Average Rate of Downlink Heterogeneous Cellular Networks

3050IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

AverageRateofDownlinkHeterogeneousCellularNetworksoverGeneralizedFadingChannels:

AStochasticGeometryApproach

MarcoDiRenzo,Member,IEEE,AlessandroGuidotti,StudentMember,IEEE,

andGiovanniE.Corazza,SeniorMember,IEEE

Abstract—Inthispaper,weintroduceananalyticalframeworktocomputetheaveragerateofdownlinkheterogeneouscellularnetworks.Theframeworkleveragesrecentapplicationofstochas-ticgeometrytoother–cellinterferencemodelingandanalysis.TheheterogeneouscellularnetworkismodeledasthesuperpositionofmanytiersofBaseStations(BSs)havingdifferenttransmitpower,density,path–lossexponent,fadingparametersanddistri-bution,andunequalbiasingfor exibletierassociation.Along–termaveragedmaximumbiased–received–powertierassociationisconsidered.ThepositionsoftheBSsineachtieraremodeledaspointsofanindependentPoissonPointProcess(PPP).Undertheseassumptions,weintroduceanewanalyticalmethodologytoevaluatetheaveragerate,whichavoidsthecomputationoftheCoverageProbability(Pcov)andneedsonlytheMomentGeneratingFunction(MGF)oftheaggregateinterferenceattheprobemobileterminal.Thedistinguishablecharacteristicofouranalyticalmethodologyconsistsinprovidingatractableandnumericallyef cientframeworkthatisapplicabletogeneralfadingdistributions,includingcompositefadingchannelswithsmall–andmid–scale uctuations.Inaddition,ourmethodcanef cientlyhandlecorrelatedLog–Normalshadowingwithlittleincreaseofthecomputationalcomplexity.TheproposedMGF–basedapproachneedsthecomputationofeitherasingleoratwo–foldnumericalintegral,thusreducingthecomplexityofPcov–basedframeworks,whichrequire,forgeneralfadingdistributions,thecomputationofafour–foldintegral.

IndexTerms—Heterogeneouscellularnetworks,aggregatein-terferencemodeling,stochasticgeometry,averagerate.

characterofthewirelesschannel[2].Forthisreason,accurateperformanceanalysisisusuallyconductedviacostly,time–consuming,andoftenproprietarysystem–levelsimulators[3].Thisapproach,however,seldomprovidesinsightfulinforma-tiononsystemdesignandonthedependencyofthesystemparameterstooptimize.Thissituationisevenexacerbatedinfuturecellulardeployments,whicharebecomingmoreheterogeneouswiththeintroductionofnewinfrastructureelements,e.g.,femto/picoBSs, xed/mobilerelays,cognitiveradios,anddistributedantennas[4]–[9]and[10]forasurvey.A.AbstractionModelsforAnalysisandDesignofCellularNetworks

Tocircumventthisproblem,communicationstheoristsusu-allyresortto“abstractions”fortractableother–cellinterfer-encemodelingandforperformanceanalysis.Theseabstrac-tionsusuallyencompasssimpli edspatialmodelsforthelocationsoftheBSs.Inparticular,threeabstractionmodelsarecommonlyadopted:i)theWynermodel[11];ii)thesingle–cellinterferingmodel[12];andiii)theregularhexagonalorsquaregridmodel[13].Theseabstractionmodels,however,areofteneitherover–simplisticorinaccurate[14].Further-more,insomecases,asfortheregularhexagonal/squaregridmodel,theystillrequireeitherintensivenumericalsim-ulationsormulti–foldnumericalintegrations.Finally,theseabstractionmodelsusuallyprovideinformationforspeci cBSsdeployments,andtypicallyfailtoprovideusefulinfor-mationformorerandom,unplanned,and/orclusteredBSsdeployments,whicharetypicalofemergingheterogeneouscellularnetworkswith,e.g.,overlaidfemtocellsandpicocells[9],[10].Motivatedbytheseconsiderations,anewabstractionmodeliscurrentlyemergingandgainingpopularity,accordingtowhichthepositionsoftheBSsaremodeledaspointsofaPoissonPointProcess(PPP)andpowerfultoolsfromappliedprobability,suchasstochasticgeometry,areleveragedtodeveloptractableintegralsandclosed–formmathematicalframeworksforimportantperformancemetrics(e.g.,coverageandaveragerate)[15]–[17].

B.StochasticGeometrybasedModelingofHeterogeneousCellularNetworks

Thestochasticgeometrybasedabstractionmodelfortheanalysisofcellularnetworksdatesbackto(atleast)1997

I.INTRODUCTION

T

HEanalyticalperformancemodelingofcellularnetworksisalong–standingopenissue[1].Themathematicalintractabilitymainlyarisesfromthedif cultyofaccuratelymodelingother–cellinterferencebytakingintoaccountthespatialpositionsoftheBaseStations(BSs)andthestochastic

ManuscriptreceivedNovember18,2012;revisedMarch18,2013.TheeditorcoordinatingthereviewofthispaperandapprovingitforpublicationwasM.Tao.

M.DiRenzoiswiththeLaboratoiredesSignauxetSyst`emes,Unit´eMixtedeRecherche8506,CentreNationaldelaRechercheScienti que–´´EcoleSup´erieured’Electricit´e–Universit´eParis–SudXI,91192Gif–sur–YvetteCedex,France(e–mail:marco.direnzo@lss.supelec.fr).

A.GuidottiandG.E.CorazzaarewiththeDepartmentofElectronics,Com-puterScience,andSystems,theUniversityofBologna,VialeRisorgimento2,40136Bologna,Italy(e-mail:{a.guidotti,giovanni.corazza}@unibo.it)mun.Conf.(ICC),Ottawa,Canada,June2012.

ThisworkwassupportedinpartbytheEuropeanCommissionundertheauspicesoftheFP7–PEOPLEMITN–CROSSFIREproject(grant317126)andtheFP7–ICTNoE–NEWCOM#project(grant318306).

DigitalObjectIdenti er10.1109/TCOMM.2013.050813.120883

c2013IEEE0090-6778/13$31.00

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3051

[3],[18].Subsequently,asimilarshotgun–based,i.e.,PPP–based,abstractionmodelwasproposedin[19],anditwasshownthat,comparedwiththetraditionalhexagonalgridmodel,theshotgunapproachprovidesupperperformancebounds.Morerecently,thePPPmodelhasbeenusedfortheanalysisofspatialandopportunisticAlohaprotocols[20],andforthecharacterizationoftheSignal–to–Interference–Ratio(SIR)of(single–tier)cellularnetworks[21].InspiteoftheseinitialandpioneeringattemptsofapplyingthePPPmodelandstochasticgeometrytotheanalysisofcellularnetworks,onlyrecentlytherandom–basedabstractionmodelforthepositionsoftheBSshasreceivedtheattentionitdeserved.Inparticular,itsemergenceandwidespreadadoptionforcellularnetworksanalysisanddesignismostlydueto[22],whereacomprehensiveframeworktocomputecoverageandaveragerateofsingle–tierdeploymentsisprovided.In[22],itisshownthatthePPPmodelisasaccurateasregulargridmodels,butithasthemainadvantageofbeingmoreanalyticallytractable.AcomprehensivestudybasedonrealBSsdeploymentsobtainedfromtheopensourceprojectOpenCellID[23]hasrevealedthatthePPPmodelcanindeedbeusedforaccuratecoverageanalysisinmajorcitiesworldwide.RecentresultsaboutthevalidationofthePPPmodelforrealBSsdeploymentsareavailablein[24],wheredatacollectedfromOfcom,i.e.,theindependentregulatorandcompetitionauthorityintheUK,isused.Fueledbytheseencouragingresults,manyresearchersarecurrentlyusingthePPP–basedabstractionmodeltostudysingle–andmulti–tiercellularnetworks,e.g.,[25]–[47]andreferencestherein.ThePPP–basedapproachisalsowidelyadoptedfornetworkinterferencemodeling,e.g.,[48]–[65].C.AnalyticalComputationoftheAverageRate:State–of–the–ArtandPaperContribution

Inthispaper,wecapitalizeontheemergingPPP–basedabstractionmodelformulti–tiercellularnetworks,andproposeanewmathematicalmethodologytocomputethedownlinkaveragerateovergeneralfadingchannels.Recentpapershavedevelopedframeworkstocomputetheaveragerateforsingle–tierdownlink[22],[39],multi–tierdownlink[26],[43],downlinkmulti–cellcoordination[36],[44],andsingle–tieruplinkcellularnetworks[30].Allthesepapersusethesametwo–stepmethodologicalapproachtocomputetheaveragerate,whichwasoriginallyintroducedin[20]andexploitsthePlancherel–Parsevaltheorem:i) rst,theCoverageProbability(Pcov)iscomputed;andii)then,theaveragerateisobtainedbyintegratingPcovoverthepositiverealaxis[20,Eq.(2.12)].Throughoutthispaper,thismethodologyisdenotedbyPcov–basedapproach.Althoughthistechniqueavoidssystem–levelsimulations,itrequires,forgeneralfadingchannels,thecomputationofafour–foldintegral[22,AppendixC].Forthisreason,manyauthorsoftenlimittheanalysistoRayleighfad-ingchannelsand/ortointerference–limitednetworks,wheresimpli edframeworkscanbeobtained.FurtherdetailsaboutthecomputationalcomplexityofthePcov–basedapproachareavailableinSectionIII-G.

Toovercomethislimitation,weproposeanewanalyticalframeworkwhich,atthesametime,reducesthenumberofintegralstobecomputed,and,similartothePcov–based

approach,is exibleenoughforapplicationtoarbitraryfadingdistributions(includingcorrelatedcompositechannelmodels).Theframeworkleveragestheapplicationofrecentresultsonthecomputationoftheergodiccapacityinthepresenceofinterferenceandnoise[66].ItavoidsthecomputationofPcov,andneedsonlytheMomentGeneratingFunction(MGF)oftheaggregateinterferenceattheprobemobileterminal.Throughoutthispaper,thisframeworkisdenotedbyMGF–basedapproach.Weshowthatitisapplicabletomulti–tiercellularnetworkswithlong–termaveragedmaximumbiased–received–powertierassociation,andthateitherasingleoratwo–foldnumericalintegralneedtobecomputedforarbitraryfadingchannels.

D.PaperOrganization

Theremainderofthispaperisorganizedasfollows.InSectionII,thesystemmodelisdescribed.InSectionIII,theMGF–basedapproachisintroducedforsingle–tiercellularnetworks.InSectionIV,theproposedmethodologyisappliedtomulti–tiercellularnetworkswith exible(biased)tierasso-ciation.InSectionV,extensiveMonteCarlosimulationsareshowntosubstantiatetheproposedmathematicalframeworkforvariousfadingchannelmodelsandcellularnetworksdeployments.Finally,SectionVIconcludesthispaper.II.SYSTEMMODELANDPROBLEMSTATEMENT

Weconsideradownlinkheterogeneouscellularnetworksmodelsimilarto[31],[43],and[46].However,thefollowingdifferenceshold.In[31]and[46],thetierassociationpolicyisbasedontheinstantaneousSignal–to–Interference–plus–Noise–Ratio(SINR).Ontheotherhand,similarto[43],weconsiderabiasedlong–termaveragedtierassociationpolicy,paredwith[31],[43],and[46]theanalyticalmethodologytocomputetheaveragerateisnotbasedonthePcov–basedapproachbutontheMGF–basedapproach.

Notation:E{·}denotestheexpectationoperator.ME{exp( sX)}istheMGFofrandomvariableXX(s)=.fX(·)denotestheProbabilityvariableX.Γ(x)= Density+∞

Function(PDF)ofrandom

0exp( t)tx 1dtisthefunction.erfc(x)=(2/√π) +∞

gamma

xexp ξ2dξisthecom-plementaryGausserrorfunction.Sa,b(·) isthe functionde nedin[67,Sec.7.5.5].Gm,n Lommel(ap,q(·) p

)

(bq)

istheMeijerG–functionde nedin[68,Sec.2.24].Δ(n,x)=[x/n,(x+1)/n,...,,(x+n 1)/n],withnbeingapositiveintegerandxarealnumber.Thei–thentryofΔ(n,x)isde-notedbyΔi(n,x).2F1(·,·,·,·)istheGausshypergeometricfunctionde nedin[69,Ch.15].1F1(·,·,·)isthecon uenthypergeometricfunctionde nedtorialoperator.Γ(z,x)= +∞in[69,Ch.13].(·!)isthefac-xtz

1exp{ xt}dtistheupper–incompletegammafunction.γ(z,x)=0tz 1

exp{ t}dtisthelower–incompletegammafunction.δ(·)istheDiracdeltafunction.Iν(·)isthemodi edBesselfunctionofthe rstkindandorderνde nedin[69,Sec.9.6].H(·)istheHeavisidefunction,i.e.,H(x)=1ifx≥0andH(x)=0ifx<0.pFq(·,·,·)isthegeneralized√hypergeometricfunctionde nedin[70,Ch.IV].j= 1istheimaginaryunit.card{·}denotesthecardinalityofaset.

