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Multiuser Diversity - Enhanced Geographic Transmissions in Wireless Channels

Permanent Link: http://ufdc.ufl.edu/UFE0024965/00001

Material Information

Title: Multiuser Diversity - Enhanced Geographic Transmissions in Wireless Channels
Physical Description: 1 online resource (95 p.)
Language: english
Creator: Goswami, Tathagata
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: adhocnetworks, channelfading, geographictransmissions, multiuserdiversity, nodeactivation, optimumpower, transmissiondistance, transportcapacity, wirelesschannels
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: In wireless multi-hop packet radio networks, the conventional packet forwarding scheme is to pre-select the next-hop receiver for a packet based on knowledge of the network topology. However, when the nodes experience fading that changes on the order of the packet duration, the conventional routing approach will often offer poor performance because the pre-selected receiver may not be able to recover the packet because of fading. An alternative approach is geographic transmission, in which the packet is transmitted in the direction of the destination, but the next-hop receiver is not pre-selected. Multiuser diversity benefit can be exploited in such a scenario because the different receivers in the direction of the destination are likely to experience independent fading channels. This approach could significantly improve the probability of the packet being correctly received by the next-hop receiver. In the first part of this work, we show that such a benefit can maximize the expected value of the maximum transmission distance, which is one routing metric we consider for geographic transmissions. To provide an application of our findings, we design geographic transmission schemes that provide multiuser diversity gain in a Rayleigh fading channel. However, this approach places significant burden on the energies of the receiving nodes if the forwarding scheme requires that all of the next-hop neighbors of the transmitter (that are in the direction of the destination) attempt to receive a packet. This is because in a wireless multihop packet radio network, the nodes are limited in battery life. Thus, in the second part of this dissertation, we consider geographic transmission schemes that provide multiuser diversity with a fixed energy constraint. Towards that end, our approach is to provide energy efficiency by limiting the energy used in reception (which depends on the number of nodes that activate to receive a transmission). In determining which nodes should activate, an intuitive approach is to turn off all nodes located either very close to the transmitter or those very far away. This is because nodes very close to the transmitter are likely to decode a packet successfully but do not achieve a large transmission distance. On the other hand, nodes at very large transmission distances have low probabilities of decoding a packet successfully. Thus we propose node-activation-based-on-link-distance (NA-BOLD) schemes in which the probability that a node will activate/turn on to try to receive a packet is a function of its distance from the transmitter. With the goal of maximizing transmission distance, we analyze the optimum NA-BOLD scheme under a constraint on the number of nodes that activate. We also consider the maximization of transport capacity -- another useful metric used in geographic transmissions. Transport capacity can be considered to be maximum transmission distance weighted by the maximum achievable rate of information transmission. Under a total energy constraint, i.e., a constraint on the sum of the energies used in transmission and reception, we consider the joint design of node-activation functions and transmission rates to maximize transport capacity. We optimize the allocation of energy between transmission and reception when nodes activate using our NA-BOLD approach. We have evaluated our NA-BOLD approach in the Nakagami-m fading channel, where the parameter m is used to indicate fading severity and can model a large variety of wireless channels.
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 Tathagata Goswami.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Shea, John M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024965:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024965/00001

Material Information

Title: Multiuser Diversity - Enhanced Geographic Transmissions in Wireless Channels
Physical Description: 1 online resource (95 p.)
Language: english
Creator: Goswami, Tathagata
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: adhocnetworks, channelfading, geographictransmissions, multiuserdiversity, nodeactivation, optimumpower, transmissiondistance, transportcapacity, wirelesschannels
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: In wireless multi-hop packet radio networks, the conventional packet forwarding scheme is to pre-select the next-hop receiver for a packet based on knowledge of the network topology. However, when the nodes experience fading that changes on the order of the packet duration, the conventional routing approach will often offer poor performance because the pre-selected receiver may not be able to recover the packet because of fading. An alternative approach is geographic transmission, in which the packet is transmitted in the direction of the destination, but the next-hop receiver is not pre-selected. Multiuser diversity benefit can be exploited in such a scenario because the different receivers in the direction of the destination are likely to experience independent fading channels. This approach could significantly improve the probability of the packet being correctly received by the next-hop receiver. In the first part of this work, we show that such a benefit can maximize the expected value of the maximum transmission distance, which is one routing metric we consider for geographic transmissions. To provide an application of our findings, we design geographic transmission schemes that provide multiuser diversity gain in a Rayleigh fading channel. However, this approach places significant burden on the energies of the receiving nodes if the forwarding scheme requires that all of the next-hop neighbors of the transmitter (that are in the direction of the destination) attempt to receive a packet. This is because in a wireless multihop packet radio network, the nodes are limited in battery life. Thus, in the second part of this dissertation, we consider geographic transmission schemes that provide multiuser diversity with a fixed energy constraint. Towards that end, our approach is to provide energy efficiency by limiting the energy used in reception (which depends on the number of nodes that activate to receive a transmission). In determining which nodes should activate, an intuitive approach is to turn off all nodes located either very close to the transmitter or those very far away. This is because nodes very close to the transmitter are likely to decode a packet successfully but do not achieve a large transmission distance. On the other hand, nodes at very large transmission distances have low probabilities of decoding a packet successfully. Thus we propose node-activation-based-on-link-distance (NA-BOLD) schemes in which the probability that a node will activate/turn on to try to receive a packet is a function of its distance from the transmitter. With the goal of maximizing transmission distance, we analyze the optimum NA-BOLD scheme under a constraint on the number of nodes that activate. We also consider the maximization of transport capacity -- another useful metric used in geographic transmissions. Transport capacity can be considered to be maximum transmission distance weighted by the maximum achievable rate of information transmission. Under a total energy constraint, i.e., a constraint on the sum of the energies used in transmission and reception, we consider the joint design of node-activation functions and transmission rates to maximize transport capacity. We optimize the allocation of energy between transmission and reception when nodes activate using our NA-BOLD approach. We have evaluated our NA-BOLD approach in the Nakagami-m fading channel, where the parameter m is used to indicate fading severity and can model a large variety of wireless channels.
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 Tathagata Goswami.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Shea, John M.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024965:00001


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MULTIUSERDIVERSITY{ENHANCEDGEOGRAPHICTRANSMISSIONSIN WIRELESSCHANNELS By TATHAGATAD.GOSWAMI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c 2009TathagataD.Goswami 2

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Idedicatethisthesistomyparents,inparticulartomyfatherwhopassedawayduring thenalyearofmyPh.D.. 3

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ACKNOWLEDGMENTS Thisthesiscouldnothavebeenpossiblewithoutthepersonalandprofessional supportofseveralpeople.Iwouldliketoexpressmydeepestandsincerestgratitude tomyadvisorDr.JohnM.Shea,forhisguidance,expertise,andcontinuedpatiencein myresearch.IenjoyedworkingwithhimimmenselysinceIstartedmyPh.D.workin 2005.Hehasalwaysbeenhighlyapproachableandveryunderstandingofmyconcernsin researchandpersonallife. IthankDr.Rao,Dr.Glover,andDr.Dixonfortheirguidance,suggestionsand interestinmywork.MysincerestgratitudegoestoDr.RaoandDr.Gloverfortheir invaluablecontributionsthathaveresultedinseveralpublicationsrelatedtotheworkdone inthisthesis.SpecialthankstoDr.Shea,Dr.RaoandDr.Gloverfortheirjointeorts throughthenumerousmeetingsattheNewEngineeringBuildingDept.OfElectricaland ComputerEngineeringandLittleHallDept.OfMathematicsthathaveresultedinthis collaborationbeingsuccessful.IgreatlyappreciatetheeortsofDr.Gloverforproviding nancialsupportduringthelastsemesterofmyPh.D.. IamgratefultoallmyfriendsintheWINGgroupwhomademydaysandnights enjoyablewhileworkinginthelab. Iamthankfultomyentirefamilyforbeingverysupportiveinmycareer,particularly mysisterandmyuncle.Finally,Iamforeverindebtedtomyparentswhochoseto undergogreatstruggleinordertoraisemeandgivemethebestinlife.Myfather,who passedawayduringthenalyearofmyPh.D.,wasaconstantsourceofinspirationand encouragementthroughoutmylife.HewantedmetofullhisdreamofbecomingaPh.D., andthisthesisisatestimonytowardsthatdreamcomingtrue. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION..................................12 1.1MultiuserDiversityinAdHocNetworks...................13 1.2OutlineofDissertation.............................17 2MAXIMUMTRANSMISSIONDISTANCEOFGEOGRAPHICTRANSMISSIONS ONRAYLEIGHCHANNELS............................19 2.1Introduction...................................19 2.2SystemModel..................................19 2.3Analysis.....................................22 2.3.1MaximumTransmissionDistance...................22 2.3.1.1FadingChannel........................25 2.3.1.2NonfadingChannel......................25 2.3.2OutageProbabilityandCriticalTransmissionDistanceforLarge Networks.................................26 2.3.2.1FadingChannel........................26 2.3.2.2NonfadingChannel......................29 2.4Results......................................29 2.5Conclusions...................................34 3DISTANCE{BASEDNODEACTIVATIONFORGEOGRAPHICTRANSMISSIONS INFADINGCHANNELS..............................36 3.1SystemModel..................................37 3.2NA-BOLD:NodeActivationBasedOnLinkDistance...........39 3.2.1ComputationallyOptimumNA-BOLDStrategy: )]TJ/F15 11.9552 Tf 5.479 -9.684 Td [(NA-BOLDC .41 3.2.2OptimumNA-BOLDStrategy: )]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(NA-BOLDO ...........43 3.2.2.1ComputingOptimalNode-ActivationFunctionfromOptimal Measure............................56 3.2.2.2MinimumRadioDensityforNA-BOLDOStrategy...56 3.3OutageProbabilityforGeographicTransmissionSchemes..........57 3.4GeographicTransmissionwithEqualNodeActivationProbabilities....57 3.5Results......................................59 3.6Conclusions...................................62 5

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4MAXIMIZINGTRANSPORTCAPACITYANDTRANSMISSIONDISTANCE FORGEOGRAPHICTRANSMISSIONSWITHENERGYCONSTRAINTS..66 4.1SystemModel..................................67 4.2DesignofNodeActivationFunctions.....................68 4.2.1OptimumTransportCapacity.....................68 4.2.2RINGOTransmissionStrategy.....................75 4.3Results......................................76 4.4Conclusions...................................82 5CONCLUSIONSANDFUTUREWORK......................84 5.1Conclusions...................................84 5.2FutureWork...................................87 REFERENCES.......................................90 BIOGRAPHICALSKETCH................................95 6

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LISTOFTABLES Table page 3-1ExpectedValueofMaximumTransmissionDistanceforvariousnodeactivation schemes........................................61 7

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LISTOFFIGURES Figure page 2-1Geographicregionconsideredintheanalysis:sectorofangle ofanannular ringwithinteriorradius R 1 andexteriorradius R 2 .................20 2-2Expectedvalueofthemaximumtransmissiondistanceasafunctionofthenormalized nodedensity......................................29 2-3Criticaldistanceoftransmissionvs.thenormalizednodedensityforanoutage probabilityof0 : 05,forapathlossexponent n =2.................31 2-4Criticaldistanceoftransmissionvs.thenormalizednodedensityforanoutage probabilityof0 : 05,forapathlossexponent n =4.................31 2-5Regionmarkingzonesofoperationforfadingandnonfadingchannelforanoutage probabilityof0 : 05...................................32 2-6Expectedvalueofthemaximumtransmissiondistancevs.thenormalizednode densityforapathlossexponent n =4 ; =0 : 95...................33 2-7Probabilitythatnotevenasinglereceiverinthetransmissiondistancereceived themessagecorrectly,forapathlossexponent n =4 ; =0 : 95..........33 3-1Geographictransmissionregion:annulusofasectorwithinnerradius R 1 ,outer radius R 2 withtransmitteratcenter.........................37 3-2DISCstrategy:allnodesareactivatedinsideasectorwithradius R d .......57 3-3DISCOstrategy:nodesareactivatedwithxedprobability p insideasector withradius R d .....................................58 3-4Nodeactivationprobabilityfor =10nodesperunitsector,withexpected numberofactivenodes =3.............................60 3-5Expectedvalueofthemaximumtransmissiondistance E [ V max ]vs.theexpected numberofactivenodes ...............................61 3-6OutageprobabilitiesfortheNA-BOLDandconventionalschemesversusthe expectednumberofactivenodes .........................62 3-7FlowchartillustratingNA-BOLDCalgorithmtodesignnodeactivationfunction ^ x ...........................................64 3-8FlowchartillustratingNA-BOLDOalgorithmtodesignoptimalnodeactivation function ^ x ......................................65 4-1Nodeactivationregion:Transmitteratcenter.Whitecirclesindicatenodesthat areasleep,blackcirclesindicateactivenodes....................67 8

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4-2VariationofexpectedvalueoftransportcapacitywithtargetSNR for =10 : 76 4-3VariationofexpectedvalueoftransportcapacitywithtargetSNR for =35 : 77 4-4Variationofexpectedvalueoftransportcapacitywithrespectto << inaRayleighfadingchannelfor =5 : .......................78 4-5Variationofexpectedvalueoftransportcapacitywithrespectto << inaRayleighfadingchannelfor =25 : ......................78 4-6Optimumtransportcapacityasafunctionof ...................79 4-7Optimum asafractionof .............................80 4-8NodeactivationprobabilityofNA-BOLD.Red-solidand-dottedlinesindicate m =1 : 0,blue-solidand-dottedlinesindicate m =2 : 0...............81 4-9NodeactivationprobabilityofRINGO.Red-solidand-dottedlinesindicate m =1 : 0,blue-solidand-dottedlinesindicate m =2 : 0...............82 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy MULTIUSERDIVERSITY{ENHANCEDGEOGRAPHICTRANSMISSIONSIN WIRELESSCHANNELS By TathagataD.Goswami August2009 Chair:JohnM.Shea Major:ElectricalandComputerEngineering Inwirelessmulti-hoppacketradionetworks,theconventionalpacketforwarding schemeistopre-selectthenext-hopreceiverforapacketbasedonknowledgeofthe networktopology.However,whenthenodesexperiencefadingthatchangesonthe orderofthepacketduration,theconventionalroutingapproachwilloftenoerpoor performancebecausethepre-selectedreceivermaynotbeabletorecoverthepacket becauseoffading.Analternativeapproachisgeographictransmission,inwhichthe packetistransmittedinthedirectionofthedestination,butthenext-hopreceiverisnot pre-selected.Multiuserdiversitybenetcanbeexploitedinsuchascenariobecausethe dierentreceiversinthedirectionofthedestinationarelikelytoexperienceindependent fadingchannels.Thisapproachcouldsignicantlyimprovetheprobabilityofthepacket beingcorrectlyreceivedbythenext-hopreceiver. Intherstpartofthiswork,weshowthatsuchabenetcanmaximizetheexpected valueofthemaximumtransmissiondistance,whichisoneroutingmetricweconsiderfor geographictransmissions.Toprovideanapplicationofourndings,wedesigngeographic transmissionschemesthatprovidemultiuserdiversitygaininaRayleighfadingchannel. However,thisapproachplacessignicantburdenontheenergiesofthereceiving nodesiftheforwardingschemerequiresthatallofthenext-hopneighborsofthe transmitterthatareinthedirectionofthedestinationattempttoreceiveapacket.This isbecauseinawirelessmultihoppacketradionetwork,thenodesarelimitedinbattery 10

