Minimum Energy to Send
Bits Through the Gaussian Channel With and Without Feedback
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IEEE Transactions on Information Theory
Volume 57 Issue 8, August 2011
IEEE Press Piscataway, NJ, USA
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2011 Article
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Save to Binder90Power constrained and delay optimal policies for scheduling transmission over a fading chan
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90Power constrained and delay optimal policies for scheduling transmission over a fading chan
PowerConstrainedandDelay；MunishGoyal,AnuragKumar,；DepartmentofElectricalCo；Email:munish,anurag,vino；Weconsideranoptimalpower；Keywords:Powerandratecon；II.INTRODUCTION；Incommunicatio
PowerConstrainedandDelayOptimalPoliciesforSchedulingTransmissionoveraFadingChannelMunishGoyal,AnuragKumar,VinodSharmaDepartmentofElectricalCommunicationEnggIndianInstituteofScience,Bangalore,IndiaEmail:munish,anurag,vinod@ece.iisc.ernet.inI.ABSTRACTWeconsideranoptimalpowerandrateschedulingproblemforasingleusertransmittingtoabasestationonafadingwirelesslinkwiththeobjectiveofminimizingthemeandelaysubjecttoanaveragepowerconstraint.Thebasestationactsasacontrollerwhich,dependinguponthetransmitterbufferlengthsandthesignalpowertointerferenceratio(SIR)ontheuplinkpilotchannel,allocatestransmissionrateandpowertotheuser.Weprovidestructuralresultsforanaveragecostoptimalstationarypolicyunderalongrunaveragetransmitterpowerconstraint.WeobtainaclosedformexpressionrelatingtheoptimalpolicywhentheSIRisthebest,totheoptimalpolicyforanyotherSIRvalue.Wealsoobtainlowerandupperboundsfortheoptimalpolicy.Keywords:Powerandratecontrolinwirelessnetworks,QualityofserviceinwirelessnetworksII.INTRODUCTIONIncommunicationsystems,manyfundamentalproblemsinvolvetheoptimalallocationofresourcessubjecttoperfor-manceobjectives.Inawirednetwork,thecrucialresourcesarethetransmissiondataratesavailableonthelink.Techniquessuchas?owcontrol,routingandadmissioncontrolareallcenteredaroundallocatingtheseresources.Weconsideraresourceallocationproblemthatarisesinmobilewirelesscom-municationsystems.Severalchallenginganalyticalproblemsarisebecauseofthespeciallimitationsofawirelesslink.Oneisthetimevaryingnatureofthemultipathchannel,andanotheristhelimitedbatterypoweravailableatatypicalwirelesshandset.Itisdesirabletoallocatetransmissionratestoausersuchthattheenergyusedtotransmittheinformationismini-mizedwhilekeepingerrorsundercontrol.Mostapplications,however,alsohavequalityofservice(QoS)objectivessuchasmeandelay,delayjitter,andthroughput.ThusthereisaneedforoptimalallocationofwirelessresourceswhichprovidessuchQoSguaranteessubjecttotheabovesaiderroranden-ergyconstraints.Variousmethodsforallocatingtransmissionresourcesarepartofmostthirdgenerationcellularstandards.Theyincludeadjustingthetransmissionpower,changingthecodingrateandvaryingthespreadinggaininaCDMAbasedsystem.ThesystemmodelinourworkisgiveninFig.1andisexplainedbelow.Weassumeaslottedsystemwherethehigher0-/03/$17.00 (C) 2003 