3052A.HeterogeneousCellularNetworksModel

LetusconsiderthePPP–basedabstractionmodelforthepositionsoftheBSsinabi–dimensionalplane.Then,aheterogeneouscellulardeploymentcanbemodeledasaT–tiernetworkwhereeachtiermodelstheBSsofaparticularclass.EachclassofBSsisdistinguishedbyitsspatialdensity(λtfort=1,2,...,T),transmitpower(Ptfort=1,2,...,T),path–lossexponent(αt>2fort=1,2,...,T),biasingfactor(Bt>0fort=1,2,...,T),andfadingparametersanddistribution.TheBSsofeachPPPareassumedtohavethesametransmitpower,thesamepath–lossexponent,thesamebiasingfactor,andtheirfadingchannelsareindependentandidenticallydistributed(i.i.d.).Theextensiontocorrelatedandidenticallydistributed(c.i.d.)fadingisdiscussedinSectionIII-F.However,formathematicalgenerality,weassumethatthefadingdistributionoftheserving(tagged)BSisdifferentfromthefadingdistributionoftheintra–tierinterferingBSs.TheBSsofeachtierareassumedtobespatiallydistributedaccordingtoahomogeneousPPP(Φtfort=1,2,...,T).TheTPPPsareassumedtobespatiallyindependent.Ouranalysisappliestoatypicalmobileterminal,aspermissibleinanyhomogeneousPPPaccordingtotheSlivnyak–Mecke’stheorem[17,vol.1,Theorem1.4.5].Withoutlossofgen-erality,thetypicalMobileTerminal(MT0)isassumedtobelocatedattheoriginofthebi–dimensionalplane.Theb–thBSofthet–thtierisdenotedbyBSt,b.ThedistancefromBSt,btoMT0isdenotedbydt,b.Thestandardpath–lossfunction

l(dt,b)=d t,bαt

isconsidered.Thepowerchannelgainofthe

BSt,b–to–MT0linkisdenotedbygt,b=|ht,b|2

,whereht,bistherelatedcomplexamplitudechannelgain.Forafaircom-parisonamongfadingchannelswithdifferentdistributions,normalizationconstraintE{g}=E

t,b|ht,b|2

the=Ω=1isassumedforeverybandfort=1,2,...,T.

Theframeworksdevelopedinthepresentpaperareap-plicabletosingle–input–single–outputtransmissionsystems.Inotherwords,BSsandMT0areequippedwithasingletransmitandreceiveantenna,respectively.Thegeneralizationoftheproposedanalyticalmethodologytomoreadvancedtransmissiontechnologiesiscurrentlyunderinvestigation,butitisbeyondthescopeofthepresentpaper.Theinterestedreadercan,however, ndpreliminaryresultstotheanalysisofmulti–antennareceiversanddual–hoprelayingin[71]and[72],respectively.Themainlimitationof[71]and[72]isthatcellassociationisnotconsideredandthatthedistancefromservingBStoprobemobileterminalisassumedtobe xed.Finally,wementionthattheaveragerateiscomputedunderthesameassumptionsasin[22,Sec.IV],i.e.,theinterferenceistreatedasnoiseandthetypicalmobileterminalusesadaptivemodulation/codingsuchthattheShannonbound,fortheoperatinginstantaneousSINR,canbeachieved.B.BiasedLong–TermAveragedTierandBSAssociationWeassumethattheBSsofeachtieroperateinopenaccessmodeforMT0[8].Asaconsequence,MT0isallowedtoaccesstoanytierswithoutanyrestrictions.Inamulti–tiercellularnetworksmodel,bothtierandBSassociationshavetobeproperlyde ned.Similarto[43,Sec.II–A],throughoutthis

IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

paperweconsideralong–termaveragedmaximumbiased–received–powerassociationpolicy.Letdt=min{dt,b}fort=1,2,...,TbethedistancefromMT0tothenearestBSofthet–thtier.LetBStfort=1,2,...,TbetheTnearestBSs.Then,MT0isassociated(tagged)tothetiert de nedasfollows:

t =argmax Pαt

(1)t=1,2,...,T

td t

Btandthetagged(serving)BSisdenotedbyBSt =BS0.

Inotherwords,MT0isconnectedtotheBSthatoffersthehighestaveragereceivedpowertoit.Accordingly,theBS0–to–MT0linkistheusefulsignal,whilealltheotherBSsineverytieractasinterferers.SincethepositionsoftheBSsarerandom,theBS0–to–MT0distanceisarandomvariableaswell[22].

Thebiasingfactor,Bt>0fort=1,2,...,T,modi esthecoveragerangeofeachtierforabetterof oadingstrategy.Forexample,ifBt>1thecoveragerangeofthet–thtierisincreased.Throughoutthispaper,weassume,similarto[43],thatalltheBSsarefully–loaded(i.e.,theirqueuesarefulland,thus,theyhavealwaysdatatotransmit).Theanalysisofheterogeneouscellularnetworkswithpartially–loadedBSsispostponedtofutureresearch,forexampleeitherusingtheconditionallythinningapproachproposedin[35]ortherecentresultsin[45]and[47].

C.ProblemStatement

Themainobjectiveofthispaperistocomputetheaverage(ergodic)rateofaheterogeneouscellularnetwork,whichismodeledasthesuperpositionofTindependentPPPs.Accordingto[22]and[43],theaverageratecanbewrittenasfollows:

R=

TAtRt(2)t=1

where:i)AtistheprobabilitythatMT0isassociatedtothe

t–thtier;andii)RtistheaveragerateofMT0conditionedonitsassociationtothet–thtier.ForthetierassociationpolicyintroducedinSectionII-B,Atisavailablein[43,Lemma1].Ontheotherhand,Rtisde nedasfollows[22,Sec.IV],[43,Eq.(46)]:

+∞

Rt=Rt(ξ)fdt,0(ξ)dξ(3)

where:i)dt,0isthedistanceofMT0fromitsservingBSby

conditioningonMT0beingtaggedtothet–thtier;ii)fdisthePDFoftherandomdistancedt,0(·)t,0,whichisgivenin[43,Lemma3]:

f2πλ dt

2 t,0(ξ)=Aξexp π TλPqBq αqξ2ααqttqq=1

PtBt

(4)

andiii)Rt(ξ)istheaveragerateofMT0conditionedonthisterminalbeingtaggedtothet–thtierandondt,0beingequaltodt,0=ξ.From[22,Sec.IV]and[43,Eq.(14)],Rt(·)can

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3053

t=2πλtR R=

+∞

ξexp π

T q=1

λq

PqBq

PtBt

2 α

q

ξ

2αt

αq

Ptgt,0ξ αt

Eln1+2dξ

σN+Iagg(ξ)

(6)

GI(y)

dyy0

+∞

α1αα y GI(y)= ξ2 1exp{ πλZI(SNRy)ξ}exp yξ2dξ

[1 M0(SNRy)]

+∞

(8)

ZI(SNRy)2ZI(SNRy)0

bewritten

asfollows:

Rt(ξ)=E{ln(1+SINRt(ξ))}

=Eln1+Ptgt,0ξ αt σ2

(5)

TN+I agg(ξ)

Iagg(ξ)=

Pq=1qgq,bd αqb∈Φq,bq{\BSt,0(ξ)}

where:i)σ2

Nisthenoisepower;ii)BSt,0(ξ)istheservingBSatdistancedt,0=ξandgt,0istheBSt,0–to–MT0powerchannelgain;andiii)Iagg(ξ)istheaggregateinterferenceconditionedondt,0=ξ,whichisgeneratedbyallBSsexcepttheservingBS1BSt,0.From(2)–(5),thetoR= Tmaint=1R twithR averageratereduces

tgivenin(6)atthetopofthispage.Theobjectiveofthenextsectionsistointroduce

anewMGF–basedapproachtoef cientlycomputeR

tin(6)forarbitraryfadingchannels.Themaincontributionistoavoidthecomputationalcomplexityofthestate–of–the–artPcov–basedapproach[20],[22],[43].Tothisend,weintroducethesimpli ednotationasfollows,whichoriginatesfromtheassumptionofidenticallydistributedfadingineachtier:i)ft,0(·)andMt,0(·)arePDFandMGFofgt,0in(6),respectively;ii)ft,I(·)andMt,I(·)arePDFandMGFofgt,bin(5),respectively;iii)MIagg(·;ξ)istheMGFofIagg(ξ)in(5);andiv)I Mq,Iagg(·;ξ)q,agg(ξ)= istheMGFof

α

b∈Φq{\BSt,0(ξ)}Pqgq,bdqq,b,i.e.,theper–tieraggregateMGFin(5).

Finally,wementionthattheaverageratein(6)providesanestimateofthemeandatarateoveracellthatisachievablebyatypicalmobileterminal[22,Sec.IV].ThisinterpretationimmediatelyfollowsfromthevalidationprocedureofthePPP–basedabstractionmodelagainstconventionalgrid–basedabstractionmodels,asdiscussedindetailin[22,Sec.V–A].

III.SINGLE–TIERCELLULARNETWORKS

TobetterintroducetheproposedMGF–basedanalyticalmethodologytocomputetheaverageratein(6),westartbyconsideringthesingle–tierreferencescenariowithT=1.Inthiscase,(6)simpli esasfollows:R= 2πλ

+∞

×ξexp πλξ2

Eln1+

Pg0ξ α

σ2N+Iagg(ξ)

dξ(7)

1Throughout

thispaper,theservingBSisdenotedbyBSinequations,andbyBSt,0(ξ)whenused

t,0whenusedinthetext.

withIagg(ξ)= inb∈(7)Φ{\theBSsubscriptξ)}Pgbd α

0(.SinceT=1,easeofnotation,tb

forthatdenotesthetierisdropped.Likewise,thesubscripttisdroppedinf0(·),M0(·),fI(·),andMI(·)aswell.

ByusingtheMGF–basedapproach,anintegralclosed–formexpressionof(7)isgiveninTheoremTheorem1:LetSNR=P σ2

1.(SNR),thentheaveragerate,NbetheSignal–to–Noise–RatioR,ofasingle–tiercellularnetworkovergeneralizedfadingchannelsisgivenin(8)atthetop

ofthispage,where: ZI(y)=M I(y)+2 T+ ∞I(y) TI(y)=Γ1 yk+1M(k) I(y)Γ2 2

1M(k)

α

k=0α+kI(y)=Egkb+1exp{ yg b}

(9)

Proof:SeeAppendixA. Theframeworkin(8)and(9)iscalledMGF–basedap-proachbecauseRcanbedirectlycomputedfromofusefulandinterferencelinks.Infact,M(k)

theMGFs

obtainedM(k)fromthe(kk+1

+1)–thderivativeofI(·)canbeMI(·),i.e.,I(y)=( d/dy)

MI(y)[73,Eq.(1.1.2.9)].Inthesequel,weshowthatM(k)

I(·)canbeexplicitlycomputedinclosed–formformanyfadingchannelmodels.Furthermore,closed–formexpressionsofM0(·)andMI(·)areavailablein[75,Sec.2.2],[76,TablesII–IV],and[77,TablesII–V]paredwiththePcov–basedapproachin,e.g.,[20],[22],and[43],theframeworkin(8)reducesthenumberoffoldintegralstobecomputedfromfourtotwo.

Bycarefullylookingat(9),someimportantconclusionsaboutthesystembehaviorasafunctionoftheBSsdensity,λ,canbedrawn,assummarizedinRemark1.

Remark1:SincetheintegrandfunctionofGI(·)in(8)isalwaysgreaterthanzero,itfollowsthatRisamonotonicallyincreasingfunctionofλ.Furthermore,Risupper–boundedasfollows:

+∞

R≤lim∞R(λ)=R(λ∞)1 M0(z)dz

λ→+=

MI(z)+TI(z)z(10)

Theanalyticalderivationof(10)isavailableinAppendixB.From(10), weobservethat:i)R(λ∞)theSNR=Pσ2

isindependentof

thetransmit–powerN.Thus,forverydenseBSsdeploymentsincreasingdoesnothelpinincreasingtheaveragerate;andii)theexistenceofa niteupper–boundforincreasingλcon rmsthatthedeploymentofmanyBSsisnotsuf cienttoachieveveryhighdataratesbutmoreadvancedinterferencemanagementtechniquesseemtobeneeded.