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life.Thus,inthesecondpartofthisdissertation,weconsidergeographictransmission schemesthatprovidemultiuserdiversitywithaxedenergyconstraint.Towardsthat end,ourapproachistoprovideenergyeciencybylimitingtheenergyusedinreception whichdependsonthenumberofnodesthatactivatetoreceiveatransmission. Indeterminingwhichnodesshouldactivate,anintuitiveapproachistoturnoall nodeslocatedeitherveryclosetothetransmitterorthoseveryfaraway.Thisisbecause nodesveryclosetothetransmitterarelikelytodecodeapacketsuccessfullybutdonot achievealargetransmissiondistance.Ontheotherhand,nodesatverylargetransmission distanceshavelowprobabilitiesofdecodingapacketsuccessfully.Thuswepropose node-activation-based-on-link-distanceNA-BOLDschemesinwhichtheprobabilitythat anodewillactivate/turnontotrytoreceiveapacketisafunctionofitsdistancefromthe transmitter.Withthegoalofmaximizingtransmissiondistance,weanalyzetheoptimum NA-BOLDschemeunderaconstraintonthenumberofnodesthatactivate.Wealso considerthemaximizationoftransportcapacity{anotherusefulmetricusedingeographic transmissions.Transportcapacitycanbeconsideredtobemaximumtransmissiondistance weightedbythemaximumachievablerateofinformationtransmission.Underatotal energyconstraint,i.e.,aconstraintonthesumoftheenergiesusedintransmissionand reception,weconsiderthejointdesignofnode-activationfunctionsandtransmission ratestomaximizetransportcapacity.Weoptimizetheallocationofenergybetween transmissionandreceptionwhennodesactivateusingourNA-BOLDapproach.We haveevaluatedourNA-BOLDapproachintheNakagamim fadingchannel,wherethe parameter m isusedtoindicatefadingseverityandcanmodelalargevarietyofwireless channels. 11

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CHAPTER1 INTRODUCTION Recentadvancesinwirelesscommunicationtechnologieshavecreatedaubiquitous presenceforwireless-baseddevicesinourdailylives.Whilethesedevicesaregetting smaller,lessexpensiveandmoreenergyecient,theyarebecomingmoreconvenient andpowerful.Theexplosivegrowthinsuchtechnologieshavecausedcellphones,PDAs, wirelessrouters,etc.toacquireimmenseimportanceinourdailylives.Battery-operated, low-powersensorsarealsoincreasinglyusedforsensing/monitoringtheenvironmentin wildlifehabitats,nuclearplants,chemicalfactories,etc.Atthesametime,criticaland emergingmission-specicneedshavealsoplaceddemandsformoresophisticatedwireless devicestobeusedinvariousmilitaryapplicationsanddisasterrecoverysituations. Inspiteofitswidespreadpopularity,itmightbelessknownthatunliketheirwireline counterpart,auniquecharacteristicofthewirelessmediumisthemultipathnatureof thewirelesschannel.Specically,obstructionsintheenvironmentoftenresultinmultiple copiesofatransmittedsignalarrivingatthereceiverafterbeingreectedovarious obstacles.Allthesecopiesoftheoriginalsignalarriveatthereceiverviadierentpaths, undergodierenttimedelays,andaddvectoriallyatthereceiver.Thisgivesriseto channelinducedsignalfading or,simply, fading inwirelesscommunications.Ifthevarious copiesofthesignaladdupdestructively,theoriginalsignalcouldbecompletelylost. Fadingistypicallymodeledasarandommultiplicativecomponentontheenvelopeof thetransmittedsignal.Forhighcarrierfrequencies,verysmallchangesinthepositions ofthecommunicatingterminalscancausesignicantchangesintheamountofreceived powerduetofading. Duetotheunpredictablenatureoffading,itisimpossibletoguaranteereliable communicationsoverafadingwirelesschannel,andthetraditionalviewofwireless communicationshasbeentoconsiderfadingasahindrancetocommunications.Until thelatenineties,techniquesdevelopedbywirelesssystemengineersweretomitigate 12

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theeectsoffading.Thebasicideaofmostofthesetechniquesistoconvertafading channelintoaconstantnon-fadingchannel.Recentlyithasbeenshownthatfadingcan actuallyprovetobeabenettoimprovetheperformanceofwirelesscommunication systems[1,2,3].Thisradicalperspectiveofviewingfadingasanopportunitythat canbeproperlyexploitedhasinitiatedacompletelynewlineofthinkinginwireless communicationsresearch{ opportunisticcommunicationsinfadingchannels .This thinkinghassimultaneouslyinitiatedaparadigmshiftfromfocusingonthecommunication performanceofpoint-to-pointlinkstothatoffocusingoncommunicationnetworkswhere thereexistmultiplecommunicatinglinksbetweenradioswhicharealsoreferredtoas users,nodesorterminals.Asdierentlinksexperiencedierentfadingconditions,itis likelythatthelinkbetweensomepairofcommunicatingusersexperiencesgoodquality, i.e.peaksintheinstantaneouschannelquality,atanyinstantoftime.Thefundamental premiseofopportunisticcommunicationsistotakeadvantageofthesepeaksinorderto improvecommunicationperformanceinfadingchannels.Theprobabilityofndingsuch peaksincreaseswiththenumberofusers.Thiseectisthewell-known multiuserdiversity eect inwirelesscommunications. 1.1MultiuserDiversityinAdHocNetworks Multiuserdiversitywasrstproposedforapplicationtocellularnetworks[1],where thepresenceofcentralizedcontrolfromabasestationcontrollermakesiteasiertoobtain themultiuserdiversitybenet.Thisisbecauseamobilecanestimatethechannelbetween thebasestationanditselfandcanfeedbackthisinformationtothebasestationthrough afeedbackchannel.Suchanideahasalreadybeenimplementedincurrentcellular systems[4,5],e.g.intheschedulingalgorithmofQualcomm'sEvolution-Data-Optimized EV-DOtechnologythatispartofthenextgeneration3Gcellularstandardsforvoice anddatacommunications.Acomprehensivesurveyonschedulingalgorithmscanbe foundin[6,7]. 13

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However,inseveralsituationsthereisinsucienttimetodeploycellularnetwork infrastructure:forexample,intacticalmilitarycommunicationsandduringdisaster-recovery operations.ThesenetworksareclassiedaswirelessmobileadhocnetworksMANETs or,multi-hoppacketradionetworksMPRNs.Mobileradioscommunicateona peer-to-peerbasisinthesenetworks[8].Duetothishighlydistributednatureofaccessing thewirelessmediumamongstthepeers,achievingthebenetofmultiuserdiversityis extremelydicultinMANETs.Thisworkaddressestheproblemofdesigningecient communicationschemesthatprovidemultiuserdiversitybenetinMANETs. Inwirelessnetworks,morethanoneoftheneighborsofthetransmittermaybean acceptablereceiverforamessage.Forinstance,inmulti-hopcommunicationbetween asourceandadestination,intermediateradiosactasrouterstodirectpacketsfrom thesourcetothedestination.Thegoalisforthemessagetoreachthedestinationin anecientmanner.Sincetheremaybemanyalternativeroutesavailablebetweenany source-destinationpair,therearemanypossibilitiesforintermediateradiosbetweenthe sourceanddestination.Ifthechannelsarechangingbecauseofmultipathfading,thenthe neighborsofaradiomaychangeovertimeevenwhenthepositionsoftheradioshavenot signicantlychangedthephysicaltopologyofthenetwork. PreviousworkonachievingmultiuserdiversityinMANETsmaybeclassiedas usingeither opportunistictransmissionOpTx or opportunisticreceptionOpRx .OpTx schemes,suchasthoseproposedin[9,10,11],whicharebasedontheapproachin[1]. AnothertypeofOpTxisproposedin[2],inwhichmobilityisusedtomovepackets closertotheirdestinations.OpRxschemesaredescribedin[12,13,14,15,16,17].These schemesgeneralizeroutingbyallowinganyofagroupofreceiverstoactasthenext-hop forwardingagentforapacket.OpTxschemesaredependentonhavingpacketsavailable formanynext-hopradiosandonhavingpacketsthatcantolerateadditionallatency. ManytypesoftraccannottoleratesuchdelaysandhenceOpRxschemesarefavoredin suchsituations.Weconsidersuchasituationinthisthesis. 14

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OneoftherstOpRxschemesdescribedintheliteratureisthealternaterouting schemedescribedin[18,SectionIV.D],inwhichapacketmaybeforwardedbyanode otherthanthatselectedbytheroutingtableafteracertainnumberoftransmission failuresoccurs.Morerecently,Larssondescribedasimilarscheme,calledselection diversityforwardingSDF[12,19],inwhichanacceptablelistofforwardingagentsis pre-determinedusingchannelstateinformationfedbackfromneighboringnodes.In[20], theauthorsconsidermaximizinginformationeciencytheproductofexpectedprogress andspectraleciencyanddesignroutingschemesthattakeadvantageofmultiuser diversity.In[16,15],aMACprotocolthatissimilartoalternateroutingwasinvestigated forcommunicationoverfadingchannels. Sincedierentchannelsexperiencedierentandoftenindependentchannel conditions,atransmittercanchoosetocommunicatewithareceiverthatisexperiencing goodchannelconditions.Inconventionalrouting,asinglereceiverispreselectedtoact asthenextforwardingagentforapacket.However,thisrequireschannelstatefeedback fromthepotentialreceiverswhichmayproducesignicantoverhead,particularlyforuse inMANETs.Inaddition,thiscanleadtoseveraldroppedpacketsandcorrespondingly lowerthroughputsinfadingchannelsthatchangeveryfast,typicallyovertheduration ofapacket.Analternativeapproachistousegeographictransmissions,inwhichthe packetistransmittedinthedirectionofthedestinationbutthenext-hopreceiverisnot pre-selected.Rather,inthisapproach,multiplereceiversinthedirectionofthedestination attempttoreceivethepacket,andoneofthereceiversthatisabletorecoverthepacket willbeselectedasthenext-hoprelay aposteriori .Anexampleofsuchaschemeis ExtremelyOpportunisticRoutingExOR[17].InExOR,thecandidateforwardersare chosenbasedontheproximityofthereceiverstothedestinationhowever,stillwithin normalconnectivityrangeofthetransmitter. Geographictransmissionschemesmakeuseofgeographicallocationoftheradiosfor deliveryofinformationinpacketradionetworks.Thetypicalassumptionisthatnodes 15

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knowtheirownlocationviaGPSorsomeotherlocalizationmechanism[21].Basedon itsownlocationandpossiblythatofitsneighbors,anodedecidestodecode/forward apacket.Theaddressofthedestinationisusuallycontainedinthepacketheader. Geographicroutingapproaches[22,23,24,25,26]eliminatetheneedfortable-based routingandcanprovidemultiuserdiversityagainstchannelfading[27].Energyeciency isoftenveryimportantinthedesignofcommunicationprotocolsforMANETs,which motivatessleepschedulingofnodescf.[28]inconjunctionwithgeographictransmission,[29, 30].Therehavebeenseveralworksthatconsideroptimizationofsleepschedules fornon-fadingchannels.Forinstance,in[31,32,33,34]severalMACprotocolsare proposedforuseinsensornetworks.Akeypartoftheseprotocolsisthecoordination ofsleepschedulesamongnodesinanaareatoprovideecientowofinformation acrossasensornetwork.Someothersignicantworksontopologycontrolforsensor networksinclude[35,36]seealso[37].However,alloftheseworksassumethatthe channelsatthenodesdonotchangerapidlywithtime,whichisconsistentwithsensor networkapplicationsbutnotmobileadhocnetworks.In[13,38],GeographicRandom ForwardingGeRaFisproposedtoovercomerandomsleepschedules.GeRaFisapplied tofadingchannelsin[3].Intheseworks,nodesareassumedtofollowarandomsleeping pattern.In[39],theauthorsintegratehybrid-ARQtechniquesintoGeRaFanddesign theHARBINGERprotocolthatprovidesabetterenergy-latencytradeo.In[40,41], Deng etal. proposesleepingschemesinwhichtheprobabilitythatanodesleepsisa linearfunctionofitsdistancefromaclusterhead.Thelinearfunctionisnotshowntobe optimalandisdesignedtocompensateforadditionaltransmitpowerneededbythenodes tocommunicatewiththeclusterhead.Onlyexponentialpathlossisconsidered. Theaimofgeographicschemesistoprovidespatialreuse[42]inconjunctionwith suitablyoptimizingsomeperformancemetricofinterest.Oneofthewidelyusedmetrics is expectedforwardprogress ,denedbyKleinrockandSilvesterasearlyas1978intheir classicpaper[43].Theydened expectedforwardprogress asthedistancetraveledbythe 16

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transmissioninthedirectionofthedestination,projectedontoalinejoiningthesource andthedestination.Insomescenarios,aprespecieddestinationradiomightnotexistif thetransmitterdoesnothavetocommunicatewithaxeddestination.Forsuchscenarios, thegoalofthetransmittercouldbetodispersethemessageasfaraspossibleintothe networkinaspecieddirection.Alternatively,axeddestinationmightexistbutbefar awayfromthetransmitter,sothatitissucientforthetransmittertosendthemessage inthedirectionofthedestinationwithoutneedingtoconsiderminimizingtheresidual distancecf.[13,38,3].Thus,itmaybeassumedthatthetransmitterbroadcaststhe messagetoreceiverslocatedwithinsomesectorcenteredonthelinetothedestination.If thissectorissmall,thenthecommongoalofmaximizingtheexpectedprogress[43,44] iscloselyapproximatedbymaximizingthetransmissiondistance.Wehaveshownthat multiuserdiversitybenetcanbeutilizedtomaximizetransmissiondistanceinMANETs in[27].SincethereceiversinaMANETareconstrainedinbatterylife,wehavedesigned node-activationschemes[45,46,47]withagoalofmaximizingtransmissiondistanceunder aconstraintonthenumberofnodesthatactivatetorecoverapacket.Theproblemof maximizingexpectedforwardprogressisconsideredin[48]. Anotherusefulmetricusedingeographictransmissionsis transportcapacity whichistheproductofthetransmissiondistanceandthemaximumachievablerateof informationtransmission[49].Weconsiderthejointdesignofnode-activationschemesand transmissionratestomaximizetheexpectedvalueoftransportcapacityin[50]. 1.2OutlineofDissertation Thisdissertationisorganizedasfollows.InChapter2,weconsidergeographic transmissionschemesthatdonotrequireanyfeedbackfromthereceivingnodesto thetransmitterorcooperationamongthereceivingnodes.Weassumethatevery next-hopneighboringradioattemptstoreceiveatransmittedpacket.However,this requiresadditionalenergyattheactivereceivingradios.InChapter3,wedevelop node-activationschemestodeterminewhichnodesshouldactivateinordertotryto 17

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receiveapacketunderanenergyconstraintthatlimitsthenumberofactiveradios. InChapter4weconsidermaximizingthetransportcapacity{anotherusefulmetric usedingeographictransmissions.Transportcapacitycanbeconsideredtobemaximum transmissiondistanceweightedbythemaximumachievablerateofinformationtransmission. Weconsiderthejointdesignofnode-activationschemesandtransmissionratesto maximizetheexpectedvalueoftransportcapacityoveraNakagamim channelunder atotalenergyconstraint,i.e.,aconstraintonthesumoftheenergiesusedintransmission andreception.Weoptimizetheallocationofenergybetweentransmissionandreception whennodesactivateusingourproposednode-activationapproach.Finally,wepresentour conclusionsinChapter5. 18