IEEElayerpresentsthedata,thatarrivesoveraslot,tothelinklayerattheendofeachslot.Thelinklayerisassumedtohaveanin?nitecapacitybuffertoholdthedata.Weassumethatthechannelgainandanyotherinterferencetothesystemremain?xedoveraslotandvaryindependentlyfromslottoslot.Overamini-slot(shownasshadedinFig.1),thebufferlengthinformationiscommunicatedtothereceiver/controller,andtheusertransmitspilotbitsata?xedpowerlevelwhichwerefertoasapilotchannel.Thereceiverestimatesthesignaltointerferenceratio(SIR)onthepilotchannel.Weassumethattheestimatesareperfect.DependingontheSIRestimatesandthebufferlengthinformation,thereceiverevaluatestheoptimaltransmissionrateandpowerforthecurrentslotandcommunicatesitbacktothetransmitter.Inpractice,therearesomerestrictionsonhowmuchthesecontrolscanvary.Inthispaperweassumethatthetransmittercantransmitatanyarbitraryrateandpowerlevel.Thetransmitterremovesthatmuchamountofdatafromthebufferandencodesitattheallocatedrate.AllthisexchangeofinformationandtheencodingisassumedtobecompletedwithinthetimeslotshownasshadedintheFig.1.Afterthisthetransmitterstartstotransmittheencodeddata.GoldsmithandVaraiya[3]areprobablythe?rsttoobtaintheoptimalpowerallocationpolicyforasinglelinkfadingwirelesschannel.Theiremphasiswasontheoptimalphysicallayerperformance,whileignoringthenetworklayerperfor-mancesuchasqueueingdelay.Inrecentwork,BerryandGallager[1]haveconsideredaproblemsimilartoours.Theyobtainedstructuralresultsexhibitingatradeoffbetweenthenetworklayerandphysicallayerperformance,i.e.theoptimalpowerandmeandelay.Theyshowttheoptimaldelaycurveisconvex,andastheaveragepoweravailablefortransmissionincreases,theachievablemeandelaydecreases.Theyalsoprovidesomestructuralresultsfortheoptimalpolicythatachievesanypointonthepowerdelaycurve.Inthiswork,weimproveupontheresultsobtainedin[1].Weprovetheexistenceofastationaryaverageoptimalpolicy,andgiveaclosedformexpressionfortheoptimalpolicyforanySIRvalueintermsoftheoptimalpolicywhentheSIRisone,i.e.,thebestSIR.Wealsoprovidelowerandupperboundsfortheoptimalrateallocationpolicy,notobtainedbyBerryandGallager.Thispaperisorganizedasfollows.InSectionIII,wegivethemodelofthesystemunderconsiderationandformulate311DataArrives Exchange of Contol fromhigher layerinformationa[n?1]a[n]Exchange of Contol informationnτHigherLayer1100Data transmission1100(n+1)τData transmission(n+2)τ1100TimeTRANSMITTER QUEUEAWGNI[n]Controller(as receiver)s[n]r[n] Channelh[n]Fig.1.Systemmodel2thecontrollerobjectiveasaconstrainedoptimizationproblem.Thenusingtheresultfrom[4],weconvertitintoafamilyofunconstrainedoptimizationproblems.ThisunconstrainedproblemisaMarkovdecisionproblem(MDP)withtheaveragecostcriterion.Weshowtheexistenceofstationaryaveragecostoptimalpolicieswhichcanbeobtainedasalimitofdiscountedcostoptimalpolicies,inSectionIV.InSectionV,weobtainstructuralresultsforthediscountedcostoptimalpolicies.WeobtainstructuralresultsfortheaverageoptimalpolicyinSectionVI.Finally,inSectionVII,we?ndconditionsunderwhichthehypothesisoftheTheoremstatedinSectionIII,holdsandhencetheexistenceofaLagrangemultiplier,andthecorrespondingoptimalpolicywhichisalsooptimalfortheoriginalconstrainedMDP.III.SYSTEMMODELANDPROBLEMFORMULATIONWeconsideradiscretetime(slotted)?uidmodelforouranalysis,andwilllatercommentonhowourresultsmaybeusedforapacketizedmodel.Theslotlengthisτ(timeunits),andthenthslotistheinterval[nτ,(n+1)τ),n≥0.Thedatatobetransmittedarrivesintothesystemfromahigherlayerattheendofeachslotandisplacedintoabufferofin?nitecapacity(SeeFigure1).ThearrivalprocessA[n]isassumedtobeanindependentandidenticallydistributed(iid)letF(a)beitsdistributionfunction.ThechannelpowergainprocessH[n]isassumedtoremain?xedoveraslotandvaryindependentlyfromslottoslot.AnyotherinterferencetothesystemismodelledbytheprocessI[n]whichstaysconstantoveraslotandisiidfromslottoslot.Wefurtherassumethatthereceivercancorrectlyestimatethesignaltointerferenceratio(SIR)γonthe“uplink”usingapilotchannel.Letσ2bethereceivernoisepower.Withoutlossofgenerality,weassumethepilottransmitterpoweris?xedtoσ2units.Duringthenthslot,theSIRγ[n]canbewrittenintermsofthecurrentchannelgain(h[n]∈(0,1]),thereceivernoisepower,andthecurrentwhereζ[n]constitutestheadditivewhiteGaussiannoiseandtheotherusers’interferencesignal.Inthismodelweassumetheexternalinterferencetobeindependentofthesystembeingmodelled.Let,forn∈{0,1,2,???},s[n]betheamountof?uidinthebufferatthenthdecisionepochandγ[n]betheSIRinthenthslot(i.e.,theinterval[nτ,(n+1)τ)).Letthestateofthesystemberepresentedasx[n]:=(s[n],γ[n]).Atthenthdecisioninstant,thecontrollerdecidesupontheamountof?uidr[n]tobetransmittedinthecurrentslotdependingontheentirehistoryofstateevolution,i.e.,x[k]fork={0,1,2,???,n}.Leta[n],n∈{0,1,2,???}betheamountof?uidarrivinginthenthslot.Sincetheamountof?uidtransmittedinaslotshouldbelessthantheamountinthebuffer,i.e.,r[n]≤s[n],foralln,theevolutionequationforthebuffercanbewrittenass[n+1]=s[n]?r[n]+a[n].σh[n]otherinterference(i[n]∈[0,∞)),asγ[n]=σ2+i[n].ThustheprocessΓ[n]isiidwiththedistributionfunctionG(γ).Further,fromtheabovede?nitionγ[n]∈(0,1].Thereceiveractsasacontrollerwhich,giventhebufferstateandtheSIR,obtainsanoptimumtransmissionschedulethatminimizesthemeanbufferdelaysubjecttoalongrunaverageˉ.ThebufferstateisavailabletotransmitterpowerconstraintPthereceiveratthebeginningofeachslot.ThemeasurementofSIRandthecontroldecisionaretakenwithinamini-slotshownasshadedandconveyedbacktothetransmitteralsointhesamemini-slot.Basedonthesedecisions,thetransmitterremovesthedatafromthebuffer,encodesitandtransmitstheencodeddataoverthechannel.Accordingtoourmodel,ifinaframentheusertransmitsasignalys[n],thenthereceivergets??yr[n]=h[n]ys[n]+ζ[n],312DenotebyX[n],S[n],R[n],n≥0,thecorrespondingrandomprocesses.Thecostofservingrunitsof?uidinaslotisthetotalamountofenergyrequiredfortransmission.WeassumeNcNisrelatedtothechannelband-widthviaNyquist’stheorem.WhenNissuf?cientlylarge,thepowerP,requiredtotransmitreliably(i.e.,withzeroprobabilityofdecodingerror),isrelatedtothetransmissionofrunitsofdatainNchannelsymbols,whentheSIRasde?nedaboveisγ,byShannon’sformula[2]fortheinformationtheoreticcapacity,i.e.,γP??1??r=ln1+2,θσwhereθ=2ln(2)N.Thuswhenthesystemstateisx,thepowerrequiredtotransmitrunitsof?uidisσ2θr(e?1).P(x,r)=γSince,inpractice,Nis?nite,thereispositiveaprobabilityofdecodingerror.InsectionVIII-A,wewillcommentonhowtheproblemgetsmodi?edbyincorporatingthiserrorprobability.SincedelayisrelatedtotheamountofdatainthebufferbyLittle’sformula[7],theobjectiveistominimizethemeanbufferlength.Givenx[0]=x,thecontroller’sproblemisthustoobtaintheoptimalr(?)thatminimizes??n1limsupnnEk=0S[k],subjectto,??nˉlimsupn1Ep(X[k],R[k])≤P.nk=0πKxˉ>0,denotebyΠPGivenPˉthesetofalladmissiblecontrolpoliciesπ∈Πwhichsatisfythelongruntransmitterpowerπˉ.Thenthecontrollerobjectivecanbe≤PconstraintKxrestatedasaconstrainedoptimizationproblem(CP)de?nedas,πsubjecttoπ∈ΠP(CP):MinimizeBxˉ1π??