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Control Techniques in Dynamic Communication Networks

Permanent Link: http://ufdc.ufl.edu/UFE0044591/00001

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Title: Control Techniques in Dynamic Communication Networks
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Subramanian, Sankrith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: cdma -- mac -- nonlinear-control -- wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Power control in the Physical Layer of a communication network is used to ensure that each link achieves its target signal-to-interference plus-noise ratio (SINR) to effect communication in the reverse link (uplink) of a wireless cellular communication network. In cellular systems using direct-sequence code-division multiple access (CDMA), the SINR depends inversely on the power assigned to the other users in the system, creating a nonlinear control problem. Due to the spreading of bands in CDMA based cellular communication networks, the interference in the system is mitigated. The non-linearity now arises by the uncertain random phenomena across the radio link, causing detrimental effects to the signal power that is desired at the base station. Mobility of the terminals, along with associated random shadowing and multi-path fading present in the radio link, results in uncertainty in the channel parameters. To quantify these effects, a nonlinear MIMO discrete differential equation is built with the SINR of the radio-link as the state to analyze the behavior of the network. Controllers are designed based on analysis of this networked system, and power updates are obtained from the control law. Analysis is also provided to examine how mobility and the desired SINR regulation range affects the choice of channel update times. Realistic wireless network mobility models are used for simulation and the power control algorithm formulated from the control development is verified on this mobility model for acceptable communication. In the Medium Access Control (MAC) layer of a wireless network that uses Carrier Sense Multiple Access (CSMA), the performance is limited by collisions that occur because of carrier sensing delays associated with propagation and the sensing electronics, and hidden terminals in the network. A continuous-time Markov model is used to analyze and optimize the performance of a system using CSMA with collisions caused by sensing delays. The throughput of the network is quantified using the stationary distribution of the Markov model. An online algorithm is developed for the unconstrained throughput maximization problem. Further, a constrained problem is formulated and solved using a numerical algorithm. Simulations are provided to analyze and validate the solution to the unconstrained and constrained optimization problems. Network packet traffic in the transport layer of Internet-style networks plays a vital role in affecting the throughput in the MAC layer. Common queue length management techniques on nodes in such networks focus on servicing the packets based on their Quality of Service (QoS) requirements (e.g., Differentiated-Services, or DiffServ, networks). In Chapter 4, continuous control strategies are suggested for a DiffServ network to track the desired ensemble average queue length level in the Premium and Ordinary Service buffers specified by the network operator. A Lyapunov-based stability analysis is provided to illustrate global asymptotic tracking of the ensemble average queue length of the Premium Service buffer. In addition, arrival rate delays due to propagation and processing that affects the control input of the Ordinary Service buffer is addressed, and a Lyapunov-based stability analysis is provided to illustrate global asymptotic tracking of the ensemble average queue length of this service. Simulations demonstrate the performance and feasibility of the controller, along with showing global asymptotic tracking of the queue lengths in the Premium Service and Ordinary Service buffers.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sankrith Subramanian.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Dixon, Warren E.
Local: Co-adviser: Shea, John M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044591:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044591/00001

Material Information

Title: Control Techniques in Dynamic Communication Networks
Physical Description: 1 online resource (94 p.)
Language: english
Creator: Subramanian, Sankrith
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

Subjects / Keywords: cdma -- mac -- nonlinear-control -- wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Power control in the Physical Layer of a communication network is used to ensure that each link achieves its target signal-to-interference plus-noise ratio (SINR) to effect communication in the reverse link (uplink) of a wireless cellular communication network. In cellular systems using direct-sequence code-division multiple access (CDMA), the SINR depends inversely on the power assigned to the other users in the system, creating a nonlinear control problem. Due to the spreading of bands in CDMA based cellular communication networks, the interference in the system is mitigated. The non-linearity now arises by the uncertain random phenomena across the radio link, causing detrimental effects to the signal power that is desired at the base station. Mobility of the terminals, along with associated random shadowing and multi-path fading present in the radio link, results in uncertainty in the channel parameters. To quantify these effects, a nonlinear MIMO discrete differential equation is built with the SINR of the radio-link as the state to analyze the behavior of the network. Controllers are designed based on analysis of this networked system, and power updates are obtained from the control law. Analysis is also provided to examine how mobility and the desired SINR regulation range affects the choice of channel update times. Realistic wireless network mobility models are used for simulation and the power control algorithm formulated from the control development is verified on this mobility model for acceptable communication. In the Medium Access Control (MAC) layer of a wireless network that uses Carrier Sense Multiple Access (CSMA), the performance is limited by collisions that occur because of carrier sensing delays associated with propagation and the sensing electronics, and hidden terminals in the network. A continuous-time Markov model is used to analyze and optimize the performance of a system using CSMA with collisions caused by sensing delays. The throughput of the network is quantified using the stationary distribution of the Markov model. An online algorithm is developed for the unconstrained throughput maximization problem. Further, a constrained problem is formulated and solved using a numerical algorithm. Simulations are provided to analyze and validate the solution to the unconstrained and constrained optimization problems. Network packet traffic in the transport layer of Internet-style networks plays a vital role in affecting the throughput in the MAC layer. Common queue length management techniques on nodes in such networks focus on servicing the packets based on their Quality of Service (QoS) requirements (e.g., Differentiated-Services, or DiffServ, networks). In Chapter 4, continuous control strategies are suggested for a DiffServ network to track the desired ensemble average queue length level in the Premium and Ordinary Service buffers specified by the network operator. A Lyapunov-based stability analysis is provided to illustrate global asymptotic tracking of the ensemble average queue length of the Premium Service buffer. In addition, arrival rate delays due to propagation and processing that affects the control input of the Ordinary Service buffer is addressed, and a Lyapunov-based stability analysis is provided to illustrate global asymptotic tracking of the ensemble average queue length of this service. Simulations demonstrate the performance and feasibility of the controller, along with showing global asymptotic tracking of the queue lengths in the Premium Service and Ordinary Service buffers.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sankrith Subramanian.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Dixon, Warren E.
Local: Co-adviser: Shea, John M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044591:00001


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CONTROLTECHNIQUESINDYNAMICCOMMUNICATIONNETWORKSBySANKRITHSUBRAMANIANADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012 1

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c2012SankrithSubramanian 2

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Tomymotherandfather 3

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ACKNOWLEDGMENTS Iexpressmymostsincereappreciationtomysupervisorycommitteechairandmentor,Dr.WarrenE.Dixon.Ithankhimfortheeducation,advice,andtheencouragementthathehadprovidedmewithduringthecourseofmystudyattheUniversityofFlorida.IalsothankDr.JohnM.Sheaforlendinghisknowledgeandsupport,andprovidingtechnicalguidance.Itisagreatpriviledgetohaveworkedwithsuchfar-thinkingandinspirationalindividuals.AllthatIhavelearntandaccomplishedwouldnothavebeenpossiblewithouttheirdedication.IalsoextendmyappreciationtoDr.JessW.CurtisandDr.EduardoL.Pasiliaofortheirsupportandcollaborativeeorts,alongwithmycolleaguesatUniversityofFloridaforencouragingsomethought-provokinganalyticaldiscussions.Mostimportantly,IwouldliketoexpressmydeepestappreciationtomyparentsP.R.SubramanianandIndhumathiSubramanian,andmysisterShilpaSubramanian.Theirlove,understanding,patienceandpersonalsacricemadethisdissertationpossible. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 7 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 2POWERCONTROLFORCELLULARCOMMUNICATIONSWITHTIME-VARYINGCHANNELUNCERTAINTIES ................. 19 2.1NetworkModelandProperties ......................... 20 2.2LinearPrediction ................................ 22 2.3ControlDevelopment ............................. 25 2.3.1ControlObjective ............................ 25 2.3.2Closed-LoopErrorSystem ....................... 25 2.4StabilityAnalysis ................................ 26 2.5SimulationResults ............................... 28 2.6Power-ControlMechanism ........................... 34 3THROUGHPUTMAXIMIZATIONINCSMANETWORKS ........... 40 3.1ThroughputMaximizationinCSMANetworkswithCollisions ....... 40 3.1.1NetworkModel ............................. 40 3.1.2CSMAMarkovChain .......................... 42 3.1.3ThroughputMaximization ....................... 44 3.1.4SimulationResults ........................... 48 3.2ThroughputMaximizationinCSMANetworkswithCollisionsandHiddenTerminals .................................... 53 3.2.1NetworkModel ............................. 53 3.2.2CSMAMarkovChain .......................... 54 3.2.3ThroughputMaximization ....................... 57 3.2.4SimulationResults ........................... 59 4CONGESTIONCONTROLFORDIFFERENTIATED-SERVICESNETWORKSWITHARRIVAL-RATEDELAYS ......................... 62 4.1QueuingSystemModel ............................. 62 4.2PremiumService ................................ 64 4.2.1ControlDesign ............................. 65 4.2.2StabilityAnalysis ............................ 66 5

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4.3OrdinaryService ................................ 69 4.3.1StabilityAnalysis ............................ 70 4.4SimulationResults ............................... 72 5CONCLUSION .................................... 77 5.1SummaryofResults .............................. 77 5.2RecommendationsforFutureWork ...................... 78 APPENDIX AESTIMATIONOFRANDOMPROCESSES .................... 79 A-1GeneralMMSEbasedestimationtheory ................... 79 A-2GaussianCase .................................. 80 BORTHOGONALITYCONDITION ......................... 82 CPROOFOFP0 .................................. 84 REFERENCES ....................................... 86 BIOGRAPHICALSKETCH ................................ 94 6

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LISTOFTABLES Table page 2-1Comparisonagainstvariouspredictionwindowsizes. ............... 33 2-2Percentageofsampletimesexperiencingoutagefordierentnumberofusersinthecellofinterest.Thecontrolgainkpistunedforthesystembasedonthenumberofusers,andke=1:310)]TJ /F6 7.97 Tf 6.58 0 Td[(4.ThepredictionwindowsizesareselectedbasedontheconditiondetZ(referto Table2-1 forthebestwindowsizeselection). ....................................... 34 2-3Percentageofsampletimesexperiencingoutageforunquantized,2-bitand3-bitpower-controlcommands. .............................. 38 3-1Optimalvaluesofthemeantransmissionratesfora2-linkcollisionnetworkforvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ). ......................................... 50 3-2Optimalvaluesofthemeantransmissionratesfora3-linkcollisionnetworkforvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ). ......................................... 51 3-3Averagenumberofcollisionsfora2-linkand3-linkcollisionnetworksforvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ). 52 3-4Optimalvaluesofthemeantransmissionratesfora3-linkcollisionnetworkwithhiddenterminals(refertoFig. 3-7 )forvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{21 )-( 3{23 ). .......................... 60 7

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LISTOFFIGURES Figure page 2-1Error,channelgain,andpowerplotofaMTwithmaximumDopplerfrequency1.35Hz. ........................................ 30 2-2PredictionerroroftheMTwithmaximumDopplerfrequency1.35Hz. ..... 31 2-3Error,channelgain,andpowerplotofaMTwithmaximumDopplerfrequency31.60Hz. ....................................... 31 2-4PredictionerroroftheMTwithmaximumDopplerfrequency31.60Hz. ..... 32 2-5Uplinkpower-controlmechanism. .......................... 35 2-6ProbabilitydensityfunctionoftheSINRerrorsofalltheMTsoperatingathigh(>25Hz)maximumDopplerfrequencies ...................... 36 2-7ComparisonagainstSong'spowercontrolalgorithm. ................ 39 3-1An-linknetworkscenarioandconictgraph. ................... 41 3-2CSMAMarkovchainfora2-linkscenariowithcollisionstates. .......... 43 3-3Meantransmissionratesofnodes1,2,and3transmittingtothesamenode4.Allnodesareinthesensingregion.Theonlinealgorithmof( 3{9 )isusedwithT=100ms,K=5,andTs=0:001ms. ...................... 48 3-4Queuelengthsofnodes1and2transmittingtothesamenode3.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ).Allnodesareinthesensingregion.Ts=0:01ms,R1=6:05dataunits/ms,R2=6:49dataunits/ms,1=0:16dataunits/ms,2=0:2dataunits/ms. ................................ 51 3-5Queuelengthsofnodes1,2,and3transmittingtothesamenode4.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ).Allnodesareinthesensingregion.Ts=0:01ms,R1=6:54dataunits/ms,R2=10:19dataunits/ms,R3=11:49dataunits/ms,1=0:02dataunits/ms,2=0:05dataunits/ms,3=0:05dataunits/ms. ..................................... 52 3-6An(n+k)-linknetworkscenarioandconictgraph. ............... 54 3-7CSMAMarkovchainwithcollisionstatesfora3-linknetworkscenariowithhiddenterminals. ....................................... 55 8

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3-8Queuelengthsofnodes1,2,and4transmittingtothesamenode3.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{21 )-( 3{23 ).Allnodesareinthesensingregion,andTs=0:01ms,R1=3:78dataunits/ms,R2=3:58dataunits/ms,R3=2:23dataunits/ms,1=0:02dataunits/ms,2=0:05dataunits/ms,3=0:05dataunits/ms. .... 61 4-1SchematicofaDiServQueueingSystem. ..................... 63 4-2EnsembleaveragequeuelengthandserviceratesforPremiumServicewithoutarrival-ratedelays. .................................. 74 4-3EnsembleaveragequeuelengthandaveragearrivalratesforOrdinaryServicewithoutarrival-ratedelays. .............................. 74 4-4EnsembleaveragequeuelengthandserviceratesforPremiumServicewithaveragearrival-ratedelay. ................................... 75 4-5Ensembleaveragequeuelengthandaveragearrival-rateforOrdinaryServicewithaveragearrival-ratedelay. ........................... 76 9

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCONTROLTECHNIQUESINDYNAMICCOMMUNICATIONNETWORKSBySankrithSubramanianDecember2012Chair:WarrenE.DixonCochair:JohnM.SheaMajor:ElectricalandComputerEngineeringPowercontrolinthePhysicalLayerofacommunicationnetworkisusedtoensurethateachlinkachievesitstargetsignal-to-interference-plus-noiseratio(SINR)toeectcommunicationinthereverselink(uplink)ofawirelesscellularcommunicationnetwork.Incellularsystemsusingdirect-sequencecode-divisionmultipleaccess(CDMA),theSINRdependsinverselyonthepowerassignedtotheotherusersinthesystem,creatinganonlinearcontrolproblem.DuetothespreadingofbandsinCDMAbasedcellularcommunicationnetworks,theinterferenceinthesystemismitigated.Thenonlinearitynowarisesbytheuncertainrandomphenomenaacrosstheradiolink,causingdetrimentaleectstothesignalpowerthatisdesiredatthebasestation.Mobilityoftheterminals,alongwithassociatedrandomshadowingandmulti-pathfadingpresentintheradiolink,resultsinuncertaintyinthechannelparameters.Toquantifytheseeects,anonlinearMIMOdiscretedierentialequationisbuiltwiththeSINRoftheradio-linkasthestatetoanalyzethebehaviorofthenetwork.Controllersaredesignedbasedonanalysisofthisnetworkedsystem,andpowerupdatesareobtainedfromthecontrollaw.AnalysisisalsoprovidedtoexaminehowmobilityandthedesiredSINRregulationrangeaectsthechoiceofchannelupdatetimes.Realisticwirelessnetworkmobilitymodelsareusedforsimulationandthepowercontrolalgorithmformulatedfromthecontroldevelopmentisveriedonthismobilitymodelforacceptablecommunication. 10

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IntheMediumAccessControl(MAC)layerofawirelessnetworkthatusesCarrierSenseMultipleAccess(CSMA),theperformanceislimitedbycollisionsthatoccurbecauseofcarriersensingdelaysassociatedwithpropagationandthesensingelectronics,andhiddenterminalsinthenetwork.Acontinuous-timeMarkovmodelisusedtoanalyzeandoptimizetheperformanceofasystemusingCSMAwithcollisionscausedbysensingdelays.ThethroughputofthenetworkisquantiedusingthestationarydistributionoftheMarkovmodel.Anonlinealgorithmisdevelopedfortheunconstrainedthroughputmaximizationproblem.Further,aconstrainedproblemisformulatedandsolvedusinganumericalalgorithm.Simulationsareprovidedtoanalyzeandvalidatethesolutiontotheunconstrainedandconstrainedoptimizationproblems.Networktracinthetransportlayerofend-to-endcongestionnetworksplaysavitalroleinaectingthethroughputintheMAClayer.CommonqueuelengthmanagementtechniquesonnodesinsuchnetworksfocusonservicingthepacketsbasedontheirQualityofService(QoS)requirements(e.g.,Dierentiated-Services,orDiServ,networks).InChapter 4 ,continuouscontrolstrategiesaresuggestedforaDiServnetworktotrackthedesiredensembleaveragequeuelengthlevelinthePremiumandOrdinaryServicebuersspeciedbythenetworkoperator.ALyapunov-basedstabilityanalysisisprovidedtoillustrateglobalasymptotictrackingoftheensembleaveragequeuelengthofthePremiumServicebuer.Inaddition,arrivalratedelaysduetopropagationandprocessingthataectsthecontrolinputoftheOrdinaryServicebuerisaddressed,andaLyapunov-basedstabilityanalysisisprovidedtoillustrateglobalasymptotictrackingoftheensembleaveragequeuelengthofthisservice.Simulationsdemonstratetheperformanceandfeasibilityofthecontroller,alongwithshowingglobalasymptotictrackingofthequeuelengthsinthePremiumServiceandOrdinaryServicebuers. 11