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CHAPTER2 MAXIMUMTRANSMISSIONDISTANCEOFGEOGRAPHICTRANSMISSIONSON RAYLEIGHCHANNELS 2.1Introduction MostofthepreviousworkrefertoChapter1fordetailsonopportunisticreception requiresthatreceiversfeedbacktheirlocationand/orchannelstateinformationtothe transmitteroreventhatnodescooperatewitheachother.Thiscanresultinsignicant overheadassociatedwithmaintainingnetworkstateinformation.Asaresult,wefocus ongeographictransmissionstrategiesthateliminatetheuseoffeedbackofchannelstate informationorlocationoftheradios. Wefocusonasingletransmissioninawirelessnetworkthatusesgeographicrouting. Weanalyzethestatisticsofthemaximumtransmissiondistancethatcanbeachieved fromthetransmittertoagroupofrandomlylocatedreceiverswhohavereceivedthe transmissionsuccessfully.Thisscenariomayariseinseveralapplications.Incertain sensornetworks[51],asensingnodenearthemiddleofamonitoredareamustrelay itsinformationtowardcollectionnodesaroundtheedgesofthearea.Typicallyinsuch scenariositisassumedthatthesourceradiotransmitsaccordingtosomepredetermined schedule,duringwhichnoothertransmissionoccursinthatchannelandhenceco-channel interferencecanbeignored.Previousworkontransmissionrangeinwirelessnetworks focusedmoreonadjustingtransmissionpowerintheabsenceoffadingtoimprove throughputorcontrolnetworkconnectivitycf.[43,52,44,53]. 2.2SystemModel Westudyabroadcastcommunicationenvironmentinwhichasinglesource radiotransmittertransmitsinformationtodestinationradiosreceiversthatare distributedaroundthesourceradioaccordingtoahomogeneousPoissonpointprocess inatwo-dimensionalplaneatrate radiosperunitarea.Werstconsiderasectorof 19

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Figure2-1.Geographicregionconsideredintheanalysis:sectorofangle ofanannular ringwithinteriorradius R 1 andexteriorradius R 2 angle ofanannularringwithinnerradius R 1 andouterradius R 2 ,asshowninFig.2-1. Theprobabilitythatthereare l radiosinsidethesectorisgivenby P [ L = l ]= exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(A A l l ;l =0 ; 1 ;:::; {1 where A = R 2 2 )]TJ/F50 11.9552 Tf 12.336 0 Td [(R 2 1 = istheareaofthesector.UnderthePoissonpointprocess, thereceiverswithinsuchanannularsectoraredistributeduniformlyinarea,andhence thedistributionfunctionforthedistancetoanarbitraryreceiveris F X i x = 8 > > > > > > < > > > > > > : 0 ;x R 1 x 2 )]TJ/F25 7.9701 Tf 6.587 0 Td [(R 2 1 R 2 2 )]TJ/F25 7.9701 Tf 6.586 0 Td [(R 2 1 ;R 1 R 2 : {2 Weconsidertransmissioninwirelessenvironmentsthatcanbemodeledaseithera fadingchanneloranonfadingAWGNchannel.TheAWGNchannelmodelisusedasa referencetodetermineinwhichscenariosinafadingchannelmultiuserdiversitycan actuallyimprovethetransmissiondistance.Weassumethatallradiosinthesystem 20

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useidentical,omni-directionalantennas.Atransmissionoverasinglehopisconsidered successfuliftheinstantaneousreceivedsignal-to-noiseratioSNRatthereceiveris greaterthanorequaltoareceiversensitivity ,whichweassumeisidenticalforall receivers. Weconsideraslowlyvarying,atRayleighfadingchannel.Wealsoassumethatthe channelfadinggainsareconstantovereachperiodduringwhichamessageistransmitted fromasourceradio,andweassumethatthefadinggainsareindependentamongnodes. Thus,thesignalpowerreceivedatanarbitrarymobilereceiverdependsonlyonthe distancebetweenthebasestationandthatreceiverandthefadinggainatthereceiver duringthattransmission. Let X i denotethedistancefromthetransmittertoreceiver i andlet i denotethe Rayleighfadingchannelcoecientforthatreceiver.Then H i = j i j 2 isanexponential randomvariablewithmean1.Withoutlossofgenerality,theinstantaneousSNRat receiver i canbemodeledas i = H i X )]TJ/F25 7.9701 Tf 6.587 0 Td [(n i ; {3 where n denotesthepath-lossexponent.Weassume n 2 ; whichisareasonable assumptionforanyregionoutsideasmallneighborhoodofthetransmittingantenna. WecomparetheperformanceofthesystemwithRayleighfadingtotheperformance withnonfadingAWGNchannels.FortheAWGNchannel,theinstantaneousSNR i atthe receiver,dependsonlyontherandomdistance X i fromthetransmitterandcanthusbe modeledas i = X )]TJ/F25 7.9701 Tf 6.586 0 Td [(n i : {4 21

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2.3Analysis 2.3.1MaximumTransmissionDistance Werstconsidertransmissionintoasectorofanannularring,asdescribedin theprevioussection.Aradiolocatedinthisregioncansuccessfullyrecoveramessage if i ,or X i n p H i = .Withoutlossofgenerality,wenormalizealldistancesto thetransmissionradiusoftheAWGNchannel,whichweconsiderasunity.Hencewe let =1anddene V i = X i 1 0 ; n p H i ] X i ; {5 where 1 A istheindicatorfunctiongivenby 1 A x = 8 > > < > > : 1 ;x 2 A 0 ; otherwise : Inotherwords, V i = X i ifthereceivercansuccessfullyrecoverthemessage,and V i = 0otherwise.Then,conditionedon N randomlylocatedradiosinthesector,themaximum distance M toareceiverthatcansuccessfullyrecoveramessagefromthetransmittercan beexpressedas M = 8 > > < > > : max V 1 ;V 2 :::V N ;N =1 ; 2 ;::: 0 ;N =0 : {6 Notethatwhentherearenoreceiversinthenetwork, M 0.Lettheconditional distributionof M giventhatthereare N radiosinthenetworkbedenoted F M v j N =0. Hence, F M v j N =0=1 ; 8 v> 0.Thus,from2{6, F M v j N =0canbecompactly expressedas F M v j N = 8 > > < > > : F V i v N ;v 0 0 ;v< 0 {7 22

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for N =0 ; 1 ;::: .FortheRayleighfadingchannel,wehavederived F V i v asfollows. F V v = P V i v = P X i 1 0 ; n p H i ] X i v = E H i P X i 1 0 ; n p H i ] X i v H i = E H i P X i min f v; n p H i g H i + E H i P X i > n p H i H i Substitutinginto2{2yields, F V v = E H i 2 4 min n n p H 2 i ;v 2 o )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 R 2 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 1 R 1 ;R 2 ] min n v; n p H i o 3 5 +1 )]TJ/F79 11.9552 Tf 11.955 0 Td [(E H i n p H 2 i )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 R 2 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 1 R 1 ;R 2 ] n p H i + 1 [ R 2 ; 1 n p H i !# + E H i h 1 [ R 2 ; 1 min n v; n p H i oi : Recallthat H i isanexponentialrandomvariablewithmeanunity.Thus,aftersome algebra,for v 2 ;R 1 ] ;R 1 > 0 ;R 2 >R 1 ,wehave, F V v =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(R n 2 + 1 R 2 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 1+ 2 n ;R n 1 )]TJ/F50 11.9552 Tf 11.955 0 Td [( 1+ 2 n ;R n 2 + R 2 1 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(R n 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(R n 2 : {8 For v 2 R 1 ;R 2 ], F V v =1 )]TJ/F15 11.9552 Tf 11.291 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(R n 2 + v 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 R 2 2 )]TJ/F50 11.9552 Tf 11.956 0 Td [(R 2 1 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(v n + 1 R 2 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 1+ 2 n ;v n )]TJ/F50 11.9552 Tf 11.956 0 Td [( 1+ 2 n ;R n 2 + R 2 1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(v n )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(R n 2 : {9 Here, a;x isthelowerincompleteGammafunction, a;x = Z x 0 t a )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(t dt; 23

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\050 a;x istheupperincompleteGammafunction, \050 a;x = Z 1 x t a )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(t dt; and\050 a = a;x +\050 a;x istheusualgammafunction.Notethatthisderivationcan betriviallymodiedtoobtain F V i v forthenonfadingAWGNchannel,forwhichwe have V i =0if X i > 1and H i 1. NowsupposethatthereceiversaroundthetransmitterarePoissondistributedin two-dimensionalspaceatrate nodesperunitareaandthetransmissionisintended onlyforradiosinanannularsectorofarea A .Thenthedistributionofthemaximum transmissiondistance, F M v ,toaradiolocatedinsidethissector,canbeeasilyobtained from2{7and2{1as F M v = E N F V i v N # =exp AF V i v )]TJ/F50 11.9552 Tf 11.955 0 Td [(A : {10 Weshallnowextendouranalysisofthemaximumtransmissiondistancetoinnite networks,assumingthatalltheradiosinthenetworkareawake,i.e. R 1 0 ;R 2 1 .Considerasequenceofrandomvariables M 1 ;M 2 ;M 3 :::M i ::: where M R denotes themaximumtransmissiondistancewhen R 1 =0and R 2 = R .Let F M i t denote thecumulativedistributionfunctionoftherandomvariable M i .Also,let F M t = lim i !1 F M i t .Notethatthelimitexistsateverypoint t 2 R andthesequenceofrandom variables f M i g convergesto M indistribution.Thus M isarandomvariablewith distributionfunction F M t .Wecanusethisdistributionfunctiontoobtaintheexpected valueofthemaximumtransmissiondistanceforboththefadingandnonfadingchannelsas E [ M ]= Z 1 0 1 )]TJ/F50 11.9552 Tf 11.956 0 Td [(F M x dx: {11 24

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2.3.1.1FadingChannel Aftersubstitutingtheexpressionfor F V i v derivedin2{9in2{10andtakingthe limits R 1 0 ;R 2 !1 ,anddoingsomesimplication,weobtain F M t =exp )]TJ/F50 11.9552 Tf 13.15 8.088 Td [( n )]TJ/F73 11.9552 Tf 9.306 16.857 Td [( 2 n ;t n : {12 Thenormalizednodedensity 0 isdenedastheexpectednumberofradioswithin asectorofangle andradiusunity,andisexpressedas, 0 = 2 .Thus,wecanrewrite 2{12as F M t =exp )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 0 n )]TJ/F73 11.9552 Tf 9.307 16.857 Td [( 2 n ;t n : {13 2.3.1.2NonfadingChannel IntheAWGNchannel,alltheradioswithinacircleofradiusunityareabletoreceive themessagecorrectly.Hencethedistributionfunctionof M as R !1 ,canbeeasily derivedtobe F M t = 8 > > < > > : exp 2 t 2 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 ;t< 1 1 ; otherwise : {14 Expressing2{14intermsofthenormalizednodedensity, 0 = 2 ,wehave, F M t = 8 > > < > > : exp 0 t 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ;t< 1 1 ; otherwise : {15 Wehaveobtainedtheexpectedvalueofthemaximumtransmissiondistanceinthe fadingchannel,numerically,aswehavenotfoundaclosedformexpression.Notethatthe distributionfunctionforthenonfadingchannelgivenby2{15doesnotdependonthe 25

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pathlossexponent n becausewehavenormalizedthetransmissiontoliewithinacircleof unitradius. 2.3.2OutageProbabilityandCriticalTransmissionDistanceforLarge Networks Inmanyscenarios,itwouldbedesirablethatthemessagetravelatleastsome minimumdistancefromthetransmitter.Inotherwords,wedesirethatthefurthest receivertosuccessfullyrecoverthemessagebeatleastsomecriticaltransmission distance d c awayfromthetransmitter.Asaresult,wedenethe outageprobability P out ,astheprobabilitythatthemaximumtransmissiondistance M islessthanthe criticaltransmissiondistance d c .Thus,for d c > 0,wehave F M d c =P out : {16 Notethatthedistributionofthelimitingtransmissiondistance M forboththe Rayleighfadingandthenonfadingchannelmodelhassomemassatzero.Hence,thevalue ofP out for d c > 0,mustsatisfythecondition, P out exp )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 0 n )]TJ/F73 11.9552 Tf 9.307 16.857 Td [( 2 n {17 forthefadingchannelmodelandP out exp )]TJ/F50 11.9552 Tf 9.299 0 Td [( 0 fortheAWGNchannelmodel.Wenext obtaintheexpressionfor d c forthefadingandAWGNchannelmodels. 2.3.2.1FadingChannel For d c > 0andsatisfying2{17,byequating2{13and2{16wehave, exp )]TJ/F15 11.9552 Tf 13.151 8.088 Td [(2 0 n )]TJ/F73 11.9552 Tf 9.307 16.857 Td [( 2 n ;t n =P out : Aftersomesimplication,weenduphavingtosolve, )]TJ/F73 11.9552 Tf 7.314 16.857 Td [( 2 n ;d n c = )]TJ/F50 11.9552 Tf 15.707 8.088 Td [(n 2 0 logP out : {18 26

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Wementionherethat2{18admitssolutionsforanypathlossexponent n 2.Wehave obtainedclosed-formexpressionsinthecaseswhen n =2and4andhavetightbounds for n 2,whicharegivenbelow. Case1: n =2 )]TJ/F73 11.9552 Tf 7.315 9.683 Td [()]TJ/F15 11.9552 Tf 5.479 -9.683 Td [(1 ;d 2 c = )]TJ/F15 11.9552 Tf 13.347 8.087 Td [(1 0 logP out d c = s log 0 )]TJ/F15 11.9552 Tf 11.291 0 Td [(logP out Case2: n =4 )]TJ/F73 11.9552 Tf 7.314 16.857 Td [( 1 2 ;d 4 c = )]TJ/F15 11.9552 Tf 13.348 8.088 Td [(2 0 logP out d c = s erfc )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 )]TJ/F15 11.9552 Tf 24.521 8.088 Td [(2 0 p logP out Intheaboveexpression,erfc )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 istheinverseofthecomplementaryerrorfunction, erfc ,whichisdenedas erfc x = 2 p Z 1 x exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(z 2 dz: Case3:Upperandlowerboundsfor n 2 Wecanrewrite2{18as: 2 n ;d n c =)]TJ/F73 11.9552 Tf 21.732 16.857 Td [( 2 n + n 2 0 logP out {19 Substitutingtherighthandsideof2{19as, c =)]TJ/F73 11.9552 Tf 21.732 9.684 Td [()]TJ/F24 7.9701 Tf 7.127 -4.977 Td [(2 n + n 2 0 logP out ,and m = n 2 ,wehave 1 m ;d 2 m c = c 27

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Referringto[54],theboundontheincompletegammafunctioncanbewrittenas, 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F51 11.9552 Tf 5.48 -9.683 Td [()]TJ/F50 11.9552 Tf 9.298 0 Td [( 1 d 2 m c 1 =m c 1 m 2 m 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F51 11.9552 Tf 5.479 -9.683 Td [()]TJ/F50 11.9552 Tf 9.299 0 Td [( 2 d 2 m c 1 =m {20 where, 1 =1 ; 2 = )]TJ/F73 11.9552 Tf 9.307 16.857 Td [( 1+ 1 m )]TJ/F25 7.9701 Tf 6.587 0 Td [(m : Werstderivealowerboundon d c .Fromtherighthandinequalityin2{20, c 1 m 2 m 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F51 11.9552 Tf 5.479 -9.684 Td [()]TJ/F50 11.9552 Tf 9.299 0 Td [( 2 d 2 m c 1 =m 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F51 11.9552 Tf 5.479 -9.684 Td [()]TJ/F50 11.9552 Tf 9.299 0 Td [( 2 d 2 m c 1 =m c 1 m 2 m exp )]TJ/F51 11.9552 Tf 5.48 -9.684 Td [()]TJ/F50 11.9552 Tf 9.298 0 Td [( 2 d 2 m c 1 )]TJ/F73 11.9552 Tf 11.955 20.444 Td [(" c 1 m 2 m # m d c 2 6 6 6 4 log 2 6 6 6 4 0 B B @ 1 1 )]TJ/F73 11.9552 Tf 11.955 16.857 Td [( c 1 m 2 m m 1 C C A 1 = 2 3 7 7 7 5 3 7 7 7 5 1 = 2 m {21 Usingthelefthandequalityin2{20,wederiveanupperboundof d c as d c 2 6 6 4 log 2 6 6 4 1 1 )]TJ/F73 11.9552 Tf 11.955 16.857 Td [( c 1 m 2 m m 3 7 7 5 3 7 7 5 1 = 2 m ; {22 where c =)]TJ/F73 11.9552 Tf 22.249 9.684 Td [()]TJ/F24 7.9701 Tf 7.127 -4.977 Td [(2 n + n 2 0 logP out ,and m = n 2 .Notethatequalityin2{21and2{22holds whenthepathlossexponent n =2. 28