=limsupExp(X[k],R[k]).nnk=0n(1)Theproblem(CP)canbeconvertedintoafamilyofun-constrainedoptimizationproblemthroughaLagrangianap-proach[4].Foreveryβ>0,theLagrangemultiplier,de?neamappingcβ:K→R+by,cβ(x,r)=s+βp(x,r).De?neacorrespondingLagrangianfunctionalforanypolicyπ∈Πby,1π??πJβ(x)=limsupExcβ(X[k],R[k]).nnk=1nThefollowingtheoremgivessuf?cientconditionsunderwhichanoptimalpolicyforanunconstrainedproblemisalsooptimalfortheoriginalconstrainedcontrolproblem(CP).Theorem3.1:[4]Let,forsomeβ>0,π?∈Πbethepolicythatsolvesthefollowingunconstrainedproblem(UPβ)de?nedas,π(UPβ):MinimizeJβ(x)subjecttoπ∈Π??(2)ItcanbeseenfromtheaboveobjectivethattheproblemhasthestructureofaconstrainedMarkovdecisionproblem(MDP)[4],whichweproceedtoformulateinthenextsection.A.FormulationasaMDPFurther,ifπ?yieldstheexpressionsBπandKπaslimits?ˉ,?x,thenthepolicyπ?isoptimalforallx∈XandKπ=Pfortheconstrainedproblem(CP).Proof:(See[4])Notethateventhoughtheresultisstatedin[4]forthecountablestatespacecase,theresultholdsalsoforthemoregeneralsituationinourpapersolongaswecanprovideasolutiontoUPβwiththerequisitepropertiesstatedintheTheorem.??Inthesubsequentsections,wesolvetheproblem(UPβ)andshowinSectionVIIthatthesolutionsatis?esthehypothesisoftheTheorem3.1.Theproblem(UPβ)isastandardMarkovdecisionproblemwithanaveragecostcriterion.Foreaseofnotation,wesuppressthedependenceontheparameterβ.IV.EXISTENCEOFASTATIONARYAVERAGECOSTOPTIMALPOLICYWeconsidertheaveragecostproblem(UPβ)andde?neacorrespondingdiscountedcostMDPwithdiscountfactorα.Weintendtostudytheaveragecostproblemasalimitofdiscountedcostproblemswhenthediscountfactorαgoestoone.Forinitialstatex,de?neVα(x)=πminExπ∈Π∞????Let{X[n],n∈{0,1,2,???}}denoteacontrolledMarkovchain,withstatespaceX=R+×(0,1],andactionspaceR+,whereR+denotesthepositiverealhalfline.Thesetoffeasibleactionsinstatex=(s,γ)is[0,s].LetKbethesetofallfeasiblestate-actionpairs.ThetransitionkernelonXgivenanelement(x,r)∈KisdenotedbyQ,where????dF(a)dG(z).Q(y∈(S??,Γ??)?X|(x,r))=S???s+rΓ??θrDe?nethemappingp:K→R+byp(x,r)=σ?1).γ(eApolicyπgeneratesattimenanactionr[n]dependingupontheentirehistoryoftheprocess,i.e.,atdecisioninstantn∈{0,1,2,???},πnisamappingfromKn×Xto[0,s[n]].LetΠbethespaceofallsuchpolicies.Astationarypolicyf∈ΠisameasurablemappingfromXto[0,s].Forapolicyπ∈Π,andinitialstatex∈X,wede?netwocostfunctionsππ,thebuffercost,andKx,thepowercostby,Bx21π??πBx=limsupExS[k].nnk=0nk=0??