11.955 Tf 11.96 0 Td[(cNTC(aoptNTC)=dmax. Thefollowinganalysisisdividedintotwoparts.Intherstpart,thereasonthattheLin-NTC-APgamereachessuboptimalsolutionsoftheNTC-APinthethreeTPcongurationsinFigure G-1 isproved.Inthesecondpart,themaximumdifference TableG-3. Objectivefunctionvalues:Transmission-powerconguration3 aoptLinaoptNTC Lin-NTC-APcLin(aoptLin)cLin(aoptLin)+f3V)]TJ /F8 11.955 Tf 11.95 0 Td[(f1VNTC-APcNTC(aoptLin)cNTC(aoptLin))]TJ /F8 11.955 Tf 11.96 0 Td[(2dmax 174

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betweentheoptimalandsuboptimalsolutionsiscalculated.BothpartsarebasedontheeffectsofswitchingfromtheTPpowersgivenbyaoptLin(i.e.,dashed-lineTPs)totheonesgivenbyaoptNTC(i.e.,solid-lineTPs).TheLin-NTC-APgamereachessuboptimalsolutionsoftheNTC-APinthethreeTPcongurationsinFigure G-1 InTP-conguration1(Figure G-1A andTable G-1 ),thedashed-lineTPofthedestinationnodeofowf2partiallycoversowf1anddoesnotcoverf3,andthesolid-lineTPcoversf1completelyandpartiallycoversf3.AccordingtoEq. 4 ,thevalueofcLinchangesduetochangesonthetotalcontention(i.e.,Pfn2FcfnT(r))andcontentionvariation(i.e.,Pfn2FcfnV(r)).Thetotalcontentionisdecreasedbythevalueofthedegreeofthelinkoff1thatispartiallycoveredbythedashed-lineTP,anditisincreasedbythevalueofthedegreeofthelinkoff2thatispartiallycoveredbythesolid-lineTP1.Intheworstcasescenario(i.e.,whencNTC(aoptLin))]TJ /F5 11.955 Tf 12.89 0 Td[(cNTC(aoptNTC)ismaximum),thesetwolinkdegreesareequaltodmax,sothetotalcontentiondoesnotchange.Thecontentionvariationisincreasedbyf3V)]TJ /F8 11.955 Tf 11.54 0 Td[(f1V.Therefore,thetotaldifferencebetweencLin(aoptNTC)andcLin(aoptLin)isf3V)]TJ /F8 11.955 Tf 12.63 0 Td[(f1V(i.e.,cLin(aoptLin))]TJ /F5 11.955 Tf 12.64 0 Td[(cLin(aoptNTC)=f3V)]TJ /F8 11.955 Tf 12.63 0 Td[(f1V).Thisresultisthereasonthat,forTP-conguration1,theLin-NTC-APgameselectsthedashed-lineTP,soitreachesasuboptimalequilibriumthatdoesnotminimizecNTC. InTP-conguration2-single-ow(Figure G-1B andTable G-1 ),thetotalcontentionisdecreasedbythevaluesofthedegreesofthetwolinksoff1thatarepartiallycoveredbythedashed-lineTP,anditisincreasedbythevaluesofthedegreesofthetwolinksoff3thatarepartiallycoveredbythesolid-lineTP.Intheworst-casescenario,thelinkdegreesareequaltodmax,sothetotalcontentiondoesnotchange.Thecontentionvariationisincreasedbyf3V)]TJ /F8 11.955 Tf 12.05 0 Td[(f1V.Therefore,thetotaldifferencebetweencLin(aoptNTC)andcLin(aoptLin)isf3V)]TJ /F8 11.955 Tf 12.53 0 Td[(f1V.Thisresultisthereasonthat,forTP-conguration2-single-ow, 1SeetheexampleinFigure 4-2 fortheexplanation. 175