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2.3.2.2NonfadingChannel For0
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achievedsaturatestothemaximumtransmissionrangeof1asthenumberofnodesinthat transmissionrangeincreases.Theexpectedvalueoftheachievedtransmissiondistancefor thefadingchannelsislargerthanfortheAWGNchannelsbecauseofmultiuserdiversity. Withasfewassixaverageneighbors,fadingchannelsalongwithmultiuserdiversity resultinanincreaseintheaveragetransmissiondistanceof25% ; 50%,and87 : 5%overthe distanceontheAWGNchannelforpath-lossexponentsof4 ; 3and2respectively. Tofurtherevaluatethebenetsofmultiuserdiversity,weconsidertransmission infadingandnonfadingchannelsintermsofthecriticaltransmissiondistance d c .The resultsinFig.2-3andFig.2-4showthenormalizedcriticaltransmissiondistancefor pathlossexponents n =2and4.Wehaveplottedthesevaluesbyconsideringanoutage probabilityof5%.When n =2,fortenradiosinthenetwork,95%ofthetime,fading improvesthecriticaltransmissiondistancebyafactorof1 : 375.Similarly,thediversity gainwhenthereare10radiosand n =4is0 : 92.Notethatthisindicatesthatifthere are10radiosinthenetwork,thenthenonfadingchanneloutperformsthefadingchannel. Thus,thereisabreak-evenpointintermsofthenormalizedradiodensity,afterwhich fadingwithmultiuserdiversityoutperformsthenonfadingchannel.Wehaveplottedthe break-evenpointsforarangeofvaluesofthepathlossexponent n inFig.2-5,foran outageprobabilityof0 : 05.Fig.2-5alsoindicatestheregioninwhichtransmissionina fadingchannelisbetterthanthenonfadingchannel.Fadingoersthebestperformancein theregionabovethecurve,andnon-fadingintheregionbelow. Possibleapplicationsofourresultsingeographictransmissions Thedesignofoptimalapproachestoturningonradiosforgeographictransmissionsis thesubjectofChapter3withreceiverpowerconstraintsandChapter4withatotalpower constraintthatisthesumofpowersconsumedintransmissionandreceptionofamessage. Inthissection,wedemonstratetheutilityofouranalyticalresultsincommunication systemsthatemploygeographictransmissionsbypresentingourrstdesignofascheme 30

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Figure2-3.Criticaldistanceoftransmissionvs.thenormalizednodedensityforanoutage probabilityof0 : 05,forapathlossexponent n =2. Figure2-4.Criticaldistanceoftransmissionvs.thenormalizednodedensityforanoutage probabilityof0 : 05,forapathlossexponent n =4. thatturnsonradioswithinacertaingeographicregion,asoriginallypresentedin[27].We donotclaimthatthisschemeisinanysenseoptimal,butitdoesdemonstratehowour resultscanbeuseful. Considerthereceiversdistributedinaninnitelylargeplanearoundthetransmitter accordingtoaPoissonprocesswithnormalizednodedensity 0 ;i.e., 0 istheaverage 31

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Figure2-5.Regionmarkingzonesofoperationforfadingandnonfadingchannelforan outageprobabilityof0 : 05. numberofradiosinsideasectorofangle withradiusunity.Ourschemelimitsthe numberofradiosthatmustturnontothosethatarelikelytobenearthemaximum receptiondistance.Todothis,wesacricesomereliabilityinthesensethatlimiting thesetofradiosthatturnonmayoccasionallydecreasethemaximumtransmission distancethatcanbeachievedormaycausethemessagetonotbesuccessfullyreceived byanyradio.Westatethatourschemeis -reliableifwelimitthereceiversthatturnon accordingto F M d c 1 j M > 0= 1 )]TJ/F50 11.9552 Tf 11.955 0 Td [( 2 {23 F M d c 2 j M > 0= 1+ 2 : Thenwelimittheradiosthatturnontothosewithinanannularsector withinterior andexteriorradiigivenby d c 1 and d c 2 ,respectively.Wehavereferredtoourschemeasthe schemeinFig.2-6andFig.2-7.In2{23,theconditionaldistribution F M t j M > 0 canbederivedfromthemarginaldistributionof M givenin2{13. 32

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Figure2-6.Expectedvalueofthemaximumtransmissiondistancevs.thenormalizednode densityforapathlossexponent n =4 ; =0 : 95. Figure2-7.Probabilitythatnotevenasinglereceiverinthetransmissiondistance receivedthemessagecorrectly,forapathlossexponent n =4 ; =0 : 95. 33

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WehavecomparedourschemewithasimplerbaselineschemetheDISCscheme" inFig.2-6andFig.2-7thattransmitsto all receiverswithinasector andtransmission radius R d fromthetransmitter.Weprovideresultsfor =0 : 95andpathlossexponent n = 4.Forafaircomparisonbetweenthetwoschemes,welimitthistransmissionradius R d such thattheaveragenumberofradiospresentwithin d c 1 and d c 2 isequaltothenumberwithin thesectorofradius R d .Wecomparethesetwoschemesbasedontheexpectedvalueof themaximumtransmissiondistanceandtheprobabilitythatnotevenasinglereceiver withinthetransmissiondistancereceivesamessagefromthetransmittercorrectly,which wedenoteasP Rx .Thisprobabilitycanbeobtainedeasilyfromtheexpressionfor F V v derivedin2{9.Itisgivenby, P Rx =exp )]TJ/F50 11.9552 Tf 9.298 0 Td [( 0 1+ 2 n ;R n 2 )]TJ/F50 11.9552 Tf 11.956 0 Td [( 1+ 2 n ;R n 1 + R 2 2 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(R n 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(R n 1 TheexpectedvalueofthemaximumtransmissiondistanceisshowninFig.2-6. The -reliableperformsbetteronaveragethantheDISCschemeintransmitting themessagefurther.ThisisbecausefortheDISCscheme,toomanyradiosturnon thatareusuallyshortofthemaximumtransmissiondistance.ThevaluesofP Rx for various 0 areplottedinFig.2-7.Weseethatweareintroducingsomecostinour -reliable protocolinthatweincreasetheprobabilitythatnoneofthereceiverssuccessfullyrecovers themessage.However,asthereceiversinthesecasesareclosetothetransmitter,thisis generallynotasignicantloss. 2.5Conclusions Inthischapter,wederivedthedistributionfunctionofthemaximumtransmission distanceachievableforageographictransmissionoverachannelexposedtopathlossand Rayleighfading.Weusedthisdistributionfunctiontoobtainthecriticaltransmission distance,givensomeoutageprobability.Wealsoprovideexpressionsforthecritical transmissiondistanceforpathlossexponents n =2 ; 4.Tounderstandthescenarios wherefadingisbenecial,wealsoprovidedresultsfortransmissioninanon-fading 34

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channel.Ourresultsindicatethatwhentherearealargenumberofradiosinthe network,thenfadingcanbebenecialintransportingthemessagefromthesourceto thedestination.Wealsoprovideanexampleofanapplicationofourresultstodesignofa simpleschemeofturningonradioswithinacertaingeographicregion. 35

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CHAPTER3 DISTANCE{BASEDNODEACTIVATIONFORGEOGRAPHICTRANSMISSIONSIN FADINGCHANNELS InChapter2weshowedthatmultiuserdiversitybenetinafadingchannelprovides agreatertransmissiondistancethanthatinanon-fadingchannel.However,geographic transmissionsthatrequirealltheneighborsofasourceradiotoreceiveatransmissioncan haveadetrimentaleectonnetworklifetime.Thisisbecausemobileradioshavelimited batteryenergy,andtheenergyconsumedinreceivingmessagescanbecomparableto thatusedintransmittingmessages[55,56].Asaresult,severalauthorshavesuggested receiver-activationtechniquespoweringosomeoftheredundantreceiverstoconserve energy[40,41,13,3,38,27].Inalloftheseworks,heuristictechniquestoconserveenergy havebeendiscussedthatarenotbasedonanyoptimizationcriterion. Forgeographictransmissioninfadingchannels,theprobabilitythatanode canreceiveamessageisadecreasingfunctionofthedistanceofthatnodefromthe transmitter,ifthepowerutilizedintransmittingamessageisxed.Thus,fromthe viewpointofasingletransmitter,thereisaninherenttradeoinselectingwhichofits neighborsshouldactivatei.e.,notsleep:Ifnodesclosetothetransmitteractivate,there isahigherprobabilitythatthemessageissuccessfullyreceived,butthemessagedoesnot makemuchprogresstowardthedestination.Ifnodesfarfromthetransmitteractivate,the messagegoesfartherifitissuccessfullyreceived,buttheprobabilityofthemessagebeing successfullyreceivedislow. Inthischapter,weconsiderthedesignofschemestodeterminewhetheranodeshould activatetoreceiveatransmissionunderanenergyconstraintontheexpectednumber ofnodesthatactivate.Sincethereceiversarerandomlydistributed,weletareceiver determinewhethertoactivateaccordingtoa node-activationfunction thatdependson thedistanceofthatradiofromthetransmitter.Thus,ourapproachiscalledNA-BOLD Node-ActivationBased-on-Link-Distance.Wedesignandevaluatetheperformance oftheoptimalandapproximatelyoptimalbutcomputationallyintensiveNA-BOLD 36

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Figure3-1.Geographictransmissionregion:annulusofasectorwithinner radius R 1 ,outerradius R 2 withtransmitteratcenter. schemes.WerefertotheseschemesasNA-BOLDOandNA-BOLDCrespectively. Weconsiderxed-ratetransmission,somaximizingtheexpectedtransmissiondistance isalsoequivalenttomaximizingthetransportcapacity 1 .Thus,wewishtomaximize thetransmissiondistancefromthesourceunderaconstraintontheexpectednumber ofreceiversthatactivateinsidethetransmissionregion,toreceiveabroadcastfromthe transmitter.Weassumethatthetransmissiondistanceachievedisequaltothedistanceto themostdistantactivereceiverthatsuccessfullyrecoversthemessage. 3.1SystemModel Weconsiderasingletransmissioninawirelesscommunicationsystemthatuses geographicrouting.Thus,weassumethatthetransmitterbroadcaststhemessageto receiverslocatedwithinsomesectorofangle centeredonthelinetothedestination.If 1 Weconsiderthemoregeneraltransportcapacityproblemin[50]. 37

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issmall,thenthecommongoalofmaximizingtheexpectedprogress[43,44]isclosely approximatedbymaximizingthetransmissiondistance.Ourobjectiveisthedesignofthe nodeactivationfunction,nottheMACprotocoltoselectthebestnode,whichhasalready beenconsideredin[13,38]. Weassumethatthenodesaredistributedaccordingtoahomogeneous,isotropic Poissonpointprocessin R 2 withintensity nodesperunitsector.Ifweconsiderthe nodeswithinsomeannularsectorofinnerandouterradii R 1 and R 2 ,respectively,then thedistance X i tothe i thnodeintheannularsectorhasdensity f X i x = 2 x R 2 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 ;R 1 0 ; {2 38

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wherewetake E [ H i ]=1toprovideunitaveragepowergain.In3{2,\050 m = R 1 0 t m )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(t dt ,andtheparameter m = E [ H 2 i ] )]TJ/F15 11.9552 Tf 13.05 0 Td [(1 )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;m 1 2 : Thelatteris commonlyreferredtoasthe fadinggure ,whichcanbevariedtomodeldierentfading conditionsinwirelesslinks. Withoutlossofgenerality,wenormalizealldistancessothatthetransmission distanceintheAWGNchannelisone.Thus,undertheassumptionofunitytransmit power,thesignalpoweratreceiver i canbemodeledas i = H i X )]TJ/F25 7.9701 Tf 6.586 0 Td [(n i ,where n denotesthe path-lossexponenttypically n 1 : 5. 3.2NA-BOLD:NodeActivationBasedOnLinkDistance WeconsiderthedesignofNA-BOLDschemesthatmaximizetheexpectedvalue ofthetransmissiondistancetothefarthestsuccessfulreceiverunderaconstrainton theexpectednumberofnodesthatactivatetoreceiveatransmission.Let x bethe conditionalprobabilitythatanodeactivatesgiventhatitisatdistance x fromthe transmitter.Wecall x the nodeactivationfunction .Weuseaprotocolmodelto determineifamessageisreceivedsuccessfully.Let betheSNRthresholdforsuccessful reception.AmessageissuccessfullyreceivedataradioiftheSNRexceedsathreshold thatisthesameatallthereceivers.If i istheSNRatreceiver i ,thenthemessageis successfullyreceivedatnode i if i > .Let U i beauniformrandomvariableon[0 ; 1]. Thenwedene V i = 8 > > < > > : X i ; i > U i < X i 0 ; otherwise; {3 i.e., V i isthetransmissiondistanceifthemessageisreceivedcorrectlyandzerootherwise. Considerrsttransmissionintoaregionofnitearea A ,whichcontains @ nodes awakeandasleep.Thenthedistancetothefarthestsuccessfulreceiveris V max = 8 > < > : max f V 1 ;V 2 :::V @ g ; @ =1 ; 2 ::: 0 ; @ =0 : {4 39

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Denotethedistributionofthei.i.d.randomvariables f V i g as F V t .Then,conditioned on @ F V max t j@ canbeexpressedas F V max t j@ =[ F V t ] @ .Since @ isaPoissonrandom variablewithmean A ,thedistributionof V max is F V max t =exp A F V t )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 : {5 Notethat V max maybezeroiftherearenoreceiversinsidetheregion,noreceivers activate,ornoneofthereceiversthatactivatehaveasucientlyhighreceivedSNR. Let K denotethenumberofnodesthatactivate.Wewishtondtheoptimum x suchthattheexpectedvalue, E [ V max ]ismaximized,subjecttoaconstraintonthe expectedvalueof K .Hence,wecanexpressouroptimizationproblemas ^ X i =argmax X i E [ V max ]{6 subjectto: E [ K ]= and0 X i 1 : Since V max isnon-negative,wecanexpress E [ V max ]as E [ V max ]= Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp A F V t )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 dt: {7 Letusdenote Y i = n p H i )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 .Denotethedistributionandthecomplementary distributionfunctionof Y by F Y y and G y respectively,where G y =1 )]TJ/F50 11.9552 Tf 12.816 0 Td [(F Y y Thenfor t> 0, F V t =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(P V i >t =1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(P X i >t;X i
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3.2.1ComputationallyOptimumNA-BOLDStrategy: )]TJ/F54 11.9552 Tf 5.479 -9.684 Td [(NA-BOLDC Wehaveprovidedanumericaltechniquetoobtaintheoptimumnodeactivation functionforarbitrarynodedensity .Towardsthatend,weintroducethefollowing theoremwhichisvalidformostmeasurespaces,inparticularthesub-intervalsof R + Theorem1. Ifwedenotetheset as = f :0 1 ; Z d = c g ; where c isaxedpositivevalueand isameasurablefunctiononthenitemeasure space ; F ; ,thentheextremepointsoftheconvexset areindicatorfunctions. Proof. Considersome 2 .Let0 > > > > < > > > > > : t )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ; 0 < t; ;t< 1+ t )]TJ/F50 11.9552 Tf 11.955 0 Td [(t; ; 1+ t )]TJ/F50 11.9552 Tf 11.955 0 Td [(t< 1 ; and 2 = )]TJ/F50 11.9552 Tf 11.955 0 Td [(t 1 = )]TJ/F50 11.9552 Tf 11.955 0 Td [(t : Then,0 1 ; 2 1and t 1 + )]TJ/F50 11.9552 Tf 11.956 0 Td [(t 2 = .Nowwehave, Z 1 )]TJ/F50 11.9552 Tf 11.955 0 Td [( d' = 1 )]TJ/F50 11.9552 Tf 11.955 0 Td [(t t Z 0 < t d + Z t< 1+ t )]TJ/F25 7.9701 Tf 6.586 0 Td [(t )]TJ/F50 11.9552 Tf 11.955 0 Td [( d: {10 Supposeforsome0 0 : {11 Weshowthatasafunctionof ;t ,3{10assumesbothpositiveandnegativevalues in ; 1 ; 1 : Henceitmustvanishsomewhere.Wediscussbelowseparatelythecases when3{10assumesnegativeandpositivevalues,respectively. 41