3054IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

2(a)1 π(πλ)2ZI(SNRy)(πλ)ZI(SNRy) √GI(y)|α=4=4yexperfc 4y2y 3(SNRy)(b)1(πλ)3ZI(πλ)ZI(SNRy)

S0,1/32GI(y)|α=6=427y27y

να+1 √αNα2 (c)DΔ(α, ν) αDαNαNyNαy 1αD,αN GI(y)|α/2=α/α=Z(SNRGααDαN αN+αDαNαN,αDy) 2(πλ)να+1Zνα+2(SNRy)NDα(πλ)Z(SNRy)I 1Δ(αD,0)DII2(2π)

Intheremainderofthissection,weshowthatthetwo–foldintegralin(8)canoftenbereducedtoasingleintegral,sinceclosed–formexpressionsofGI(·)existformanypath–lossexponentsα.Also,weshowthatthein niteseriesin(9)canbecalculatedforcommonfadingdistributionsoftheinterferencechannels.

LetusconsiderthecomputationofGI(·)asafunctionofthepath–lossexponentα.ThemainresultissummarizedinCorollary1.

Corollary1:Letα=4,α=6,andα/2=αN/αDwithαNandαDbeingtwopositiveintegernumbers,thenGI(·)in(8)hasclosed–formexpressionshownin(11)atthetopofthispage,whereνα=α/2 1.

Proof:Equation(11)followsfromsomenotableintegrals.Morespeci cally:(a)from[73,Eq.(2.2.1.8)];(b)from[73,Eq.(2.2.1.13),Eq.(2.2.1.14)];(c)from[73,Eq.(2.2.1.22)].Thisconcludestheproof. Sincethecasestudyα/2=αN/αDencompassesmanyscenariosofpracticalinterest,whenreferringtoCorollary1,wewillimplicitlyassumetheclosed–formexpressionofGI(·)usingtheMeijerG–function.

Thesingle–integralexpressioninTheorem1canbeef -cientlycomputedbyusingtheGauss–Chebyshevquadraturerule,assummarizedinRemark2asfollows.

Remark2:ByusingGauss–Chebyshevintegration,RinTheorem1canbecomputedas[69,Eq.(25.4.39)]:

N GCQ

R≈

wn

[1 M0(SNRsn)]GI(sn)(12)

n=1

snwherewnandsnforn=1,2,...,NGCQareweightsand

abscissas,respectively,ofthequadraturerule[74,Eq.(22)andEq. (23)]:

π2sin 2n 1

w=2NGCQπn 4N GCQcos2π 4cos22Nn 1πGCQπ+4(13)

sπ2n n=tan

4cos 12Nπ+πGCQ4

putationofTI(·)in(9)forGeneralFadingChannelsTheorem1andCorollary1needthecomputationofTI(·),

whichdependsonthefadingdistributionoftheinterferencechannels.AsmentionedinTheorem1,TI(·)can,inprinciple,becomputedfromthederivativesofMI(·).However,closed–formexpressionscanbeobtainedformanyfadingchannelmodelsbyalsoavoidingthecomputationofthein niteseries

(11)

in(9).SomekeycasestudiesareanalyzedinPropositions1–4

forNakagami–m,Log–Normal,compositeNakagami–mandLog–Normal,andcompositeRiceandLog–Normalfading,respectively.

Proposition1:LettheinterferencelinksexperienceNakagami–mfading.Accordingly,gbfollowsaGammadistributionwithparameters(m,Ω),whichwedenoteasgb～Gamma(m,Ω)[75,Sec.2.2.1.4].Then,TI(·)in(9)hasclosed–formexpressionasfollows:

T=m

m m 2

1 m I(y)1 yy+ (m+1) ΩαΩ

(14)×2

m 1 2F1m+1,1,2 α,yy+

ΩProof:SeeAppendixC.

Proposition2:LettheinterferencelinksexperienceLog–Normalfading.Accordingly,gbfollowsbutionwith～LogN parameters(indB) μ,σ2 aLog–Normaldistri-,whichwedenoteasgbμ,σ2

[75,Sec.2.2.2].Then,TI(·)in(9)hasclosed–formexpressionasfollows:

T2 11N GHQ

√I(y)≈1 y√w n10(2σ s

n+μ)/10×exp απ√

n=1 10(2σs n+μ)/10y

(15)×2√1F11,2 2σs n

+μ)/10α,10(

ywherew nands nforn=1,2,...,NGHQareweightsandabscissas,respectively,oftheGauss–Hermitequadraturerule[69,Eq.(25.4.46)].

Proof:SeeAppendixD. Proposition3:Lettheinterferencelinksexperiencecom-positeNakagami–mandLog–Normalfading.Accordingly,gbfollowsaGammadistributionbyconditioningonitsmeanpower,which,inturn,followsaLog–Normaldistribution.denotethisdistributionasg

We b～Gamma/LogNm,μ,σ2[75,Sec.2.2.3.1].Then,TI(·)in(9)hasclosed–formexpres-sionasfollows:

T(y)≈mm+11 2

1

1Iαy√π

NGHQ

×

w nω mn(y+mω n)

(m+1)

(16)

n=1

×2 2

2F1m+1,1,α

,y(y+mω

n) 1withω n=10

(√

2σs n+μ)/10

.

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3055

1NGHQ

21 2(n) y√w nω n[y+(1+K)ω n]TI(y) TI(y)≈(1+K)exp{ K}1 απn=1

l+∞ K(1+K)l+12(n) 1 ,y[y+(1+K)ω n] 2F1l+2,1,2 TI(y)=(17)

l=0

(l!)1+K+(y/ ωn)Proof:SeeAppendixE.

Proposition4:Lettheinterferencelinksexperiencecom-positeRiceandLog–Normalfading.Accordingly,gbfol-lowsanon–centralChi–Squaredistributionbycondition-ingonitsmeanpower,which,inturn,followsaLog–Normaldistribution.2thisdistributionasgb～/LogN Wedenote

ChiSquareK,μ,σ [78,Eq.(6)],withKbeingtheRicefactor.IfK=0,TI(·)in(9)hasclosed–formexpressiongivenin(17)atthetopofthispage.IfK=0,thecompositeRiceandLog–NormalfadingreducestothecompositeNakagami–mandLog–Normalfadingwithm=1(Suzukidistribution[75,Sec.2.2.3.2])andProposition3canbeused.

Proof:SeeAppendixF. From(17),weobservethat,unliketheotherfadingdistribu-tionsinPropositions1–3,forcompositeRiceandLog–Normalfadingwestillneedtocalculateanin niteseriestocomputeTI(·)in(9).ThecomputationoftheseriescanbeavoidedassuggestedinRemark3asfollows.

Remark3:UsingthemappingbetweenthemparameterofaNakagami–mdistributionandtheKfactorofaRicedistribution[75,Eq.withm=(1+K)2

(2.26)],(17)canbeapproximatedby(16)

(1+2K).

Finally,wewouldliketoemphasizethatthefadingdis-tributionsstudiedinPropositions1–4arejustsomeselectedexamples,whichhavebeenchosenbecausetheyareoftenusedintheoreticalanalysis.However,ouranalyticalmethodologytocomputeTI(·)in(9)isapplicabletoarbitraryfadingdistributionsasdescribedinRemark4.Remark4:From(9),weobserveTI(·computation +∞ofM(k) that)needsI(y)=Egkb

+1

exp{ yg theb}=xk+1

exp{ yx}fgb(x)dx.WiththeexceptionoftheLog–Normal0

distribution,whichisstudiedinProposition2,from[75,Sec.2.2],[76,TablesII–IV],and[77,TablesII–V]wenotethattwogeneralsituationscanarise:

1)f (x;y)=exp{ yx}fg(x)=Aexp{ B(y)x},whereAisaconstantb

Inotherwords,f

andB(·)isafunctionofy.

(·;·)isstillanexponentialfunctioninx.Inthiscase,closed–formexpressionsofTI(·)canbeobtainedbyusingthesamedevelopmentasinProposition1forNakagami– 2)f)=CxυGm,n mfading.(a)

gb(xp,qDx p(bυareconstants.Inotherwords,q),whereC,D,and

thedistributionofthepowerchannelgaingbcanbecastintermsofaG–function.Accordingly,M(k)

Meijer

closed–formasanotherMeijerI(·)canbecomputedinG–functionbyusingtheMellin–Barnestheoremandthenotableintegralin[68,Eq.(2.24.3.1)].Ingeneral,inthiscaseitisnotpossibletoavoidthecomputationofthein niteseriesin(9).

α

B.Ef cientComputationoftheMeijerG–Functionin(11)Thecomputationoftheaverageratein(7)byusingTheorem1andCorollary1needs,ingeneral,thecalculationoftheMeijerG–functionin(11).Thisspecialfunctioniscommonlyusedinwirelesscommunicationstheory,e.g.,[76],[77],[79]–[81],anditisavailableinseveralstandardmathematicalsoftwarepackages.Thus,ingeneral,itscomputationcanbeperformedveryef ciently.However,in(8)theMeijerG–functionmustbecalculatedforallpositiverealvalues,anditisknownthatthenumericalcomplexityandthenumericalaccuracyofcommonalgorithmstocomputetheMeijerG–functionincreasesanddecreases,respectively,forsmallvaluesofitsargument,i.e.,fory→0in(11),see,e.g.,[82]and[83].Inordertoprovideaframeworkthatisgeneralandaccuratebutalsosimpleandstabletocompute,Corollary2providesanumericallyef cientandstablesolutiontocomputeGI(·)in(11),whichexploitsanasymptoticexpansionoftheMeijerG–functionforlargevaluesofitsargument.

Corollary2:Letα/2=αN/αDwithαNandαDbeingtwopositiveintegernumbers,thenGI(·)in(8)canbeef -cientlycomputedasshownin(18)atthetopofthenextpage,whereεisasmallpositiveconstant.Proof:SeeAppendixG. TherationalebehindCorollary2istoavoidthecalculationoftheMeijerG–functionforsmallvaluesofitsargument,andtoreplacetheMeijerG–functionwithanaccurate,simpletocompute,andnumericallystableexpansionformula.Inotherwords,GI(·)iscomputedbyusingU(·),i.e.,theexactformulainCorollary2,aslongastheargumentoftheMeijerG–functionisnosmallerthanε.Ontheotherhand,whenthisoccurstheasymptoticalexpansionU(asymptote)(·)isused,whichissimpleandfasttobecomputed.This“adaptive”approachallowsustokeepthedesiredaccuracywithoutincreasingthenumericalcomplexityandwithoutincurringinnumericalinstabilities.Thekeyparameterfortheef cientcomputationofGI(·)in(18)isε,whichdependsonthemathematicalsoftwarepackagebeingusedtocomputetheMeijerG–function.Inpractice,εisthesmallestvalueoftheargumentoftheMeijerG–functionforwhichitcanbeef cientlycomputed.Ifε=0,Corollary2reducestoCorollary1.

Finally,weclosethissectionwithRemark5andRemark6.

Remark5:From(18),wenotethataverycomputation-allyef cientframework,whichisaccurateforsuf cientlysmallvaluesofε,canbeobtainedbysimplyneglectingU(asymptote)(·),asshownin(19)atthetopofthenextpage .

Remark6:TheintegralinGI(·)belongstotheso–called“Weibull–type”integrals,sinceitcoincideswiththeintegraltobecomputedtoobtaintheMGFoftheWeibulldistribution

3056IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

να+1√α2ααNNyαD yDαN1α GI(y)≈ZI(SNRy) 2(πλ)να+1Zνα+2(SNRy)ΥHααD(πλ)αNZαN(SNRy) αN+αD 1DI I2(2π) (asymptote) Υ(z)=U(z)H (z)[1 H(z ε)] (z ε)+U H Δ(αN, να)D,αN U(z)=GαzαN,αD Δ(αD,0) ααα DDN (asymptote)Δq(αD,0) U(z)=limU(z)=Γ(Δr(αD,0) Δq(αD,0))Γ(1+Δq(αD,0) Δp(αN, να))z z→0+q=1 r=1p=1

r=q

(18)

D

ααD

αy1D,αN

GI(y)≈GααN+αDαN,αDν+1ν+2αα 1ZI(SNRy)2(πλ)ZI(SNRy)(2π)2

NαD

ααNy×H εαNαND

αα(πλ)Z(SNRy)DI

√να+1

αDαN2

NαD ααyΔ(α, ν)NαN αNαN

(πλ)ZI(SNRy) Δ(αD,0)

(19)

[84,Eq.(2)]and[85,Eq.(2)].SimilartoCorollary1,itcanbecomputedintermsofMeijerG–function[84]and[86]orintermsofgeneralizedhypergeometricfunction[85].Fur-thermore,variousclosed–formapproximationsareavailableintheliterature,suchas[87]–[90]andreferencestherein.TheinterestedreadercanconsultthesepapersandthereferencesthereintoidentifyalternativewaysofcomputingGI(·)thatavoidspecialfunctions,suchastheMeijerG–function.Ontheotherhand,tothebestoftheauthorsknowledge,theapproachproposedinCorollary2isnotavailableintheliterature. .C.Interference–LimitedScenario

Inmanypracticalsituationsofinterest,thebackgroundnoiseisoftennegligiblecomparedtotheaggregateinterfer-ence[22]and[43].Inthiscase,Theorem1simpli esasshowninCorollary3.