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CHAPTER1INTRODUCTIONTheeldofcommunicationsfacesamultitudeofchallengeswhileprovidingQualityofService(QoS)forabroadclassofapplications.Forinstance,themobilityofnodesinawirelessnetworkcauserandomshiftsinthedopplerfrequenciesofthesignalthatisbeingtransmittedthenodetoareceivernode.Inaddition,duetothepresenceofobstaclesinthepath,scatteringofthesignaltakesplace,andthereceivedsignalisthesummationoftheserandomphase-shiftedmulti-pathsignals.Thisphenomenaiscommonlyknownasmulti-pathfading,andvariousmodelsaredevelopedintheliteraturetocharacterizethephenomena.Varioustechniquessuchaspowercontrol,adaptivemodulationandcoding,symbolmappingdiversity,time/spacediversityreceptionetc.areusedtomitigatemulti-pathfading.Varioustransmitterpower-controlmethodshavebeendevelopedtodeliveradesiredqualityofservice(QoS)inwirelessnetworks[ 1 { 20 ].Earlyworkonpowercontrolusingacentralizedapproachwasinvestigatedin[ 1 ],whichintroducedtheconceptofsignal-to-interference(SIR)-balancing,whereitisdesiredthatallreceiversachievethesameSIR.In[ 2 ],theoptimalsolutiontotheSIR-balancingproblemisderivedbyreformulatingtheproblemasaneigenvalue/eigenvectorproblemandinvokingthePerron-Frobeniustheorem.Methodsweredevelopedtoreduceco-channelinterferenceforagivenchannelallocationusingtransmitterpowercontrolin[ 3 ]and[ 5 ].In[ 5 ],theperformanceofoptimumtransmit-poweralgorithmsareanalyzedintermsofoutageprobabilities.Astochasticdistributedtransmit-powerapproachwasalsoinvestigatedin[ 3 { 5 ].Thesealgorithmswereframedwithonlypathlossaectingthechanneluncertainty.Adistributedautonomouspower-controlalgorithmwasintroducedin[ 6 ],wherechannelreuseismaximized.Basedonalinearanalysisofthesystem,andconstrainingtheeigenvalues,thepowerapproachesanoptimalpowervector.Ageneralizedframeworkforuplinkpowercontrolisprovidedin[ 8 ],wherecommonpropertiesforinterference 12

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constraintsareidentied.Anupperlimitforthepowerwasimposedforeachuserintheconstrainedpower-controlalgorithmof[ 7 ].Activelinkprotection(ALP)schemeswereintroducedin[ 11 ]and[ 13 ],wheretheQoSofactivelinksismaintainedaboveathresholdlimittoprotectthelinkquality.AnoptimumpowercontrollerformulticellCDMAwirelessnetworkswasdesignedin[ 12 ],wherethechannelwasassumedtobeslowlyvaryingwithoutfading.In[ 15 { 20 ]powercontrolalgorithmsaredesignedforsystemswithradiochanneluncertaintiescausedbymobilityoftheuserterminals.Thesechanneluncertaintiesincludeexponentialpathloss,shadowing,andmultipathfading,whicharemodeledasrandomvariablesinthesignal-to-interferenceplusnoiseratio(SINR)measurements.Optimization-basedapproachesthatcanprovidefeaturessuchasoutageguarantees,robustness,andpowerminimizationinthepresenceoffadingbutthatrequireknowledgeofallchannelgainsarepresentedin[ 15 { 19 ].Adistributedpower-controlschemewassuggestedin[ 20 ];however,thefadingprocessismodeledasslowlychangingsothatthechannelgaincanbeaccuratelyestimated,andpracticallimitationsonthetransmissionpowerarenotconsidered.Multipathfadinghasthemostcriticaleectonthedesignofapower-controlsystembecauseofthetimeandamplitudescales.Multipathfadingiscausedbyreectionsintheenvironment,whichcausemultipletime-delayedversionsofthetransmittedsignaltoaddtogetheratthereceiver.Thetimeosetscausethesignalstoaddwithdierentphases,andthusmultipathfadingcanchangesignicantlyoverdistancescalesasshortasafractionofawavelength.Forinstance,forasystemusingthe900MHzcellularband,thechannelcoherencetime(thetimeforwhichthechannelisessentiallyinvariant)foraMTtravelingat30miles/hourisapproximately10ms.Toallowthepowercontrollertocompensateforfastfadinginthechannel,channelpredictionmaybeused.Linearmodels,referredtoasautoregressivemovingaverageprocesswithexogenousinput(ARMAX),wereusedin[ 21 22 ]forthepower-control 13

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process.In[ 22 ],ageneralizedpredictivecontrolmethodwasdevelopedtocounterloop-delayinclosedloopDS-CDMApowercontrol.Alinearpredictionmethodisusedin[ 23 ]topredictalinkparameter.Ashorttermfadingpredictionisdonein[ 24 25 ].Hallenetal.focusedonlong-rangefadingprediction[ 26 { 28 ]basedonthefactthattheamplitude,frequencyandphaseofeachmultipathcomponentvarymuchslowerthantheactualfadingcoecient.ThefocusofChapter 2 istodevelopaSINR-basedpower-controlalgorithmthatwouldreducetheoutageprobabilityintheradiolinkbypredictingthepowerofthechannel.Theprediction-basedpower-controlprocessisdevelopedbasedontheevolutionofradio-linkparametersfromtheSINRdynamicsandtheavailablefeedbackSINRmeasurements.InChapter 2 ,theradiochannelcharacteristicsdiscussedaboveareanalyzed,andthefadingpowerispredictedandusedinthecontroldesign.Forthispurpose,alinearminimummean-squareerror(LMMSE)predictorisusedtoobtainareliablepredictionofthefadingcoecientatthenextinstance.Inourpreviouswork[ 29 ],thepredictorusedmeasurementsofthefadingprocess.Inpractice,onlytheSINRcanbemeasureddirectly.ALMMSEpredictorisdevelopedthatusesonlySINRmeasurementsandestimatesoftheDopplerfrequencythatcanbederivedfromlocalSINRmeasurements,inclusiveofpathlossandshadowing.ThemotivationbehindusingtheSINRmeasurementsaloneisthatitisnotpossibletocalculatethefadingpowerfromtheSINRmeasurementswhenthelatterisaectedbyshadowing,pathloss,andinterferenceinadditiontofastfading.ALyapunov-basedanalysisisperformedtoprovideanultimateboundontheSINRerror,thesizeofwhichcanbereducedbychoosingappropriatecontrolgains.Inaddition,variationsinothercomponentsoftheradiochannelsuchaspathlossandlog-normalshadowingarealsoaccountedforusingthisanalysistool.ThecontrolleruseslocalSINRmeasurements[ 6 ],[ 11 ]fromthecurrentandneighboringcellstomaintaintheSINRsofMTsintheacceptablecommunicationrange,providedchannelgainsarelimitedtosomepracticalregionofoperation.Therealchannelgainsmaybearbitrarilylow,inwhich 14

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casenopowercontrolalgorithmcanachievethedesiredperformanceduetolimitsontheavailablepower.Inthesecases,thecontrollermaynotbeabletoregulatetheSINRintothedesiredrange,andoutagemayoccur,wheretheSINRfallstoolowforacceptablecommunication.Simulationisusedtoassesstheperformanceoftheproposedpredictionandpower-controlalgorithm.Theeectsofthechoiceofpredictionwindowsizeandquantizationofthepower-controlcommandareassessed.Inaddition,theperformanceiscomparedwithapreviouslyproposedup/downpowercontrolalgorithmfromtheliterature[ 30 ].IntheMediumAccessControl(MAC)layerofawirelessnetwork,collisionsduetotransmissionofpacketsbymorethanonenodetothesamereceiverresultsinpacketdropsatthereceiver.SuchabberationsoccurduetothepresenceofsensingdelaysinCarrierSenseMultipleAccess(CSMA)networks,andpresenceofHiddenterminals(HTs)inthenetwork.Inaddition,queueingconstraintsofthepacketsintheTransportLayerofanetworkcausescongestionanddelaysofpacketsinthenode.TherehasbeenasignicanteorttomodelvariousformsofCSMAprotocolsoverthepastfewyears[ 31 { 33 ].WorkonMAClayerthroughputoptimizationfocusesonmanipulatingspecicparametersoftheMAClayerincluding,forexample,windowsizesandtransmissionrates,tomaximize/optimizethethroughputinthepresenceofconstraints.Forexample,CarrierSenseMultipleAccess(CSMA)MarkovchainbasedthroughputmodelingandanalysisoftheMACalgorithmswereintroducedin[ 31 32 ],whileperformanceandthroughputanalysisoftheconventionalBinomialexponentialbackoalgorithmshavebeeninvestigatedin[ 34 35 ].Inmostcases,previousMAC-layeroptimizationalgorithmshavefocusedprimarilyonparametersandfeedbackfromtheMAClayerbyexcludingcollisionsduringtheanalysis(cf.[ 31 33 ]).InChapter 3 ,wedevelopacontinuous-timeMarkovmodelforasystemusingCSMAthatincorporatestheeectofcollisionsandallowsoptimizationofthetransmissionratesofthenetworktomaximizethroughputormeetspeciedthroughputtargets.Thepurposeofthisworkis 15

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todevelopapproachesthatwillbeusefulinfuturecross-layeroptimizationandcontrolalgorithms.PreliminaryworkonCSMAthroughputmodelingandanalysiswasdonein[ 31 ]basedontheassumptionthatthepropagationdelaybetweenneighboringnodesiszero.AcontinuousMarkovmodelwasdevelopedthatprovidedtheframeworkandmotivationforthiswork.In[ 33 ],acollision-freemodelisusedtoquantifyandoptimizethethroughputofthenetwork.Thefeasibilityofthearrivalratevectorguaranteesthereachabilityofmaximumthroughput,whichinturnsatisestheconstraintthattheservicerateisgreaterthanorequaltothearrivalrate,assumingthatthepropagationdelayiszero.Ingeneralcommunicationnetworks,eectsofpropagationdelayplayacrucialroleinmodelingandanalyzingthethroughputofthenetwork.Recenteortsattemptedvariousstrategiestoincludedelaymodelsinthethroughputmodel.Forexample,in[ 36 ],delayisintroduced,andisusedtoanalyzeandcharacterizetheachievablerateregionforstaticCSMAschedulers.Collisions,andhencedelayisincorporatedin[ 37 ]intheMarkovmodel,andthemeantransmissionlengthofthepacketsisusedasthecontrolvariabletomaximizethethroughput.Inthisdissertation,amodelforpropagationdelayisproposedandincorporatedinthemodelforthroughput.Thismodelallowsforthetransmissionratestobeselectedtomaximizethroughputinanunconstrainedoptimizationproblemandtomeetfeasiblethroughputgoalsinaconstrainedoptimizationproblem.Inaddition,collisionsduetohiddenterminalsinthenetworkarealsomodeledandanalyzed.Linkthroughputisoptimizedbyoptimizingthewaitingtimesinthenetwork.QueuelengthmanagementindynamicnetworkssuchastheInternethasbeenalongstandingresearchfocus.Severalqueueingnetworkmodelshavebeenproposedforsuchnetworkstoperformcongestioncontrol.In[ 38 ],awidelyusedframeworkwasintroducedformodelingtheInternetwhereeachowisassociatedwithautilityfunctionandtheobjectiveistomaximizetheaggregateutilitysubjecttolinkconstraints.Thereafter,in[ 39 ]and[ 40 ],areviewofaclassofprimal-dualalgorithmswasperformedanddesign 16

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guidelineswereprovidedforsuchalgorithmsthatfeaturedynamicadaptationsatbothendsofadecentralizedend-to-endcongestionnetwork.AnExponentialRandomEarlyDetection(E-RED)wasdevelopedthatmodiesthestandardTCP-RenoandRandomEarlyDetection(RED)algorithms.Theproblemofinputtracbasedmodelingofinternet-stylenetworkswasaddressedin[ 41 ]withanemphasisonqueuelengthevolutionandserverratelimitations.Multi-servicearchitecturesfortheInternetsuchasIntServandDiServarchitectures(cf.[ 42 43 ])havealsobeenanareaofrecentinterest.Thesearchitecturescharacterizethepacketsbasedontheirlossanddelayrequirements,andhenceprioritizebasedontheQoSneeded.Tipperet.al(cf.[ 41 ])developeddierentialequationmodelsbasedontheapproximatemodelof[ 44 ]thatdescribethebehaviorofthenetworkbytime-varyingprobabilitydistributionsandanonlineardierentialmodelforrepresentingthedynamicsofthenetworkintermsoftime-varyingmeanquantities(cf.[ 41 ])forcomputernetworksundernonstationaryconditions.SuchmodelsarealsoknownasFluidFlowModels(FFM).ControleortsinsuchFFMsfocusonprovidingqueuemanagementservices.Classicallinearanalysistechniqueswereemployedin[ 45 ]forAsynchronousTransferMode(ATM)congestioncontrolproblems,andtheusageofprobabilisticfeedbackshowedbetterperformanceinthesenseofreducingsteadystateoscillations.AnalyticalmodelswereintroducedforATMRoutingin[ 46 ]andcontrolandoptimizationalgorithmsweresuggested.Astochasticlinearmodelforowinnetworkswasstudiedfromacontroltheoreticperspectivein[ 47 ].Subsequently,in[ 48 { 50 ],linearanalysistechniqueswereemployedforcongestioncontrolproblems.Adaptiveowcontrollersforhighresourceutilizationweredevelopedin[ 51 ]and[ 52 ].Nonlinearowcontrollerswereintroducedin[ 53 { 55 ]forATMbasednetworksusingtheframeworkintroducedby[ 41 ]and[ 44 ].Mostofthesetechniquesintroducedwereheuristicwithelaboratesimulationstodemonstratethesystembehavior.AnIntegratedDynamicCongestionController(IDCC)wasdevelopedin[ 56 ]basedonadaptivenonlinearcontroltechniques,andLyapunov-basedcongestion 17

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controldevelopmentwasintroduced.Anultimatelyboundedstabilityresultisobtainedbyassumingthatthederivativeoftheensembleaveragearrivalrateisboundedbyaniteconstant.Aslidingmodevariablestructurecongestioncontrollerwasutilizedin[ 57 ]basedontheFFM.In[ 58 ],asecondorderslidingmodecontrollerwasintroducedthatclaimedthattheensembleaveragearrivalrateforpremiumservicewasunknownwhileusingthesameinthecontroller.Recently,anewclassofcontinuouscontrollersweredevelopedthatasymptoticallystabilizesaclassofnonlinearsystemsinthepresenceofboundedsucientlysmoothdisturbances(cf.[ 59 60 ]).ByusingtheRISEdesignapproach,acontinuouscongestioncontrolstrategyisdevelopedinChapter 4 usingonlytheerrormeasurementsbetweentheactualandthedesiredensembleaveragequeuelengthforPremiumTracServiceforDiServnetworks.Thisapproachisdierentfrom[ 57 ]inthesensethatthecontrolleriscontinuous,andglobalasymptoticregulationoftheensembleaveragequeuelengthinthePremiumServicebuerisobtained.TheinevitablepresenceofdelayinthearrivalratesduetopropagationandprocessingisaddressedinthecontroldevelopmentforOrdinaryServices,andglobalasymptoticregulationoftheensembleaveragequeuelengthintheOrdinaryServicebuerisobtained. 18

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CHAPTER2POWERCONTROLFORCELLULARCOMMUNICATIONSWITHTIME-VARYINGCHANNELUNCERTAINTIESPowercontrolinacode-divisionmultipleaccess(CDMA)basedcellularnetworkisachallengingproblembecausethecommunicationchannelschangerapidlybecauseofmultipathfading.Theserapiductuationscausedetrimentaleectsonthecontroleortsrequiredtoregulatethesignal-to-interferenceplusnoiseratios(SINRs)tothedesiredlevel.Thus,thereisaneedforpower-controlalgorithmsthatcanadapttorapidchangesinthechannelgaincausedbymultipathfading.Muchofthepreviousworkhaseitherneglectedtheeectsoffastfading,assumedthatthefadingisknown,orassumedthatallthelinkgainsareknown.Inthischapter,wemodeltheeectsoffastfadinganddeveloppracticalstrategiesforrobustpowercontrolbasedonSINRmeasurementsinthepresenceofthefading.WedevelopacontrollerforthereverselinkofaCDMAcellularsystem,anduseaLyapunov-basedanalysistoprovethattheSINRerrorisgloballyuniformlyultimatelybounded.WealsoutilizealinearpredictionlterthatutilizeslocalSINRmeasurementsandestimatesoftheDopplerfrequencythatcanbederivedfromlocalSINRmeasurementstoimprovetheestimateofthechannelfadingusedinthecontroller.Thepower-controlalgorithmissimulatedforacellularnetworkwithmultiplecells,andtheresultsindicatethatthecontrollerregulatestheSINRsofallthemobileterminals(MTs)withlowoutageprobability.Inaddition,apulse-code-modulationtechniqueisappliedtoallowthecontrolcommandtobequantizedforfeedbacktothetransmitter.SimulationresultsindicatethattheoutageprobabilitiesofalltheMTsarestillwithintheacceptablerangeifatleast3-bitquantizationisemployed.Comparisonstoastandardalgorithmillustratetheimprovedperformanceofthepredictivecontroller. 19

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2.1NetworkModelandPropertiesWeconsiderthereverselinkofacellularsystememployingCDMA.TheSINRxi(l)2Risdened(indB)foreachradiolinki=1;2;:::n,as xi(l)=10logagi(l)Pi(l) Ii(l);(2{1)wherel2Z,thefunctionlog()denotesthebase10logarithm,gi(l)2RisthechannelgainintheradiolinkbetweenMTiandtheBaseStation(BS),Pi(l)2RisthepowertransmittedbyMTitotheBS,a2Risthebandwidthspreadingfactorortheprocessinggain[ 61 ]denedastheratioofthetransmissionbandwidth(inHertz)tothedatarate(inbits/second),andIi(l)2RistheinterferencefromtheMTsinallthecells,denedas Ii(l)=Xj6=igj(l)Pj(l)+i:(2{2)In( 2{2 ),i2Rdenotesthethermalnoisepowerinlinki,whichisassumedtobeaconstantvaluegreaterthanzero.SincethenoisepowerisboundedandtheinterferencepowerfromeachMTislessthanitstransmitpower,Ii()isnon-zeroandbounded.Thechannelgaingi(l)in( 2{1 )ismodeledas[ 62 ] gi(l)=gd0di(l) d0)]TJ /F4 7.97 Tf 6.59 0 Td[(100:1i(l)jHi(l)j2:(2{3)In( 2{3 ),gd02Risthenear-eldgain(see[ 63 ]formodeldetails).Thesecondfactorin( 2{3 )istheexponentialpathloss,whichdependsonthethedistancedi(l)2RfromMTitotheBSandthepath-lossexponent,2R,whichtypicallytakesvaluesbetweentwoandve.Exponentialpathlossholdsinaregionoutsidethenear-eldregion(i.e.,theregionsatisfyingdfd0di(l),wheredfistheFraunhoferdistance).MTscannottravelwithindistanced0oftheBSandonlycommunicatewiththeBSiftheyarewithinapredeterminedradiusofcoverage,sodi()isnon-zeroandboundedwithinaparticularoperatingcell.Thefactors100:1i(l)andjHi(l)j2in( 2{3 )areusedtomodellarge-scale 20