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Negativevaluesof3{10: Considersome t forwhich3{11istrue.Then, 1 )]TJ/F50 11.9552 Tf 11.955 0 Td [(t t Z 0 < t d )]TJ/F50 11.9552 Tf 11.955 0 Td [(t f 0 < t g! 0as t 0 : Furthermore,wehave, lim 0 ;t 0 Z t<< 1+ t )]TJ/F25 7.9701 Tf 6.586 0 Td [(t )]TJ/F50 11.9552 Tf 11.955 0 Td [( d = Z 0 << 1 )]TJ/F50 11.9552 Tf 9.298 0 Td [(d< 0using3{11 : Hence,itfollowsthat3{10attainsnegativevalues. Positivevaluesof3{10: Noticethat lim t 1 Z 0 < t 1 t d = Z 0 < 1 d> 0using3{11 : {12 Also, 1 1 )]TJ/F50 11.9552 Tf 11.955 0 Td [(t Z t< 1+ t )]TJ/F25 7.9701 Tf 6.586 0 Td [(t j )]TJ/F50 11.9552 Tf 11.955 0 Td [( j d Z d 0as t 1 ; {13 becauseontheset t< 1+ t )]TJ/F50 11.9552 Tf 11.632 0 Td [(t; j )]TJ/F50 11.9552 Tf 11.632 0 Td [( j 1 )]TJ/F50 11.9552 Tf 11.632 0 Td [(t .Hence3{12and3{13implythat 3{10attainspositivevalues.Thus,if isanextremepointoftheconvexset ,then 3{11mustbefalse,i.e. Z 0 < t d =0forevery0
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A j =[ a j ;b j .Thus,wecanrewrite3{9asthefollowingconvexoptimizationproblem, max f j g Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(A Z 1 t X j j 1 A j x G x f X x dx dt # {15 subjectto: 8 > > > < > > > : A R 1 0 X j j 1 A j x f X x dx = 0 X j j =1 ; 0 < j 1 8 j: wherethearea A = R 2 2 )]TJ/F50 11.9552 Tf 12.25 0 Td [(R 2 1 ifweconsidertransmissionover[0 ; 2 inFig.3-1.Thus theoptimumnodeactivationfunctioncanbeexpressedas ^ x = X j j 1 A j x ; {16 where A j =[ a j ;b j andthe j 'scanbeobtainedbysolving3{15numerically.The completealgorithmispresentedgraphicallyinFig.3-8. 3.2.2OptimumNA-BOLDStrategy: )]TJ/F54 11.9552 Tf 5.479 -9.684 Td [(NA-BOLDO Inthissection,wendananalyticalsolutionfor ^ providedthat issuciently large.Itissimpletoseethattheobjectivefunctionin3{9isamonotonicallynon-decreasing functionof forxed .InSection3.2.2.1,wereformulatetheoptimizationproblemin 3{9asaproblemtondanoptimalmeasurethatremovestheexplicitdependenceon and f X .InSection3.2.2.1,weshowhowtocalculatetheoptimalnode-activationfunction fromtheoptimalmeasure,andwegiveanexpressionfortheminimumradiodensity for whichthisispossible.Letusconsiderasimilaroptimizationproblemto3{9, A =max Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G x dx dt {17 overallmeasures on[0 ; 1 ]and G isacontinuousfunctionon[0 ; 1 suchthat = ; 1 = ; given lim x !1 xG x =0 : {18 43

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Wewishtoobtainthemeasure thatmaximizes A asin3{17.InSection3.2.2.1,we showhowthesolutionfor dx in3{18canbeusedtondthesolutionto3{9.We startwiththecontinuityof A : Lemma1. A iscontinuous:If n weakly[59]then, A n A Proof. Now1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(x x sothat Z 1 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G x n dx dt Z 1 R Z 1 t G x n dx dt = Z 1 R x )]TJ/F50 11.9552 Tf 11.955 0 Td [(R G x n dx sup x R xG x n R; 1 sup x R xG x using3{18.Since G iscontinuousand n weakly, Z 1 t G x n dx Z 1 t G x dx exceptperhapsforcountablymainly t 's.Byboundedness,weget Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G x n dx dt Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G x dx dt: Thiscompletestheproof. Weshowthat satisfying3{17existsuniquelywiththehelpofthefollowing theorem. Theorem2. Thereisaunique with = maximizing 3{17 Proof. A isstrictlyconcaveontheclosedconvexset f : = g closedwith respecttoweaktopology.ThecontinuitywasprovedinLemma1.Wenowprovethe existence.Let besuchthat =sup f A : = g : 44

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Alsolet n besuchthat, n = lim n !1 A n = : Claim f n g aboveistight[59].Toseethis,westartwithsomesimpleinequalities.If 0
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Also, Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 25.214 8.088 Td [( ;R Z R t G x dx dt )]TJ/F73 11.9552 Tf 11.955 16.272 Td [(Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R t G x dx dt = Z R 0 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R t G x dx )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 25.213 8.088 Td [( ;R Z R t G x dx dt Z R 0 ;R )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z R t G x dx exp )]TJ/F50 11.9552 Tf 25.213 8.088 Td [( ;R Z R t G x dx dt: {22 In3{22wehaveusedthefollowinginequality.If Y>X; thenwehavethefollowing inequality: exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(X )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(Y Y )]TJ/F50 11.9552 Tf 11.955 0 Td [(X exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(Y : Alsonotethat ;R Z R t G x dx sup x G x : Usingthisin3{22,weget Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 25.213 8.088 Td [( ;R Z R t G x dx dt )]TJ/F73 11.9552 Tf 11.955 16.273 Td [(Z R 0 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R t G x dx dt ;R )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [( sup x G x Z R 0 xG x dx : {23 Nowwearereadytoprovethetightnessof f n g : Forany with = andany R such that ;R 6 =0 ; themeasure R = ;R 1 [0 ;R ] satises R = : Suppose issuch that A + : Then,using3{21weget A R A + Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R t G x dx dt + Z 1 R xG x dx + : {24 Nowusing3{23andnotingthat A R istheleftmosttermin3{23,wegetfrom 3{24, ;R )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [( k G k 1 Z R 0 xG x dx Z 1 R xG x dx + : {25 46

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Observethat3{24holdsforany suchthat A + andany R suchthat ;R 6 =0 : Furtherforany ,wehave, Z R 0 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R t G x dx dt Z R 0 dt Z R t G x dx = Z R 0 xG x dx Therefore,using3{21,wendthatforany A )]TJ/F73 11.9552 Tf 11.955 16.272 Td [(Z 1 R xG x dx Z R 0 xG x dx : {26 Since Z 1 R xG x dx sup x R xG x ; wecannd R sothat 8 Z 1 R xG x dx : Therefore,if issuchthat A + ,wegetfrom3{26with R = R )]TJ/F50 11.9552 Tf 11.955 0 Td [( )]TJ/F50 11.9552 Tf 11.955 0 Td [( Z R 0 xG x dx : Usingthisin3{25,wendforany suchthat A + and R chosenasabove, ;R )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 2 exp k G k 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 : Thisprovestightness. Wehaveprovedabovethat 9 1 unique,ofcoursesuchthat A 1 =sup f A : = g : Wenowinvestigatethepropertiesof 1 .Forconvenienceofnotation,wewrite instead of 1 : Theorem3. Let satisfy = ;A = .Then issupportedinaninterval [ R 1 ;R 2 ] ; 0
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Proof. If ;R > 0 ; then R = ;R 1 [0 ;R ] isacandidatein3{17. Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R t G x R dx dt = A R = A Z R 0 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R t G x dx dt + Z 1 R xG x dx {27 Thelastinequalityfollowsfrom3{21.Werewrite3{27as Z R 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R t G x dx )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R t G x R dx dt Z 1 R xG x dx ; andusing3{23weget ;R )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [( k G k 1 Z R 0 xG x dx Z 1 R xG x dx : {28 Notingthat = ; wegetfrom3{28, exp )]TJ/F50 11.9552 Tf 9.298 0 Td [( k G k 1 1 ;R Z R 0 xG x dx 1 R; 1 Z 1 R xG x dx : {29 From3{18,since xG x 0 ; as x !1 ; 3{29cannotholdas R !1 unless R; 1 =0forsome R Let R 2 bethesmallest R suchthat R; 1 =0 : Thenforeach R
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and A = Z R 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 t G x dx dt + Z R 2 R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 t G x dx dt: {32 Using3{31and3{32in3{30,weget, I 1 = Z R 2 R exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R 2 t G x dx )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 t G x dx dt Z R 0 exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 R G x dx )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 t G x dx dt = Z R 0 exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 R G x dx dt )]TJ/F73 11.9552 Tf 11.955 16.272 Td [(Z R 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 R G x dx exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R t G x dx dt = I 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(I 3 ; {33 wherewehavedenoted I 2 = Z R 0 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R 2 R G x dx 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R t G x dx dt; and I 3 = Z R 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 R G x dx )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 R G x dx dt: Proceedingasin3{22,wehave, I 1 Z R 2 R R;R 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 Z R 2 t G x dx exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 t G x dx dt ;R R;R 2 exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 R Gd Z R 2 R dt Z R 2 t G x dx = ;R R;R 2 exp )]TJ/F50 11.9552 Tf 29.108 8.088 Td [( R;R 2 Z R 2 R Gd Z R 2 R x )]TJ/F50 11.9552 Tf 11.956 0 Td [(R G x dx ;R R;R 2 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [( k G k 1 Z R 2 R x )]TJ/F50 11.9552 Tf 11.955 0 Td [(R G x dx ; {34 since = = ;R 2 : Now, I 2 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z R 2 R G x dx Z R 0 dt Z R t G x dx Z R 0 xG x dx : {35 49

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From3{33,3{34and3{35,weget ;R R;R 2 exp )]TJ/F50 11.9552 Tf 9.299 0 Td [( k G k 1 Z R 2 R x )]TJ/F50 11.9552 Tf 11.955 0 Td [(R G x dx Z R 0 xG x dx sup 0
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Inotherwords, isconcentratedonthemaximaof ~ G y .Thiscompletestheproof. Letusnowassumethatthesupportof isanintervaloftheform[ R 1 ;R 2 ] : Using 3{40,weobtain, G x Z x 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G s ds dt = M = C Say ;R 1 x R 2 {41 a.e. andhencebycontinuityandtheassumptionthatthesupportof is[ R 1 ;R 2 ]. Assuming 1 G isdierentiable,wegetfrom3{41, exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 x G s ds = C d dx 1 G x ;R 1 x R 2 : {42 Assumingthatlog d dx 1 G x isdierentiable,from3{42weobtain G x dx = d dx log d dx 1 G x dx: Inotherwords, dx = 1 G x d dx log d dx 1 G x dx;R 1 x R 2 : {43 Thisdetermines in[ R 1 ;R 2 ] : Westillneedtondtheexpressionsfor R 1 ;R 2 and C Substituting x = R 1 in3{41weget, G R 1 Z R 1 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G s ds dt = C; andsince isconcentratedon[ R 1 ;R 2 ],wethushave, G R 1 R 1 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 R 1 G s ds = C: {44 Substituting x = R 1 in3{42,weget exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 R 1 G s ds = C d dx 1 G x x = R 1 : {45 51

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Using3{45in3{44weget, G R 1 R 1 d dx 1 G x x = R 1 =1 or, R 1 G 0 R 1 + G R 1 =0 : {46 Wecansolve3{46for R 1 .Wealsoknowthat Z R 2 R 1 dx = andsofrom3{43, Z R 2 R 1 1 G x d dx log d dx 1 G x dx = : {47 Wecansolve3{47for R 2 .Using3{43in3{42,weget, exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z R 2 x d dt log d dt 1 G t dt = C d dx 1 G x ;R 1 x R 2 : Theaboveequationcanbefurthersimpliedinto exp log d dx 1 G x )]TJ/F73 11.9552 Tf 11.956 16.857 Td [( log d dx 1 G x x = R 2 = C d dx 1 G x or, 1 C = d dx 1 G x x = R 2 : {48 Thus,wehaveobtainedtheexpressionsfor R 1 ;R 2 and C Wenowspecifytheconditionssothat R 1 ;R 2 existuniquely.In3{43,werequire that isapositivemeasurein[ R 1 ;R 2 ] : 3{43showsthatthisisonlypossibleif d dx 1 G x isincreasingin[ R 1 ;R 2 ] ; i.e.1 =G isconvexin[ R 1 ;R 2 ]: Lemma2. Ameasure givenin 3{43 ispositivei 1 G isconvexin [ R 1 ;R 2 ] : Lemma3. Suppose G ispositive, G 0 iscontinuouson [0 ; 1 lim x !1 xG x =0 ; and Z 1 0 G x dx< 1 : {49 52

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Thentheequation, RG 0 R + G R =0{50 hassolutions.Ifinaddition, 1 G isconvex,then 3{50 hasauniquesolution. Proof. Integrationbypartsgives, Z 1 0 G x dx = )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 0 xG 0 x dx i.e. Z 1 0 [ G x + xG 0 x ] dx =0 : Therefore, G x + xG 0 x mustassumebothpositiveandnegativevalues.Bycontinuity, 3{50hassolutions. Dividingby G 2 ,wend3{50isequivalentto x G 0 x G x 2 + 1 G x =0 or, 1 G x )]TJ/F50 11.9552 Tf 11.955 0 Td [(x d dx 1 G x =0 : Now, d dx 1 G x )]TJ/F50 11.9552 Tf 11.955 0 Td [(x d dx 1 G x = )]TJ/F50 11.9552 Tf 9.298 0 Td [(x d 2 dx 2 1 G x < 0 because1 =G isconvex.Thusthefunction 1 G x )]TJ/F50 11.9552 Tf 11.955 0 Td [(x d dx 1 G x isstrictlydecreasing.Soitcanhaveatmostonezero. Nowthatwehaveshowntheuniqueexistenceof R 1 ,weneedtoshowthat R 2 exists uniquely,i.e.3{47hasauniquesolutionforeach ,orthat, Z 1 0 1 G x d dx log d dx 1 G x dx = 1 : {51 53