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theLin-NTC-APgameselectsthedashed-lineTP,soitreachesasuboptimalequilibriumthatdoesnotminimizecNTC. InTP-conguration2-multiple-ows(Figure G-1C andTable G-2 ),thetotalcontentionisdecreasedbythevaluesofthedegreesofthetwolinksoff1thatarepartiallycoveredbythedashed-lineTP,anditisincreasedbythevaluesofthedegreesofthelinkoff3andthelinkoff4thatarepartiallycoveredbythesolid-lineTP.Intheworst-casescenario,thelinkdegreesareequaltodmax,sothetotalcontentiondoesnotchange.Thecontentionvariationisincreasedbyf3V+f4V)]TJ /F8 11.955 Tf 12.58 0 Td[(f1V.Therefore,thetotaldifferencebetweencLin(aoptNTC)andcLin(aoptLin)isf3V+f4V)]TJ /F8 11.955 Tf 12.43 0 Td[(f1V.Thisresultisthereasonthat,forTP-conguration2-multiple-ows,theLin-NTC-APgameselectsthedashed-lineTP,soitreachesasuboptimalequilibriumthatdoesnotminimizecNTC. InTP-conguration3(Figure G-1D ,Figure G-1E ,andTable G-2 ),thetotalcontentionisdecreasedbythevaluesofthedegreesofallthelinksoff1thatarepartiallycoveredbythedashed-lineTPs,anditisincreasedbythevaluesofthedegreesofthelinksoff3thatarepartiallycoveredbythesolid-lineTPs.Intheworst-casescenario,thelinkdegreesareequaltodmax.GiventhatthereareatotalofPf1int+1TPsthatcoverpartiallythelinksoff1andf3,thetotaldecreaseandincreaseareeachequalto(Pf1int+1)dmax.Therefore,thecontentiondoesnotchange.Thecontentionvariationisincreasedbyf3V)]TJ /F8 11.955 Tf 12.53 0 Td[(f1V.Therefore,thetotaldifferencebetweencLin(aoptNTC)andcLin(aoptLin)isf3V)]TJ /F8 11.955 Tf 12.44 0 Td[(f1V.Thisresultisthereasonthat,forTP-conguration3,theLin-NTC-APgameselectsthedashed-lineTPs,soitreachesasuboptimalequilibriumthatdoesnotminimizecNTC.ThemaximumdifferencebetweentheoptimalsolutionoftheNTC-APandthesolutioncalculatedbytheLin-NTC-APgameisbounded ThesuboptimalequilibriumoftheLin-NTC-APgamereachesavalueforcNTCthatisgreaterthantheoptimal(i.e.,minimum)byadifferencewhichisupper-boundedasfollows. 176

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InTP-conguration1(Figure G-1A andTable G-1 ),whentheTPischangedfromthedashed-lineTPtothesolid-lineTP,themaximumcontentionexperiencedbyf1isdecreasedbythevalueofthedegreeofthelinkoff1thatispartiallycoveredbythedashed-lineTP.Themaximumcontentionexperiencedbyf3doesnotchangebecausef4isthecauseofthemaximumcontentionoff3whenthedashed-lineTPisset,andwhenthesolid-lineTPisset,thismaximumcontentionisnotexceeded,i.e.,whenthesolid-lineTPisset,thenodesoff3whoseincominglinksarepartiallycoveredbytheTPsoff4andf2experiencethesamemaximumcontention,whichisequalto2dmax.ThisisshowninFigure G-2 inwhichthenodesoff3withmaximumcontentionarehighlightedandonlytheTPsthatcausetheircontentionareincluded.Therefore,giventhatthemaximumcontentionexperiencedbyf1isdecreasedbyavalueequaltoalinkdegreeandthemaximumcontentionexperiencedbyf3isnotvaried,cNTC(aoptLin)ishigherthancNTC(aoptLin)byatmostdmax. FigureG-2. Nodesoff3withmaximumcontentioninTP-conguration1 InTP-conguration2-single-ow(Figure G-1B andTable G-1 ),themaximumcontentionexperiencedbyf1isdecreasedbythevalueofthedegreeofoneofthetwolinksoff1thatarepartiallycoveredbythedashed-lineTP.Themaximumcontentionexperiencedbyf3doesnotchangebecausef4isthecauseofthemaximumcontentionoff3whenthedashed-lineTPisset,andwhenthesolid-lineTPisset,thismaximumcontentionisnotexceeded,i.e.,whenthesolid-lineTPisset,thenodesoff3whoseincominglinksarepartiallycoveredbytheTPsoff4andf2experiencethesame 177