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log-normalshadowing(frombuildings,terrain,orfoliage)andsmall-scalemultipathfading,respectively.Foranalyticalpurposes,theshadowingisgenerallymodeledaslog-normal;i.e.,i(l)2RisaGaussianrandomprocess.ThefadingisoftenmodeledasRayleighfading,whereHi(t)isusuallytakentobeazero-mean,complex-valued,wide-sensestationaryGaussianrandomprocess[ 63 ],andthusjH(t)jisaRayleighrandomvariableforeacht.However,bothoftheseprocessesareunbounded,whichmeansthatanynon-negativechannelgainispossible,andhenceanyreceivedpowerlevelispossible.However,gi(l)cannottakearbitrarilylargevaluesinpracticebecausethereceivedpowercannotexceedthetransmittedpower.Furthermore,acellularsystemcannotpracticallytransmittooverfadeduserswhoareinverydeepfades(i.e.,whengi(l)isclosetozero)becausedoingsowouldrequireextremelylargepoweratthatuserandtheotherusers(becausethepowertransmittedtoeachusercausesinterferenceattheotherusers)[ 64 ].Hence,thesubsequentcontrol-systemdevelopmentisbasedontheassumptionthattheshadowinggain100:1i(l)andfadinggainjHi()j2arebothboundedfromaboveandbelow.However,theperformanceissimulatedin section2.5 and section2.6 forchannelsthatmayresultinarbitrarilylowsignallevels,whichmayresultinthepower-controlalgorithmfailingtoregulatetheSINRtothedesiredregion.UnderstandinghowtheSINRchangesisbenecialforthedevelopmentandanalysisofthesubsequentpower-controllaw.Takingtherstdierenceof( 2{1 )yields xi(l) Ts=[10log(agi(l+1)))]TJ /F1 11.955 Tf 11.95 0 Td[(10log(agi(l))] Ts+ui(l) Ts)]TJ /F1 11.955 Tf 13.15 8.09 Td[([10log(Ii(l+1)))]TJ /F1 11.955 Tf 11.95 0 Td[(10log(Ii(l))] Ts; (2{4) whereTsisthesamplingtimeofthenetwork,andui(l)2Rdenotesanauxiliarycontrolsignaldened8i=1;2;::::;nas ui(l),10[log(Pi(l+1)))]TJ /F1 11.955 Tf 11.96 0 Td[(log(Pi(l))];(2{5) 21

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whichisusedtodeterminethepowerupdatelaw.TheSINRatthenextupdateintervalxi(l+1)2Rcanthenbeexpressedas xi(l+1)=%i[gi(l+1);Ii(l+1)])]TJ /F3 11.955 Tf 11.95 0 Td[(%i[gi(l);Ii(l)]+xi(l)+ui(l);(2{6)wherethefunctional%i2Risdened8i=1;2;::::;nas %i(yi;zi)=10logayi zi:(2{7) 2.2LinearPredictionThedevelopmentofapowercontrollerforradiolinksinaCDMAnetworkischallengingduetorapid,largescalechangesinSINRandisexacerbatedbyaconstraintthateachlink'stransmitpowerislessthansomePmax2R.Inthischapter,weattempttoimproveperformancebyestimatingtheSINRagi(l+1)=Ii(l+1)tocompensateforthedelaysinmeasurementandcontrol.NotethatthevariouschannelcomponentsthatcontributetotheSINR,suchasfadingandshadowingpowerandpathlossarenotcomputablefromthereceivedSINR,whichmotivatesourdesignbasedontheSINR.LetXi(),gi(l)=Ii(l).Weuselinearminimummean-squareerror(LMMSE)predictionofXi(l)givenn1pastvalues,Xi(l)]TJ /F1 11.955 Tf 12.36 0 Td[(1);Xi(l)]TJ /F1 11.955 Tf 12.36 0 Td[(2);::;Xi(l)]TJ /F3 11.955 Tf 12.36 0 Td[(n1).TheLMMSEestimatoris[ 65 ] ^Xi(l)=l)]TJ /F6 7.97 Tf 6.58 0 Td[(1Xm=l)]TJ /F4 7.97 Tf 6.59 0 Td[(n1(m)ifXi(m))]TJ /F3 11.955 Tf 11.96 0 Td[(g+(2{8)wherethecoecients(m)idependonthesecond-orderstatisticsofXi(l),isthemeanoftherandomprocessXi()foralll.Letfi,vi cosibetheDopplerfrequencyofMTi,whereviisthevelocityofmotionoftheMT,iistheanglebetweenthetransmittedsignalandthedirectionofmotionoftheMT,andisthewavelengthofthetransmittedsignal.TheDopplerfrequencyoftheMTcanbeestimatedfromtheSINRmeasurements(cf.[ 66 ]).LetTpbethepredictionobservationsamplingtime,whichisselectedsuchthat 22

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itisatleasttheNyquistrate,i.e.,twicetheexpectedmaximumoftheDopplerfrequenciesoftheMTs[ 28 ].Forthesubsequentdesignofthepredictor,denebi()2Rforeachradiolinkithatquantiesthechannelwithoutfading,i.e., bi()=gd0di() d0)]TJ /F4 7.97 Tf 6.59 0 Td[(100:1i() Ii():(2{9)The(m)i'sin( 2{8 )satisfytheorthogonalitycondition[ 65 ].Deningi,h(l)]TJ /F6 7.97 Tf 6.59 0 Td[((n1)]TJ /F6 7.97 Tf 6.59 0 Td[(1))i,....,(l)iiandusingtheorthogonalityconditionyields Ti=266666666664E[bi(l)bi(l)]TJ /F3 11.955 Tf 11.96 0 Td[(n1)]EjHi(l)j2jHi(l)]TJ /F3 11.955 Tf 11.96 0 Td[(n1)j2...E[bi(l)bi(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1)]EjHi(l)j2jHi(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1)j2377777777775TZ)]TJ /F6 7.97 Tf 6.58 0 Td[(1;(2{10)whereZjk=E[bi(l)]TJ /F1 11.955 Tf 11.95 0 Td[((n1)]TJ /F3 11.955 Tf 11.96 0 Td[(j))bi(l)]TJ /F1 11.955 Tf 11.96 0 Td[((n1)]TJ /F3 11.955 Tf 11.95 0 Td[(k))jHi(l)]TJ /F1 11.955 Tf 11.95 0 Td[((n1)]TJ /F3 11.955 Tf 11.96 0 Td[(j))j2jHi(l)]TJ /F1 11.955 Tf 11.96 0 Td[((n1)]TJ /F3 11.955 Tf 11.95 0 Td[(k))j28j;k=1;2;::::;n1)]TJ /F1 11.955 Tf 11.19 0 Td[(1,andwehaveusedthefactthatbi()isindependentofjHi()j2.Here, E[bi(l)]TJ /F1 11.955 Tf 11.96 0 Td[((n1)]TJ /F3 11.955 Tf 11.95 0 Td[(j))bi(l)]TJ /F1 11.955 Tf 11.95 0 Td[((n1)]TJ /F3 11.955 Tf 11.96 0 Td[(k))]=E1 (Ii(l)]TJ /F1 11.955 Tf 11.96 0 Td[((n1)]TJ /F3 11.955 Tf 11.95 0 Td[(j)))2| {z },RdE"gd0di(l)]TJ /F1 11.955 Tf 11.96 0 Td[((n1)]TJ /F3 11.955 Tf 11.96 0 Td[(j)) d0)]TJ /F4 7.97 Tf 6.58 0 Td[()]TJ /F1 11.955 Tf 5.48 -9.69 Td[(100:1i(l)]TJ /F6 7.97 Tf 6.59 0 Td[((n1)]TJ /F4 7.97 Tf 6.59 0 Td[(j))2#| {z },RI; (2{11) sincethenumeratorin( 2{9 )isindependentofthedenominator.RIresultsfromslowvariationsinthepathlossandshadowing,andhencecanbeestimatedfromtimeaverages.Also,theinterferenceduringthedurationofthepredictionsamplingcanbetreatedasapproximatelyconstant[ 67 ],whichisareasonableapproximationwhenthespreadingfactorislarge.BasedontheseassumptionsRdandRIareapproximatedas1. 23

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TheautocovariancefunctionforjHi()j2isRjHij2(lTp)J20(2fn(lTp))[ 68 ],[ 69 ],whereJ0isthezeroth-orderBesselfunctionoftherstkind,andfnisthemaximumDopplerfrequency.Therefore,from( 2{10 ), Ti=266666664J20(2fn(Tpn1))J20(2fn(Tp(n1)]TJ /F1 11.955 Tf 11.96 0 Td[(1)))...J20(2fnTp)377777775TZ)]TJ /F6 7.97 Tf 6.59 0 Td[(1;(2{12)wherethecomponentsofZaredened8j;k=1;2;::::;n1as Zjk=Zkj=8><>:J20(2fn(Tpjj)]TJ /F3 11.955 Tf 11.96 0 Td[(kj));j6=kjHij2;j=k;(2{13)fn6=0andjHij2isthevarianceoftherandomprocessjHi()j2foralll:TheDopplerfrequencyofeachMTismeasuredperiodicallyandthisisusedtoupdatethecoecientsoftheLMMSEestimator.Notethatthecoecientsofiin( 2{12 )areboundedifthecovariancematrixin( 2{13 )isinvertible,whichwilloccurwithprobability1ifTpislessthantheNyquistrate[ 28 ])andtheeectofmeasurementnoiseisconsidered.Thus,thelinearpredictor^Xi()isbounded.Tosummarizethealgorithmforcalculatingthechannelestimate,anarrayofpreviousandcurrentSINRmeasurementsareinputstothelinearpredictorratherthanthefadingpowerjHi(l)j2.Ateveryinstant,thepredictor,basedontheavailableSINRmeasurementsandtheautocorrelationmodeloffading,givesanestimate^Xi(l)ofthequantityXi(l).Inourimplementation,themeanofthevariableXi(l)iscalculatedfrom200initialsamplesoftheSINRmeasurementsandthetransmitterpowerused,andtakingtheweightedaverageof10(0:1xi(m))=(aPi(m)).TheconstantsiandZ)]TJ /F6 7.97 Tf 6.59 0 Td[(1in( 2{12 )arecalculatedfromtheautocovariancefunctionforjHi()j2(andcanbecalculatedoineandstoredforaquantizedsetofDopplerfrequencies).Thepredictedquantity^Xi(l)in( 2{8 )isaninputtothecontroller(referto section2.3 and section2.4 forcontroldevelopment). 24

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Linearpredictionofthefadingprocessrequiresmeasurementofthexi()atthecurrentandpreviousinstances;theperformanceofthepredictorcanbeimprovedbyincreasingthenumberofmeasurementsn1usedtopredictthefadingprocessatinstancel.Practically,asthenumberoftimesamplesusedintheestimatorbecomeslarge,theperformanceofthepredictordoesnotimprovebutdegradesbecausethematrixZbecomesillconditioned. 2.3ControlDevelopment 2.3.1ControlObjectiveThenetworkQoScanbequantiedbytheabilityoftheSINRtoremainwithinaspeciedoperatingrangewithupperandlowerlimits,min;max2Rforeachlinkdened8i=1;2;::::;nas i;minxi(l)i;max;(2{14)wherei;minandi;maxdependonthequality-of-servicerequirementsofmobilestationi.KeepingtheSINRabovetheminimumthresholdeliminatessignaldropout,whereasremainingbelowtheupperthresholdminimizesinterferencetoadjacentcells.ThecontrolobjectiveforthefollowingdevelopmentistoregulatetheSINRtoatargetvaluei2Rsuchthati;minii;max,whileensuringthattheSINRremainsbetweenthespeciedlowerandupperlimitsforeachchannel.Toquantifythisobjective,aregulationerrorei(l)2Risdenedas ei(l)=xi(l))]TJ /F3 11.955 Tf 11.96 0 Td[(i;8i=1;2;::::;n:(2{15) 2.3.2Closed-LoopErrorSystemBytakingtherstdierenceof( 2{15 ),using( 2{3 ),( 2{6 ),and( 2{7 ),andpropertiesofthelog()function,theopen-looperrordynamicsforeachlinkcanbedeterminedas ei(l)=gi(l+1))]TJ /F3 11.955 Tf 11.96 0 Td[(gi(l)+ui(l);(2{16) 25

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wheretheauxiliaryfunctiongi()2Risdened8i=1;2;::::;nas gi()=xi())]TJ /F1 11.955 Tf 11.95 0 Td[(10log(aPi())(2{17)wherer nPi=12gi()isboundedbasedontheexplanationin section2.1 .Basedon( 2{16 )andthesubsequentstabilityanalysis,theauxiliarypowercontrollerui(l)isdesignedas ui(l)=)]TJ /F1 11.955 Tf 11.3 0 Td[((kp+ke)ei(l))]TJ /F1 11.955 Tf 13.73 3.02 Td[(^Yi(l+1)+gi(l);(2{18)where^Yi(l+1)2Risdened8i=1;2;::::;nas ^Yi(l+1)=10logn^Xi(l+1)o;(2{19)and ^Xi()6=0:(2{20)where^Xi()aregivenin( 2{8 ),andthepredictionobservationsamplingrateischosentobeatleasttheNyquistratefor( 2{20 )tohold.From( 2{5 ),( 2{18 ),and( 2{19 ),thepowerupdatelawforeachradiochannelisobtained8i=1;2;::::;nas [Pi(l+1)]dB=)]TJ /F1 11.955 Tf 11.3 0 Td[((kp+ke)ei(l))]TJ /F1 11.955 Tf 11.96 0 Td[(10logna^Xi(l+1)o+xi(l):(2{21) 2.4StabilityAnalysisTheorem1:Thepowerupdatelawin( 2{21 )ensuresthatallclosedloopsignalsarebounded,andthattheSINRregulationerrorapproachesanultimatebound2R,whichcanbedecreasedwithincreasingkpin( 2{18 )uptothemaximumpowerlimitsanddecreasingthesamplingintervalsuptopracticallimits,providedkein( 2{18 )isselectedas 0
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Proof. LetV(e;l):D[0;1)!Rbeapositivedenitefunctiondenedas V(e;l)=nXi=11 2e2i(l):(2{23)Takingtherstdierenceof( 2{23 ),byusingthefactthat(ab)=ab+ba+ab,andsubstitutingfor( 2{16 )yields V=nXi=1ei(l)[gi(l+1))]TJ /F3 11.955 Tf 11.96 0 Td[(gi(l)+ui(l)]+e2i(l);(2{24)whereei(l)istheerrorbetweenthesamplingtimeforradiolinki,andnPi=1e2i(l)isboundedbyaconstantc,thesizeofwhichcanbecontrolledbythesamplingtime.Ananalysisforthisclaimcanbedevelopedasin[ 70 ],thoughthesubsequentsimulationiscarriedoutbychoosingahigh(andfeasible)samplingrate.Substituting( 2{18 )into( 2{24 )yields VnXi=1)]TJ /F3 11.955 Tf 11.96 0 Td[(kee2i(l)+vuut nXi=1e2i(l)(gi(l+1))]TJ /F1 11.955 Tf 13.73 3.02 Td[(^Yi(l+1))2+nXi=1)]TJ /F3 11.955 Tf 11.96 0 Td[(kpe2i(l)+c:(2{25)NotethatkpisusedtodampoutnPi=1(gi(l+1))]TJ /F1 11.955 Tf 14.51 3.02 Td[(^Yi(l+1))2in( 2{25 )whilekeistheproportionalgainusedbythecontrollerwhere0
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Basedon( 2{23 )and( 2{27 ),anupperboundfortheSINRerrorcanbedevelopedas nXi=1e2i(l)nXi=1e2i(l0)(1)]TJ /F3 11.955 Tf 11.95 0 Td[(ke)l+ 1)]TJ /F1 11.955 Tf 11.95 0 Td[((1)]TJ /F3 11.955 Tf 11.95 0 Td[(ke)l ke!25c2 kp+c:(2{28)Theassumptionthatgi(l)2L1,thefactthat^Yi(l)2L1from section2.2 .,( 2{19 ),and( 2{20 ),andthefactthatei(l)2L1from( 2{28 )canbeusedtoconcludethatui(l)2L1from( 2{18 ),andhencePi(l+1)2L1from( 2{21 ).Basedon( 2{28 ),asl!1,thenorm-squaredSINRerrorisultimatelyboundedas((25c2)=(kekp))+(c=ke).Theultimateboundcanbedecreasedbyincreasingkp;however,themagnitudeofkpispracticallyrestrictedbytheconstraintthat0
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reusepattern.TenMTsweresimulatedineachcell.TheRandom-WaypointmodelisusedtosimulatethemobilityoftheMTs,withtheinitialtopologydrawnfromthesteadystate(stationary)distribution(cf.[ 72 ],[ 73 ]).Themobilevelocityateachwaypointisrandomlychosenfromauniformdistributionbetween2km/hrand48km/hr.Thus,theprobabilitydensityfunctionofthevelocityisgivenby[ 73 ]fi(v)=Ch vf0Vjh(v),wheref0Vjh(v)=1 48km/hr)]TJ /F6 7.97 Tf 6.59 0 Td[(2km/hr=1 46km/hrandCh=14:47isanormalizationconstant.ThesubscripthisusedtodenotethephaseoftheMT[ 73 ].ThevelocityforeachoftheMTsisobtainedusingtheinversetransformmethod[ 74 ]as v=exp(3:179r+0:6931);(2{29)whererisuniformlydistributedbetween0and1.Thepurposeofthesimulationsectionistodetailtheperformanceofthecontroller,andthisisdonebyincludingtheplotoftheworst-casescenariooftheradio-link,i.e.,whentheDopplerfrequencyishigh(referto Figure2-3 ).Thesimulationswererepeated10times(MonteCarloSimulations)operating70MTs(10MTsineachofthetypicalseven-cellreusepattern)ineachsimulationsothatthedatacollectedforthesubsequentanalysisissucient.Also,eachsimulationwascarriedoutwithxedcontrolgainskpandke.TheaveragevalueoftheoutageprobabilitiesoftheMTsoperatingineachofthefourmaximumDopplerfrequencyrangesaretabulated(referto Table2-1 )alongwiththefeasiblewindowsizeforvariousrangesoftheDopplerfrequencies.Pathloss,withfreespacepropagationeectsandlog-normalshadowing,ismodeled[ 63 ]asshownin( 2{3 ).Theangleismeasuredperiodically,andtheDopplerfrequencyisobtainedfrom( 2{29 ),whichisusedtogeneratetheRayleighfadingandupdatethecoecientsoftheLMMSEpredictor.Thechannelsamplingtime(Ts)andpredictionobservationsamplingtime(Tp)arebothsetto1:7ms,basedonperformingacontinuoustimeSINRerroranalysis[ 70 ].ThetargetSINR,wassetto8dB,withadesiredoperatingrangebetween6and10dB,whichisdenedin subsection2.3.1 .Thermalnoise, 29