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Since 1 G isconvexandpositive,and G y 0as y !1 : Z y 1 G x d dx log d dx 1 G x dx 1 G y log d dx 1 G x y forany and y<: Since d dx h 1 G x i isincreasing,3{51isproved.Wehavethusshown that R 2 existsuniquely. Wenowhave, Theorem5. Suppose R 1 0 G x dx< 1 ;G;G 0 continuousand 1 G convex.Then R 1 ;R 2 satisfying 3{46 and 3{47 existuniquely.Therefore,themeasure in 3{43 satises 3{42 andalsosatisesTheorem4. Proof. Onlythelasttwoassertionsneedbeestablished.CallthefunctioninTheorem4 ~ G x : ~ G x = G x Z x 0 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 R 1 G s ds dt: {52 Then,using3{45in3{52,for0 x R 1 ~ G x = G x Z x 0 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 R 1 G s ds dt = xG x C d d 1 G = R 1 = CxG x 1 R 1 G R 1 : {53 From3{42and3{46,for R 1 x R 2 ~ G x = G x Z R 1 0 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 t G s ds dt + Z x R 1 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 t G s ds dt = G x CR 1 d d 1 G = R 1 + C G x )]TJ/F50 11.9552 Tf 24.524 8.088 Td [(C G R 1 # C: {54 54

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Finallyfor x R 2 ; ~ G x = G x Z R 1 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G s ds dt + Z R 2 R 1 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G s ds dt + Z x R 2 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 R 1 G s ds dt = G x CR 1 d d 1 G = R 1 + C 1 G R 2 )]TJ/F15 11.9552 Tf 26.214 8.088 Td [(1 G R 1 + x )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 # ; = G x C G R 2 + x )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 : {55 3{54showsthat isconcentratedonthesetwhere ~ G = C .Tocompletetheproof,we mustshowthat C =max x 0 ~ G x Toseethis,recallthatlim x !1 xG x =0 : From3{55and3{53weseethat ~ G x tendstozeroas x 0and x !1 : Atanymaximaof ~ G ~ G 0 mustvanish.Usingthefact that R 1 istheuniquesolutionto3{46weseethat ~ G 0 cannotvanishintheopenintervals ;R 1 and R 2 ; 1 .Thisconcludestheproof. Finallyweneedtoshowthat denedby3{43indeedmaximizes A denedin 3{17: Theorem6. Foreverymeasure on [0 ; 1 ; with = ;A A : Proof. TheproofiscontainedinTheorem4,butwespelloutthedetails. Let beanyothermeasurewith = .Thefunction, B = Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 t G x dx + )]TJ/F50 11.9552 Tf 11.955 0 Td [( dx dt isstrictlyconcavein0 1 : Itsderivativeat =0is, Z 1 0 G x Z x 0 exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 t G s ds dt dx )]TJ/F50 11.9552 Tf 11.955 0 Td [( dx 0 55

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because Z 1 0 G x Z x 0 exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G s ds dt dx = Z 1 0 ~ G x dx max x ~ G x = C = Z 1 0 ~ G x dx asprovedinTheorem5. 3.2.2.1ComputingOptimalNode-ActivationFunctionfromOptimalMeasure Theoptimalmeasure isabsolutelycontinuouswithrespecttotheLebesguemeasure andhasdensity 0 x = 1 G x d dx log d dx 1 G x ;R 1 x R 2 {56 andsatises Z R 2 R 1 0 x dx = : {57 Here, R 1 canbeobtainedbysolving3{46,and R 2 canbeobtainedbysolving3{47. Withthehelpof3{1,3{9,3{57andsubstitutingtheactivationarea A = R 2 2 )]TJ/F50 11.9552 Tf 11.96 0 Td [(R 2 1 weget Z R 2 R 1 0 x 2 x 2 xdx = : {58 Wecanthuswritetheoptimalnode-activationas ^ x = 0 x 2 x ; {59 where needstobesucientlylargetoguarantee0 < 1 : ThecompleteNA-BOLDO algorithmfordesigningtheoptimalnodeactivationfunction ^ x isillustratedinFig.3-8. 3.2.2.2MinimumRadioDensityforNA-BOLDOStrategy ThenodeactivationfunctionisaconditionalprobabilityasdenedinSection3.2. Hence,theoptimalnodeactivationfunctiongivenin3{59hastosatisfy0 < 1.This 56

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Figure3-2.DISCstrategy:allnodesareactivatedinsideasectorwithradius R d issatisedif 1 2 max R 1
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Figure3-3.DISCOstrategy:nodesareactivatedwithxedprobability p insideasector withradius R d beyondthat,asshowninFig.3-3.Considerrstasimplestrategyinwhich all nodes insideasectoraroundthetransmitteruptoaxeddistanceareawaketoreceivethe transmission;i.e. P i =1= p 1 8 i asshowninFig.3-2.Werefertothisstrategyas theDISCstrategy. Amoresophisticatedstrategyistoturnonnodeswithaxedprobability p outtoa largerradiusthantheDISCscheme,wherethisradiusisobtainedtosatisfytheconstraint thattheexpectednumberofnodesthatactivateis .Letting x = p; 0 > < > > : R 2 d p = 0 p 1 : 58

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WerefertothisschemeastheO ptimized-DISC strategyDISCO.FortheRayleighfading channel,theobjectivefunctionin3{62is ^ p =argmax p Z R d 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 n n p p 2 n ;R n d )]TJ/F50 11.9552 Tf 11.955 0 Td [( 2 n ;t n dt # ; where a;x = Z x 0 t a )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(t dt istheincompletegammafunction.TheoutageprobabilityfortheDISCOschemeis p out =exp )]TJ/F15 11.9552 Tf 9.298 0 Td [(2 n n p 2 p 2 n ;R n d : 3.5Results Inthissection,wepresenttheperformanceoftheDISC,DISC-O,andNA-BOLD schemes.Weprovideresultsfor m =1,commonlyknownastheRayleighfadingchannel model.ThereceiveSNRthresholdisassumedtobeunity,i.e. =1.Thepath-loss exponent n =4andthenodedensityis =10nodesperunitsector.TheNA-BOLD approachturnsonnodeslocatedinsideasectorannulus,asshowninFig.3-1.The expectednumberofnodesthatturnonis .FortheNA-BOLDOscheme,weobtained theoptimumnodeactivationfunctionusingapiece-wiselinearapproximationwiththe helpofthe j 'snumericallyobtainedbysolving3{15. Thenodeactivationfunctionsforthevariousschemesareshownasafunctionof distancefromthetransmitterinFig.3-4for =3.FortheNA-BOLDOscheme, R 1 = 0 : 7071forany ,andwenumericallyfound R 2 =0 : 9771for =3.Thecorresponding valuesfortheNA-BOLDCschemeare0 : 63and0 : 84respectively. Theexpectedvaluesofthemaximumtransmissiondistancetothefarthestreceiver thatsuccessfullyreceivedthetransmissionareshowninFig.3-5.Wehavealsoprovided theperformanceofasub-optimalNA-BOLDSscheme,thedetailsofwhicharediscussed in[46].SincetheNA-BOLDSschemerequires > 1,wehaveplottedfor2 10. 59

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Figure3-4.Nodeactivationprobabilityfor =10nodesperunitsector,withexpected numberofactivenodes =3. Forthisrangeof ,theNA-BOLDschemessignicantlyoutperformtheconventional schemesthatemployxednodeactivation.WeprovideTable3-1inordertocompare theirperformancerelativetotheoptimalscheme.Inparticular,theNA-BOLDapproachis betterthantheDISCschemebymorethan100%.TheDISCOschemeiswithin9 : 5%to 5 : 5%ofNA-BOLDfor =3to10.InapracticalMPRN,thenumberofnodesthatturn oninistypicallylessthan6 : 0[43]andhenceNA-BOLDschemesmightbebettersuited forsuchscenarios. TheoutageprobabilitiesareplottedinFig.3-6.Sinceoutageislesslikelytooccur whenmorenodesarepresent,thecurvesaredecreasingfunctionsof .TheDISCschemes haverelativelyloweroutageprobabilitiesastheyturnonmorenodesthatarelocated closertothetransmitter.TheNA-BOLDSshowsaslightdegradationinoutage 60

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Figure3-5.Expectedvalueofthemaximumtransmissiondistance E [ V max ]vs.the expectednumberofactivenodes Table3-1.ExpectedValueofMaximumTransmissionDistanceforvariousnodeactivation schemes Constantnodeactivationapproach NA-BOLDApproach DISC DISCO NA-BOLDS NA-BOLDC NA-BOLDO 2 0.1714 0.5455 0.5976 0.5981 0.5983 3 0.2437 0.6718 0.7243 0.7246 0.7247 4 0.3025 0.7580 0.8091 0.8097 0.8098 5 0.3519 0.8201 0.8707 0.8713 0.8716 6 0.3950 0.8677 0.9179 0.9186 0.9189 7 0.4335 0.9049 0.9556 0.9565 0.9566 8 0.4686 0.9351 0.9866 0.9875 0.9876 9 0.5012 0.9602 1.0127 1.0135 1.0136 10 0.5316 0.9814 1.0350 1.0356 1.0359 61

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Figure3-6.OutageprobabilitiesfortheNA-BOLDandconventionalschemesversusthe expectednumberofactivenodes performancewithrespecttoNA-BOLDC.However,thisdegradationisnominal{ whenthenumberofactivenodesisashighas10,theoutageprobabilitiesare4 : 5%and 3 : 6%,fortheNA-BOLDCandtheNA-BOLDSschemesrespectively. 3.6Conclusions Inthischapter,weproposednodeactivationschemesthatusedistancetoincreasethe expectedtransmissiondistanceforgeographictransmissionsoverfadingchannels.Optimal andsuboptimalnodeactivationschemesaredevelopedandcompared.Ourresultsshow thatNA-BOLDschemesoersignicantlybetterperformancethanconventionalscheme thatjustturnsactivatearadio'sneighborsouttosomeradiuswithequalprobabilities. Wehaveadoptedameasure-theoreticframeworktosolvefortheoptimalnodeactivation function.Weinvestigatedthepropertiesoftheoptimalmeasureandderiveconditionson whenitexistsuniquely.Wederivedtheoptimalmeasureandmappedittotheoptimal 62

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distance-basednodeactivationfunctionthatreceiverscanusetodeterminewhetherto activatetotrytoreceiveatransmission. 63

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Figure3-7.FlowchartillustratingNA-BOLDCalgorithmtodesignnodeactivation function ^ x 64

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Figure3-8.FlowchartillustratingNA-BOLDOalgorithmtodesignoptimalnode activationfunction ^ x 65

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CHAPTER4 MAXIMIZINGTRANSPORTCAPACITYANDTRANSMISSIONDISTANCEFOR GEOGRAPHICTRANSMISSIONSWITHENERGYCONSTRAINTS Inwirelesscommunications,thereisalwaysatradeobetweentransmissionrateand transmissiondistance:increasingthetransmissionraterequiresahighersignal-to-noise ratioSNRatthereceiver,whichmeansthatthereceivergenerallyhastobeclosertothe transmitter.Oneapproachtocombiningbothofthesemeasuresis transportcapacity [49], whichistheproductofthetransmissionrateandthedistancetraveledforamessage. Consequently,transmissiondistancecanbeviewedastransportcapacity[49]withaxed transmissionrate.InChapter3,wedesignedschemesthatmaximizetransmissiondistance withaconstraintontheenergyusedinreceptionwhichdependsonthenumberof nodesthatactivatetoreceiveatransmission.Inthischapter,weconsideratotalenergy constraint,i.e.,aconstraintonthesumoftheenergiesusedintransmissionandreception. Underaxedtotalenergyconstraint,anincreasedtransmissionpowerresultshigher averagereceivedpowersbutcausesfewerreceiverstobeturnedontoreceivethemessage ifnodesactivatewithaxedprobability.Ontheotherhand,asmallertransmission powermakesthemessagetravelashorterdistanceyetallowsmorereceiverstoactivate, thusprovidinggreatermultiuserdiversitybenet.Therefore,intelligentpoliciesfor powerallocation,involvingbothtransmissionandreceptionneedtobeconsidered.In thischapter,weaddressthisproblemandoptimizetheallocationofenergybetween transmissionandreceptionwhennodesactivatedierentlybasedontheirdistancefrom thetransmitter. Werstshowthattheobjectiveofmaximizingtransportcapacitydecomposesinto twoseparateobjectivesofselectingtheoptimaltransmissionrateandselectinganoptimal nodeactivationfunctiontomaximizethetransmissiondistance.Thenweconsiderthe jointdesignofnode-activationfunctionsandtransmissionratetomaximizethetransport capacity.Moreover,wealsointroduceasimplernodeactivationschemethatactivates 66

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Figure4-1.Nodeactivationregion:Transmitteratcenter.Whitecirclesindicatenodes thatareasleep,blackcirclesindicateactivenodes. nodeswithequalprobabilitieswithintheactivationregion.Ourresultsrevealthatthis schemeperformsveryclosetotheoptimal,distance-basedapproach. 4.1SystemModel Weconsidergeographictransmissioninawirelessnetworkinwhichthechannelsfrom atransmittertotheneighboringradiosaresubjecttoexponentialpathlossandfading. Wemodelthepositionsoftheradiosusingatwo-dimensionalhomogeneousPoissonpoint processwithintensity nodesperunitarea.Thus,theprobabilitydensityfunctionforthe distancetoanarbitraryreceiver,denoted f X x isgivenby f X i x = 2 x R 2 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 2 1 ;R 1
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thusweassumeasimilarmodelhereasshowninFig.4-1.Weassumethatnodesknow theirgeographicallocationandthuscandeterminetheirdistancesfromaradiothat isscheduledtotransmitatacertaintime.Weconsiderthedesignofdistributednode activationfunctionsinwhichtheradiosindependentlydecidewhethertheyshouldkeep theirreceiversactiveorturnthemo. 4.2DesignofNodeActivationFunctions 4.2.1OptimumTransportCapacity Ourgoalistooptimizethewayinwhichradiosactivatetomaximizetransport capacityachievedbygeographictransmissionsunderatotalenergyconstraint,i.e.the sumoftheenergiesconsumedintransmittingandreceivingamessage.Weassumethat thetransmitterhasnoknowledgeoftheneighboringradiosandtransmitsmessageswith somepower P t .Theradiosknowtheirdistancefromthetransmitterandthepower P t usedintransmission.Foragiventransmissionrate,thereisatargetSNR ,which isidenticalateveryreceiver.AnodeisasuccessfulreceiverifthereceivedSNRis greaterthanatargetSNRandalsoifitisactiveturnsonitsreceivertorecoverthe transmission.ThemaximumachievablerateoftransmissionoveracomplexGaussian channelatSNR isgivenby s =log 2 + ;> 0 : {2 Denotethedistancetoanarbitraryreceiverbytherandomvariable X i .Let i =1 denotetheeventthatnode i locatedatdistance X i fromthetransmitterisactive.Dene x ,thenodeactivationfunction,as x P i =1 j X i = x .Furthermore,given that X i = x; thenodeactivationfunction dependsonthetargetSNR aswellas thetransmissionpower P t .Intuitively,thehigherthetargetSNRisorthehigherthe transmissionpoweris,thelesslikelyanodelocatedatsomedistancefromthetransmitter 68

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isgoingtogetactivated.Hence,wehave, i = 8 > > < > > : 1 ; withprobability x;;P t ; 0 ; withprobability )]TJ/F50 11.9552 Tf 11.955 0 Td [( x;;P t : {3 Node i decidestostayawakeprobabilisticallyif U i < X i ;;P t ,where U i isa uniformlydistributedrandomvariableon[0 ; 1].Ifweignoretheimpactofinterference onthereceivedsignal,thesignalpoweratreceiver i dependson X i ,thechannel fadinggain H i ,andthetransmissionpower P t .Thus,theinstantaneousreceived SNRis i = P t H i X )]TJ/F25 7.9701 Tf 6.586 0 Td [(n i ,wherewehavedenotedthepath-lossexponentas n .Letus substitute Y i ;P t = n p H i P t )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 anddenotethecomplementarydistributionfunctionas G y ; ;P t = P Y ;P t >y .Thenusing3{2,wend G y ; ;P t = 1 \050 m Z 1 mP )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 t y n t m )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(t dt;y> 0 : {4 Let V i ;;P t denotethedistancetoanodeifitisawake i =1andhassucient receivedSNR i > ;otherwiselet V i ; =0.Thus,wehave, V i ;;P t = 8 > > < > > : X i ;X i t =1 )]TJ/F73 11.9552 Tf 11.955 16.273 Td [(Z 1 t x;;P t G x ; ;P t f X x dx: {6 Letusdenotethetransmissiondistancetothefarthestsuccessfulreceiverbythe randomvariableas V max ;;P t andconditiononatotalof L nodesawakeand asleepinthegeographicregion.Wedenetransportcapacity,denotedbytherandom variable T ;;P t astheproductofthemaximumachievablerateoftransmissionand 69