2

=0,thentheaveragerate,R,in(7)Corollary3:LetσN

simpli esasfollows:

+∞

1 M0(z)dz(SNR∞)

(20)R|σ2=0=R=

NMI(z)+TI(z)z0Proof:Byusing ofvariableSNRy=zin(8), thechange 1

itfollowsthatGISNRz=1/ZI(z)sinceSNR→∞if2

=0.Thisconcludestheproof. σN

From(20),interestingconsiderationsabouttheaverageratecanbemade,assummarizedinRemark7.

Remark7:Similarto[22,Eq.(8)],(20)con rmsthatforinterference–limitedcellularnetworkstheaveragerateisindependentofthedensityofBSsaswellasofthetransmit–power.Thus,increasingeithertheBSsdensityorthetransmit–powerarenoteffectivesolutionstoincreasetheaveragerate.Moreadvancedinterferencemanagementmechanismsareneeded.Furthermore,ourframeworkshowsthatthesetwotrendsholdregardlessofthefadingchannelmodel,andthattheyseemtobemainlyrelatedtothePPPspatialmodeloftheBSs.Finally,bycomparing(20)with(10)weobservethatR≤R(λ∞)=R(SNR∞).ThisimpliesthattheaveragerateofacellularsystemwithunboundedBSsdensityand nite

transmit–poweristhesameastheaveragerateofacellularsystemwithunboundedtransmit–powerand niteBSsdensity.

D.High–SNRScenario

InCorollary3,wehavestudiedtheaveragerateintheabsenceofbackgroundnoise.InCorollary4,westudythescenariowithsmallbutnon–zeronoise,i.e.,thehigh–SNRsetup.

Corollary4:AsafunctionoftheSNR,theaveragerate,R,in(7)isupper–andlower–boundedasfollows:

α 1 α/2(SNR 1)(SNR∞)

=R (πλ)Γ1+R

2SNR +∞

1 M0(z)(21)×dz1+(α/2)

0[MI(z)+TI(z)]≤R(SNR)≤R(SNR∞)Proof:Equation(21)immediatelyfollows,withthesameanalyticalsteps,from(39)inAppendixB,andbyusingtheidentity(α/2)Γ(α/2)=Γ(1+α/2)in(40).Thisconcludestheproof. From(21),interestingconsiderationsabouttheaverageratecanbemade,assummarizedinRemark8.

Remark8:Bydirectinspectionof(21),weobservethat:i)thelower–bound,R(SNR 1),isthehigh–SNRapproximationoftheaverageratesincelimSNR→+∞R(SNR 1)=R(SNR∞);ii)theaveragerateincreaseswiththeSNRbyapproachingtheupper–boundR(SNR∞)withlinearconvergencerate;iii)theaveragerateincreaseswiththeBSsdensitybyapproachingtheupper–boundR(SNR∞)=R(λ∞)with(α/2)–ordercon-vergencerate;andiv)thelargerthepath–lossexponent,α,thefastertheconvergencespeedtoR(SNR∞)=R(λ∞)asafunctionofλ. E.FrequencyReuse

Inthissection,westudytheimpactoffrequencyreuseontheaveragerate.Inparticular,weconsideracellularnetwork

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3057

withFB≥1frequencybands.ThesetupwithFB=1correspondstotheuniversalfrequencyreusecasestudiedinTheorem1.Also,similarto[22],weassumethateachinterferingBSpicksatrandomoneoftheFBfrequencybandswhentransmitting.TheaveragerateisgiveninCorollary5.Corollary5:Theaveragerate,R(FB),ofasingle–tiercellularnetworkwithFB≥1availablefrequencybandsandrandomfrequencyreusecoincideswithRin(20),and(21)byreplacingZI(·)in(9)withZ(F(8),)

(10),

IB(z)=(FB 1)+ MI(z)+TI(z),andλ(whenavailable)withλ(FB)=λFB.Proof:TheprooffollowsbytakingintoaccountthatforFB≥1:i)theaverageratein(5)becomesR(FB)(ξ)=(1/FB)E{ln(1+SINR(ξ))};ii)onaverage,theinterferenceoriginatesfromaPPPwithBSsdensityequaltoλ(FB)=λ/FB;andiii)thetierandBSassociationPDFisindependentofFB[22,Sec.VI–A].Accordingly,theproofproceedsalongthesamelinesasTheorem1.Thisconcludestheproof. FromCorollary5,interestingconsiderationsabouttheav-erageratecanbemade,asgiveninRemark9.

Remark9:BydirectinspectionofR(FB)inCorollary5,itfollowsthattheaveragerateismaximizedforFB=1,i.e.,foruniversalfrequencyreuse.Also,fordenselydeployedBSs(10)andforinterference–limitedcellularnetworks(20),theaverageratelinearlydecreaseswiththenumberofavailablefrequencybandsFB.Thesetrendsareinagreementwith[22,Sec.VI–B]andholdforgeneralfadingchannelmodels. F.CorrelatedLog–NormalShadowing

TheMGF–basedapproachintroducedsofarcanbeappliedtoawidevarietyofchannelconditions,notablycompositefadingchannelsthataccountforLog–Normalshadowing(seeProposition3andProposition4).TheaveragerateinTheorem1isapplicable,however,onlytoi.i.d.fast–fadingandLog–Normalshadowing.Itiswell–known,ontheotherhand,thatshadowingcorrelationseverelyaffectstheperformanceofcellularnetworks[93]–[97].Inthissection,weprovideasimplemethodologytoextendtheframeworkinTheorem1toc.i.d.fadingchannels.Thereasonofrestrictingtheanal-ysistoequi–correlatedfadingoriginatesfromthestochasticgeometryapproachforother–cellinterferencemodelingusedinthepresentpaper,whichisapplicableonlytoidenticallydistributedfading.Themethodologyusedtoobtainequi–correlatedLog–NormalrandomvariablesexploitstheOwenandSteckmethodforthegenerationofequi–correlatedmul-tivariateNormaldistributions[98].

Asanillustrativeexample,letthegenericdownlinkchan-nelexperiencecompositeNakagami–mfast–fadingandLog–Normalshadowing,asdescribedinProposition3.Thepro-posedmethodologyisreadilyapplicabletootherfadingchannelmodelswithcorrelatedLog–Normalshadowing,aswellastomulti–tiercellularnetworksbyapplyingthesamemethodologytotheframeworkdiscussedinSectionIV.Morespeci cally,weassumei.i.d.fast–fadingandc.i.d.shadowing.Accordingly,card{Φ}channelpowergainsgbforb∈Φwithcorrelationcoef cientρandparameters(m,μ,σ2)canbeobtainedasfollows[98]:

Step1:Generatecard{Φ}equi–correlatedvariableswhereS

¯asXb=σ√ρS¯+σ√

Normalrandom

1 ρSb+μforb∈Φ,

andSbforb∈Φareasetofi.i.d.Normalrandomvariableswithzeromeanandunitvariance.Thecard{Φ}randomvariablesXbhavemeanμandvarianceσ2,forb∈Φ,regardlessofthecorrelationcoef cientρ.

Step2:Convertthesetofcard{Φ}equi–correlatedNor-malrandomvariablesintoasetofcard{Φ}equi–correlatedLog–NormalrandomvariablesasYb=10Xb/10forb∈Φ.

Step3:Generatecard{Φ}independentGammarandom

variablesgbwithfadingseveritymandmeanvalueYbforb∈Φ.

-positeNakagami–mandLog–Normalfading,itfollowsthatthecard{Φ}randomvariablesing upontherandomvariable¯gbare,bycondition-,i.i.d.withm,μ+σ√ρS,¯σ2

(1 ρ) Sparameters.Accordingly,weproposethefol-lowingapproachtocomputethedownlinkaveragerateofcellularnetworks:

Step1:TheframeworkinTheorem1isappliedditioningupontherandomvariableS¯bycon-andbysub-stitutingμ→μ+σ√ρS¯

andσ→σ√resultingaveragerateisdenotedbyR 1S

¯ ρ.The

.Step2:TheconditioninguponthestandardNormalrandom

variableS¯isremovedbyaveragingoveritsPDFf¯

√ S(x)=12πexp x2 2

.Informulas,thedownlinkaveragerateoverc.i.d.fadingchannelscanbecomputedasfollows:

+∞NR=R(x)f(a)¯(x)dx1

GHQ

w √ ∞S≈√

πηR2 sηη=1

(22)

where(a)isobtainedbyapplyingGauss–Hermitequadratures,andR(x)istheaveragerateinTheoremetersm(x)=m,μ(x)=μ+σ√

1withfadingρx,andσ(x)=σ√param-1 ρ.Insummary,therationalebehindtheproposedapproachtodealwithshadowingcorrelationconsistsin:i) rst,generatingasetofcorrelatedLog–Normalrandomvariablesthatareconditionallyindependentand,thus,applyingtheframeworkforindependentshadowing;andii)then,removingthecon-ditioningviaasinglenumericalintegration.Accordingly,shadowingcorrelationcanbetakenintoaccountwithonlyasingleextranumericalintegral,whichcanbeef cientlycomputedusingGauss–Hermitequadraturesasshownin(22).G.Pcov–vs.MGF–basedApproach:AComparisonInSectionI-C,wehavestatedthatbothPcov–andMGF–basedapproachescanbeappliedtogeneralfadingdistribu-tions.However,Pcov–andMGF–basedapproachesneed,ingeneral,thecomputationofafour–andatwo–foldnumericalintegral,respectively.Inbothcases,theintegralsmayinvolvethecomputationofspecialfunctions,which,however,areef cientlyimplementedincommerciallyavailablesoftwarepackages.Duetothereductionofthenumberoffoldintegralstobecomputed,theMGF–basedapproachisexpectedtobemorecomputationallyef cient.Theaimofthissectionis

3058IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

+∞ +∞

(a) 2¯ =RT1(r,t)drdtrexp πλr2πλ 00 +∞ t 1 (b) 12 α¯1(r,t)=¯2(r,s)dsM0 2πre 1Texp 2πσNjs(2πjs)js 1T

∞ +∞ (c) 2/α ¯2(r,s)=expπλr2 2πλα 1(2πjs) Tx2/αΓ 2/α,2πjsr αx Γ( 2/α)fI(x)dx

(23)

tobettercomparestrengths,weaknesses,andcomputational

complexityofthesetwoapproaches.

Tobetterconductthiscomparison,wesummarizein(23),shownatthetopofthispage,thefour–foldintegralexpressionoftheaverageratethatisobtainedfromthePcov–basedapproach.Morespeci cally,(23)isobtainedfrom[22]asfollows:(a)originatesfrom[22,AppendixC];(b)originatesfrom[22,AppendixB];and(c)originatesfrom[22,Theorem4,Eq.(4)].Forconsistencyandeaseofcomparison,thesamenotationasfortheMGF–basedapproachisused.

BycomparingtheMGF–basedapproachin(8)withthePcov–basedapproachin(23)thefollowingcommentscanbemade:

Bothapproachesmayneedthecomputationofsomespecialfunctions.Morespeci cally,theMGF–basedapproachinvolvesthecomputationofhypergeometricfunctionsinTI(·),andthePcov–basedapproachinvolvestheT

¯computationoftheincompleteGammafunctionin2(·,·). BothapproachesmayneedtouseGauss–Hermitequadra-turestocomputef0(·),M0(·),fI(·),andMI(·)forcompositechannelmodels.ThisneedoriginatesfromtheanalyticalintractabilityofLog–NormalshadowinganditisindependentofeitherthePcov–ortheMGF–basedapproachbeingused.

ByusingCorollary1,thetwo–foldintegralin(8)maybereducedtoasingle–integralforsomepath–lossex-ponents.Likewise,byusingtheMellin–Barnestheoremin[68,Eq.(2.24.2.1)]andtheMeijerG–functionrep-resentationoftheupper–incompleteGammafunction[68,Ea.(8.4.16.2)],aclosed–formexpressionofT

¯in

2(·,·)in(23)maybeobtained.Asaconsequence,theMGF–basedapproachreducesthenumberoffoldintegralstobecomputedandavoidsthecomputationofcomplexintegrals.