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Figure2-1. Error,channelgain,andpowerplotofaMTwithmaximumDopplerfrequency1.35Hz. ,wassetto)]TJ /F1 11.955 Tf 9.3 0 Td[(83dBm.TheinitialpowerlevelforallMTswaschosenas10dBm.Also,thepredictionwindowsizeisupdatedonlinetoavoidanill-conditionedmatrixZ.Startingataspeciedmaximumpredictionwindowsize,thesizeofthewindowisconsecutivelyreducedby1untildetZ10)]TJ /F6 7.97 Tf 6.59 0 Td[(5.TheresultsinFigs. 2-1 2-4 areobtainedwithkp=0:65,ke=1:310)]TJ /F6 7.97 Tf 6.59 0 Td[(4,andthespreadingfactoraischosenas512,whichisthemaximumforWidebandCDMAsystems.Notethatthesamevaluesofthecontrolgainsandspreadingfactorarealsousedinthesubsequentsimulations.Thecontrolgainsweretunedusingsimulationswithadierentsetofrandomseedsthanthoseusedintheperformanceevaluation.Theoutputofthelinearpredictorislimitedto^Xmax=47dBforthereasonsexplainedin section2.2 Figure2-1 showstheSINRerror,channelgainandpowerplotsofaMTthathasamaximumDopplerfrequencyof1:35Hz.NotethattheDopplerfrequenciesinsimulationsaregeneratedfromtheaforementionedtopologymodel.ADopplerfrequency 30

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Figure2-2. PredictionerroroftheMTwithmaximumDopplerfrequency1.35Hz. Figure2-3. Error,channelgain,andpowerplotofaMTwithmaximumDopplerfrequency31.60Hz. 31

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Figure2-4. PredictionerroroftheMTwithmaximumDopplerfrequency31.60Hz. of1:35HzrepresentsaMTwithlowmobility.ThepredictionerrorforthisMTisshownin Figure2-2 Figure2-1 indicatesthatthepowercontrollerregulatestheSINRoftheMTwithinthedesiredrange(minxi()max)withlowoutageprobability. Figure2-3 showstheSINRerror,channelgainandpowerplotsofaMToperatingwithamaximumDopplerfrequencyof31:60Hz.ADopplerfrequencyof31:60HzrepresentsaMTwithhighmobility1.Thedottedlinesnotetheregionsofdeepfades,whichresultinlargepredictionerrors,asshownin Figure2-4 .Theinaccuracyofthelinearpredictorandthelimitsonmaximumtransmitpower(and,correspondingly,controleort)inthedeepfadedzonescauseoutageattheMTatthosetimes.TheSINRofthisradiolinkoperatingwith 1MTswithhighervelocitiescanrelyontimediversity,ratherthanfading,tooperateinafadingchannel. 32

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Table2-1. Comparisonagainstvariouspredictionwindowsizes. Average%ofsamplessuchthatximinMax.Dopplerfrequencyrange(Hz)BestwindowsizesuchthatdetZMax.Pred.windowsizeof1Max.Pred.windowsizeof2Max.Pred.windowsizeof3Max.Pred.windowsizeof4 0)]TJ /F1 11.955 Tf 11.96 0 Td[(10210:625:19)]TJ 85.81 0 Td[()]TJ /F1 11.955 Tf -453.61 -14.44 Td[(10)]TJ /F1 11.955 Tf 11.95 0 Td[(202;315:624:016:91)]TJ /F1 11.955 Tf -453.61 -14.45 Td[(20)]TJ /F1 11.955 Tf 11.95 0 Td[(30319:9413:427:29)]TJ /F1 11.955 Tf -453.61 -14.45 Td[(30)]TJ /F1 11.955 Tf 11.95 0 Td[(403;422:989:887:005:07 amaximumDopplerfrequencyof31:60Hzisintheacceptablecommunicationrangeatallothertimes,andtherequiredpowerisintheimplementablerange.Simulationswerecarriedoutforprediction-basedpower-controlalgorithmswithdierentpredictionwindowsizesbasedonthesametopologymodelwithtenMTsinacelltocomparetheresults. Table2-1 showstheaverage%outagesfordierentrangesofthemaximumdopplerfrequency(cf.[ 15 75 ])oftheMTswhenthesimulationiscarriedoutusingdierentpredictionwindowsizes.Theaverage%outagesfortheMTswerecomputedbyrunning5-10simulationsandclassifyingtheMTsbasedontheirmaximumdopplerfrequencies(column1in Table2-1 ).ThebestwindowsizeisthemaximumvalueofthewindowsizesothatthematrixZisnotill-conditioned(i.e.,detZ),andthecorrespondingaverage%outageisenteredinbold.Themaximumdopplerfrequencyismeasuredfrequently(cf.[ 66 ]andthereferencestherein),i.e.,400Tsinthissimulation,andthemeasuredvaluesareusedtocalculatethelinearcoecientsmi,8m=1;2;::::;n1)]TJ /F1 11.955 Tf 12.17 0 Td[(1.Itcanbeinferredthattheseboldedvaluesfallwithinthethresholdlevelforvoicecommunications.Forvoicecommunications,thetypicaloutagetargetis10%[ 76 ].Theresultsin Table2-2 showtheperformanceofthepredictivecontrolalgorithmfordierentnumbersofuserspercell.Outageprobabilitieslessthan10%canbeachievedfor10,20,or40userspercell.However,thecontrolgainkpmustbeincreasedasthenumber 33

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Table2-2. Percentageofsampletimesexperiencingoutagefordierentnumberofusersinthecellofinterest.Thecontrolgainkpistunedforthesystembasedonthenumberofusers,andke=1:310)]TJ /F6 7.97 Tf 6.58 0 Td[(4.ThepredictionwindowsizesareselectedbasedontheconditiondetZ(referto Table2-1 forthebestwindowsizeselection). Average%ofsampleswhereximinDopplerfreq.range(Hz)10users20users40users 0)]TJ /F1 11.955 Tf 11.96 0 Td[(101:32:22:610)]TJ /F1 11.955 Tf 11.95 0 Td[(202:14:15:020)]TJ /F1 11.955 Tf 11.95 0 Td[(305:16:16:730)]TJ /F1 11.955 Tf 11.95 0 Td[(405:58:49:5 Bestkp0:650:71Avg.TransmitPower)]TJ /F1 11.955 Tf 9.3 0 Td[(16:47dBm)]TJ /F1 11.955 Tf 9.3 0 Td[(15:51dBm)]TJ /F1 11.955 Tf 9.29 0 Td[(14:75dBm ofuserstoachievethisoutageprobability,andthisresultsinanincreaseintheaveragetransmittedpowerperuser. 2.6Power-ControlMechanismInpractice,thenumberofbitsthatcanbesentforpowerupdatestothemobileterminalislimited.Thus,thissectionconsidersthedesignofapower-controlmechanismthatselectsfromanitesetofpoweradjustments.Variousresultsintheliteraturefocusondevelopingquantizedpower-controlalgorithms[ 14 21 77 ].Apower-controlalgorithmwithaxedstepsizewasintroducedin[ 14 ].Duetothetime-varyingnatureoftheradiochannel,theperformanceofthismechanismislimited.Apulse-code-modulationrealizationwasdevelopedin[ 77 ]toreducetheoutageprobabilitybyvaryingtherangeofthepowerupdates.Inthissection,apower-updatemechanismbasedonthepulse-code-modulationrealizationisusedtoupdatethetransmitterpoweratthemobileterminal,andtheoutageprobabilitiesoftheradiolinksarecomparedwiththeoutageprobabilitieswithoutquantizationobtainedin section2.5 .Therealizationofthepower-controlcommandisbasedontheerrorsignalgeneratedattheBS,asshownin Figure2-5 .Thequantizationoftheerrorsignalisdonebyanalyzingtheprobabilitydensityfunction( Figure2-6 )oftheworstcaseunquantized 34

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Figure2-5. Uplinkpower-controlmechanism. errorsignals(cf. section2.5 ),i.e.,theradiolinksoperatingatthehighDopplerfrequency.Notethattheprobabilitydensityfunctionoftheerrorsignale(l)isrepresentedasfE(e).Weassumethatapowercontrolcommandisonlyissuediftheerrorsignalislarge.Thepresenceofapowercontrolcommandisusuallysignaledbyaseparatecontrolbit(asinIS-95/cdma2000).Thus,fork-bitquantization,2k+1levelscanbeused,whereonelevelmapstoazerocommand.TheerroristhenquantizedbypartitioningtheempiricaldensityoftheerrorsignalsthatoperateathighmaximumDopplerfrequenciesthatareobtainedfromaseparatesimulationoftheunquantizedsystem(toavoidover-training),shownin Figure2-6 ,intobinsofequalprobability.Thequantizedvalueofthecorrespondingcontrolisthendenedasthemediangiventhatthesignalliesinthatbin,asthatisfoundtooerbetterperformancethanothermeasures,suchastheconditionalmean.Thequantizationschemedependsonthenumberofbitsusedfor 35

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Figure2-6. ProbabilitydensityfunctionoftheSINRerrorsofalltheMTsoperatingathigh(>25Hz)maximumDopplerfrequencies quantization.For3-bitquantization,thequantizederrorsignal(indB)isgivenby upcmi(l)=8>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>:3:18ifui(l)2(2:08;1)1:47ifui(l)2(1:10;2:08]0:77ifui(l)2(0:44;1:10]0:17ifui(l)2()]TJ /F1 11.955 Tf 9.29 0 Td[(0:07;0:44]0ifei(l)2()]TJ /F1 11.955 Tf 9.3 0 Td[(0:035;0:035))]TJ /F1 11.955 Tf 9.3 0 Td[(0:31ifui(l)2()]TJ /F1 11.955 Tf 9.3 0 Td[(0:56;)]TJ /F1 11.955 Tf 9.29 0 Td[(0:07])]TJ /F1 11.955 Tf 9.3 0 Td[(0:81ifui(l)2()]TJ /F1 11.955 Tf 9.3 0 Td[(1:09;)]TJ /F1 11.955 Tf 9.29 0 Td[(0:56])]TJ /F1 11.955 Tf 9.3 0 Td[(1:47ifui(l)2()]TJ /F1 11.955 Tf 9.3 0 Td[(1:98;)]TJ /F1 11.955 Tf 9.29 0 Td[(1:09])]TJ /F1 11.955 Tf 9.3 0 Td[(3:00ifui(l)2(;)]TJ /F1 11.955 Tf 9.3 0 Td[(1:98]:(2{30) 36

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For2-bitquantization,thequantizederrorsignal(indB)isgivenby upcmi(l)=8>>>>>>>>>><>>>>>>>>>>:2:08ifui(l)2(1:10;1)0:44ifui(l)2()]TJ /F1 11.955 Tf 9.29 0 Td[(0:07;1:10]0ifei(l)2()]TJ /F1 11.955 Tf 9.3 0 Td[(0:035;0:035))]TJ /F1 11.955 Tf 9.3 0 Td[(0:56ifui(l)2()]TJ /F1 11.955 Tf 9.3 0 Td[(1:09;)]TJ /F1 11.955 Tf 9.29 0 Td[(0:07])]TJ /F1 11.955 Tf 9.3 0 Td[(1:98ifui(l)2(;)]TJ /F1 11.955 Tf 9.3 0 Td[(1:09]:(2{31)Thethresholdsontheerrorwhennopowercontrolcommandisissuedistuned(to0:035dB,inthiscase)basedonrepeatedsimulationoftheunquantizedsystem,quantizingthecontrolsignal,simulationofthequantizedsystem,andperformanceanalysisintermsofoutageprobability.MonteCarloSimulationswerecarriedoutonthenetworktopologyasdescribedin section2.5 ,usingthe2-bit(22=4combinations)and3-bit(23=8combinations)quantizederrorsignalstodeterminethen-bitpowercontrolcommanddecisionthatisprovidedtotheMT.Resultswereobtainedbyrstsimulatingusingtheunquantizedpowercontroller(i.e.,powercontrollerwithinnitefeedbackbandwidth).Anothersimulationiscarriedoutbyseedingtheprecedingsimulationusingthesamerandomseeds,butnowusinga2-bitfeedback.Similarly,resultsareobtainedfora3-bitfeedback.Then,10newsimulationsareexecutedusingtheunquantizedcontroller,andtheabovementionedprocessisrepeatedfor2-bitand3-bitfeedback.Dataiscollected,storedandtabulatedin Table2-3 Table2-3 showstheaverageoutageprobabilityofthevariousschemes(unquantized,2-bit,and3-bitpowercontrolcommand)obtainedfromsuchrepeatedsimulationstocompareandchoosethebest(intermsofreducingtheoutageprobability)possiblequantizationschemebasedonthebandwidthconstraints.From Table2-3 ,a3-bitpowercontrolcommandsignalprovidesperformancethatfallsintheacceptableregionforvoicecommunication,andhencethisschemecanbeusedinconjunctionwiththecontrollertodeliverthedesiredQoSforeachradiolink.Notethatthecontrolgainskpandkearexedthroughoutthecourseofthesimulations. 37

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Table2-3. Percentageofsampletimesexperiencingoutageforunquantized,2-bitand3-bitpower-controlcommands. Average%ofsampleswhereximinDopplerfreq.range(Hz)Unquantizedcontrolcommand3-bitcommand2-bitcommand 0)]TJ /F1 11.955 Tf 11.95 0 Td[(101:31:41:510)]TJ /F1 11.955 Tf 11.95 0 Td[(202:12:73:820)]TJ /F1 11.955 Tf 11.95 0 Td[(305:17:311:530)]TJ /F1 11.955 Tf 11.95 0 Td[(405:59:813:6 Wecomparedtheperformanceofourcontrolalgorithmwiththeup/downpowercontrolalgorithmdescribedandanalyzedin[ 30 ].Theup/downpowercontrolalgorithmuses1-bitfeedbacktodeterminewhethertoadjustthepowerupordownbyaxed0:5dB.Wecomparetheperformanceoftheup/downpowercontrollertothepowercontrolalgorithmdevelopedinthischapterbothwithandwithoutchannelprediction.Theresultsareillustratedin Figure2-7 .Theresultsshowthattheuseof3-bitfeedbackwithourcontrolalgorithmprovidessubstantialgainsoverthe1-bitup/downcontrolalgorithmforallmobilevelocities.ForDopplerfrequenciesover10Hz,theuseofchannelpredictionprovidesasignicantadditionalperformancegain,especiallyathighDopplerfrequencies.ForinstanceformobileradioswithDopplerfrequenciesbetween30Hzand40Hz,theup/downpowercontrollerhasoutageprobabilityover0.22.Usingthepowercontrolalgorithmdevelopedinthischapter,butwithoutchannelprediction,lowerstheoutageprobabilitytolessthan0.19.Theadditionofchannelpredictionfurtherlowerstheoutageprobabilitytolessthan0.1,therebysatisfyingthetypicaltargetoutageprobabilityformobilevoicecommunications.Thus,thebenetsofusingchannelpredictionandmulti-bitfeedbackaredemonstrated. 38

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Figure2-7. ComparisonagainstSong'spowercontrolalgorithm. 39

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CHAPTER3THROUGHPUTMAXIMIZATIONINCSMANETWORKS 3.1ThroughputMaximizationinCSMANetworkswithCollisionsInMAClayerofawirelessnetworkthatusesCSMA,theperformanceislimitedbycollisionsthatoccurbecauseofcarriersensingdelaysassociatedwithpropagationandthesensingelectronics.Inthischapter,acontinuous-timeMarkovmodelisusedtoanalyzeandoptimizetheperformanceofasystemusingCSMAwithcollisionscausedbysensingdelays.ThethroughputofthenetworkisquantiedusingthestationarydistributionoftheMarkovmodel.Anonlinealgorithmisdevelopedfortheunconstrainedthroughputmaximizationproblem.Further,aconstrainedproblemisformulatedandsolvedusinganumericalalgorithm.Simulationsareprovidedtoanalyzeandvalidatethesolutiontotheunconstrainedandconstrainedoptimizationproblems. 3.1.1NetworkModelConsideraninfrastructurenetwork,suchasawirelesslocalareanetwork(WLAN),consistingofanaccesspointandnmobilestations.TherearenlinksconnectingthestationstotheAP,asshowninFig. 3-1 .Allofthenodesinthenetworkareassumedtosensethetransmissionsofalloftheothernodes,providedthatthetransmissionsdonotbeginwithinaxedsensingdelay,Ts.IftwoormorenodesinitiatepackettransmissionwithinTs,therewillbeacollision,andallofthepacketsinvolvedinthetransmissionareassumedtobelost.InatypicalCSMAnetwork,thetransmitterofnodekbacksoforarandomperiodbeforeitsendsapackettoitsdestinationnode,ifthechannelisidle.Ifthechannelisbusy,thetransmitterfreezesitsbackocounteruntilthechannelisidleagain.Itisassumedthatthebackotime,orthewaitingtimeofeachlinkkisexponentiallydistributedwithmean1=Rk.TheobjectiveinthischapteristodeterminetheoptimalvaluesofthemeantransmissionratesRk,k=1;2;:::;n,sothatthethroughputinthenetworkiseithermaximized(ifallofthenodesareassumedtohavethesametracrequirements)orsothatthethroughputrequirementsofthenodesaremet(iffeasible). 40

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Figure3-1. An-linknetworkscenarioandconictgraph. Forthispurpose,aMarkovianmodelisused,anditsstates,denedasxi2f0;1gn,representsthestatusofthenetworkwhere1representsanactivelink,and0representsanidlelink.Forexample,ifthekthlinkinstateiisactive,thenxik=1.Twosetsofindicesaredenedbelowforthecollision-freetransmissionstates,A,andthecollisionstates,C:A=ijnPk=1xik=1C=ijnPk=1xik>1wherexik=8><>:1iflinkkinstateiisactive;0otherwise:Previousworkinthiseldassumedthatthepropagationdelaybetweenneighboringnodesiszero(cf.[ 31 33 ]),andhence,themotivationbehindthischapteristomaximizethethroughputinthenetworkinthepresenceofsensingdelays,andconsequently 41