αcβ(X[k],R[k])k313astheoptimaltotalexpecteddiscountedcostfordiscountfactorα,0<α<1.Vα(x)iscalledthevaluefunctionforthediscountedcostMDP.Thefollowinglemma[6]provestheexistenceofstationarydiscountedcostoptimalpolicies.WewillneedConditionsW.W1.Xisalocallycompactspacewithacountablebase.W2.R(x),thesetoffeasibleactionsinstatex,isacompactsubsetofR(theactionspace),andthemultifunctionx→R(x)isuppersemicontinuous.W3.Qiscontinuousinrwithrespecttoweakconvergenceinthespaceofprobabilitymeasures.W4.c(x,r)islowersemi-continuous.Lemma4.1:[[6],Proposition2.1]UnderConditionsW,thereexistsadiscountedcoststationaryoptimalpolicyfαforeachα∈(0,1).??Nowwestatearesultrelatedtotheexistenceofstationaryaverageoptimalpolicieswhichcanbeobtainedaslimitofdiscountedcostoptimalpoliciesfα.De?newα(x)=Vα(x)?infVα(x).x∈XwhenthesystemstartswithanemptybufferandthebestSIR.Alsowhenthebufferisempty,thesetoffeasibleactionsis{0}.Thusascβ(x0,0)=0,wehave,??Vα(y)Q(dy|(x0,0)).Vα(x0)=αXInaddition,consideringthepolicyr(x)=sforallx∈X,weget,??βσ2θs(e?1)+αVα(x)≤s+Vα(y)Q(dy|(x,s))γXButfromthede?nitionofQ,itfollowsthatQ(dy|(x,s))=Q(dy|(x0,0)).Thusweget,βσ2θs(e?1)+Vα(x0).Vα(x)≤s+γBythede?nitionofwα(x)wehavewα(x)=Vα(x)?Vα(x0)andhenceβσ2θs(e?1)<∞forx∈X.wα(x)≤s+γThusalltheconditionsofTheorem4.1aresatis?ed.HencewehaveprovedtheexistenceofastationaryaverageoptimalpolicywhichcanbeobtainedasalimitofdiscountoptimalpoliciesasdescribedinTheorem4.1.V.ANALYSISOFTHEDISCOUNTEDCOSTMDPInthissection,weobtainsomestructuralresultsfortheα-discountedoptimalpolicyforeachα∈(0,1).Asαis?xedintheanalysisthatfollowsinthissection,wesuppresstheexplicitdependenceonα.Forastate-actionpair(x=(s,γ),r)de?neu:=s?r,i.e.,u∈[0,s)istheamountofdatanotservedwhenthesystemstateisx.Itfollowsfromthede?nitionofQthatQ(dy|(x,r))=Q(dy|u).ThuswecanrewritetheDCOE(Equation3)intermsofuas,????βσ2θ(s?u)(e?1)+αH(u),V(x)=mins+γu∈[0,s]??∞??10Theorem4.1:[[6],Theorem3.8]SupposethereexistsapolicyΨandaninitialstatex∈XsuchthattheaveragecostJΨ(x)<∞.Letsupα<1wα(x)<∞forallx∈XandtheConditionsWhold,thenthereexistsastationarypolicyf1whichisaveragecostoptimalandtheoptimalcostisindependentoftheinitialstate.Alsof1islimitdiscountoptimalinthesensethat,foranyx∈Xandgivenanysequenceofdiscountfactorsconvergingtoone,thereexistsasubsequence{αm}ofdiscountfactorsandasequence??xm→xsuchthatf1(x)=limm→∞fαm(xm).Remark:InTheorem4.1,thesubsequenceofdiscountfactorsdependsuponthechoiceofx.FirstweverifytheConditionsW.ConditionsW1holdstruesincethestatespaceisasubsetofR2whichislocallycompactwithacountablebase.ThesetR(x)=[0,s]iscompactandthemappingx(=(s,γ))→[0,s]iscontinuous,thustheconditionW2holds.ConditionW3followsfromthede?nitionofthetransitionkernelQ(?)sincefordistributionsonR2,weakconvergenceisjustconvergenceindistribution.Asthefunctionciscontinuous,theconditionW4follows.Thisimpliestheexistenceofstationarydiscountedcostoptimalpoliciesfα.The?rsthypothesisofTheorem4.1shouldholdinmostpracticalproblemsbecauseotherwisethecostisin?niteforanychoiceofthepolicy,andthusanypolicyisoptimal.Toverifythatsupα<1wα(x)<∞forx∈X,x=(s,γ)wewritethediscountedcostoptimalityequation(DCOE)as??????Vα(y)Q(dy|(x,r))(3)Vα(x)=mincβ(x,r)+α0≤r≤sX(4)wherethefunctionH(u)isde?nedas,H(u):=Lemma5.1:0?V(u+a,γ)dF(a)dG(γ).(5)(i))H(u)isconvex,andhenceH??(u)isincreasing.(ii))1≤H??(u)≤1+θeσ2βηeθu,whereη=.??∞????Givenγ,Vαs,γisclearlyincreasinginssincethelargeristheinitialbufferthelargerwillbethecosttogo.Thusarginfx∈XVα(x)=(0,1)=:x0,i.e.,thein?mumisachieved0???10eθadF(a)dG(γ)γProof:SeetheAppendix.??314A.StructureoftheoptimalpolicyNowitcanbeseenthatforeachx,therighthandsideoftheDCOEisaconvexprogrammingproblem.Usingstandardtechniquesforsolvingaconstrainedconvexprogrammingproblem,weobtainedthefollowingresultfortheu(x)thatachievestheminimuminEquation4.???r(s,1)asr(s),andV(s,1)asV(s).ThusfromEquation4,wehave,????V(s)=mins+βσ2(eθ(s?u)?1)+αH(u).