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thetransmissiondistancetothefarthestsuccessfulreceiver.Hence, T ;;P t = s V max ;;P t {7 Using4{2in4{7,andtakingexpectation,wehave, E [ T ;;P t ]=log 2 + E [ V max ; ;;P t ] : {8 Letusnowassumethatthenumberofnodesthatareawaketoreceiveatransmission insidetheactivationregionasshowninFig.4-1is @ @ L where @ isaPoissonrandom variable.Ifwenowconstrainthatthetotalpowerusedintransmissionandreceptionofa messageis c ,wecanexpressouroptimizationproblemas, max s ; x;;P t ;P t E [ T ;;P t ] subjectto: 8 > > > > > > > > > > < > > > > > > > > > > : P t > 0 > 0 P t + P r E [ @ ]= c 0 < x;;P t 1 ; {9 where P r isthepowerexpendedinreceivingasingletransmissionandisassumedtobe known apriori .Letusdenotetheratioofthepowerusedintransmissionandreceptionof amessageas ,i.e. = P t P r .Wefurthereliminatethedependenceof P r inourapproach bydenotingtheratiooftotalpower c to P r as = c P r .Henceforth,whereverapplicablein thischapter,weuse ; forfuturenotationinsteadof P t ;c respectively.Consequently,the transportcapacity E [ T ;;P t ]canbereplacedby E [ T ;; ]andthenusing4{8in 70

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4{9,weget max s ; x;; ; log 2 + E [ V max ; ;; ] subjectto: 8 > > > > > > > > > > < > > > > > > > > > > : > 0 > 0 + A R R 2 R 1 x;; f X x dx = 0 < x;; 1 : {10 Notethat, and aredimensionlessquantities,sincetheydenotethetransmitted powerperunitreceivedpowerandthetotalpowerconsumedperunitreceivedpower respectively. If L nodesarelocatedinsidetheactivationregion,theconditionaldistributionof V max ;; ,denoted F V max t ; ;; j L is F V max t ; ;; j L =[ F V t ; ;; ] L .Since L isPoissonwithmean A ,wehave, F V max t ; ;; =exp )]TJ/F51 11.9552 Tf 8.136 -9.683 Td [()]TJ/F50 11.9552 Tf 11.956 0 Td [(A )]TJ/F50 11.9552 Tf 11.955 0 Td [(F V t ; ;; : {11 Thus,using4{6in4{11,wendtheexpectedvalueofthetransmissiondistancetothe farthestsuccessfulreceiver E [ V max ]dependson ; and as E [ V max ; ;; ]= Z 1 0 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(A Z 1 t x;; G x ; ; f X x dx dt: {12 Stretching/compressingdistancesbythefactor n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 > 0 ;> 0 Wendthat4{10isdiculttosolvebecauseofthedependenceoftheobjective functionon and ,whereeven dependson and .However,wecansimplifythis problembyexpressingtheobjectivefunctionasaproductoftwofunctions:suchthatone functionexplicitlydependson x; andtheotherexclusivelyon .Startingfrom4{4, somealgebraicmanipulationyields E [ V max ;; ]= n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 E [ V max ; = ]. 71

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Werstsubstitute z = n p )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 x in4{12.Thenweget E [ V max ; ;; ]= Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(A Z 1 t n p )]TJ/F18 5.9776 Tf 5.756 0 Td [(1 z n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;; G z n p )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 ; ; :f X z n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 dz n p )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 dt: {13 Nownotefrom4{1that f X z n p )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 = 1 n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 f X z : Also,4{4reveals G x ; ; = G z ; = when z = n p )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 x .Letusnowsubstitute v = t n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 intheouterintegral.Then,wehave E [ V max ; ;; ]= Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.298 0 Td [(A Z 1 v z n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;; G z ; = :f X z dz n p 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 dv: {14 Letusdenote e z;; = 1 n p 2 )]TJ/F24 7.9701 Tf 6.587 0 Td [(2 z n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 ;; : Thus,withoutlossofgenerality, E [ V max ; ;; ]= 1 n p )]TJ/F24 7.9701 Tf 6.587 0 Td [(1 Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F50 11.9552 Tf 9.299 0 Td [(A Z 1 v e z;; G z ; = f X z dz dv: {15 Clearly,thefunction e z;; thatmaximizestheobjectivefunctionin4{10isthe sameasthefunction x; = ; thatmaximizes E [ V max ; ; = ; ].Therefore,4{10 72

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canbedecomposedas max n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 log 2 + max x; = ; n p E [ V max ; ; = ; ] subjectto: 8 > > > > > > > > > > < > > > > > > > > > > : > 0 > 0 A R R 2 R 1 x;; f X x dx = 0 < x;; 1 : {16 Wecanthusconsiderthefollowingindependentoptimizationproblems: max n p )]TJ/F24 7.9701 Tf 6.586 0 Td [(1 log 2 + subjectto: > 0{17 and, max ; x; = ; n p E [ V max ; ; = ; ] subjectto: 8 > > < > > : + A R R 2 R 1 x; = ; f X x dx = 0 < x; = ; 1 : {18 Thesolutionto4{17canbeobtainedbysolvingtherootsof, n )]TJ/F15 11.9552 Tf 11.955 0 Td [(+ log e + =0 ;> 0 : {19 Wecast4{18intoanalternatemeasure-theoreticformulationasfollows.Letusconsider thefollowingoptimizationproblem A ; =sup ; n p Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.273 Td [(Z 1 t G x ; = dx dt subjectto: 8 > > < > > : + R 1 0 x dx = ; given > 0 : {20 73

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overallmeasures on[0 ; 1 ]and G isacontinuousfunctionon[0 ; 1 suchthatforany ; wehave, lim x !1 xG x ; ; =0 : Wecanfurtherrewrite4{20as A ; =sup n p sup Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G x ; = dx dt subjectto: 8 > > < > > : + R 1 0 x dx = ; given > 0 : {21 Werstx andexpresstheinnermostoptimizationin4{21as A ; =max Z 1 0 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(exp )]TJ/F73 11.9552 Tf 11.291 16.272 Td [(Z 1 t G x ; = dx dt subjectto: Z 1 0 x dx = )]TJ/F50 11.9552 Tf 11.955 0 Td [(; given : {22 Wesolve4{22foraxed ,usinganapproachsimilartothatdetailedinSection3.2.2. Asaresult,hereafterwerefertotheoptimalsolutionof4{22astheNA-BOLD approach.TheoptimalLebseguemeasure isabsolutelycontinuousandhasdensity 0 x = 1 G x ; = d dx log d dx 1 G x ; = ;R 1 x R 2 ; {23 andsatises Z R 2 R 1 0 x dx = )]TJ/F50 11.9552 Tf 11.955 0 Td [(; where, R 1 canbesolvedbysolving R 1 G 0 R 1 ; = + G R 1 ; = =0 : {24 Foraxed R 2 canbeobtainedfrom, Z R 2 R 1 1 G x ; = d dx log d dx 1 G x ; = = )]TJ/F50 11.9552 Tf 11.956 0 Td [(: {25 74

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Havingsolved4{21foraxed or,equivalently,4{22,wenowproceedtoobtainthe optimum .Using4{23intheobjectivefunctionof4{21andaftersomealgebra,we get A ; = n p R 2 )]TJ/F50 11.9552 Tf 11.955 0 Td [(R 1 G 0 R 1 ; = G 0 R 2 ; = G R 2 ; = G R 1 ; = 2 + G R 1 ; = 2 G 0 R 2 ; = 1 G R 2 ; = )]TJ/F15 11.9552 Tf 53.149 8.088 Td [(1 G R 1 ; = # : {26 Wenallyobtaintheoptimum bydierentiating4{26withrespectto Using4{1,substituting A = R 2 2 )]TJ/F50 11.9552 Tf 12.213 0 Td [(R 2 1 in4{18,alongwith4{22,wederivethe optimalnode-activationfunction x; = ; connedontheinterval[ R 1 ;R 2 ],for axed as x; = ; = 0 x 2 x ; {27 providedthenodedensityperunitsector islargeenoughtoguarantee x; = ; 1.Wecanobtaintheminimumnodedensityforaxed as 1 2 max R 1
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4.3Results WeevaluateperformanceintheNakagamim channelwhichcanmodelawidearray ofwirelesschannels,wheretheparameter m indicatesthefadingseverity,withvalues rangingfrom m =0 : 5mostsevere,half-Gaussianto m = 1 nofading.Forthese results,wehavechosenthepathlossexponent n =4.SinceourNA-BOLDapproach signicantlyoutperformsconventionalschemesc.f.[50],[46],and[47],wecomparethe performanceofNA-BOLDwiththeRINGOtransmissionschemeinthischapter.Wehave assumedthenodedensityperunitsector =10fortheRINGOscheme.Wehaveproved inSection4.2.1thattheNA-BOLDschemerequiresthedensityofnodesdistributedin thetransmissionregion,Fig.4-1exceedacertainminimumvalue.Inordertoobtainour plots,wehavechosennodestobedistributedwithnodedensityperunitsector =50000 fortheNA-BOLDscheme.WehavepresentedourresultsintermsofthetargetSNRat thereceivers ,thetransmittedpowerperunitreceivedpower ,andthetotalpower consumedperunitreceivedpower Figure4-2.VariationofexpectedvalueoftransportcapacitywithtargetSNR for =10 : 76

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Figure4-3.VariationofexpectedvalueoftransportcapacitywithtargetSNR for =35 : InFig.4-2,wehaveplottedthetransportcapacityasafunctionofthethreshold SNR indBwhenthefadingguresare m =1Rayleighfadingchanneland m =5 : 0. Forthisplot,wehavexed =10andmaximizedthetransportcapacityoverall allowable 'sthatsatisfytheconstraints.Theoptimum is4 : 6070and4 : 6178forthe NA-BOLDandRINGOschemesrespectively,when m =1 : 0.When m =5 : 0 ; the correspondingvaluesare5 : 5057and5 : 5556respectively.Forboththeschemes,wend theoptimumtargetSNRonlydependsonthepath-lossexponent n ,andequals16 : 9dB when n =4.Thiscorrespondstoanoptimumtransmissionrateof5 : 65bit/s/Hz.Fig.4-2 alsoindicatesthatthetransportcapacitiesofNA-BOLDandRINGOschemesarevery similaruntil =20dBand =17dBonchannelswith m =1 : 0 ; 5 : 0respectively.For greaterthantheabovevalues,wenoticethetransportcapacityofNA-BOLDdegrades withincreasein moregracefullythanthatofRINGO.For =10 ; ahighertransport capacityisobtainedinaless-fadedchannelwithastrongLOScomponent m =5 : 0in comparisontoaseverely-fadedchannel m =1 : 0.However,asseenfromFig.4-3,this 77

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trendreverseswhen =35,becausegreatermultiuserdiversitybenetisobtainedina severely-fadedchannelincomparisontoaless-fadedchannel. Figure4-4.Variationofexpectedvalueoftransportcapacitywithrespectto << inaRayleighfadingchannelfor =5 : Figure4-5.Variationofexpectedvalueoftransportcapacitywithrespectto << inaRayleighfadingchannelfor =25 : 78

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Figure4-6.Optimumtransportcapacityasafunctionof Wehaveplottedthetransportcapacityasafunctionofthetransmittedpowerper unitreceivedpower onaRayleighfadingchannelinFig.4-4andFig.4-5for =5 ; 25 respectivelywhentheoptimumthresholdSNR =16 : 9dB.FromFig.4-4wendthat theperformanceofRINGOandNA-BOLDareverysimilartoeachotherfor > 1 : 5. Thishappensas > 11inFig.4-5.SincethetransportcapacityplotsofbothRINGO andNA-BOLDFig.4-5areskewedtotheleft,wegethighertransportcapacitiesfor bothschemesusinglarger 's,when =25.Ontheotherhand,when =5,theplots areskewedtotheright,andweobtainlowertransportcapacitiesusinglarger 's.This impliesthatwhenthetotalpowerperunitreceivedpower islarge,itismorebenecial toallocatemorepowertowardstransmissionratherthanreception,ofamessage,ona Rayleighfadingchannel.However,forsmall ,thereverseappearstobetrue. Fig.4-6showsthemaximizedexpectedvalueoftransportcapacityasafunctionof thetotalpowerconsumedperunitreceivedpower when m =1 : 0 ; 5 : 0.Thetransport capacitiesofNA-BOLDandRINGOareverysimilarinaseverely-fadedchannel m = 1 : 0andalsoinaless-fadedchannelwithastrongLOScomponent m =5 : 0.This isbecausebothschemesobtainsimilarmultiuserdiversitybenetsinafadingchannel. 79

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But,notethatfor > 21,thetransportcapacitiesarehigherinaseverely-fadedchannel thaninaless-fadedchannel.Thisisbecauseaseverely-fadedchannelprovidesagreater multiuserdiversitybenetcomparedtoaless-fadedchannel,inthepresenceofalarge numberofreceivers.Inaseverely-fadedchannel,greaterpowerisallocatedtowards activatingreceiverswhen islarge,incomparisontoaless-fadedchannel.Fig.4-6 alsoindicatesthattheslopeoftheoptimaltransportcapacitydecreaseswith .This isbecausemultiuserdiversitybenetisobtainedaccordingtothelawofdiminishing marginalreturns"andfurtherveriesourearlierobservationofallocatinggreater transmissionpowerinFig.4-5when issignicantlylarge. Figure4-7.Optimum asafractionof Theoptimumvaluesof asafractionofthetotalpowerallocatedperunitreceived power ,when m =1 : 0 ; 5 : 0areplottedinFig.4-7.Thereisnoobservabledierence betweentheoptimumpowerallocationsofRINGOandNA-BOLDschemes,overtherange of usedintheplots.ItisalsointerestingtonotethatFig.4-7indicatesthevaluesof whenmore/lesspowershouldbeallocatedtowardstransmission/receptionofamessage. Inaless-fadedchannel m =5 : 0,for < 8,itisbenecialtoallocatemorepower 80

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Figure4-8.NodeactivationprobabilityofNA-BOLD.Red-solidand-dottedlinesindicate m =1 : 0,blue-solidand-dottedlinesindicate m =2 : 0. towardsreception,ratherthantransmission,and,when < 14inaseverely-fadedchannel m =1 : 0.Further,thisobservationisalsoinagreementwithourearlierresultsinthis section.Wealsondif isxed,agreatertransmissionpowerisrequiredinaless-faded channel m =5 : 0thaninaseverely-fadedchannel m =1 : 0,inordertomaximize transportcapacitywithaconstraintonthesumofpowersusedintransmissionand receptionofamessage. Tofurtherinvestigatetheeectsofthetotalpowerandthechannelfadingparameters indecidingwhichnodestoactivate,weevaluatethenodeactivationprobabilityofthe NA-BOLDandRINGOschemesinFig.4-8andFig.4-9respectively,forlarge = 20andsmall =10valuesof onchannelshavingfadinggures m =1 : 0and m =2 : 0.FromFig.4-8andFig.4-9,weobservethatbothschemesareconservative underseverefading m =1 : 0andturnsonreceiversclosertothetransmitterthanfor less-severefading m =2 : 0,when issmall.Also,theactivationregionofbothschemes 81