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collisions.Althoughcollisionsduetohiddenterminalsarepossible,thischapterfocusesoncollisionsduetosensingdelay.Nevertheless,theformulationsinthischaptercanbeextendedtohiddenterminalsaswellusingtheformulationsoftherateoftransitionsin[ 31 ].ThefollowingsectionexplainsthecontinuousCSMAMarkovchainindetail. 3.1.2CSMAMarkovChainFormulationsofMarkovmodelsforcapturingtheMAClayerdynamicsinCSMAnetworksweredevelopedin[ 31 32 ].Thestationarydistributionofthestatesandthebalanceequationsweredevelopedandusedtoquantifythethroughput.Recently,acontinuoustimeCSMAMarkovmodelwithoutcollisionswasusedin[ 33 ]todevelopanadaptiveCSMAtomaximizethroughput.Collisionswereintroducedin[ 37 ]intheMarkovmodel,andthemeantransmissionlengthofthepacketsareusedasthecontrolvariabletomaximizethethroughput.Sincemostapplicationsexperiencerandomlengthofpackets,thetransmissionrates(packets/unittime),Rk,k=1;2;:::;n;ofthelinksareusedasapracticalmeasureinthischapter.ThemodelforthewaitingtimesisbasedontheCSMArandomaccessprotocol.TheprobabilitydensityfunctionofthewaitingtimeTkisgivenby fTk(tk)=8><>:Rkexp()]TJ /F3 11.955 Tf 9.3 0 Td[(Rktk)fortk0;0fortk<0:(3{1)Duetothesensingdelayexperiencedbythenodesinthenetwork,theprobabilitythatlinkkbecomesactivewithinatimedurationofTsfromtheinstantlinklbecomesactiveis pck,1)]TJ /F1 11.955 Tf 11.95 0 Td[(exp()]TJ /F3 11.955 Tf 9.3 0 Td[(RkTs)(3{2)bythememorylesspropertyoftheexponentialrandomvariable.Thus,therateoftransistionGitooneofthenon-collisionstatesintheMarkovchaininFig. 3-2 isdenedas Gi=NXk=1 xikRkYl6=k(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)!8i2A:(3{3) 42

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Figure3-2. CSMAMarkovchainfora2-linkscenariowithcollisionstates. TherateoftransistionGitooneofthecollisionstatesisgivenby Gi=NXk=1 xikRkYl6=k(pcl)xil(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)!8i2C:(3{4)Thestate(1;1)inFig. 3-2 representthecollisionstate,whichoccurswhenalinktriestotransmitwithinatimespanofTsfromtheinstantanotherlinkstartstransmitting.TheprimaryobjectiveofmodelingthenetworkasacontinuousCSMAMarkovchainistomaximizetheprobabilityofbeinginthecollision-freetransmissionstates.Forthispurpose,thestationarydistributionofthecontinuoustimeMarkovchainisdenedas p)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xi,exp(ri) Pjexp(rj);(3{5) 43

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where ri,8>>>>>>>>>><>>>>>>>>>>:ln8><>:nPk=1 xikRkQl6=k(1)]TJ /F4 7.97 Tf 6.59 0 Td[(pcl)(1)]TJ /F12 5.978 Tf 5.75 0 Td[(xil)! nPk=1xikk9>=>;fori2A;nPk=1 xikRkQl6=k(pcl)xil(1)]TJ /F4 7.97 Tf 6.58 0 Td[(pcl)(1)]TJ /F12 5.978 Tf 5.76 0 Td[(xil)! minm:xim6=0(m)fori2C;1otherwise:(3{6)where1=iisthemeantransmissionlengthofthepacketsifthenetworkisinoneofthestatesinsetA.ThesetA,Ccn(0;0)Trepresentthesetofallcollision-freetransmissionstateindices,wheretheelementsinthesetCrepresentthecollisionstateindices,andtheelementsinthesetCcrepresentthenon-collisionstateindices.In( 3{6 ),thedenitionsfortherateoftransitionsin( 3{3 )and( 3{4 )areused,and( 3{5 )satisesthedetailedbalanceequation(cf.[ 78 ]). 3.1.3ThroughputMaximizationToquantifythethroughput,alog-likelihoodfunctionisdenedasthesummationoverallthecollision-freetransmissionstatesas F(R),Xi2Aln)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(p)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(xi:(3{7)Byusingthedenitionforp(xi)in( 3{5 ),thelog-likelihoodfunctioncanberewrittenas F(R)=nXk=1lnRk k)]TJ /F1 11.955 Tf 11.96 0 Td[((n)]TJ /F1 11.955 Tf 11.95 0 Td[(1)nXk=1RkTs)]TJ /F3 11.955 Tf 11.96 0 Td[(nln"nXk=1Rk kYl6=kexp()]TJ /F3 11.955 Tf 9.3 0 Td[(RlTs)+Xi2Cexp0@1 minm:xim6=0(m)NXk=1 xikRkYl6=k(pcl)xil(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)!1A+exp(1)]: (3{8) Forexample,thelog-likelihoodfunctionin( 3{7 )fora2-linkscenariocanbeexpressedas 44

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F(R1;R2)=lnR1 1)]TJ /F3 11.955 Tf 11.96 0 Td[(R2Ts+lnR2 2)]TJ /F3 11.955 Tf 11.96 0 Td[(R1Ts)]TJ /F1 11.955 Tf 9.3 0 Td[(2lnexplnR1exp()]TJ /F3 11.955 Tf 9.3 0 Td[(R2Ts) 1+explnR2exp()]TJ /F3 11.955 Tf 9.29 0 Td[(R1Ts) 2+exp1 min(1;2)(R1(1)]TJ /F1 11.955 Tf 11.96 0 Td[(exp()]TJ /F3 11.955 Tf 9.3 0 Td[(R2Ts))+R2(1)]TJ /F1 11.955 Tf 11.95 0 Td[(exp()]TJ /F3 11.955 Tf 9.3 0 Td[(R1Ts))))+exp(1)]:ThefunctionF(R)in( 3{8 )isconcave,sincenaturallogarithmsandsummationofconcavefunctionsisaconcavefunction(cf.[ 79 ]).Inaddition,F(R)0,sinceln(p(xi))0fromthedenitionofp(xi)in( 3{5 ).Theoptimizationproblemisdenedas maxR(F(R)):(3{9)TakingthepartialderivativewithrespecttoRkin( 3{8 )yields @F(R) @Rk=1 Rk)]TJ /F1 11.955 Tf 11.96 0 Td[((n)]TJ /F1 11.955 Tf 11.96 0 Td[(1)Ts)]TJ /F3 11.955 Tf 14.68 8.09 Td[(n D(1 kYl6=kexp()]TJ /F3 11.955 Tf 9.3 0 Td[(RlTs))]TJ /F10 11.955 Tf 11.3 20.44 Td[( Xm:m6=kRm mYl6=m;kexp()]TJ /F3 11.955 Tf 9.3 0 Td[(RlTs)!Tsexp()]TJ /F3 11.955 Tf 9.3 0 Td[(RkTs)+Xxi2C24exp0@1 minm:xim6=0(m)NXk=1xikRkYl6=k(pcl)xil(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)1A0@1 minm:xim6=0(m) xikYl6=k(pcl)xil(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)+ Xm:m6=kximRmYl6=m;k(pcl)xil(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.58 0 Td[(xil)@ @Rk(pck)xik(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pck)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xik); (3{10) 45

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k=1;2;:::;n)]TJ /F1 11.955 Tf 11.96 0 Td[(1,where D,nXk=1exp0B@ln0B@RkQl6=kexp()]TJ /F3 11.955 Tf 9.3 0 Td[(RlTs) k1CA1CA+Xi2CexpNPk=1 xikRkQl6=k(pcl)xil(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)! minm:xim6=0(m)+exp(1): (3{11) Anonlinegradient-basedalgorithmisusedtosolvetheproblemin( 3{9 ).Thegradientlawisdenedas lnRk(t+T)=lnRk(t)+K@F(R) @Rk;(3{12)k=1;2;:::;n)]TJ /F1 11.955 Tf 12.35 0 Td[(1,whereK2Risthestepsize,Tisthetimeintervalbetweenupdates,and@F(R)=@Rkisdenedin( 3{10 ).Thecalculationof@F(R)=@Rkatthetransmitteroflinkkisdeterminedasfollows.Thetransmittingnodeoflinkkcalculatesthesteady-stateprobabilitiesofthestatesp(xi);8i2AeveryTunittime.Thetransmittingnodeoflinkkcalculatesthesteady-stateprobabilitiesofthecollision-freetransmissionstatesalone,sincethesearesucienttoestimatethemeantransmissionratesRm;m6=kusing( 3{5 ).Foran-linkcase,thetransmitteroflinkkneedstosolvethefollowingsetofindependentnonlinearequations(aftermanipulationsof( 3{5 )), Rlexp()]TJ /F3 11.955 Tf 9.3 0 Td[(RlTs)=P(Onlylinklisactive) exp()]TJ /F3 11.955 Tf 9.3 0 Td[(RkTs)Rk P(Onlylinkkisactive);(3{13)8l6=k.NotethatlinkkcanuseitscurrentvalueofthemeantransmissionrateRktosolve( 3{13 ).ThevalueofTcanbechosensucientlylargesothatp(xi);8i2Acanbemeasuredaccurately.Further,largeTaectsidenticationofthecollision-freetransmissionstatesbythetransmitteroflinkkusingtheCarrierSense(CS)protocol.ThemaximumsensingdelayTsandthemeantransmissionlengths1=k;k=1;2;:::;n)]TJ /F1 11.955 Tf 11.99 0 Td[(1areassumedtobeknownatallthetransmittingnodes.Hence,thealgorithmisdistributed. 46

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Inadditiontomaximizingthelog-likelihoodfunction,certainconstraintsmustbesatised.TheservicerateS(R)ateachtransmitterofalinkneedstobeequaltothearrivalrate,andthechosenmeantransmissionratesRk,k=1;2;:::;n;needtobenon-negative.Thus,theoptimizationproblemcanbeformulatedasmaxR(F(R))subjectto ln)]TJ /F1 11.955 Tf 11.95 0 Td[(lnS(R)=0;(3{14)and R0;(3{15)whereR2Rn;S(R)2Rn)]TJ /F6 7.97 Tf 6.59 0 Td[(1;and2Rn)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Theservicerateforalinkistherateatwhichapacketistransmitted,andisquantiedasSk(R),exp ln RkQl6=kexp()]TJ /F4 7.97 Tf 6.58 0 Td[(RlTs) k!! D;k=1;2;:::;n)]TJ /F1 11.955 Tf 12.55 0 Td[(1;andDisdenedin( 3{11 ).Notethatlnk)]TJ /F1 11.955 Tf 12.55 0 Td[(lnSk(R)=0;k>0isconcaveforallk.Theoptimizationproblemdenedaboveisaconcaveconstrainednonlinearprogrammingproblem,andobtainingaanalyticalsolutionisdicult.Therearenumericaltechiquesadoptedintheliteraturewhichhaveinvestigatedsuchproblemsindetail[ 79 { 82 ].Inthiswork,asuitablenumericaloptimizationalgorithmisemployedtosolvetheoptimizationproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ).Thefollowingsectionanalyzesthenumericalresultsobtainedbysolvingtheunconstrainedproblemof( 3{9 ),andcomparesthemeantransmissionratesobtainedonlinefromthedistributedalgorithmof( 3{12 )withtheoptimalvalues.Further,numericalanalysisoftheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 )isperformed. 47

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Figure3-3. Meantransmissionratesofnodes1,2,and3transmittingtothesamenode4.Allnodesareinthesensingregion.Theonlinealgorithmof( 3{9 )isusedwithT=100ms,K=5,andTs=0:001ms. 3.1.4SimulationResultsACSMAplatformisdevelopedusingMATLABthatusesthestandardcarriersensechannelaccessprotocol.Aslottimeof10sisused,andthemeantransmissionlengthsofthepackets,1=k,k=1;2;:::;n;aresetto1ms.AnupdatetimeofT=100msandastepsizeofK=5areused.Thedistributedalgorithmin( 3{12 )isusedtogeneratetherateupdatesforeachtransmittingnodek=1;2;:::;n)]TJ /F1 11.955 Tf 12.57 0 Td[(1.Thetransmitteroflinkkcalculatesthesteady-statedistributionofthestatesp(xi);8i2AeveryTunittime,andestimatesthemeantransmissionratesoftheothertransmittingnodesRm;m6=kusing( 3{13 )tocalculate( 3{10 ).Anonlinearequationsolver(MATLABbuilt-infunctionfzero)canbeusedtosolve( 3{13 ).Themeantransmissionrateupdatescanthusbecalculatedfrom( 3{12 ).Fora3-linknetworkwithsensingdelayof0:001ms,themeantransmissionratesconvergenceisshowninFig. 3-3 48

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Theoptimalvalueforthemeantransmissionratesfora3-linknetworkiscalculatedoinefortheunconstrainedproblemof( 3{9 )forcomparisonpurposes.TheMATLABbuilt-infunctionfminuncisusedforthispurpose,andtheoptimalvalueforthemeantransmissionrateswereobtainedas R1opt=R2opt=R3opt=19:84dataunits/ms. (3{16) Fig. 3-3 indicatesthatmeantransmissionratesobtainedfromtheonlinedistributedalgorithmof( 3{12 )convergetotheoptimalvalues,denedin( 3{16 ).Theonlinealgorithm( 3{12 )doesnottakeintoaccounttherateconstraintdenedin( 3{15 ).Theconstrainedconcavenonlinearprogrammingproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 )issolvedbyoptimizingthemeantransmissionratesRk,k=1;2;:::;n;ofthetransmittingnodesinthenetworkofFig. 3-1 byasuitablenumericaloptimizationalgorithm.AMATLABbuilt-infunctionfminconisusedtosolvetheoptimizationproblembyconguringittousetheinteriorpointalgorithm(cf.[ 83 84 ]).Oncethemeantransmissionratesareoptimized,theyarexedinaCSMAplatform(developedinMATLAB)thatusesthecarriersensechannelaccessprotocol.Thefunctionfminconsolvestheoptimizationproblemonlyforasetoffeasiblearrivalrates.Aslottimeof10sisused,andthemeantransmissionlengthsofthepackets,1=k,k=1;2;:::;n;aresetto1ms.Further,astable(andfeasible)setofarrivalrates,inthesensethatthequeuelengthsatthetransmittingnodesarestable,arechosenbeforesimulation.A2-linkcollisionnetworkissimulatedusingtheplatformexplainedabove.Theoptimalvaluesofthemeantransmissionrates,R1andR2,areobtainedandtabulatedasshowninTable 3-1 fordierentvaluesofthesensingdelayTs.Notethatthecapacityofthechannelisnormalizedto1dataunit/ms.Themeantransmissionlengthsofthepackets,1=1=1=2=1ms. 49

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Table3-1. Optimalvaluesofthemeantransmissionratesfora2-linkcollisionnetworkforvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ). Max.FeasibleArrivalRateOpt.MeanTXrateSensingDelay12R1R2 0:0010:40:64:156:220:010:390:426:056:490:10:20:391:332:35 ACSMAsystemwithcollisionsisimplementedinMATLAB.Fig. 3-4 showstheevolutionofthequeuelengthsofthenodes1and2(refertoFig. 3-1 )forasensingdelayofTs=0:01ms.Theoptimalmeantransmissionrates(R1=6:05dataunits/ms,R2=6:49dataunits/ms)aregeneratedbyfmincon,andthestablearrivalratesof1=0:16dataunits/msand2=0:2dataunits/msareused.A3-linkcollisionnetworkissimulatedsimilarly,andtheoptimalvaluesofthemeantransmissionrates,R1,R2andR3,areobtainedandtabulatedasshowninTable 3-2 fordierentvaluesofthesensingdelayTs.Fig. 3-5 showstheevolutionofqueuelengthsofthenodes1,2,and3(refertoFig. 3-1 )forasensingdelayofTs=0:01ms.Themeantransmissionlengthsofthepackets,1=1=1=2=1=3=1ms.Theoptimalmeantransmissionrates(R1=6:54dataunits/ms,R2=10:19dataunits/ms,R3=11:49dataunits/ms)aregeneratedbyfmincon,andthestablearrivalratesof1=0:01dataunits/ms,2=0:05dataunits/ms,3=0:02dataunits/msareused.Thesimulationsarerepeated10timesforeachof2-linkand3-linkcollisionnetworks,andtheaverage(arithmeticmean)ofthenumberofcollisionsiscalculatedforeachcase.Table 3-3 showstheaveragenumberofcollisionswhenasetofoptimizedvalueofthemeantransmissionratesareused.Thepacketcollisionsinthenetworkarereducedtolessthan0:2%forthesensingdelayslistedinthetable. 50

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Figure3-4. Queuelengthsofnodes1and2transmittingtothesamenode3.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ).Allnodesareinthesensingregion.Ts=0:01ms,R1=6:05dataunits/ms,R2=6:49dataunits/ms,1=0:16dataunits/ms,2=0:2dataunits/ms. Table3-2. Optimalvaluesofthemeantransmissionratesfora3-linkcollisionnetworkforvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ). Max.FeasibleArrivalRateOpt.MeanTXrateSensingDelay123R1R2R3 0:0010:220:310:37:3110:289:950:010:130:210:246:5410:1911:490:10:120:120:12:262:261:94 51

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Figure3-5. Queuelengthsofnodes1,2,and3transmittingtothesamenode4.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ).Allnodesareinthesensingregion.Ts=0:01ms,R1=6:54dataunits/ms,R2=10:19dataunits/ms,R3=11:49dataunits/ms,1=0:02dataunits/ms,2=0:05dataunits/ms,3=0:05dataunits/ms. Table3-3. Averagenumberofcollisionsfora2-linkand3-linkcollisionnetworksforvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{9 ),( 3{14 ),and( 3{15 ). Av.collisionsusingoptimizedMeanTXrates,in%SensingDelay,Ts;inms2-link3-link 0:0010:060:160:010:080:170:10:020:04 52