(6)u∈[0,s]u(x)=0for{x∈X:u(x)=sforElseu(x)is0<u(x)<s.ThissolutionisdepictedinFigure2.βθσ2θs≤H??(0)}.αγe2{x∈X:H??(s)≤βθσαγ}??2θsθuαH(u)thesolutionofσe=eγβθ,andTheobjectivefunction,beingsumofastrictlyconvexandaconvexfunction,isstrictlyconvex.Thusithasauniqueminimizerforeachs.Nowweobtainsomestructuralresultsforu(s).Notethatr(s)=s?u(s).Thefollowingtheoremsgivestructuralresultsforthediscountedcostoptimalpolicyu(s)obtainedasasolutiontoEquation6.Theorem5.1:Theoptimalrateallocationpolicyr(s)=s?u(s)isnondecreasingins.Proof:Weshowthisbycontradiction.Lettherebes1ands2suchthats1r(s2).Thusr(s2)>r(s1)≤s1<s2andhenceapolicywhichusesr(s2)instates1andr(s1)instates2isfeasible.Sincer(?)isoptimal,itfollowsthats1+βσ2(eθr(s1)?1)+αH(s1?r(s1))<s1+βσ2(eθr(s2)?1)+αH(s1?r(s2))s2+βσ2(eθr(s2)?1)+αH(s2?r(s2))<s2+βσ2(eθr(s1)?1)+αH(s2?r(s1)),wherethestrictinequalityholdsduetouniquenessoftheminimizer.Nowbyaddingthetwoequationsweget,H(s1?r(s2))?H(s1?r(s1))>H(s2?r(s2))?H(s2?r(s1))whichcontradictstheconvexityofH(?)asprovedinLemma5.1.??ObservethatTheorem5.1impliesthatforanypair(s1,s2)satisfyings1<s2,wehaves1?u(s1)≤s2?u(s2),i.e.,u(s2)?u(s1)≤s2?s1.Theorem5.2:Theoptimalpolicyu(s):=s?r(s)isnondecreasingands11?ln(κ2)≤u(s)≤s?ln(κ1),22θθwhereκ1andκ2areconstants.Remark:Wehaveu(s)+r(s)=s,andweseethatbothr(s)andu(s)arenondecreasingins.Proof:Wearguebycontradiction.Lettherebes1ands2suchthats1u(s2).Thusapolicywhichusesu(s2)instates1andu(s1)instates2isfeasible.Sinceu(?)isoptimal,itfollowsfromtheuniquenessoftheminimizerthats1+βσ2(eθ(s1?u(s1))?1)+αH(u(s1))<s1+βσ2(eθ(s1?u(s2))?1)+αH(u(s2))s2+βσ2(eθ(s2?u(s2))?1)+αH(u(s2))<s2+βσ2(eθ(s2?u(s1)))?1)+αH(u(s1)).1111SERVENOTHING000γ111111SERVEEVERYTHING00SFig.2.CharacterizationoftheoptimalpolicyforthediscountedcostMDPObservations:1)ItisoptimalnottoserveanythingwhentheSIResti-1islarge).matedislow(i.e.,γ2)WhentheSIRestimatedatthereceiverishigh,itisoptimaltoserveeverythinguntilavalueofthebuffersizethatincreaseswithr.3)InthelowSIRregion,assincreasesitbecomesoptimaltoservedataasthedelaycostthenexceedsthepowercost.B.AstatespacereductionNowweshowthattheoptimalpolicyforanySIRγcanbecalculatedbyknowingtheoptimalpolicywhentheSIRis1.NotethatH(u)isafunctionthatdoesnotdependuponthepolicy,andβ,θ,αaregivenconstants.Considertwostatesx1andx2.ItfollowsfromtheresultinSectionV-Athatexceptforthecasewhenu(x)=s,thecontrolsu(x1)andu(x2)arethesameif11θs1e=eθs2γ1γ2Thuswecancomputetheoptimalpolicyu(x)foranyx∈Xbyknowingtheoptimalpolicyforthecasewhenγis?xedtooneandonlysisallowedtovary.Inordertocomputeu(s,γ),1andthentheoptimalwe?rstobtains1suchthateθs1=eθsγpolicyu(s,γ)canbewrittenintermsofu(?,1)as,????1??1??????,1.u(s,γ)=mins,us+lnθγThusforsubsequentanalysis,wecanconcentrateonlyontheevaluationofu(s,1)andwehenceforthcallittheoptimalpol-icy.Forthenotationalconvenience,wewriteu(s,1)asu(s),315三亿文库包含各类专业文献、生活休闲娱乐、幼儿教育、小学教育、高等教育、各类资格考试、应用写作文书、专业论文、文学作品欣赏、90Power constrained and delay optimal policies for scheduling transmission over a fading chan等内容。　
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