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iswiderwhen islargeandthechannelisseverelyfaded,thanwhenitisless-faded.This illustratestheadaptivityofboththeNA-BOLDandRINGOapproachestothetotal powerperunitreceivedpower ,aswellastheseverityofthechannelfading. Figure4-9.NodeactivationprobabilityofRINGO.Red-solidand-dottedlinesindicate m =1 : 0,blue-solidand-dottedlinesindicate m =2 : 0. 4.4Conclusions Weinvestigatedthedesignofschemestomaximizethetransportcapacityonfading channelswithgeographictransmissionandconstraintsonthesumofenergiesconsumed intransmissionandreceptionofapacket.Inadditiontoextendingourpreviously developedoptimalnodeactivationbasedonlinkdistanceNA-BOLDapproachof maximizinglinkdistancetothemorecomplicatedproblemofmaximizingtransport capacityunderaconstraintonthesumoftheenergiesusedintransmissionandreception. Weintroducedanodeactivationschemethatactivatesreceiverswithinanannularregion locatedawayfromthetransmitter,accordingtoaxedprobability.Wendthatthis schemeoerssimilarperformancetothemore-sophisticated,optimalNA-BOLDapproach. 82

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Forbothschemes:ourresultsindicatethatinafadingchannel,whenthetotalpower availableislarge,itismorebenecialtoallocategreaterpowertowardstransmission, ratherthanreceptionofamessage.Moreover,sinceaseverely-fadedchannelprovides greatermultiuserdiversitybenetinthepresenceofalargenumberofreceivers,for optimalpowerallocation,greaterpowerisallocatedtowardsactivatingreceiversthan intransmissionofamessage,comparedtoaless-fadedchannel,whenthetotalpower availableislarge. 83

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CHAPTER5 CONCLUSIONSANDFUTUREWORK 5.1Conclusions Inthiswork,wehaveaddressedtheproblemofenhancingtheperformanceof geographictransmissionsbyexploitingmultiuserdiversityinwirelessadhocnetworks. Wehaveoptimizedtwoimportantgeographictransmissionmetrics{expectedvalue oftransmissiondistanceandexpectedtransportcapacity,whenthewirelesschannelis subjectedtorandomfadingandexponentialpathloss.Ourndingsrevealthatinthe presenceofalargenumberofusers,thebenetsofmultiuserdiversitycanbeexploited toenhancethesemetricswithoutanyfeedbackfromthereceivingnodesbacktothe transmitter.Wehavesuggestedlink-layerschemestoillustratehowourresultsmightbe utilizedinprotocoldesign. Inthepresenceofalargenumberofusers,geographictransmissionschemesthat maximizetransmissiondistanceinafadingchannelachieveagreatertransmissiondistance thanthoseschemesinaconstantnonfadingchannelinwhichthereisnodiversity.Wend thatinaRayleighfadingchannelwithanaverageofsixnodes,multiuserdiversitybrings aboutanincreaseintheaveragetransmissiondistanceby25% ; 50%,and87 : 5%over thedistanceinthenonfadingchannelforpath-lossexponentsof4 ; 3and2,respectively. Thisisbecause,inanonfadingchannel,allusershaveidenticalchannelgains,andwhen thetransmissionissubjectedtopath-loss,thetransmissioncannottravelbeyondaxed transmissionrange.However,inafadingchannel,usershaverandomgoodorbad channelgains,whicharelikelytocausesomeuserwhoisfarawayfromthetransmitter toreceiveatransmissionsuccessfully.Thislikelihoodincreaseswiththenumberofusers. Alternatively,thismeansthatwhenthenumberofnodesisnotsucientlyhigh,ahigher transmissiondistancecanbeobtainedinanon-fadingchannelthaninafadingchannel. Sincemultiuserdiversitybasedschemesimposesignicantburdenonthebattery lifeofthenodesifallnodesareawaketoreceiveapacket,wedesignedgeographic 84

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transmissionschemesthatprovidemultiuserdiversitybenetwithaxedenergy constraint.Ourschemesprovideenergyeciencybyemployingdistributednode activationtechniquestopowerosomeoftheredundantnodes.Accordingtoourschemes, nodesdecideindependentlywhethertoturnon/owithoutanykindofcoordinationor control,whichinturnalsoreducestheburdenofmaintaining/exchangingnetworkstate information.Ournodeactivationtechniquemakesuseofnodelocationinformation,which isconsideredtobeavailableatthenodesforgeographictransmissionschemes. Withthegoalofmaximizingtransportcapacityandtransmissiondistance,wehave designedoptimumandsub-optimumnodeactivationschemesthatwouldenableanode todeterminewhethertoattempttoreceiveapacketbasedonitslinkdistancefromthe transmitter,withaconstraintonthenumberofnodesthatactivate.Ourworkonthese problemsisalsopublishedin[45,50,46,47].Wehavecomparedourschemeswithseveral conventionalschemesthatmakeuseofconstantsleepschedulingalgorithmsfornodes locatedwithinaxeddistancefromthetransmitter.Ourresultsindicatethatforthe sameconstraintonenergy,ournodeactivationschemesprovideahighertransmission distance/transportcapacitythanconventionalschemesthatactivatenodeswithina xeddistancefromthetransmitter.Inaddition,wealsointroducedasub-optimalnode activationschemethatactivatesnodeswithinanannularactivationregion,withaxed probabilityforeverynodeintheactivationregion.Wendthatthisschemeoers nearlyidenticalperformancetotheoptimalapproachyetdoesnotinvolvecomplicated mathematics. Sincebothtransmissionaswellasreceptionofapacketcostsenergy,wehave investigatedthedesignofschemestomaximizethetransportcapacityonfadingchannels withgeographictransmissionswithaconstraintonthesumofenergiesconsumed intransmissionandreception.Underaxedtotalenergyconstraint,anincreased transmissionpowerresultsinhigheraveragereceivedpowers,butcausesfewerreceivers tobeturnedontoreceivethemessage.Ontheotherhand,asmallertransmissionpower 85

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makesthemessagetravelashorterdistanceyetallowsmorereceiverstogetactivated. Thus,withtheobjectiveofmaximizingtransportcapacity,wehaveoptimizedallocationof energiesbetweentransmissionandreceptionundervariousfadingconditionswhennodes activatesimilarlyanddierentlybasedontheirdistancefromthetransmitter.Transport capacityisaproductofthedistanceoverwhichatransmissiontravelsandtherateof transmission.Inthegeographicscenariothatweconsider,theoptimumtransmission ratedoesnotdependonthenodeactivationtechnique;i.e.itisidenticalforboththe distance-basedschemesaswellasthexedschemes. Insummarywehavedesigneddistance-basedgeographicapproachesthatprovide multiuserdiversitybenetwithaconstraintontheenergiesusedintransmission andreceptionofamessage.Ourstrategiesaredesignedtobalancebalancebetween transmissionandreceptioninanoptimummanner.Inaddition,wehavealsodesigned anodeactivationschemethatactivatesreceiverswithinanannularregionlocated awayfromthetransmitter.Thisschemeissub-optimumandinvolveslesscomplicated mathematics;yetoersnearlyidenticalperformancetothemore-sophisticated,optimal NA-BOLDapproach.Forbothoptimalandsub-optimalapproaches,inafadingchannel, whenthetotalpoweravailableislarge,wenditismorebenecialtoallocategreater powertowardstransmission,ratherthanreceptionofamessage.Moreover,sincea severely-fadedchannelprovidesgreatermultiuserdiversitybenetinthepresenceofa largenumberofreceivers,foroptimalpowerallocation,greaterpowerisallocatedtowards activatingreceiversthanintransmissionofamessage,comparedtoaless-fadedchannel, whenthetotalpoweravailableislarge.Notonlydoournodeactivationschemesprovide multiuserdiversitybenetinanenergy-ecientmannerwhenlargenumberofnodes arepresent,italsoadherestoachannel-dependenti.e.moreconservativeinaseverely fadedchannelmechanismofturningonnodeswhenfewerreceiversarepresent.This demonstratestheeectivenessandadaptabilityofourschemestoavailableresources transmissionpowerandnumberofreceiversandchannelfadingconditions. 86

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5.2FutureWork Untilnow,wehaveconsideredexpectedvalueoftransmissiondistanceandexpected transportcapacityasthegeographictransmissionmetric.Amorepracticalmetricto considerwouldbethe expectedforwardprogressperhop. The expectedforwardprogress perhop ,isthedistancetraveledbythetransmissionalongthedirectionofthedestination, toareceiverthatobtainedthetransmissionsuccessfully,projectedontoalinejoining thetransmitterandthedestination.Theperformanceofaroutingprotocolutilizingthis metricwasrstanalyzedintheclassicpapers[44],[43]aswellasinsubsequentpapers. Itwouldbeinterestingtoaddresstheproblemofoptimumnodeactivationwiththe goalofmaximizingexpectedforwardprogressperhop.Thisisadicultproblemtosolve sinceforwardprogressdependsontheangularosetofthereceiversinadditiontotheir distancefromthetransmitter.Wehavedesignedasimpler,sub-optimalapproach[48] thatturnsonnodeswithinasmalleranglebyextendingourpreviouslydeveloped distance-dependentNA-BOLDschemetoincorporatespatialdispersion,whenthe receiversareconstrainedinenergyandtransmissionsareconnedwithinanangle. Futureworkwithinourframeworkcanbedirectedtowardssolvingthefollowingopen problems: Designofnodeactivationstrategiesforothergeographicmetricsincluding maximizinginformationeciencyandresidualeciency Informationeciency [60,20],isdenedastheproductoftheexpectedforwardprogress andthespectraleciency.Wedeneanewmeasure, residualeciency ,astheproductof thespectraleciencyandtheoriginalsource-destinationseparationminustheresidual distancetothedestination.Ineachcase,thenodeactivationfunctionswilldependnot onlyonanode'sdistancefromthesourcebutalsoitsangularosetfromthelinejoining thesourceandthedestination. 87

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Designofnodeactivationstrategieswithaconstraintonenergiesinreception, whenthenodesknoweachothers'locations Inmanywirelessnetworks,thelocationsofthenodeschangerelativelyslowlyalthough thefadinggainsmaychangequicklyincomparisontotherateofnetworkupdates.For suchscenarios,thenetworkmodelusedinouranalysiswithrandomnodelocationsis notappropriate,andthedesignofnodeactivationfunctionswillbedierent.Forexample, considerasysteminwhichthetransmitterknowsthelocationsoftheclosest L nodes.In particular,considerthemaximizationoftransmissiondistanceandequivalently,transport capacity,whichdependsonlyonthenodedistances X = X 1 ;X 2 ;:::;X L ,wherewithout lossofgenerality,wetake X i X j if i j .Wecanuseorderstatistics[61]tondthe jointdistributionordensityofthedistances.Theproblemofndinganactivation functiontomaximizethetransmissiondistancecanbesimpliedto max Y2X X i =1 Y i P H i Y )]TJ/F25 7.9701 Tf 6.586 0 Td [(n i > 1 suchthat: jYj = {1 Theproblemformulationaboveisproblematicforuseinsensornetworksinwhichthe nodesarexedandthechannelchangesrelativelyslowly.Thisisbecausethesamesetof nodeswillgetactivated,thusdepletingtheenergyofthosenodes.Onewayof overcomingthisproblemistodealwiththeenergyissuesbypruningthenodeswithleast energyfromthelist,beforeselectingwhichnodestoactivate. Designofnodeactivationstrategiesbyallowingpacketretransmission, possiblyalongwithpacketcombiningatreceivers Ifthechannelchangesbetweentwotransmissions,thendiversityisachievedbothinspace acrossnodesandtime.Thesimplestapproachtoincorporateretransmissionwith opportunisticreceptionistokeepthesamereceiversactivefortheretransmissionsofa packet.Inordertoimprovetheenergyeciency,itmaybeusefultoallowasubsetofthe 88

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receiversthatareactiveinthersttransmissionstobeactiveinlatertransmissions,where theselectionofthesubsetcanbedoneadaptivelybasedonthechannelqualityduring previoustransmissions. Wehaveconsideredsingletransmissionsonlyfortheworkinthisthesis.Extensionsof ourworkcanfurtherincludetransmissionstraversingmultiplehopsandtothedesignof energy-ecientMACprotocolsforgeographictransmissions. 89

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REFERENCES [1]R.KnoppandP.A.Humblet,Informationcapacityandpowercontrolinsingle-cell multiusercommunications,"in Proc.1995IEEEInt.Conf.Commun. ,vol.1,Seattle, June1995,pp.331{335. [2]M.GrossglauserandD.Tse,Mobilityincreasesthecapacityofadhocwireless networks," IEEE/ACMTrans.Networking ,vol.10,pp.477{486,August2002. [3]M.ZorziandR.R.Rao,Energy-ecientforwardingforadhocandsensornetworks inthepresenceoffading,"in Proc.2004IEEEInt.Conf.Commun. ,vol.7,June 2004,pp.3784{3789. [4]P.Black,M.Grob,R.Padovani,N.Sindhushyana,andS.Viterbi,CDMA/HDR: abandwidthecienthighspeedwirelessdataservicefornomadicusers," IEEE Commun.Mag. ,vol.38,pp.70{77,July2000. [5]R.Ferrus,L.Alonso,A.Umbert,X.Reves,J.Perez-Romero,andF.Casadevall, Cross-layerschedulingstrategyforumtsdownlinkenhancement," IEEECommunicationsMagazine ,vol.43,no.6,pp.S24{S28,June2005. [6]H.FattahandC.Leung,Anoverviewofschedulingalgorithmsinwireless multimedianetworks," WirelessCommunications,IEEE ,vol.9,no.5,pp.76{83, Oct.2002. [7]Y.CaoandV.Li,Schedulingalgorithmsinbroadbandwirelessnetworks," Proc. IEEE ,vol.89,no.1,pp.76{87,Jan.2001. [8]C.Perkins,Ed., AdHocNetworking .Addison-Wesley,2000. [9]X.Liu,E.K.P.Chong,andN.B.Shro,Opportunistictransmissionscheduling withresource-sharingconstraintsinwirelessnetworks," IEEEJ.Sel.AreasCommun. vol.19,no.10,pp.2053{2064,Oct.2001. [10]B.Sadeghi,V.Kanodia,A.Sabharwal,andE.Knightly,Opportunisticmediaaccess formultirateadhocnetworks,"in Proc.MobiCom'02 .NewYork,NY,USA:ACM, 2002,pp.24{35. [11]J.Wang,H.Zhai,Y.Fang,andJ.M.Shea,OMAR:Utilizingdiversityinwirelessad hocnetworks," IEEETrans.MobileComputing ,vol.5,pp.1764{1779,Dec.2006. [12]P.Larsson,Selectiondiversityforwardinginamultihoppacketradionetwork withfadingchannelandcapture,"in ACMSIGMOBILEMobileComputingand CommunicationReview ,Oct.2001,pp.47{54. [13]M.ZorziandR.R.Rao,GeographicrandomforwardingGeRaFforadhocand sensornetworks:Multihopperformance," IEEETrans.MobileComputing ,vol.2, no.4,pp.337{348,Oct.-Dec.2003. 90

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BIOGRAPHICALSKETCH TathagataD.GoswamireceivedtheBachelorsofTechnologydegreeinElectrical Engineeringin2003fromtheIndianInstituteofTechnologyIITinRoorkee,India.He receivedhisM.S.degreeinElectricalandComputerEngineeringin2005,andhisPh.D. degreeinElectricalandComputerEngineeringinAugust2009,bothattheUniversityof Florida,Gainesville.Hisresearchinterestsincludewirelesscommunications,cross-layer design,andgeographictransmissionsinwirelessnetworks. 95