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3.2ThroughputMaximizationinCSMANetworkswithCollisionsandHiddenTerminalsTwosourcesoffailureinthecarrier-sensingmechanismaredelaysinthecarriersensingmechanismandhiddenterminals,inwhichanongoingtransmissioncannotbedetectedataterminalthatwishestotransmitbecausethepathlossfromtheactivetransmitterislarge.CollisionsduetosensingdelayswasmodeledinSection 3.1 .Inthissection,theeectofthesecarrier-sensingfailures(bothduetosensingmechanismandhiddenterminals)ismodeledusingacontinuous-timeMarkovmodel.ThethroughputofthenetworkisdeterminedusingthestationarydistributionoftheMarkovmodel.Thethroughputismaximizedbyndingoptimalmeantransmissionratesfortheterminalsinthenetworksubjecttoconstraintsonsuccessfullytransmittingpacketsataratethatisatleastasgreatasthepacketarrivalrate. 3.2.1NetworkModelConsideran(n+k)-linknetworkwithn+k+1nodesasshowninFig. 3-6 ,wherenetworkAconsistsofnlinksandnetworkBconsistsofklinks.Assumethatallnodescansenseallothernodesinthenetwork.However,thereisasensingdelay,sothatiftwonodesinitiatepackettransmissionwithinatimedurationofTs,therewillbeacollision.Let(n+k)denotethetotalnumberoflinksinthenetwork.InatypicalCSMAnetwork,thetransmitterofnodembacksoforarandomperiodbeforeitsendsapackettoitsdestinationnode,ifthechannelisidle.Ifthechannelisbusy,thetransmitterfreezesitsbackocounteruntilthechannelisidleagain.Thisbackotime,orthewaitingtime,foreachlinkmisexponentiallydistributedwithmean1=Rm.TheobjectiveinthischapteristodeterminetheoptimalvaluesofthemeantransmissionratesRm,m=1;2;:::;n+k,sothatthethroughputinthenetworkismaximized.Forthispurpose,aMarkovianmodelisusedwithstatesdenedasxi:A!f0;1gn+k,wherei2Arepresentsthestatusofthenetwork,whichtakesthevalueof1foranactivelinkand0representsanidlelink.Forexample,ifthemthlinkinstateiisactive,thenxim=1. 53

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Figure3-6. An(n+k)-linknetworkscenarioandconictgraph. Previousworkassumesthatthepropagationdelaybetweenneighboringnodesiszero(cf.[ 31 33 ]).Sincepropagationdelaysenablethepotentialforcollisions,thereexistsmotivationtomaximizethethroughputinthenetworkinthepresenceofthesedelays.Additionally,collisionsduetohiddenterminalsarepossible,andthissectioncapturestheeectofhiddenterminalsintheCSMAMarkovchaindescribedinthefollowingsection. 3.2.2CSMAMarkovChainFormulationsofMarkovmodelsforcapturingtheMAClayerdynamicsinCSMAnetworksweredevelopedin[ 31 32 ].Thestationarydistributionofthestatesandthebalanceequationsweredevelopedandusedtoquantifythethroughput.Recently,acontinuoustimeCSMAMarkovmodelwithoutcollisionswasusedin[ 33 ]todevelopanadaptiveCSMAtomaximizethroughput.Collisionswereintroducedin[ 37 ]intheMarkovmodel,andthemeantransmissionlengthofthepacketsareusedasthecontrolvariabletomaximizethethroughput.Sincemostapplicationsexperiencerandomlengthofpackets, 54

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Figure3-7. CSMAMarkovchainwithcollisionstatesfora3-linknetworkscenariowithhiddenterminals. thetransmissionrates(packets/unittime),Rm,m=1;2;:::;n;provideaapracticalmeasure.ThemodelforwaitingtimesisbasedontheCSMArandomaccessprotocol.TheprobabilitydensityfunctionofthewaitingtimeTmisgivenby( 3{1 ).Duetothesensingdelayexperiencedbythenetworknodes,theprobabilitythatlinkmbecomesactivewithinatimedurationofTsfromtheinstantlinklbecomesactiveisgivenin( 3{2 ).Thus,therateoftransitionGitooneofthenon-collisionstatesintheMarkovchaininFig. 3-7 isasdenedin( 3{3 ).TherateoftransitionGitooneofthecollisionstatesisgivenin( 3{4 ).Forexample,thestate(1;1;0)inFig. 3-7 representsthecollisionstate(fornetworkA),whichoccurswhenalinktriestotransmitwithinatimespanofTsfromtheinstantanotherlinkstartstransmitting. 55

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TheprimaryobjectiveofmodelingthenetworkasacontinuousCSMAMarkovchainisthattheprobabilityofcollision-freetransmissionneedstobemaximized.Forthispurpose,therateriisdenedas ri,8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:ln8>>>><>>>>:nPu=1 xiuRuQl6=u(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)! nPu=1xiuu9>>>>=>>>>;;i2ATnPu=1 xiuRuQl6=u(pcl)xil(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.58 0 Td[(xil)! minm:xim6=0(m);i2AC1;i2AI;(3{17)sothatthestationarydistributionofthecontinuoustimeMarkovchaincanbedenedasin( 3{4 )as p(i),exp(ri) Pjexp(rj);(3{18)where,in( 3{17 ),1=misthemeantransmissionlengthofthepacketsifthenetworkisinoneofthestatesinsetATinsensingregionA.ThesetAT,AcCn(0;0)Trepresentsthesetofallcollision-freetransmissionstates,wheretheelementsinthesetACrepresentsthecollisionstates,andtheelementsinthesetAcCrepresentsthenon-collisionstates.ThesetAIrepresentstheinactivestate,i.e.,xi=(0;0;0).In( 3{17 ),thedenitionsfortherateoftransitionsin( 3{3 )and( 3{4 )areused,and( 3{18 )satisesthedetailedbalanceequation(cf.[ 78 ]).Inaddition,ifthereareHiddenTerminals(HT)inthenetworkasshowninFig. 3-7 ,thenricanbedenedforthesensingregionBinasimilarwayasdenedforsensingregionAin( 3{17 ).LetsetsBT,BC,andBIrepresentthecollision-freetransmissionstates,collisionstates,andtheinactivestatesrespectively.Basedonthetransmission,collisionandidlestatesofthelinksinthesensingregionsAandB,ibelongstooneofthe 56

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combinationsofthesetsAT,AC,AI,BT,BC,andBI.Therefore(cf.[ 31 ]),ri,8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:FAFB;i2AT[BTGAFB;i2AC[BTFB;i2AI[BTFAGB;i2AT[BCGAGB;i2AC[BCGB;i2AI[BCFA;i2AT[BIGA;i2AC[BI1;i2AI[BI;whereFA,ln8>>>><>>>>:nPu=1 xiuRuQl6=u(1)]TJ /F3 11.955 Tf 11.96 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)! nPu=1xiuu9>>>>=>>>>;;GA,nPu=1 xiuRuQl6=k(pcl)xil(1)]TJ /F3 11.955 Tf 11.95 0 Td[(pcl)(1)]TJ /F4 7.97 Tf 6.59 0 Td[(xil)! minm:xim6=0(m):FBandGBcanbedenedsimilarlyfornetworkBinFig. 3-6 3.2.3ThroughputMaximizationToquantifythethroughput,alog-likelihoodfunctionisdenedasthesummationoverallthecollision-freetransmissionstatesas F(R),Xi2(AT[BI)[(AI[BT)ln(p(i)):(3{19) 57

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Byusingthedenitionforp(i)in( 3{5 ),thelog-likelihoodfunctionin( 3{19 )canberewrittenas F(R)=nXu=1lnexplnRu u)]TJ /F1 11.955 Tf 11.96 0 Td[((n)]TJ /F1 11.955 Tf 11.95 0 Td[(1)nXu=1RuTs+k+nXv=1+1lnexplnRv v)]TJ /F1 11.955 Tf 11.96 0 Td[((k)]TJ /F1 11.955 Tf 11.96 0 Td[(1)k+nXv=n+1RvTs)]TJ /F1 11.955 Tf 11.29 0 Td[((n+k)ln"Xi2AT[BTexp(FAFB)+Xi2AC[BTexp(GAFB)+Xi2AI[BTexp(FB)+Xi2AT[BCexp(FAGB)+Xi2AC[BCexp(GAGB)+Xi2AI[BCexp(GB)+Xi2AT[BIexp(FA)+Xi2AC[BIexp(GA)+Xi2AI[BIexp(1)#: (3{20) ThefunctionF(R)in( 3{8 )isconcave,sincenaturallogarithmandsummationofconcavefunctionsisaconcavefunction(cf.[ 79 ]).Inaddition,F(R)0sinceln(p(xi))0fromthedenitionofp(xi)in( 3{18 ).Theoptimizationproblemisdenedas maxR(F(R)):(3{21)Inadditiontomaximizingthelog-likelihoodfunction,certainconstraintsmustbesatised.TheservicerateS(R)ateachtransmitterofalinkneedstobeequaltothearrivalrate,andthechosenmeantransmissionratesRk,k=1;2;:::;n;needtobenon-negative.Thus,theoptimizationproblemcanbeformulatedasmaxR(F(R))subjectto ln)]TJ /F1 11.955 Tf 11.95 0 Td[(lnS(R)=0;(3{22)and R0;(3{23) 58

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whereR2Rn;S(R)2Rn)]TJ /F6 7.97 Tf 6.59 0 Td[(1;and2Rn)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Theservicerateforalinkistherateatwhichapacketistransmitted,andisquantiedforsensingregionAasSm(R),exp ln RkQl6=mexp()]TJ /F4 7.97 Tf 6.59 0 Td[(RlTs) m!! Pjexp(rj);m=1;2;:::;n)]TJ /F1 11.955 Tf 12.64 0 Td[(1;andthedenominatorisdenedin( 3{17 ).ServiceratesforsensingregionBcanbedenedsimilarly.Notethatlnm)]TJ /F1 11.955 Tf 12.99 0 Td[(lnSm(R)=0;andm>0isconcaveforallm.Theoptimizationproblemdenedaboveisaconcaveconstrainednonlinearprogrammingproblem,andobtainingananalyticalsolutionisdicult.Therearenumericaltechiquesadoptedintheliteraturewhichhaveinvestigatedsuchproblemsindetail[ 79 { 82 ].AsdetailedinSection 3.2.4 ,asuitablenumericaloptimizationalgorithmisemployedtosolvetheoptimizationproblemdenedin( 3{21 )-( 3{23 ). 3.2.4SimulationResultsTheconstrainedconcavenonlinearprogrammingproblemdenedin( 3{21 )-( 3{23 )issolvedbyoptimizingthemeantransmissionratesRm,m=1;2;:::;n+k;ofthetransmittingnodesinthenetworkofFig. 3-6 .AMATLABbuilt-infunctionfminconisusedtosolvetheoptimizationproblembyconguringittousetheinteriorpointalgorithm(cf.[ 83 84 ]).Oncethemeantransmissionratesareoptimized,theyarexedinasimulation(developedinMATLAB)thatusestheCSMAMACprotocol.Thefunctionfminconsolvestheoptimizationproblemonlyforasetoffeasiblearrivalrates.Aslottimeof10sisused,andthemeantransmissionlengthsofthepackets,1=m,m=1;2;:::;n+k;aresetto1ms.Further,astable(andfeasible)setofarrivalrates,inthesensethatthequeuelengthsatthetransmittingnodesarestable,arechosenbeforethesimulation.ThecollisionnetworkofFig. 3-6 issimulatedusingtheplatformexplainedabove.Theoptimalvaluesofthemeantransmissionrates,R1,R2,andR3,areobtainedandtabulatedasshowninTable 3-4 fordierentvaluesofthesensingdelayTs(Notethat 59

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Table3-4. Optimalvaluesofthemeantransmissionratesfora3-linkcollisionnetworkwithhiddenterminals(refertoFig. 3-7 )forvariousvaluesofsensingdelays.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{21 )-( 3{23 ). Max.FeasibleArrivalRateOpt.MeanTXrateSensingDelay123R1R2R3 0:0010:20:20:13:943:941:960:010:180:170:113:783:582:230:10:120:120:12:562:561:65 inthescenarioofFig. 3-6 ,thesensingdelayappliestothenodesinnetworkA).Thecapacityofthechannelisnormalizedto1dataunit/ms.Themeantransmissionlengthsofthepacketsare1=1=1=2=1=3=1ms.AsimulationofaCSMAsystemwithcollisionsisimplementedinMATLAB.Fig. 3-8 showstheevolutionofthequeuelengthsofnodes1,2,and4(refertoFig. 3-6 )forasensingdelayofTs=0:01ms.Theoptimalmeantransmissionrates(R1=3:78dataunits/ms,R2=3:58dataunits/ms,R3=2:23dataunits/ms)aregeneratedbyfmincon,andthestablearrivalratesof1=0:05dataunits/ms,2=0:05dataunits/ms,and3=0:01dataunits/msareused. 60

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Figure3-8. Queuelengthsofnodes1,2,and4transmittingtothesamenode3.Theoptimumvaluesofthemeantransmissionratesarethesolutiontotheconstrainedproblemdenedin( 3{21 )-( 3{23 ).Allnodesareinthesensingregion,andTs=0:01ms,R1=3:78dataunits/ms,R2=3:58dataunits/ms,R3=2:23dataunits/ms,1=0:02dataunits/ms,2=0:05dataunits/ms,3=0:05dataunits/ms. 61

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CHAPTER4CONGESTIONCONTROLFORDIFFERENTIATED-SERVICESNETWORKSWITHARRIVAL-RATEDELAYSNetworkpackettracinthetransportlayerplaysavitalroleinaectingthethroughputofInternet-stylenetworks.CommonqueuelengthmanagementtechniquesonnodesinsuchnetworksfocusonservicingthepacketsbasedontheirQualityofService(QoS)requirements(e.g.,Dierentiated-Services,orDiServ,networks).Inthischapter,continuouscontrolstrategiesaresuggestedforaDiServnetworktotrackthedesiredensembleaveragequeuelengthsinmultiplequeues.ALyapunov-basedstabilityanalysisisprovidedtoillustrateglobalasymptoticregulationoftheensembleaveragequeuelengthofthePremiumServicebuer.Inaddition,arrivalratedelaysduetopropagationandprocessingthataectsthecontrolinputoftheOrdinaryServicebuerareaddressed,andaLyapunov-basedstabilityanalysisisprovidedtoillustrateglobalasymptoticregulationoftheensembleaveragequeuelengthofthisservice.Simulationsdemonstratetheperformanceandfeasibilityofthecontroller,alongwithshowingglobalasymptoticregulationclosetothedesiredvaluesofthequeuelengthsinthePremiumServiceandOrdinaryServicebuers. 4.1QueuingSystemModelDiServarchitecturesareexamplesofhigh-speednetworkarchitecturesusedinTCP/IPandATMtechnologies.In[ 56 ],inspiredby[ 43 ],theincomingtractoanodeinanetworkisclassiedintoPremiumTracService,OrdinaryTracService,andBestEortTracService.PremiumTracServiceisdesignedforapplicationssuchasvideoconferencing,audio,andvideoondemand,whicharecharacterizedbystringentlossanddelayconstraints.OrdinaryTracServicehavesomeexibilityintermsofdelayrequirements.Examplesofsuchapplicationsincludewebbrowsing,email,andftp.Finally,BestEortTracServicearedesignedfortheclassofapplicationsthatdonothaveanydelayorlossconstraints.Inthischapter,thecontrolobjectiveisdenedforapplicationsthatusePremiumTracServiceandOrdinaryTracService(refertoFig. 4-1 ). 62

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Figure4-1. SchematicofaDiServQueueingSystem. ThesubsequentdevelopmentisbasedonaFluidFlowModel(FFM)commonlyusedinnetworkperformanceanalysis(cf.[ 41 44 ]).Suchmodelsaregeneral,anddescribeawiderangeofqueueingandcontentionsystems(cf.[ 85 { 87 ]).Assumingnopacketdrops,theowconservationprincipleforasinglequeue(cf.[ 41 56 ])isusedtodenetheevolutionoftheensembleaverageofthequeuelength,q(t)2R+,inthesystemas _q=)]TJ /F3 11.955 Tf 9.3 0 Td[(uG(q)+;(4{1)whereq(0)=q0,andG:R+![0;1)istheoeredload,alsoknownastheensembleaverageutilizationofthequeueattimet,andthecontrolinputu(t)2R+isthequeueservercapacity.In( 4{1 ),(t)2R+istheensembleaveragearrivalratedenedas(t)=E[a]; 63

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whereE[a]denotestheexpectedvalueofthearrivalrate,a(t)2R+.Thequeueingmodelbasedon[ 56 ]usesM/M/1(i.e.,Markovianinput,Markovianoutput,singleserver)assumptions(cf.[ 41 44 ])toobtaintheensembleaveragequeuelengthevolution.Inaddition,thepresenceofpossibledelays(cf.[ 88 { 90 ])inthearrivalratestothePremiumService,andmorecriticallytheOrdinaryServicebuerswillaectthecontrolsignicantly.Suchdelaysariseduetoprocessingandpropagation(forinstance,theIDCCschemein[ 56 ]canpotentiallycausedelays).Inthiswork,weaddresstime-varyingarrivalratedelays;hence,theensembleaveragequeuelengthevolutioncanbeexpressedas _qi=)]TJ /F3 11.955 Tf 9.29 0 Td[(uiqi 1+qi+i(t)]TJ /F3 11.955 Tf 11.96 0 Td[(i(t));(4{2)wherei2fp;rgandsubscriptspandrrepresentPremiumServiceandOrdinaryServicerespectively.Itisassumedthat0r(t)rmax,andj_r(t)jdrmax<1,wherermaxanddrmaxareknownpositiveconstants.Theassumptionfor_r(t)indicatesthatthetime-delaymustbeslowlytime-varying.Itisalsoassumedthatj_p(t)jdpmaxandjp(t)jdpmax,wherepmaxanddpmaxareknownpositiveconstants.Themodelin( 4{2 )isvalidfor0qi(t)qbuersizeand0u(t)userver,whereqbuersizeisthemaximumpossiblequeuesize,anduserveristhemaximumallowableserverrate. 4.2PremiumServiceTheunknownaveragearrivalrateofthePoissonarrivalprocessisdenotedbyap(t),p(t)]TJ /F3 11.955 Tf 11.95 0 Td[(p(t))2R+,andup(t)2R+isthequeueservercapacitythatactsasthecontrolvariable.Itisassumedthat8t,ap(t)isupperboundedbytheallowablerateforincomingPremiumTrac,denotedbyapmax,whichin-turnisboundedbyuserver[ 56 ].Inadditiontop(t)beingbounded,itsrstandsecondderivativesareassumedtobebounded[ 56 ].Since_p(t)andp(t)areassumedtobebounded,therstandsecondtimederivativesofap(t)canbeboundedfromitsdenitionas _ap_ap;apap:(4{3) 64