Ininterference–limitedscenarios,theMGF–basedap-proachinCorollary3offersasigni cantreductionofthecomputationalcomplexityandtheaverageratecanbecalculatedfromthesimplesingleintegralin(20).Ontheotherhand,thecomputationalcomplexityofthePcov–basedapproachisnotsigni cantlyaffectedinthisexp scenario.In 2πσ2 fact,theNjs=1inT

¯onlysimpli cationin(23)is1(·,·),whichdoesnotleadtofurtherreductionofthenumberoffoldintegralstobecomputed.

ThedesiredformoftheaveragerateofferedbytheMGF–basedapproachleadstosimpleandintuitiveun-derstandingoftheperformanceofcellularnetworksforavarietyofspecialoperatingscenarios,suchasdensecellularnetworksdeployments(Remark1),interference–

dominatedenvironments(Remark7andRemark8),frequencyreusestrategies(Remark9).Ontheotherhand,littleinsightcanbegainedfrom(23)forgeneralfadingdistributions.However,(23)canbesigni cantlysimpli edforRayleighfadingchannelsandinterestingdesignguidelinescanbeinferredfromit[22].

Theconsiderationsaboveoriginatefromthedirectinspec-tionof(8)and(23),andprovideaqualitativecomparisonofthereductionofcomputationalcomplexitythatcanbeexpectedbyusingtheMGF–insteadofthePcov–basedapproach.TobetterunderstandtheadvantagesoftheMGF–basedapproach,wehavealsoconductedsomenumericaltestswiththegoalofprovidingamorequantitativeassessmentofthecomputationalcomplexity.Theconventionalapproachthatisoftenusedtoconductthesetestsittoconsideracasestudyforwhich,withfurtheranalyticalmanipulations,theintegralexpressionsin(8)and(23)canbesimpli edorevencomputedinclosed–form,andtocomparetheiraccuracyandcomputa-tionaltimewithoutapplyinganymathematicalmanipulations.Followingthislineofthought,wehaveconsideredRayleighfadingasabenchmarkandhaveimplementedinMathematicatheformulasin(8)and(23)astheyappearinthepresentpaper.Infact,simpleclosed–formexpressionsforRayleighfadingareavailablein[22].AsfarastheMGF–basedapproachisconcerned,theouterintegralin(8)iscomputedusing(12)withNGCQ=2000.ThehighvalueofNGCQischosenasaworstcasesetupfortheMGF–basedapproach.Variouscom-binationsofpath–lossexponents,={2.05,2.2,2.5,anddensitiesofBSs,λ= 10 α6,10 4,10 2,10

1

3,4,5},,havebeenconsidered.Thechosenpath–lossexponentscovertypicalpropagationenvironmentsforcellularapplications[99,Table2.2],[100,Ch.2,Sec.5],andthechosendensitiesofBSscoversparse,normal,anddensecellulardeployments[22],[42].Thetestshavebeenexecutedinalaptopcomputer.Inalltheanalyzedscenarios,theMGF–basedapproachin(8)hasbeenabletoprovideaccurateestimatesoftheaveragerateinlessthan ve/sixsecondsforeachSNRpointtobecomputed.Ontheotherhand,thePcov–basedapproachin(23)hasnotbeenabletoprovideanynumericalestimatesafter veminutesi.e.,σ2

ofcomputation.Ininterference–limitedscenarios,

=0,thecomputationalbasedN

complexityoftheMGF–approachisfurtherreduced,whilethecomputationalcomplexityofthePcov–basedapproachisnotaffected.Theseoutcomescon rmtheadvantagesoftheproposedMGF–basedapproachforanalysisanddesignofcellularnetworks.IV.MULTI–TIERCELLULARNETWORKS

Inthissection,weextendtheanalyticalframeworktogenericmulti–tiercellularnetworks.Theanalyticaldevelop-mentis,inmanyways,similartoSectionIII.Thus,onlythe

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3059

+∞

(t)(y) G I t=2πλt Rdy[1 M(SNRy)]t,0t y0

+∞T αt (t)(y)= (t,q)(y)ξ2αqexp{ yξαt}dξ GZξexp πλ qII

q=1

(24)

t= R

(t)(y)G

dy[1 Mt,0(SNRty)]Iy0

+∞ αα αy1(t)(t) 1 22 GZ (y)=ξexp πλ(y)ξexp yξdξtI I(t)(t)+∞

(27)

Z I(y)2Z I(y)0

mostimportantanalyticaldetailsarereportedinwhatfollows.

Thedepartingpointis(6)andthemainresultissummarizedinTheorem2.

Theorem2: tfort=1,2,...,Tin(6).LetSNRt=P Lettσ2

RNbetheexpressionofR

SNRofthet–thtier,anexplicitclosed–form

tforarbitraryfadingchannelsisgivenin(24)atthetopofthispage,where:

Z (t,q) 2/αqPqBq

1

I

(y)=PqBqPtBMq,ISNRqy t 2/αq P tB t

+PqBqPqB1

PBtTq,IqPSNRtBt

qy

t(25)

T(y)=Γ1

2

q,Iαq

+× ∞yk+1M(k)

2 1(26)q,I(y)Γ2 k=0

α+k

q

andM(k)E g

q,I(s)=kq,b+1

exp{ sgq,b}.

Proof:TheprooffollowsbyusingthesamestepsasinAppendixAandbytakingintoaccountthat,thankstothespatialandchannelindependenceofthePPPs,theMGFoftheaggregate Tinterferenceforthegenerict–thtierisMIagg(s;ξ)=q=1Mq,Iagg(s;ξq)whereMq,Iagg(·;ξq)isgivenin(36)andcanbecomputedinclosed–formfromλq,

Mq,I(·),Tq,I(·),andξq=(Pq/Pt)1/αq(Bq/Bt)1/α

qξαt/αq[43,Eq.(42)].Thisconcludestheproof. Theorem2providesaverygeneralexpressionfortheaveragerateofmulti–tiercellularnetworksthat,ingeneral,needsthecomputationofatwo–foldnumericalintegralbutisapplicabletotiershavingdifferentpath–lossexponentsandfadingdistributions.Furthermore,closed–formexpressionsforTq,I(·)in(26)canbeobtainedfromPropositions1–4,similartothesingle–tiercase.TheextensiontocorrelatedLog–NormalshadowingfollowsimmediatelyfromSectionIII-F.Eventhoughgeneral,Theorem2providesaframeworkthatislessanalyticallytractablethanTheorem1andCorollary1.Asimplerandmoreinsightfulanalyticalframeworkcanbeobtainedbyassumingthatallthetiershavethesamepath–lossexponent,i.e.,αt=αfort=1,2,...,Twhilestillkeepingtheassumptionthattheper–tierfadingdistributionisdifferent.TherelatedframeworkisgiveninCorollary6.

Corollary6:Letαt=αfort=1,2,...,T,thenR

tin(6)canbeexplicitlycomputedasshownin(27)atthetopofthis page,where: Z (t)(y)= T λq (t,q) IZI(y)

q=1λt 2/α P qB 1 Z (t,q)I(y)=PqBq PtBtMq,Iq

PSNRtBtqy +PqBq 2/α 1 PTPq,IqBqSNRqyt

Bt

Pt

B

t

(28)

Proof:Theprooffollowsdirectlyfrom(24)andfromsomealgebraicmanipulationssimilartoAppendixA.Thisconcludestheproof. FromCorollary6,thefollowingimportantremarkcanbemade.

Remark10:BycomparingCorollary6andCorollary1,weobservethatthetwoformulashavethesamestructure.speci cally,R

More

tin(27)canbeobtainedfromRin(8)bysimplyreplacingZI(·)withZ (t)(·).Furthermore,Theorem1reducestoCorollary6,asexpected,I

forT=1. Remark10allowsustoeasilygeneralizemanyimportantresultsobtainedforthesingle–tiersetuptothemulti–tiercase.Inparticular,Corollary7generalizesCorollary5formulti–tiercellularnetworkswithrandomfrequencyreuse;Corollary8generalizesRemark1byinvestigatingtheimpactofdenseBSsdeployments;andCorollary9generalizesCorollary3andCorollary4bystudyinginterference–limitedmulti–tiercellularsystemsandhigh–SNRoperatingconditions.

Corollary7:Theaveragerateofthet–thtiercellularnetworkwithαt=αandwithF(tiert)

ofamulti–

frequencybandsandrandomfrequencyreuseB≥1availableforevery1,2,...,Tcanbeobtainedfrom(27)byreplacingZ tier(t,q)t=

(·)in(28)with:

I

2/Z (t,q)PqBqα (q) I(y)= PFtBt

B 1 +PqBq 2/αPqBq 1

P tBtMq,ISNRqy

PtBt

+PqBq 2/α PqB 1

PtBtTq,IqPSNRtBt

qy

(29)

3060IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

t≤R

=

λ→+∞

+∞0

(λ∞) t(λ)=RlimRt

T q=1

(t)κqFB(q)κtFB

PqPt

1 Mt,0(z)

2/α

BqPqBq 1Pq(q)

FB 1+Mq,ISNRqz+Tq,I

BPBP

t

t

t

Bq

tBt

1

SNRqz

dzz

(30)

(SNR∞) (SNR 1)≈RRtt

×

0+∞

π

λtFB

(t)

α/2

α 1

Γ1+

2SNRt

Bq2/α(q)

FB 1+Mq,I

B

t

T

q=1

(t)

κqFB(q)κtFB

PqPt

1 Mt,0(z)

PqBq

PtBt

1

SNRqz

+Tq,I

PqBqPtBt

1

1+(α/2)

SNRqz

dz

(31)

fort,q=1,2,...,T,andλtwithλt=λt/FBfort=1,2,...,T.

Proof:ItfollowsfromCorollary5andRemark10.Thisconcludestheproof. Corollary8:Letusconsideramulti–tiercellularnetwork

(t)

withαt=αandwithFB≥1availablefrequencybandsandrandomfrequencyreuseforeverytiert=1,2,...,T.Also,letλt=κtλandSNRt=χtSNRfort=1,2,...,T.Ifλ→+∞,thentheaveragerateinCorollary7isupper–boundedasshownin(30)atthetopofthispage.

Proof:TheprooffollowsimmediatelyfromRemark1andRemark10.Thisconcludestheproof. Corollary9:Letusconsideramulti–tiercellularnetwork

(t)

withαt=αandwithFB≥1availablefrequencybandsandrandomfrequencyreuseforeverytiert=1,2,...,T.Also,letλt=κtλandSNRt=χtSNRfort=1,2,...,T.

2IfσN=0(i.e.,SNR→+∞),whichimpliesthatthesystemisinterference–limited,thentheaveragerateinCorollary7is

(SNR∞)=R (λ∞),whereR (λ∞) t≤Rupper–boundedasRttt

2

isgivenin(30).Furthermore,ifσNissmallbutnon–zero,i.e.,SNR 1,thentheaveragerateinCorollary7canbeapproximatedasshownin(31)atthetopofthispage.

Proof:TheprooffollowsfromCorollary3,Corollary4,andRemark10.Thisconcludestheproof. V.NUMERICALANDSIMULATIONRESULTS

Inthissection,weshowsomenumericalexamplesinordertoverifytheaccuracyoftheproposedanalyticalmethodol-ogyagainstMonteCarlosimulations,aswellastoshowtheimpactofdifferentfadingparametersanddistributionsontheaveragerate.Forafaircomparisonamongdifferentfadingdistributions,themeansquarevalueofeachfadingdistributionisnormalizedandsetequaltoone.ThisimpliesΩ=1forRayleighandNakagami–mdistributions,and 2

μ= ln(10)σ20forLog–Normal,compositeNakagami–mandLog–Normal,andcompositeRiceandLog–Normaldistributions.Furthermore,usefulandinterferencelinksareassumedtohavethesamefadingdistribution.Theanalyticalframeworkisimplementedasdescribedinthecaptionsofeach gure.AsfarasthecompositeRiceandLog–Normalfadingmodelisconcerned,bothframeworksinProposition4andRemark3areimplemented.Wehaveveri edthatboth

FB

(t)

(t)

frameworksprovidethesameaccuracy.Thus,theapplicationofRemark3isrecommendedsinceitissimplertocompute.

a)MonteCarloSimulations:AsfarasMonteCarlosim-ulationsareconcerned,wehaveusedthefollowingmethod-ology[46,AppendixF].

Step1:A nitecircularareaof(normalized)radiusRA

aroundtheorigin,i.e.,wheretheprobemobileter-minalislocated,isconsidered.Theradiusischosensuf cientlylargetominimizetheerrorcommit-tedinsimulatingthein nitebi–dimensionalplane.Intheconsideredsetup,theradiusRAischo-2

≥100,whereλmin=sensuchthatλminRA

min{λ1,λ2,...,λT}.Forexample,RA=100ifλmin=10 1andλmin=10 2,RA=1000ifλmin=10 4,andRA=10000ifλmin=10 6.Step2:Foreachtier,thenumberofBSsisgenerated

followingaPoissondistributionwithdensityλtand

2

areaπRA.