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maximumcontention,whichisequalto2dmax.Therefore,giventhatthemaximumcontentionexperiencedbyf1isdecreasedbyavalueequaltoonelinkdegreeandthemaximumcontentionexperiencedbyf3isnotvaried,cNTC(aoptLin)ishigherthancNTC(aoptLin)byatmostdmax. InTP-conguration2-multiple-ow(Figure G-1C andTable G-2 ),themaximumcontentionexperiencedbyf1isdecreasedbythevalueofthedegreeofoneofthetwolinksoff1thatarepartiallycoveredbythedashed-lineTP.Themaximumcontentionsexperiencedbyf3andf4donotchangebecausef5isthecauseofthemaximumcontentionsoff3andf4whenthedashed-lineTPisset,andwhenthesolid-lineTPisset,thesemaximumcontentionsarenotexceeded,i.e.,whenthesolid-lineTPisset,thenodesoff3andf4whoseincominglinksarepartiallycoveredbytheTPsoff5andf2experiencethesamemaximumcontention,whichisequalto2dmax.Therefore,giventhatthemaximumcontentionexperiencedbyf1isdecreasedbyavalueequaltoonelinkdegreeandthemaximumcontentionsexperiencedbyf3andf4arenotvaried,cNTC(aoptLin)ishigherthancNTC(aoptLin)byatmostdmax. InTP-conguration3(Figure G-1D ,Figure G-1E ,andTable G-3 ),themaximumcontentionexperiencedbyf1isdecreasedbytwicethevalueofthedegreeofthelinkoff1thatispartiallycoveredbytwodashed-lineTPs.Themaximumcontentionexperiencedbyf3doesnotchangebecausef2andf4arethecauseofthemaximumcontentionoff3whenthedashed-lineTPsareset,andwhenthesolid-lineTPsareset,thismaximumcontentionisnotexceeded,i.e.,whenthesolid-lineTPsareset,thetwonodesoff3whoseincominglinksarepartiallycoveredbytheTPsoff2andf4andbytheTPsoff5andf6respectivelyexperiencethesamemaximumcontention,whichisequalto3dmax.Therefore,giventhatthemaximumcontentionexperiencedbyf1isdecreasedbyavalueequaltotwolinkdegreesandthemaximumcontentionexperiencedbyf3isnotvaried,cNTC(aoptLin)ishigherthancNTC(aoptNTC)byatmost2dmax. 178

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Inthegeneralcase,i.e.,whenthemaximumcontentionexperiencedbytwoormoreowsarenotminimizedbyaoptLin,theTP-congurationforeachoftheseowscorrespondstooneofthethreepossibleTP-congurations.ThemaximumpossibledifferencebetweencNTC(aoptLin)andcNTC(aoptNTC)isachievedwhenTP-congurations1or2single-owarerepeatedasmanytimesaspossible.Thereasonisthatthesecongurationsaretheonesthatusethelessnumberofows.Therefore,theyaretheonesthatcanbereplicatedthehighestnumberoftimes.Foreachreplication,thecNTC(aoptLin))]TJ /F5 11.955 Tf 12.14 0 Td[(cNTC(aoptNTC)valueisincreasedbydmax.ThisisshowninFigure G-3 forTP-conguration2-single-owwhichisrepeatedthemaximumnumberoftimes,i.e.,jFj)]TJ /F6 7.97 Tf 18.13 0 Td[(2 2times. FigureG-3. Worst-casescenario 179

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BIOGRAPHICALSKETCH GustavoVejaranoearnedhisBachelorofScience(BS)degreefromUniversidaddelValle,Cali,Colombiain2005,andhisMasterofScience(MS)andDoctorofPhilosophy(PhD)degreesfromUniversityofFlorida,Gainesville,Florida,UnitedStatesin2009and2011respectively.Gustavo'sresearchinterestsareWirelessMultihopNetworks(WMN),Cyber-PhysicalSystems:AdaptationMechanismsforWMNstoHumanActivityandBehavior,Self-OrganizedCommunicationNetworks,CognitiveRadiosandNetworking,MultimediaCommunications,Cross-LayerDesign,and3rdand4thGenerationCellularNetworks.HehasbeenmemberoftheWirelessandMobileSystemsLaboratory,underthedirectionofDr.JaniseMcNair,since2007. 188