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4.2.1ControlDesignTofacilitatethesubsequentanalysisofthequeueingsystem,anensembleaveragequeuelengtherrorep(t)2Risdenedas ep,qp)]TJ /F3 11.955 Tf 11.95 0 Td[(qpd;(4{4)whereqpd(t)2Risthedesiredensembleaveragequeuelengthprovidedbythenetworkoperator.Itisassumedthattherstandsecondderivativesofthedesiredensembleaveragequeuelengthareknownandbounded[ 56 ].Tofacilitatethesubsequentanalysis,alteredtrackingerrorisdenedas rp,_ep+pep;(4{5)wherep2R+denotesaconstantcontrolgain.Thelteredtrackingerrorisonlyintroducedtofacilitatethesubsequentanalysisandisnotassumedtomeasurable.Takingthetimederivativeof( 4{4 )andusing( 4{5 )yields rp=_qp)]TJ /F1 11.955 Tf 14.12 0 Td[(_qpd+ep=)]TJ /F3 11.955 Tf 9.29 0 Td[(upqp 1+qp+ap(t))]TJ /F1 11.955 Tf 14.11 0 Td[(_qpd+ep: (4{6) FromM/M/1queueingformulas,theensembleaverageutilizationofthequeueisdenedasG(qp),qp=(1+qp)andisassumedtobeknown(cf.[ 56 57 ]).Hence,thecontrollawforpremiumserviceisdenedas up,qp 1+qp)]TJ /F6 7.97 Tf 6.58 0 Td[(1;(4{7)where(t)isasubsequentlydesignedauxiliarycontroller.Aftersubstituting( 4{7 )into( 4{6 ),thelteredtrackingerrorcanberewrittenas rp=)]TJ /F3 11.955 Tf 9.3 0 Td[(+ap(t))]TJ /F1 11.955 Tf 14.12 0 Td[(_qpd+ep:(4{8) 65

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Tofacilitatethedesignof(t),thetimederivativeof( 4{8 )isobtainedas _rp=)]TJ /F1 11.955 Tf 11.52 0 Td[(_+_ap(t))]TJ /F1 11.955 Tf 12.82 0 Td[(qpd+_ep:(4{9)Basedon( 4{9 )andthesubsequentstabilityanalysis,theauxiliarycontrolterm(t)isdenedas ,_qpd(0))]TJ /F1 11.955 Tf 14.11 0 Td[(_qpd(t)+;(4{10)where(ep)2RistheFilippovsolutiontothefollowingdierentialequation _=(kp1+kp2+p)rp+sgn(ep)+ep;v(0)=0;(4{11)wherekp,2R+areconstantcontrolgains.TheexistenceofsolutionscanbeestablishedusingFilippovtheoryofdierentialinclusions(cf.[ 91 { 94 ])for_2K[h](ep;rp;t)whereh(ep;rp;t)2Risdenedintheright-handsideof_in( 4{11 ),andK[h],\>0\Sm=0coh(B(v;))]TJ /F3 11.955 Tf 11.95 -.01 Td[(Sm);whereTSm=0denotestheintersectionofallsetsSmofLebesguemeasurezero,codenotesconvexclosure,andB(v;)denotestheopenballofradiusaroundv.Theclosed-looperrorsystemisobtainedbysubstitutingthetimederivativeof( 4{10 )into( 4{9 )as _rp=_ap(t))]TJ /F1 11.955 Tf 11.96 0 Td[(((kp1+kp2+p)rp+sgn(ep)+ep)+_ep:(4{12) 4.2.2StabilityAnalysisTheorem4.1:Thecontrollerdesignedin( 4{7 )and( 4{11 )ensuresglobalasymptoticensembleaveragequeuelengthregulationinthePremiumServicebuerprovidedthecontrolgainsareselectedaccordingtothesucientconditions kp2>2p 2;p<2;(4{13) 66

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and >_ap+1 ap;(4{14)where_apandapareintroducedin( 4{3 ). Proof. Letyp(t)2R3bedenedasyp(t),zTp(t)p PTwherezp(t)2R2isdenedaszp(t),ep(t)rp(t)T,andtheauxiliaryfunctionP(ep;t)2RistheFilippovsolutiontothefollowingdierentialequation _P(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(rp_ap)]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(ep); (4{15) P(ep(t0);t0)=jep(0)j)]TJ /F3 11.955 Tf 17.93 0 Td[(ep(0)_ap(0): (4{16) ExistenceofsolutionsforP(ep;t)canbeestablishedusingFilippovtheoryofdierentialinclusionsinamannersimilartothedevelopmentin( 4{11 ).Providedthesucientconditionin( 4{14 )issatised,theconditionthatP(t)0canbeproven(refertoAppendix C ).LetVa(yp;t):R3[0;1)!Rbearegularandacontinuouslydierentiablefunctioninyp,denedas Va(yp;t)=1 2r2p+1 2e2p+P:(4{17)Thetimederivativeof( 4{17 )existsalmosteverywhere(a.e.),i.e.,foralmostallt2[t0;tf],and_Va(yp;t)a:e2Vp(yp;t),where:~Vp(yp;t)=\2@Vp(yp;t)TK_ep_rp1 2P)]TJ /F11 5.978 Tf 7.78 3.26 Td[(1 2_P;where@Vp(yp)isthegeneralizedgradientofVp(yp;t)[ 95 ].SinceVp(yp;t)isacontinuouslydierentiablefunctioninyp, :~VprVpK[]T;(4{18)whererVp=heprp2p Pi: 67

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UsingthecalculusofK[]from[ 96 ]andbyusing( 4{12 )and( 4{15 ),( 4{15 )canbeexpressedas :~Vprpn_ap)]TJ /F1 11.955 Tf 11.95 0 Td[((kp1+kp2+p)rp)]TJ /F3 11.955 Tf 11.96 0 Td[(K[sgn(ep)])]TJ /F3 11.955 Tf 11.95 0 Td[(ep+p_epo+epfrp)]TJ /F3 11.955 Tf 11.96 0 Td[(pepg)]TJ /F3 11.955 Tf 20.59 0 Td[(rp_ap)]TJ /F3 11.955 Tf 11.95 0 Td[(K[sgn(ep)]; (4{19) whereK[sgn(ep)]=(ep)[ 96 ]suchthat(ep)=8>>>><>>>>:1;ifep(t)>0[)]TJ /F1 11.955 Tf 9.3 0 Td[(1;1];ifep(t)=0)]TJ /F1 11.955 Tf 9.3 0 Td[(1;ifep(t)<0:Using( 4{5 ),( 4{19 )canberewrittenas :~Vpa:e:)]TJ /F3 11.955 Tf 23.83 0 Td[(kp1jrpj2)]TJ /F3 11.955 Tf 11.96 0 Td[(kp2jrpj2)]TJ /F3 11.955 Tf 11.95 0 Td[(pjrpj2+prpfrp)]TJ /F3 11.955 Tf 11.96 0 Td[(pepg)]TJ /F3 11.955 Tf 9.3 0 Td[(pjepj2; (4{20) wherethesetin( 4{19 )reducestothescalarinequalityin( 4{20 )sincetheRHSiscontinuousa.e.,i.e.,theRHSiscontinuousexceptfortheLebesguenegligiblesetoftimeswhenep=0[ 97 98 ].ApplyingYoung'sInequality,( 4{20 )canberewrittenas:~Vpa:e:)]TJ /F3 11.955 Tf 23.83 0 Td[(kp1jrpj2)]TJ /F3 11.955 Tf 11.96 0 Td[(kp2jrpj2)]TJ /F3 11.955 Tf 11.95 0 Td[(pjrpj2+pjrpj2+2p 2jepj2+2p 2jrpj2)]TJ /F3 11.955 Tf 11.95 0 Td[(pjepj2::~Vpa:e:)]TJ /F3 11.955 Tf 23.83 0 Td[(kp1jrpj2)]TJ /F10 11.955 Tf 11.96 16.85 Td[(kp2)]TJ /F3 11.955 Tf 13.15 9.16 Td[(2p 2jrpj2)]TJ /F10 11.955 Tf 11.95 16.85 Td[(p)]TJ /F3 11.955 Tf 13.15 9.16 Td[(2p 2jepj2:Iftheconditionin( 4{13 )issatised :~Vpa:e:)]TJ /F3 11.955 Tf 23.84 0 Td[(Wp(yp);(4{21)whereWp(yp),pkzpk2wherep,minnkp1;kp2)]TJ /F4 7.97 Tf 13.15 6.86 Td[(2p 2;p)]TJ /F4 7.97 Tf 13.15 6.86 Td[(2p 2o.Sinceep(t),rp(t)2L1,standardlinearanalysismethodscanbeusedtoprovethat_ep(t)2L1from( 4{5 ).Sinceep(t),rp(t)2L1,theassumptionthatqpd(t),_qpd(t)existandare 68

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boundedcanbeusedalongwith( 4{4 )and( 4{5 )toconcludethatqp(t),_qp(t)2L1.Therefore,from( 4{7 )and( 4{10 ),up(t),(t)2L1.Sinceep(t),rp(t),_ep(t),_ap(t)2L1,( 4{12 )indicatesthat_rp(t)2L1.ThedenitionofWp(y)andzp(t)canbeusedtoprovethatWp(y)isuniformlycontinuous.Therefore,Barbalat'slemma[ 99 ]canbeinvokedtoconcludethat kzp(t)k2!0ast!1:(4{22)Fromthedenitionofzp(t),( 4{22 )canbeusedtoshowthatrp(t)!0andep(t)!0ast!1: 4.3OrdinaryServiceTheevolutionoftheensembleaveragequeuelengthfortheOrdinaryServiceisgivenin( 4{2 )withi=r,i.e., _qr=)]TJ /F3 11.955 Tf 9.29 0 Td[(urqr 1+qr+r(t)]TJ /F3 11.955 Tf 11.96 0 Td[(r(t));(4{23)whereur(t)=userver)]TJ /F3 11.955 Tf 11.96 0 Td[(up(t)isknown,andthecontrolvariable(ensembleaveragearrivalrate)r(t)]TJ /F3 11.955 Tf 11.96 0 Td[(r(t))2Rneedstobedesigned.LettheensembleaveragequeuelengtherrorfortheOrdinaryServicequeueingsystemer(t)2Rbedenedas er,qr)]TJ /F3 11.955 Tf 11.96 0 Td[(qrd;(4{24)whereqrd(t)2Risthedesiredensembleaveragequeuelengthforthisservice.Tofacilitatethesubsequentanalysis,thelteredtrackingerrorrr(t)isdenedas rr,_er+rer+r(t))]TJ /F3 11.955 Tf 11.96 0 Td[(r(t)]TJ /F3 11.955 Tf 11.96 0 Td[(r(t)):(4{25) 69

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Basedonthesubsequentstabilityanalysis,thecontrollerforOrdinaryService,i.e.,r(t),isdesignedas r(t),)]TJ /F3 11.955 Tf 9.3 0 Td[(kr8<:1+r krer+tZ0r+1 krer()+r())]TJ /F3 11.955 Tf 11.96 0 Td[(r(t)]TJ /F3 11.955 Tf 11.96 0 Td[(r())d9=;+urqr 1+qr+_qrd+kr1+r krer(0))]TJ /F3 11.955 Tf 9.3 0 Td[(ur(0)qr(0) 1+qr(0))]TJ /F1 11.955 Tf 14.11 0 Td[(_qrd(0): (4{26) Aftertakingthederivativeof( 4{24 ),using( 4{23 ),andsubstitutingfor_erin( 4{25 )yields rr=)]TJ /F3 11.955 Tf 9.29 0 Td[(urqr 1+qr)]TJ /F1 11.955 Tf 14.11 0 Td[(_qrd+rer+r(t):(4{27)Anadditionalderivativeof( 4{27 )istakentofacilitatethesubsequentanalysis.Hence,byusing( 4{26 ),thederivativeof( 4{27 )canbeexpressedas _rr=d dt)]TJ /F3 11.955 Tf 9.3 0 Td[(urqr 1+qr)]TJ /F1 11.955 Tf 12.81 0 Td[(qrd+r_er)]TJ /F3 11.955 Tf 11.95 0 Td[(kr[_er+rer+r(t))]TJ /F3 11.955 Tf 11.96 0 Td[(r(t)]TJ /F3 11.955 Tf 11.96 0 Td[(r(t))]+d dturqr 1+qr+qrd)]TJ /F3 11.955 Tf 11.96 0 Td[(r_er)]TJ /F3 11.955 Tf 11.95 0 Td[(er:=)]TJ /F3 11.955 Tf 9.29 0 Td[(krrr)]TJ /F3 11.955 Tf 11.96 0 Td[(er: (4{28) 4.3.1StabilityAnalysisTheorem4.2:Thecontrollerdesignedin( 4{26 )ensuresglobalasymptoticensembleaveragequeuelengthregulationintheOrdinaryServicebuerprovidedthecontrolgainsareselectedaccordingtothesucientconditions r>1 2;2!(1)]TJ /F1 11.955 Tf 13.86 0 Td[(_r) 2!+1>r;(4{29)where!2R+isasubsequentlydenedcontrolgain.Proof:Letyr(t)2R3bedenedasyr(t),zTr(t)p QTwherezr(t)2R2isdenedaszr(t),er(t)rr(t)T.LetQ_r;t;r2RdenotetheLyapunov-Krasovskii 70

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functional,denedas Q,!tZt)]TJ /F4 7.97 Tf 6.59 0 Td[(r(t)0@tZs_r()2d1Ads;(4{30)where!2R+isaknownpositiveconstant.LetVr(yr;t):R3[0;1)!Rbeapositive-denitefunctiondenedas Vr=1 2e2r+1 2r2r+Q:(4{31)Takingthederivativeof( 4{31 ),andusing( 4{25 ),( 4{28 )and( 4{30 )yields _Vr=er(rr)]TJ /F3 11.955 Tf 11.96 0 Td[(rer)]TJ /F3 11.955 Tf 11.95 0 Td[(ea)+rr()]TJ /F3 11.955 Tf 9.3 0 Td[(krrr)]TJ /F3 11.955 Tf 11.95 0 Td[(er)+!r_r2)]TJ /F3 11.955 Tf 11.96 0 Td[(!(1)]TJ /F1 11.955 Tf 13.86 0 Td[(_r)tZt)]TJ /F4 7.97 Tf 6.59 0 Td[(r(t)_r()2d;(4{32)whereea,r(t))]TJ /F3 11.955 Tf 11.96 0 Td[(r(t)]TJ /F3 11.955 Tf 11.95 0 Td[(r())=tZt)]TJ /F4 7.97 Tf 6.59 0 Td[(r(t)_r()d:UsingYoung'sinequality, jerjjeajjerj2 2+jeaj2 2:(4{33)Using( 4{33 )andbyutilizingthefactthat_r(t)2tZt)]TJ /F4 7.97 Tf 6.58 0 Td[(r(t)_r()2d;jeaj2rtZt)]TJ /F4 7.97 Tf 6.58 0 Td[(r(t)_r()2d;theexpressionin( 4{32 )canbeupperboundedas_Vr)]TJ /F3 11.955 Tf 28.56 0 Td[(rjerj2)]TJ /F3 11.955 Tf 11.96 0 Td[(krjrrj2+jerj2 2+r 2tZt)]TJ /F4 7.97 Tf 6.59 0 Td[(r(t)_r()2d+!rtZt)]TJ /F4 7.97 Tf 6.58 0 Td[(r(t)_r()2d)]TJ /F3 11.955 Tf 9.3 0 Td[(!(1)]TJ /F1 11.955 Tf 13.87 0 Td[(_r)tZt)]TJ /F4 7.97 Tf 6.59 0 Td[(r(t)_r()2d=)]TJ /F10 11.955 Tf 11.29 16.86 Td[(r)]TJ /F1 11.955 Tf 13.15 8.09 Td[(1 2jerj2)]TJ /F3 11.955 Tf 11.96 0 Td[(krjrrj2)]TJ /F10 11.955 Tf 11.96 13.27 Td[(!(1)]TJ /F1 11.955 Tf 13.87 0 Td[(_r))]TJ /F3 11.955 Tf 11.96 0 Td[(!r)]TJ /F3 11.955 Tf 13.15 8.09 Td[(r 2tZt)]TJ /F4 7.97 Tf 6.59 0 Td[(r(t)_r()2d: 71

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If( 4{29 )issatised,then _Vr)]TJ /F3 11.955 Tf 21.91 0 Td[(Wr(yr);(4{34)whereWr(yr),rkzrk2,forsomepositiveconstantr2R+.Theinequalityin( 4{34 )canbeusedtoshowthater(t),rr(t)2L1.Theclosed-looperrorsystemcanbeusedtoshowthattheremainingsignalsarebounded.ThedenitionofWr(y)andzr(t)canbeusedtoprovethatWr(y)isuniformlycontinuous.Therefore,Barbalat'slemma[ 99 ]canbeinvokedtoconcludethat kzr(t)k2!0ast!1:(4{35)Fromthedenitionofzr(t),( 4{35 )canbeusedtoshowthatrr(t)!0ander(t)!0ast!1: 4.4SimulationResultsNumericalsimulationsareperformedinMatlabtodemonstratetheperformanceofthedevelopedcontrollerfortheDiServnetwork.Sincethemodelin( 4{2 )isvalidfor0u(t)userver,thecontrollersimplementedinthesimulationsforthePremiumServiceandOrdinaryServiceare up(t),max"0;min(userver;qp 1+qp)]TJ /F6 7.97 Tf 6.58 0 Td[(1)#(4{36)and ur(t)=max[0;userver)]TJ /F3 11.955 Tf 11.95 0 Td[(up(t)];(4{37)respectively.Hence,theinitialparametersarechosenappropriatelybasedonthedomainofoperationoftheDiServsystem.Themaximumallowableserverrate,userver,ischosentobe200dataunitsperunittime,where1unittimeisequalto100ms.InitialensembleaveragequeuelengthforboththePremiumService,qp(0),andtheOrdinaryService,qr(0),arechosentobe100dataunits.TheinitialserverrateforPremiumService,up(0),andtheinitialauxiliarycontrol,(0),arebothchosentobe50dataunitsperunittime. 72