Step3:TheBSsofeachtieraredistributedfollowinga

uniformdistributionoverthecircularregionofarea2.πRA

Step4:Independentchannelgainsaregeneratedforeach

BSofeverytier.

Step5:ThetierandBSassociationpolicydescribedin

SectionII-Bisapplied,andusefulandinterferencelinksareidenti ed.

Step6:GiventheassociatedtieranditstaggedBS,the

SINRiscomputedasshownin(5).

Step7:Therateofthegeneric MonteCarlotrialiscom-(t)

putedasRmc=(1/FB)ln(1+SINRt ),wheret isthetaggedtier.

Step8:Finally,theaveragerateiscomputedbyrepeating

Step1–Step7forNmctimesandeventuallycalculat- NmcingR=(1/Nmc)mc=1Rmc.Inoursimulations,wehaveconsideredNmc=106.

InSectionIII-G,wehavecomparedthecomputationalcomplexityofPcov–andMGF–basedapproaches,andwehaveshownthattheproposedanalyticalmethodologyturnsouttobemorecomputationalef cientforgeneralfadingdis-tributions.AsfarasthecomputationalcomplexitycomparisonwithMonteCarlosimulationsisconcerned,ourexperimentshaverevealedthateachsimulationcurveshowninthissectioncanbeobtainedinacomputationtimeoftheorderofafew

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Fig.1.Averagerateofasingle–tiercellularnetworkoverRayleighfading(FanalyticalB=1).framework,MarkersshowwhichMonteiscomputedCarlosimulations.byusingTheoremSolidlines1,Corollaryshowthe2withε=0.05,Proposition1with(m=1,Ω=1),andRemark2withN(2.8)].

GCQ=2000.Furthermore,M0(·)=MI(·)areobtainedfrom[75,Eq.Fig.2.Averagerateofasingle–tiercellularnetworkoverNakagami–mfading(FtheanalyticalB=framework,1)ingTheoremSolid1,linesCorollaryshow2withε=0.05,Proposition1with(m=2.5,Ω=1),andRemark2withN(2.22)].

GCQ=2000.Furthermore,M0(·)=MI(·)areobtainedfrom[75,Eq.minutes( vetotenminutesdependingonthesetup)byusingtheMGF–basedapproach.Ontheotherhand,thesamecurvecanbeobtainedintensofhours(tentosixtydependingonthesetup)ofcomputationtimebyusingMonteCarlosimulations.Inadditiontothelongersimulationtimeandtothemoreresourcesforthecomputation,itisimportanttomentionthatMonteCarlosimulationstendtobelessaccuratefor:i)sparsecellularnetworks;ii)lowpath–lossexponents;andiii)high–SNR.ThereasonisthatintheseoperatingscenariosRAandNmcmustbeincreasedinordertoaccountfortheinterferingBSsthatarefarfromtheprobemobileterminal,andwhich,inthesecases,cannolongerbeneglected.

b)FrameworkValidationforSingle–TierCellularNet-works:InFigs.1–5,theaveragerateofRayleigh,Nakagami–

Fig.3.Averagerateofasingle–tiercellularnetworkoverLog–Normalfading(FtheanalyticalB=framework,1)ingTheoremSolid1,Corollarylinesshow2withε=0.05,Proposition2with(σ=6dB,μ= ln(10)σ2NM·)=are5obtained,andRemark/20dB)andGHQfrom2[75,withEq.N(2.54)]GCQ=with2000N.Furthermore,M0(·)=I(GHQ=5.

Fig.4.Averagerateofasingle–tiercellularnetworkovercompositeNakagami–mandLog–Normalfading(Fsimulations.SolidlinesshowtheanalyticalB=framework,1).MarkerswhichshowisMontecomputedCarlobyusingTheorem1,Corollary2withε=0.05,Proposition3with(m=2.5,σN=6dB,μ= ln(10)σ2/20dB)andNGHQ=5(2.58)]with2000N.Furthermore,MM,andRemark2withGCQ=0(·)=I(·)areobtainedfrom[75,Eq.GHQ=5.

m,Log–Normal,compositeNakagami–mandLog–Normal,andcompositeRiceandLog–Normalfadingisshown,respec-tively,forasingle–tiercellularnetwork.Overall,weobserveaverygoodaccuracyoftheproposedMGF–basedapproach.Furthermore,weobserve,asexpected,thattheaveragerate:i)increaseswiththeBSsdensity;ii)dependsonthefadingdistribution;andiii)increaseswiththepath–lossexponent.InFigs.1–5,wehaveconsidereddensecellularnetworks(λ≥0.1)andlargepath–lossexponents(α≥4).ThereasonofthischoiceismainlyduetothelongtimeneededtoobtainedMonteCarlosimulationsforlessdensecellularnetworksandforsmallerpath–lossexponents.However,itisimportanttoverifytheaccuracyoftheproposedMGF–basedapproachformorepracticaldensitiesofBSsandforawider

3062Fig.5.Averagerateofasingle–tiercellularnetworkovercompositeRiceandLog–Normalfading(FlinesshowtheanalyticalB=framework,1)ingTheoremSolid1,Corollary2with2ε=0.05,Proposition4with(K=10,σ=6dB,μ2000= ln(10)σ/20dB)andNGHQ=5,andRemark2withNGCQ=Theapplication.InPropositionofRemark4,the3seriesprovidesin(17)theissametruncatedresultandtotheaccuracy, rst100butterms.withlesscomputationalcomplexity.Furthermore,Mfrom(46)withN0(·)=MI(·)areobtainedGHQ=5.

rangeofpath–lossexponents.Ingeneral,practicaldensitiesformacroBSsdeploymentsareoftheorderofλ≈10 6[22],[42],[43].Thus,totestnumericalaccuracyandstabilityoftheMGF–basedapproach,weconsiderinthesetλ= 10

6,10 4,10 2,10

1

densitiesofBSs,inordertostudysparse,medium,anddensedeployments.Asfarasthepath–lossexponentisconcerned,weconsidervaluesinthesetα={2.05,2.2,2.4,3,4,5}>2,whichcovertypicalprop-agationenvironmentsforcellularapplications[99,Table2.2],[100,Ch.2,Sec.5].TheresultsofthisstudyareshowninFigs.6–8.

InFigs.6and7,wecompareMonteCarlosimulationswiththeMGF–basedapproach.Inparticular,numericalresultsareobtainedbyusingboththe(exact)two–foldintegralinTheorem1andthe(approximated)single–integralinCorollary2inordertotestcomplexityandaccuracyofbothmethods.AsfarasMonteCarlosimulationsareconcerned,itisworthmentioningthatonlysomeSNRpoints(markers)areshowninthe gures.ThemissingSNRpointsarenotshownbecauseofthelongsimulationtimeandtheneedtoconsiderverylargesimulationareastogetaccurateestimatesoftheaveragerate.Thesimulationtimeincreases,ingeneral,formoresparsecellularnetworksandforsmallerpath–lossexponents.TheSNRpointsshowninthe guresarethoseforwhichaccurateestimatescanbeobtainedwiththesimulationsetupdescribedabove.Thenumericalexamplescon rmtheverygoodaccuracyandthenumericalstabilityoftheMGF–basedapproachforallcellularnetworksetups.BycomparingthecurvesobtainedusingTheorem1andCorollary2,wenoticethatthelatterisveryaccurateexceptinthetransition(corner)regionfromnoise–tointerference–limitedoperatingconditions,wheretheaverageratereachestheasymptotecalculatedinCorollary3(furthercommentsareavailablebelowwherethehigh–SNRscenarioisdiscussed).Thereason

IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

Fig.6.Averagerateofasingle–tiercellularnetworkovercompositeNakagami–mandLog–Normalfading(Fsimulations.SolidlinesshowtheanalyticalB=framework,1).MarkerswhichshowisMontecomputedCarlobyusingTheorem1,Corollary2withε=0.05,Proposition3with(m=2.5,σN=6dB,(2.58)]with2000μ=N. Furthermore,ln(10)σ2/M20dB(·))and=MNGHQ=5,andRemark2withGCQ=0I(·)arecomputingthetwo–fold5integral.TheblackinTheoremdashed1lineswithoutareobtainedobtainedfromusingtheby[75,MeijerdirectlyEq.GHQ=G–functioninCorollary2.

Fig.7.Averagerateofasingle–tiercellularnetworkovercompositeNakagami–mandLog–Normalfading(Fsimulations.SolidlinesshowtheanalyticalB=framework,1).MarkerswhichshowisMontecomputedCarlobyusingTheorem1,Corollary2withε=0.05,Proposition3with(m=2.5,σN=6dB,μ=(2.58)]withN. Furthermore,ln(10)σ2/M20dB·))and=MNGHQ=5,andRemark2withGCQ=20000(I(·)areobtainedfrom[75,putingtheGHQtwo–fold=5integral.TheblackinTheoremdashed1lineswithoutareobtainedusingthebyMeijerdirectlyG–functioninCorollary2.

ofthis(inpracticenegligible)numericalinaccuracyoriginatesfromthenon–smoothtransitionintroducedbytheadoptionoftheHeavisidefunctioninCorollary3.

InFig.8,wecomparetheaveragerateasafunctionofthepath–lossexponent.OnlynumericalresultsobtainedfromtheMGF–basedapproachareshowninthis gure.Thecurvesareobtainedbyusingthetwo–foldintegralinTheorem1.AsfarastheapplicationofCorollary2isconcerned,itsaccuracyforα≥2.4isshowninFigs.6and7.Forα<2.4,itislesspracticaltousetheMeijerG–functioninCorollary2since

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3063

Fig.8.Averagerateofasingle–tiercellularnetworkovercompositeNakagami–mandLog–Normalfading(FbyusingtheanalyticalframeworkandBMonte=1).CarloThecurvessimulationsareobtainedarenotshown.Morespeci cally,thecurvesareobtainedbycomputingthetwo–foldintegralinTheorem1,Proposition3with(m=2.5,σ=6dB,μ= Furthermore,ln(10)σ2/M20dB)andNGHQ=5,andRemark2withNGCQ=2000.N0(·)=MI(·)areobtainedfrom[75,Eq.(2.58)]withGHQ=5.

forsuchvaluesofαwewouldhaveαN 1andαD 1,and,thus,computationtimeandnumericalaccuracywouldhighlydependonthespeci cimplementationoftheMeijerG–function.Ontheotherhand,Theorem1providesveryaccurateestimatesinafewseconds(foreachSNRpoint),asdiscussedinSectionIII-G.Figure8clearlyshowsthatTheorem1providesreliablenumericalestimatesfortheconsideredsetofpath–lossexponents,includingα≈2.Asfarastheperformancetrendisconcerned,Fig.8showsaverydifferentbehaviorfordense(λ=10 2)andmedium/sparse(λ=10 4)cellularnetworks.Indensecellularnetworks,thehigherthepath–lossexponentthebettertheaveragerateregardlessoftheoperatingSNR.Ontheotherhand,inmedium/sparsecellularnetworkstwoSNRregionscanbeidenti ed:i)forlow–SNR(noise–limitedregime),thelowerthepath–lossexponentthehighertheaveragerate.Thisisduetothatfactthattheusefulsignalundergoesalowerattenuationandthattheaggregateinterferenceisnegligiblecomparedtotheadditivenoise;andii)forhigh–SNR(interference–limitedregime),thehigherthepath–lossexponentthehighertheaveragerate.ThisisduetothefactthattheadditivenoiseisnegligiblecomparedtotheaggregateinterferenceandthattheinterferingBSsundergoalargerattenuation,whichhasamorepronouncedeffectontheaverageratethanthelargerattenuationundergonebytheusefulsignal.Theseresultsareinagreementwithintuitionandcon rmtheusefulnessoftheproposedMGF–basedapproachforcellularnetworksanalysisanddesign.

ForeaseofcomparisonwithMonteCarlosimulations,inthefollowingonlylargepath–lossexponentsanddensecellularnetworksdeploymentsareconsidered.

c)ImpactofFadingModelandFadingParameters:InFigs.9–12,theaveragerateofNakagami–m,Log–Normal,compositeNakagami–mandLog–Normal,andcompositeRiceandLog–Normalfadingisshown,respectively,forasingle–tiercellularnetworkandfordifferentchoicesofthefading

Fig.9.Averagerateofasingle–tiercellularnetworkoverNakagami–mfading(FSolidlinesBshow=1,SNRtheanalytical=10dB).framework,MarkersshowwhichMonteiscomputedCarlosimulations.byusingTheorem1,Corollary2withε=0.05,Proposition1withΩ=1,andRemark2withNfrom[75,Eq.(2.22)].

GCQ=2000.Furthermore,M0(·)=MI(·)areobtainedFig.10.Averagerateofasingle–tiercellularnetworkoverLog–Normalfad-ing(FlinesshowB=the1,SNRanalytical=10framework,dB)ingTheoremSolid1,Corollary2withε=0.05,Proposition2withμ= ln(10)σ2andN/20dBMGHQ=·)5,areandobtainedRemarkfrom2with[75,Eq.NGCQ(2.54)]=with2000N.Furthermore,0(·)=MI(GHQ=5.

parameters.Alsointhiscase,theframeworkprovidesveryaccurateestimates.Morespeci cally,weobservethat:i)theaveragerateisslightlysensitivetomandKfadingparametersofNakagami–m,compositeNakagami–mandLog–Normal,andcompositeRiceandLog–Normaldistributions,aswellasthatitincreasesforlessseverefading(mincreases)andinthepresenceofastrongerline–of–sightcomponent(Kincreases);andii)theaverageratestronglydependsontheshadowingstandarddeviationσofLog–Normal,compositeNakagami–mandLog–Normal,andcompositeRiceandLog–Normaldistributions,aswellasthatitdecreasessigni cantlyformoresevereshadowing(σincreases).

d)CorrelatedLog–NormalShadowing:InFigs.13and14,numericalexamplesinthepresenceofshadowingcorre-

3064Fig.11.Averagerateofasingle–tiercellularnetworkovercompositeNakagami–mandLog–Normalfading(Fλalytical=0.25framework,).MarkerswhichshowMonteiscomputedCarlobysimulations.B=1usingTheoremSolid,SNR=10dB,and1,linesCorollaryshowthe2withan-εand=Remark0.05,Proposition2withN3withμ= ln(10)σ2/20dBandNGHQ=5,obtainedfrom[75,Eq.GCQ(2.58)]=with2000N.Furthermore,M0(·)=MI(·)areGHQ=5.

Fig.12.Averagerateofasingle–tiercellularnetworkovercompositeRiceandLog–Normalfading(FshowMonteCarlosimulations.B=Solid1,SNRlines=10showdB,theandanalyticalλ=0.25framework,).MarkerswhichiscomputedbyusingTheorem21,Corollary2withε=0.05,Proposition4withμ= ln(10)σ2withN/20dBandNGHQ=5,andRemarktothe rstGCQ100=terms.2000.TheInPropositionapplication4,oftheRemarkseries3inprovides(17)isthetruncatedsameresultandaccuracy,butwithlesscomputationalcomplexity.Furthermore,M0(·)=MI(·)areobtainedfrom(46)withNGHQ=5.

lationoveracompositeNakagami–mandLog–Normalfadingchannelareshown.WeobservethattheproposedapproachforcorrelatedLog–Normalshadowingisveryaccuratefordifferentchoicesofthecorrelationcoef cient.Furthermore,bycomparingFigs.13and14withFig.4,wenotethattheframeworkinSectionIII-Freducestotheindependentcaseforρ=0.Morespeci cally,the guresshowthefollowingperformancetrends:i)forhigh–SNR,theaveragerateisindependentofthedensityofBSsregardlessoftheshad-owingcorrelationcoef cient;ii)forlow–SNR,theaveragerateslightlydecreasesiftheshadowingcorrelationcoef cient

IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

Fig.13.Averagerateofasingle–tiercellularnetworkovercorrelatedcompositeNakagami–mandLog–Normalfading(FMonteCarlosimulations.SolidlinesshowtheanalyticalB=framework,1).Markerswhichshowiscomputedbyusing(22)withN0α.05,Proposition3with(m=GHQ2.5,=σ5=,Theorem6dB,μ1,=Corollary2withε=M=(4·))and=MN,andRemark2withN ln(10)σ2/20dB,GHQ=5GCQ=1000.Furthermore,0I(·)areobtainedfrom[75,Eq.(2.58)]withNGHQ=5.

Fig.14.Averagerateofasingle–tiercellularnetworkovercorrelatedcompositeNakagami–mandLog–Normalfading(FMonteCarlosimulations.SolidlinesshowtheanalyticalB=framework,1).Markerswhichshowiscomputedbyusing(22)withN0α.05M=,5Proposition)and=MN3with5,(andm=GHQRemark2.5,=σ5,Theorem1,Corollary2withε=[75,2=with6dBEq.N,μ= ln(10)σ2/20dB,GHQ=(2.58)]GCQ=with1000N.Furthermore,0(·)I(·)areobtainedfromGHQ=5.

increases;andiii)forhigh–SNR,theaveragerateincreasesiftheshadowingcorrelationcoef cientincreases.Eventhoughitmayseemcounterintuitivethattheaveragerateincreaseswithshadowingcorrelation,thisresultseemstooriginatefromtheBSsassociationpolicyadoptedinthepresentpaper(theprobemobileterminalisassociatedwiththeclosestBS).Finally,weemphasizethat(apparently)counterintuitivetrendsinthepresenceofLog–Normalshadowinghavebeenobservedinotherpapers,e.g.,[22]and[97]whereitisshownthatthecov-erageprobabilityincreasesandthattheblockingprobabilityisnotalwaysincreasingwiththeshadowingstandarddeviation,respectively.Thiscon rms,onceagain,theimportanceoftakingintoaccountcorrelatedLog–Normalshadowingfor

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3065

Fig.15.Averagerateofasingle–tiercellularnetworkwithfrequencyreuseovercompositeNakagami–mandLog–Normalfading(λ=0.25).MarkersshowMonteCarlosimulations.Solidlinesshowtheanalyticalframework,whichiscomputedbyusingTheorem1,Corollary2withε=0.05,Proposition3with(m=2.5,σ=6dB,μ= ln(10)σ2andN/20dB)M(·)GHQ=M=5,andRemark2withNGCQ=2000.Furthermore,0I(·)areobtainedfrom[75,Eq.(2.58)]withNGHQ=5.

Fig.16.Averagerateofatwo–tiercellularnetworkovercomposite

Nakagami–mandLog–Normalfading(FSNR=SNRB(t)

=1fort=lines1,2,showandtheanalyticalframework,1).MarkersshowMonteCarlosimulations.SolidwhichiscomputedbyusingCorollary6,Corollary2withε=0.05,Proposition3with(m=2.5,σ=6dB,μ= Furthermore,ln(10)σ2/M20dB)andNGHQ=5,andRemark2withNGCQ=2000.(2.58)]withNt,0(·)=5.

Mt,I(·)fort=1,2areobtainedfrom[75,Eq.GHQ=accurateperformanceprediction.

e)ImpactofFrequencyReuse:InFig.15,theimpactoffrequencyreuseontheaveragerateofasingle–tiercellularnetworkovercompositeNakagami–mandLog–Normalfadingisstudied.Ournumericalanalysiscon rmsthe ndingsinSectionIII-E,andthatfrequencyreuseisdetrimentalfortheaveragerate.Assuggestedin[22],frequencyreusesignif-icantlyimprovesthecoverageprobability.Theanalysisofthistrade–offisoutofthescopeofthepresentpaperbutiscurrentlybeinginvestigatedbytheauthors.

f)FrameworkValidationforMulti–TierCellularNet-works:InFig.16andFig.17,theaveragerateofatwo–

Fig.17.Averagerateofatwo–tiercellularnetworkovercompositeRiceLog–Normalfading(FB(t)

and

=1fort=1,2,andSNR=SNR1).MarkersshowMonteCarlosimulations.Solidlinesshowtheanalyticalframework,whichiscomputedbyusingCorollary6,Corollary2withε=0.05,Proposition4with(K=10,σ=6dB,μ= ln(10)σ2N/20dB)andseriesGHQin=(17)5,isandtruncatedRemarkto2thewith rstN100GCQterms.=2000Theapplication.InPropositionofRemark4,the3providesthesameresultandaccuracy,butwithlesscomputationalcomplexity.M5.

t,0(·)=Mt,I(·)fort=1,2areobtainedfrom(46)with=cellularnetworkovercompositeNakagami–mandLog–andcompositeRiceandLog–Normalfadingisana-respectively.Asanillustrativeexample,weconsiderthewheretheBSsofeverytiertransmitwithapowerisinverselyproportionaltotheirspatialdensity.Thisisreasonablechoiceif,e.g.,the rsttierisusedtomodelBSsandthesecondtierisusedtomodelfemtoBSs.observe,asexpected,thattheaverageratesigni cantlywhentheBSsdensityofthelowertierincreases.frameworkprovidesaverygoodaccuracy.Furthermore,noteanegligibledifferencebetweenthetwofadingmodelsthechosensetofparameters.

g)High–SNRScenario:Finally,weobservethat,inall gures,theaveragerateincreaseswiththeSNRtendingtowardsahorizontalasymptoteforhigh–SNR.Bydirectinspection,thereadercanverifythatthehorizontalasymptotecoincideswiththeaverageratethatcanbecomputedusingCorollary3andCorollary8withR (SNR∞)=R (λby

∞)inCorollary9.Inotherwords,theSNRtregionwheret

theaveragerateis atcorrespondstotheinterference–limitedoperatingregime.Asaconsequence,weconcludethatthesimpleformulasgiveninthesecorollariesarequiteaccurateforSNR≥SNR ,whereSNR isthecornerpointwheretheaverageratestartsapproachingthehorizontalasymptote.Forexample,SNR ≈20dBforα=4inFig.1.Ingeneral,SNR dependsonthedensityofBSs,thepath–lossexponent,andthefadingchannelmodel.ThereadercanidentifySNR fordifferentcellularsetupsbydirectinspectionofallthe guresshowninthepaper.Figure8showsmanycellularnetworksdeploymentsofinterest.

Byobservingtheasymptoticbehavioroftheaveragerateforhigh–SNR,aninterestingproblemtoinvestigateiswhetherSNR islowerorgreaterthanthetypicaloperatingSNRof

3066currentcellularnetworksdeployments.Thisquestionisinter-estingbecausemanypapersassume,foranalyticaltractability,thattheadditivenoiseisalwaysnegligibleandthat,asaconsequence,cellularnetworksareinterference–limited.Allthe guresshowninourpapercon rmthattheinterference–limitedassumptionisaccurateonlyiftheoperatingSNRisgreaterthanSNR .InordertoassesswhethertheoperatingconditionSNR≥SNR isusuallyveri ed,weinvestigate,asanexample,thesamesetupasin[43],whichholdsfortypicalcellularMorespeci cally,SNR=P σ2networks.N;ii)σ2

weconsider:i)N=W/L 104dBm=4·10 140;iii)W=kBT0B0=Wattisthenoisepower,wherekB=1.38·10 23Joule/KelvinistheBoltzmann’sconstant,T0=290Kelvinisthenoisetemperature,B0=10MHzisthereceiverbandwidth;andiv)L0= 38.5dB=1.41·10 4isthepath–lossatareferencehaveSNR=P σ2distanceofonemeter.Accord-ingly,weN=(L0/W)P=3.52·109P.ByassumingtypicaltransmitpowersequaltoPMacro=40Watt,PMicro=6.3Watt,PPico=0.13Watt,andPFemto=0.05Wattformacro,micro,pico,andfemtoBSs[42],[101],weobtainSNRMacro=111.50dB,SNRMicro=103.45dB,SNRPico=86.60dB,andSNRFemto=82.45dB,respec-tively. BycomparingthesetypicaloperatingSNRswiththeSNRshowninour gures,weconcludethattheconditionSNR≥SNR iswellsatis edformanysystemsetupsanalyzedinthepaper.Forexample,letusconsider thesetupsshowninFig.8.WeobservethatSNR≥SNRisalwaysveri edindense(λ=10 2)cellularnetworksforeverypath–lossexponents,andinmedium/sparse(λ=10 4)cellularnetworksforlowpath–lossexponents(α<4inthe gure).Ontheotherhand,forlargerpath–lossexponentstheconditionmaynotbeveri edforsometypesofBSs.Thiscon rmsthattypicalcellularnetworksdeploymentscanbeapproximated,inmostcases,tobeinterference–limited,andthatthesimpleframeworksinCorollary3andCorollary8canbeusedfor rst–orderperformanceanalysis,design,andoptimization.

VI.CONCLUSION

Inthispaper,wehaveintroducedacomprehensivemath-ematicalframeworkfortheanalysisoftheaveragerateof

multi–tiercellularnetworkswhoseBSsareassumedtoberandomlydistributedaccordingtoaPPPspatialdistribution.Theframeworkisapplicabletogeneralfadingchannelmodelswitharbitraryfadingparameters.Numericallyef cientandstablealgorithmstocomputesometranscendentalfunctions,suchastheMeijerG–function,havebeenproposed.Theframeworkneedsthecomputationofeithersingle–ortwo–foldintegralsforgeneralfadingdistributionsandarbitrarypath–lossexponents.Furthermore,shadowingcorrelationcanbetakenintoaccountwithanotherextranumericalinte-gral.Theframeworkcanhandlerandomfrequencyreuse,anditsimpli essigni cantlyforinterference–limitedcellularnetworksandforhigh–SNRsetups.ExtensiveMonteCarlosimulationshavecon rmedtheaccuracyoftheproposedanalyticalmethodologyforvariousfadingdistributionsandcellulardeployments.

IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

APPENDIXAPROOFOFTheorem1

By using [66,Lemma

1]withN=M=1,wehave:Eln1+

X

Y+1+∞

=

MY(z) MX,Y(z)

exp{ z}dz(32) 0z

(=a)+∞MY(z)[1 MX(z)]exp{ z}dz

0zwhere:i)XandYarearbitraryables;ii)M X,Y(z)=Ee z(X+non–negativeY)

randomvari-istheMGFofrandomvariableX+Y;andiii)(a)holdsifXFrom(32)withX=(Pg0ξ α) andσ2

Yareindependent. 2

,bytakingintoaccountthat,conditioningN,Y=Iagg(ξ)σanduponξ,XandN

Yareindependent,theexpectationin(7)canbere–writtenasfollows: Eln1+Pg0ξ α

σ2

N+Iagg(ξ)

+∞

=

exp{ z}

M(z;ξ) 1 M0 SNRξ αz Iaggdz0

z(33)Wenotethattheidentityin(33)avoidstheneedofcomputingPcovandmakesouranalyticaldevelopmenttotallydifferentfromcurrentpractice[20],[22],and[43].

From(33),itisapparentthataclosed–formexpressionofMIagg(·;ξ)isneeded.ThisistheMGFoftheaggregateinterference,whichisgeneratedbyalltheinterferersthatlieoutsideadiskofradiusξ.Inotherwords,duetothetierandBSassociationpolicy,MT0hasanexclusionzonearounditwherenointerferingBSsarelocatedandareallowedtotransmit.Thisiscalledexclusionregion[5].TheMGFofIagg(·)hasbeenstudiedin[91]foragenericannularregionwithradiiAandB.Inparticular,MIagg(·;ξ)canbeobtainedfrom[91,Eq.(6)]bylettingA→ξandB→+∞,asfollows:

M)=exp

Iagg(s;ξ πλξ2exp πλξ2M I×exp πλs2/α

Γ1 2E sξ α

g2/α α

b

×expπλs2/αEg2/α2

bΓ1 α

,sgbξ

α

(34)In[91],theexpectationoverthefadingdistributionoftheinterferencechannelsisnotcomputedinclosed–formandonlyboundsforRayleighfadingareprovided.Tothebestofourknowledge,thereisnoanalyticalframeworkthatprovidesanexactandclosed–formexpressionoftheMGFin(34)forgeneralfadingchannels.Inwhatfollows,weprovideageneralmethodologytothisend.Thisisacontributionofthispaper.Byusing[69,Eq.(6.5.3)],[69,Eq.(6.5.4)],and[69,Eq.(6.5.29)],wehave:

Γ(z,x)=Γ(z) γ(z,x)

+=Γ(z) Γ(z)xz

exp{ x}

∞xk

k=0

Γ(z+k+1)

(35)

DIRENZOetal.:AVERAGERATEOFDOWNLINKHETEROGENEOUSCELLULARNETWORKSOVERGENERALIZEDFADINGCHANNELS...3067

+∞ 1ααz 1¯22R(z;λ)=dξξξexp{ πλZI(z)ξ}exp SNR0SNR

+∞ α (a)α(b)α11 12Γ[πλZI(z)] 2≤ξexp{ πλZI(z)ξ}dξ=(38)

SNR0Bysubstituting(35)in(34),MIagg(·;ξ)simpli esM

2 exp πλξ2M to:Iagg(s;ξ)=expπλξIsξ×exp πλξ2T α

Isξ α(36)whereTI(·)isgivenin(9).Closed–formexpressionsofTI(·)areavailableinPropositions1–4.

Finally,bysubstituting(36)in(33),theaverageratein(7)simpli esto(8)byusingthechangeofvariabley=zξ αandbyapplyingtheintegrationbypartstotheintegralinξ.Thisconcludestheproof. APPENDIXB

PROOFOFRemark1

From(8),byusingthechangeofvariableSNRy=z,wehave:

+∞

R=

1 M0(z)dzα +∞1 M0(z0

ZI(z)z )20ZI(z)R¯

(z;λ)dz(37)

whereR¯(·;·)isgivenin(38)atthetopofthis

page, and(a)follows bytakingintoaccountthat0≤exp (z/SNR)ξα/2≤1foreverychoiceoftheparame-ters,while(b)followsfrom[73,Eq.(2.1.1.1)].

Asaconsequence,theaverageratein(37)isupper–andlower–bound asfollows:+∞1 M0(z)dz +∞

1 M0(z)dz0ZI(z)z R¯(λ)≤R≤0

ZI(z)(39)

zwhere:

R

¯(λ)=1αΓ α (πλ) α

2 SNR+∞

22×[1 M0(z)]Z (40)I(α2+1)(z)dz

Equation(10)thatlimλ→+∞R

¯immediatelyfollowsfrom(40)byobserving

(λ)=0foreveryα>2.Thisconcludestheproof.

APPENDIXC

PROOFOFProposition1

Byusing[75,Eq.(2.21)]and[73,Eq.(2.2.1.2)],M(k)

follows:I(·)in(9)canbecomputedasM(k)

1 m m m (m+k+1)I(s)=Γ(m)Ωs+Ω

Γ(m+k+1)(41)

Bysubstituting(41)inTI(·)in(9)andbyusingtheidentity[69,Eq.(15.1.1)]:+ ∞Γ(k+A)sx+k

+C)z+k

k=0Γ(k+B)(sx=sΓ(A) (42)(s+C)z

Γ(B)

2F1A,1,B,s

s+CSNR2

withA,B,C,x,andsbeingpositiveconstants,weeventually

obtain(14)withsomealgebraicmanipulationsandusingtheidentityΓ(z+1)=zΓ(z).Thisconcludestheproof.

APPENDIXD

PROOFOFProposition2

ByusingtheapproximateexpressionofthePDFofLog–Normal(1/√π) randomvariablesin[77,TableIV],NGHQ √n=1w

nδx 10(2σ sn+μ)/10

i.e.,f,M(k)gb(x)≈I(·)in(9)canbecomputedasfollows:

(k)(s)≈√1N GHQ

MI

(k+1)(√2σs n+μ)/10

πw n10

n=1√

(43)

×exp 10(2σs n+μ)/10s

Bysubstituting(43)inTI(·)in(9)andbyusingtheidentity

[69,Eq.(13.1.2)]:

+ ∞sx+k

Γ(k+B)=sxΓ(B)

1F1(1,B,s)

(44)

k=0

withBandzbeingpositiveconstants,weobtain(15)after

somealgebra.Thisconcludestheproof.

APPENDIXE

PROOFOFProposition3

SimilartoAppendixD,inordertohaveananalyti-callytractableexpressionofPDFandMGFofcompositeNakagami–mandLog–Normalfading,weusetheapproxi-mationoftheLog–NormaldistributionthatisbasedontheGauss–Hermitequadraturein[75,Eq.(2.58)]and[76,V].Accordingly,M(k)

Table

I(·)in(9)canbecomputedasfollows:

M(k)(s)≈√

1mm

I

πΓ(m)

N

GHQ

×

w nω mn(s+mω n) (m+k+1)Γ(m+k+1)

n=1

(45)

Bysubstituting(45)inTI(·)in(9),from(42)weeventually

get(16).Thisconcludestheproof.

APPENDIXFPROOFOFProposition4SimilartoAppendixD,inordertohaveananalyticallytractableexpressionofPDFandMGFofcompositeRice

andLog–Normalfading,weusetheapproximationoftheLog–NormaldistributionthatisbasedontheGauss–Hermitequadraturein[77,TableIV].Inparticular,from[78,Eq.(6)]weobtain(46)shownatthetopofthenextpage,obtainedbyusing[75,Eq.(2.17)].Thus,M(k)where(a)isI(·)in(9)can

3068IEEETRANSACTIONSONCOMMUNICATIONS,VOL.61,NO.7,JULY2013

NGHQ 1 f(x)≈(1+K)exp{ K}√w nω nexp{ (1+K)ω nx}I02K(1+K)ω nx gbπ NGHQ (a)K/ ω1+K1 n exp sw n Mgb(s)≈√π1+K+(s/ ωn)1+K+(s/ ωn)n=1

n=1

(46)

(k)

MI

NGHQ +∞ 1

(s)≈(1+K)exp{ K}√ nxk+1exp{ [s+(1+K)ω n]x}I02K(1+K)ω nxdxw nω

πn=10

NGHQ

(a)K(1+K)1 (k+2)=(1+K)exp{ K}√w nω nΓ(k+2)[s+(1+K)ω n]1F1k+2,1,

πn=11+K+(s/ ωn)

(47)

12

√TI(s)≈(1+K)exp{ K}Γ1 απ

NGHQ+∞k+1 Γ(2+k)sK(1+K)

× nFw nωk+2,1,1121+K+(s/ ωn)Γ2 α+k[s+(1+K)ω n]2+kn=1k=0

(48)

becomputedasshownin(47)atthetopofthispage,where

(a)isobtainedbyusing[69,Eq.(6.631)].Bysubstituting(47)inTI(·)in(9),weobtain(48)shownatthetopofthispage.Sincethein niteseriesin(48)isnotfastconverging,weelaboratefurtherTI(·)inordertoobtainabetterexpressionforsimplenumericalcomputation.ForK=0,thiscanbeobtainedby rstreplacing1F1(·,·,·)withitsseriesexpansionin[69,Eq.(13.1.2)],andthencomputingthein niteseriesinkbyusing(42).Eventually,weobtain(17)withsomealgebraicmanipulationsandusingtheidentityΓ(z+1)=zΓ(z).ThecaseK=0canbeobtainedbynotingthat1F1(A,1,0)=1foreveryA,andbyapplyingthesameprocedureasinAppendixEwithm=1.Thisconcludestheproof.

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1–6.MarcoDiRenzo(S’05–AM’07–M’09)wasborninL’Aquila,Italy,in1978.HereceivedtheLaurea(cumlaude)andthePh.D.degreesinElectricalandInformationEngineeringfromtheDepartmentofElectricalandInformationEngineering,UniversityofL’Aquila,Italy,inApril2003andinJanuary2007,respectively.

FromAugust2002toJanuary2008,hewaswiththeCenterofExcellenceforResearchDEWS,UniversityofL’Aquila,Italy.FromFebruary2008toApril2009,hewasaResearchAssociatewiththe

TelecommunicationsTechnologicalCenterofCatalonia(CTTC),Barcelona,Spain.FromMay2009toDecember2009,hewasanEPSRCResearchFellowwiththeInstituteforDigitalCommunications(IDCOM),TheUniversityofEdinburgh,Edinburgh,UnitedKingdom(UK).

SinceJanuary2010,hehasbeenaTenuredResearcher(“Charg´edeRechercheTitulaire”)withtheFrenchNationalCenterforScienti cResearch(CNRS),aswellasafacultymemberoftheLaboratoryofSignalsand

Systems(L2S),ajointresearch´d’Electricit´´e(SUPELEC),´laboratoryoftheCNRS,theEcoleSup´erieure

andtheUniversityofParis–SudXI,Paris,France.Hismainresearchinterestsareintheareaofwirelesscommunicationstheory.HeisaPrincipalInvestigatorofthreeEuropean–fundedresearchprojects(MarieCurieITN–GREENET,MarieCurieIAPP–WSN4QoL,andMarieCurieITN–CROSSFIRE).

Dr.DiRenzoistherecipientofthespecialmentionfortheoutstanding ve–year(1997–2003)academiccareer,UniversityofL’Aquila,Italy;theTHALESCommunicationsfellowshipfordoctoralstudies(2003–2006),UniversityofL’Aquila,Italy;theBestSpin–OffCompanyAward(2004),AbruzzoRegion,Italy;theTorresQuevedoawardforresearchonultrawidebandsystemsandcooperativelocalizationforwirelessnetworks(2008–2009),MinistryofScienceandInnovation,Spain;the“D´erogationpourl’EncadrementdeTh`ese”(2010),UniversityofParis–SudXI,France;the2012IEEECAMADBestPaperAwardfromtheIEEECommunicationsSociety;andthe2012ExemplaryReviewerAwardfromtheIEEEWIRELESSCOMMUNICATIONSLETTERSoftheIEEECommunicationsSociety.HecurrentlyservesasanEditoroftheIEEECOMMUNICATIONSLETTERS.

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