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From( 4{37 ),ur(0)=950dataunitsperunittime.TheensembleaveragearrivalrateforOrdinaryService,r(0),ischosentobe100dataunitsperunittime.ThedesiredensembleaveragequeuelengthforPremium,qdp(t),andOrdinaryService,qdr(t),arechosentobe100and50dataunitsrespectively.ThearrivalrateattheinputofthePremiumServicequeueischosenasp(t)=30+0:05cos 5tdataunits/unittime.Thecontrolgainsarechosenaskp1=0:05;kp2=0:05;p=0:501;=0:1forPremiumService,andkr=0:1forOrdinaryService.TheOrdinaryServicecontrollerusesthetechniqueoffeedbacklinearizationwithoutarrival-ratedelays(see[ 100 ]).TheimplementedcontrollerforPremiumServiceisobtainedbyusing( 4{11 )andsubstituting( 4{10 )into( 4{36 ).Fig. 4-2 showstheensembleaveragequeuelengthplotandthecorrespondingserverratesforPremiumServiceswithoutarrival-ratedelays.ThequeuelengthforPremiumServiceasymptoticallyconvergesclosetothedesiredvalueasshowninFig. 4-2 .Fig. 4-3 showstheensembleaveragequeuelengthplotandthecorrespondingaveragearrivalratesforOrdinaryServiceswithoutarrival-ratedelays.ThequeuelengtherrorforOrdinaryServiceexponentiallyconvergestozero.Withdelays,theensembleaveragequeuelengthplotandthecorrespondingserverratesforPremiumServicesisshowninFig. 4-4 withatime-varyingdelayp(t)=0:5+0:1sin 2tseconds:ItcanbeinferredfromFig. 4-2 andFig. 4-4 thatthearrival-delaydelayactsasadisturbanceforPremiumServicebuer,andhencetheconvergenceoftheensemble 73

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Figure4-2. EnsembleaveragequeuelengthandserviceratesforPremiumServicewithoutarrival-ratedelays. Figure4-3. EnsembleaveragequeuelengthandaveragearrivalratesforOrdinaryServicewithoutarrival-ratedelays. 74

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Figure4-4. EnsembleaveragequeuelengthandserviceratesforPremiumServicewithaveragearrival-ratedelay. averagequeuelengthinthePremiumServicebuerwithoutdelayissimilartotheconvergencewithdelay,providedthedelayboundsestablishedinSection 4.1 aresatised.TheensembleaveragequeuelengthinOrdinaryServicebuerasymptoticallyconvergesclosetozerowitharrival-ratedelay,unlikethecasewithnodelaywhereweobtainexponentialconvergence(seeFig. 4-3 ).Thecontrolgainsarechosenaskp1=0:5;kp2=0:12;p=0:15;=0:1forPremiumService,andkr=0:3;r=0:01forOrdinaryService.Fig. 4-5 showstheensembleaveragequeuelengthplotandthecorrespondingaveragearrivalrateplotforOrdinaryServicewithtime-varyingdelay,r(t)=0:1+0:1sin 2tseconds: 75

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Figure4-5. Ensembleaveragequeuelengthandaveragearrival-rateforOrdinaryServicewithaveragearrival-ratedelay. 76

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CHAPTER5CONCLUSION 5.1SummaryofResultsInChapter 2 ,ALMMSEprediction-basedpower-controlalgorithmwasdevelopedforawirelessCDMA-basedmultiplecellularnetworkedsystemdespiteuncertainmultipathfading.ThepredictoruseslocalSINRmeasurementsatthepreviousandcurrenttimeinstances,alongwiththeDopplerfrequency(whichcanalsobeestimatedfromtheSINRmeasurements)toestimatethechanneluncertainties.ALyapunov-basedanalysisisusedtodevelopthecontrollerandaresultingultimateboundforthesampledSINRerror,whichcanbedecreaseduptoapointbyincreasingthecontrolgains.SimulationsindicatethattheSINRsofalltheradiolinksareregulatedintheregionminxi()maxwithanoutageprobabilityoflessthan10%,andpowerrequirementsofalltheMTswereintheimplementablerange.Outagesatsomesamplesweredeterminedtobeduetolimitationsofthelinearpredictor,andthishighlightstheneedformoresophisticatedpredictionandcontroldevelopmenttoolstoaddressthisissue.Simulationsarealsodoneusing2-bitand3-bitcontrolfeedback,andtheresultsshowthattheperformanceisstillwithintheacceptableoutagerangeifatleasta3-bitpowercontrolcommandisused.Comparisonagainstastandardpowercontrolalgorithmfromtheliteratureisdonetodemonstratetheadvantagesofusingchannelpredictionandmulti-bitfeedback.InChapter 3 ,amodelforcollisionsisdevelopedandincorporatedinthecontinuousCSMAMarkovchain.Anonlinedistributedalgorithmformaximizingthecollision-freetransmissionstatesisdevelopedthatestimatestheratesfromthesteady-statedistributionoftheMarkovstates.Toaccountfortherateconstraints,aconstrainedoptimizationproblemisdened,andanumericalsolutionissuggested.Simulationresultsinferthattheaveragenumberofcollisionsbyusingtheoptimizedparametersisreducedtolessthan0:2%.Inaddition,amodelforcollisionscausedduetohiddenterminalsisdevelopedandincorporatedinthecontinuousCSMAMarkovchain.Aconstrainedoptimization 77

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problemisdened,andanumericalsolutionissuggested.SimulationresultsareprovidedtodemonstratethestabilityofthequeuesforagivenstablesetofarrivalratesInChapter 4 ,acontinuouscontrolstrategyissuggestedforaDiServnetworktotrackthedesiredensembleaveragequeuelengthlevelspeciedbythenetworkoperator.ALyapunov-basedstabilityanalysisisprovidedtoillustrateglobalasymptotictrackingofthequeuelengthsinthePremiumServicebuer.Inaddition,arrivalratedelaysduetopropagationandprocessingthataectsthecontrolinputoftheOrdinaryServicebuerisaddressed,andaLyapunov-basedstabilityanalysisisprovidedtoillustrateglobalasymptotictrackingoftheensembleaveragequeuelengthofthisservice.Simulationsdemonstratetheperformanceandfeasibilityofthecontroller,alongwithshowingglobalasymptotictrackingofthequeuelengthsinthePremiumServiceandOrdinaryServicebuers. 5.2RecommendationsforFutureWorkFutureeortswillfocusextendingtheresultinChapter 3 todesigncross-layerthroughputmaximizationandtopologyrecongurationalgorithmstoaddressmobility,energy,andqueuelengthconstraintsattheterminals.Further,serviceratelimitationsinChapter 4 remainsanopenproblemthatcouldbefurtherexplored. 78

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APPENDIXAESTIMATIONOFRANDOMPROCESSES A-1GeneralMMSEbasedestimationtheoryLetW(l)besomerandomprocessthatneedstobeestimated.Theproblemofndingtheestimatesofthezeromeangaussianrandomvariablescanbedenedas"2min=min^W(l)EW(l))]TJ /F1 11.955 Tf 15.36 3.02 Td[(^W(l)2givenW(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(3);::=min^W(l)EhW2(l))]TJ /F1 11.955 Tf 11.96 0 Td[(2^W(l)W(l)+^W2(l)igivenW(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(3);::=min^W(l)EW2(l))]TJ /F1 11.955 Tf 11.96 0 Td[(2^W(l)E[W(l)]+^W2(l)givenW(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(3);::TondtheminimumvalueoftheestimateofW,d d^W(l)nEW2(l))]TJ /F1 11.955 Tf 11.95 0 Td[(2^W(l)E[W(l)]o=0givenW(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::=)0)]TJ /F1 11.955 Tf 11.96 0 Td[(2E[W(l)]+2^W(l)=0givenW(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::Theestimateisgiven[ 65 ]as^W(l)=E[W(l)jW(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::]:TheconditionalestimateisgivenbyE[W(l)jW(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::];whereW(l);W(l)]TJ /F1 11.955 Tf 12.25 0 Td[(1);W(l)]TJ /F1 11.955 Tf 12.25 0 Td[(2);W(l)]TJ /F1 11.955 Tf 12.24 0 Td[(3);::arealljointlygaussianandW(l)]TJ /F1 11.955 Tf 12.25 0 Td[(1);W(l)]TJ /F1 11.955 Tf -448.44 -23.9 Td[(2);W(l)]TJ /F1 11.955 Tf 11.11 0 Td[(3);::::arethepastvaluesoftherandomvariableWthatareusedtoestimatethecurrentvalueW(l). 79

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A-2GaussianCaseTheconditionalprobabilitydensityfunctionisgivenby[ 101 ]fW(l)[W(l)jW(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(3);::]=fW(l);W(l)]TJ /F6 7.97 Tf 6.59 0 Td[(1);W(l)]TJ /F6 7.97 Tf 6.59 0 Td[(2);:::[W(l);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::] fW(l)]TJ /F6 7.97 Tf 6.58 0 Td[(1);W(l)]TJ /F6 7.97 Tf 6.58 0 Td[(2);:::[W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::];wherethenumeratoranddenominatorarejointdensityfunctionsofthezero-meangaussianrandomvariablesWuptoinstantslandl)]TJ /F1 11.955 Tf 12.92 0 Td[(1respectively.TheCovarianceMatricesKnandKn)]TJ /F6 7.97 Tf 6.58 0 Td[(1aredenedasKn=Eh[Yl]:[Yl]Ti;andKn)]TJ /F6 7.97 Tf 6.59 0 Td[(1=Eh[Yl)]TJ /F6 7.97 Tf 6.58 0 Td[(1]:[Yl)]TJ /F6 7.97 Tf 6.59 0 Td[(1]Ti;whereYl=W(l)]TJ /F3 11.955 Tf 11.96 0 Td[(s)W(l)]TJ /F1 11.955 Tf 11.96 0 Td[((s)]TJ /F1 11.955 Tf 11.96 0 Td[(1))::W(l)T;andYl)]TJ /F6 7.97 Tf 6.59 0 Td[(1=W(l)]TJ /F3 11.955 Tf 11.96 0 Td[(s)W(l)]TJ /F1 11.955 Tf 11.96 0 Td[((s)]TJ /F1 11.955 Tf 11.96 0 Td[(1)):W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1)T:SincethemeansoftherandomvariablesWarezeroatanylfW(l)[W(l)jW(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(3);::]=exp)]TJ /F6 7.97 Tf 10.5 4.71 Td[(1 2YTlK)]TJ /F6 7.97 Tf 6.59 0 Td[(1nYl (2)n 2jKnj1=2:(exp)]TJ /F6 7.97 Tf 10.5 4.71 Td[(1 2YTl)]TJ /F6 7.97 Tf 6.59 0 Td[(1K)]TJ /F6 7.97 Tf 6.58 0 Td[(1n)]TJ /F6 7.97 Tf 6.58 0 Td[(1Yl)]TJ /F6 7.97 Tf 6.59 0 Td[(1 (2)(n)]TJ /F11 5.978 Tf 5.76 0 Td[(1) 2jKn)]TJ /F6 7.97 Tf 6.59 0 Td[(1j1=2))]TJ /F6 7.97 Tf 6.58 0 Td[(1: (A-1)SinceW(l)isazero-meangaussianrandomprocess,theMMSEestimateisalinearestimate,i.e.,E[W(l)jW(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(2);W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(3);::]canbeobtainedbymanipulatingEquation A-1 .Forasimplecasewithonlyonegivenvalue,thelinearMMSEestimationis 80

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givenbyE[W(l)jW(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1)]=W(l)+W(l)W(l)]TJ /F6 7.97 Tf 6.58 0 Td[(1)W(l) W(l)]TJ /F6 7.97 Tf 6.59 0 Td[(1))]TJ /F3 11.955 Tf 5.48 -9.68 Td[(W(l)]TJ /F1 11.955 Tf 11.95 0 Td[(1))]TJ /F3 11.955 Tf 11.95 0 Td[(W(l)]TJ /F6 7.97 Tf 6.59 0 Td[(1)=W(l)W(l)]TJ /F6 7.97 Tf 6.59 0 Td[(1)W(l) W(l)]TJ /F6 7.97 Tf 6.58 0 Td[(1)W(l)]TJ /F1 11.955 Tf 11.96 0 Td[(1); (A-2)whereW(l)W(l)]TJ /F6 7.97 Tf 6.59 0 Td[(1)istheautocorrelationfunction,W(l)andW(l)]TJ /F6 7.97 Tf 6.58 0 Td[(1)arethevariances. 81

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APPENDIXBORTHOGONALITYCONDITIONLetY,X1,X2,X3,....,XNbegaussianrandomvariableswithzeromeans.TheMMSEestimateistheconditionalmean,givenbyE[YjX1;X2;X3;::::;XN]=NXk=1aiXi:TherandomvariablesY)]TJ /F4 7.97 Tf 15.89 11.36 Td[(NPk=1aiXi,X1,X2,X3,....,XNarejointlygaussian.Sincethersttermisuncorrelatedwithalltherest,itcanbeinferredthattherandomvariableY)]TJ /F4 7.97 Tf 15.89 11.35 Td[(NPk=1aiXiisuncorrelatedwithX1,X2,X3,....,XN.Therefore,E" Y)]TJ /F4 7.97 Tf 16.8 14.94 Td[(NXk=1aiXi!jX1;X2;X3;::::;XN#=E" Y)]TJ /F4 7.97 Tf 16.8 14.94 Td[(NXk=1aiXi!#=E[Y])]TJ /F4 7.97 Tf 16.8 14.94 Td[(NXk=1aiE[Xi]=0;sinceE[Y]=E[Xi]=0.Thecondition E" Y)]TJ /F4 7.97 Tf 16.8 14.94 Td[(NXk=1aiXi!jX1;X2;X3;::::;XN#=0(B-1)isknownastheOrthogonalityCondition,whichcanalsobewrittenasY)]TJ /F3 11.955 Tf 11.95 0 Td[(aTX?X;whereX=X1X2::XNT:Theai'scanbeobtainedfromtheorthogonalitycondition.Note:FromEquation B-1 ,wegetE[YjX1;X2;X3;::::;XN])]TJ /F4 7.97 Tf 16.81 14.94 Td[(NXk=1aiE[XijX]=0: 82

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=)E[YjX1;X2;X3;::::;XN])]TJ /F4 7.97 Tf 16.81 14.95 Td[(NXk=1aiXi=0=)E[YjX1;X2;X3;::::;XN]=NXk=1aiXi:Thus,theconditionalmeanofazero-meangaussianrandomvariableYisgivenbyalinearestimateofthegivenvariablesXis. Calculationofai's.FromtheOrthogonalityconditioninEquation B-1 [ 65 ]E" Y)]TJ /F4 7.97 Tf 16.8 14.94 Td[(NXk=1aiXi!jXp#=0;1pk=)E[YXp]=NXk=1aiE[XkXp];1pk: =)kYX=aTKYY;(B-2)wherea,a1a2::aNT;kYX,E[YX1]E[YX2]E[YX3]::E[YXN]=KYX1KYX2KYX3::KYXN;andthecovariancematrixKXX=EXXT:FromEquation B-2 aT=kYXKTXX: 83

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APPENDIXCPROOFOFP0Lemma:Thesolutiontothedierentialequation _P(t)=)]TJ /F3 11.955 Tf 9.3 0 Td[(rp_ap)]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(ep); (B-1) P(ep(t0);t0)=jep(0)j)]TJ /F3 11.955 Tf 17.94 0 Td[(ep(0)_p(0) (B-2) satisestheconditionP(ep;t)0ifsatisesthecondition >_ap+1 pap:(B-3) Proof. Byusing( 4{5 ),integratingbyparts,andregroupingyields tZ0rp_ap())]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(ep())d=tZ0_ep_ap())]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(ep())d+tZ0pep_ap())]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(ep())d=_ap(t)ep(t))]TJ /F1 11.955 Tf 13.74 3.15 Td[(_ap(0)ep(0))]TJ /F3 11.955 Tf 11.96 0 Td[(jep(t)j+jep(0)j)]TJ /F4 7.97 Tf 18.39 18.66 Td[(tZ0pep 1 p@_ap() @!d+tZ0pep_ap())]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(ep())d: (B-4) From( 2{13 ),theexpressionin( B-4 )canbeupperboundedbytZ0rp_p())]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(ep())djep(t)jh_p)]TJ /F3 11.955 Tf 11.95 0 Td[(i+jep(0)j)]TJ /F1 11.955 Tf 19.72 3.16 Td[(_p(0)ep(0)+tZ0pjep()j_ap+1 pap)]TJ /F3 11.955 Tf 11.95 0 Td[(d: 84

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Therefore,iftheconditionin( B-3 )issatised,then tZ0rp_ap())]TJ /F3 11.955 Tf 11.96 0 Td[(sgn(ep())djep(0)j)]TJ /F1 11.955 Tf 19.72 3.15 Td[(_ap(0)ep(0)P(ep(t0);t0): (B-5) Integrating( B-1 )onbothsides,andusing( B-2 )yieldsP(ep(t);t)=jep(0)j)]TJ /F3 11.955 Tf 17.93 0 Td[(ep(0)_ap(0))]TJ /F4 7.97 Tf 18.39 18.66 Td[(tZ0rp()_ap())]TJ /F3 11.955 Tf 11.95 0 Td[(sgn(ep())d;whichindicatesthatP(ep(t);t)0from( B-5 ). 85

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BIOGRAPHICALSKETCH SankrithSubramanianwasborninChennai,Indiain1985.HereceivedhisBachelorofEngineeringdegreeininstrumentationandcontrolengineeringfromSt.Joseph'sCollegeofEngineering(AnnaUniversity),India,in2006,andhisMasterofSciencedegreeinelectricalandcomputerengineeringfromtheUniversityofFloridain2009.Hisinterestslieintheeldofnetworkedcontrolsystemsandcross-layerdesign.SankrithpursuedhisDoctorofPhilosophyintheDepartmentofElectricalandComputerEngineering.HewasaGraduateResearchAssistantintheNonlinearControlsandRoboticsgroupattheUniversityofFlorida,underthesupervisionofDr.WarrenE.Dixon.Hewasco-advisedbyDr.JohnM.SheaoftheWirelessInformationNetworkingGroupattheUniversityofFlorida.HewasalsocollaboratingwithAirForceResearchLaboratory,EglinAirForceBase,Florida,whileworkingtowardshisPhD.Theprimaryfocusofhisresearchwastoaddressthechallengesfacedinvariouslayersofdynamiccommunicationnetworks,anddesigncontrolalgorithmsandcrosslayerschemes. 94