DSpace at VNU: Soft computing methods for WiMax Network Planning on 3D Geographical Information Systems

Contents lists available atScienceDirectwww.elsevier.com/locate/jcss aInstitute of Research and Development, Duy Tan University, Danang, Viet Nam bVNU University of Science, Vietnam Nati

Contents lists available atScienceDirect

www.elsevier.com/locate/jcss

aInstitute of Research and Development, Duy Tan University, Danang, Viet Nam

bVNU University of Science, Vietnam National University, Hanoi, Viet Nam

cDivision of Data Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

dFaculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Viet Nam

Particle Swarm Optimization

Parallel Random Forest

Soft computing

WiMax Network Planning

Inthispaper, wepresent an applicationofsoft computingmethodsfor theproblem ofWiMaxNetworkPlanningon3DGeographicalInformationSystems(3DGIS)thatoptimizesbothperformanceofthenetwork(CoverageandQuality-of-Service)andinvestmentcosts(the number of base stations and sectors) A pre-processing procedure using latestresults of parallel Random Forest classification algorithm to determine valid positions

of base stations on a terrain of 3D GIS is proposed Based upon those positions, wedesign ageneralized mathematical model takingintoaccount 3Dobstacles in pathlosscalculationprocess.Inordertogenerateoptimalsolutionsofthemodel,ahybridalgorithmbetweengreedyBTPandimprovedParticleSwarmOptimizationincorporatedwithparallelcomputingispresented Experimentalvalidationofthe proposedmethodincomparisonwithotherrelevantonesisperformed

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1 Introduction

With thegrowing demands of Internet accesses forscientific andcommercial uses nowadays,there is a strong need

ofhigh-qualitynetworkinfrastructures thatensurelargebandwidthsandhighnetworkspeedsbetweenaccesspointsinageographic area likeatown oraprovince.The problemofWiMax Network Planning on 3D Geographical Information Systems

(3D GIS)isindeedoneofthemostpopulartopicsincurrentresearches.Thisproblemaimstodeterminetheoptimalnumber

ofWiMaxBaseStations(BSs),theirsuitablelocationsonagiventerrainof3DGISandtheirconfigurationsofsectorsforthebestresultsofboththeperformanceofthenetwork(i.e.CoverageandQuality-of-Service)toall fixedusersontheterrainandtheinvestmentcosts (i.e.theminimal numbersofBSsandsectors).It isamulti-objectiveoptimizationproblemthatinvolvessomenetworkparametersandgeographicconstraintsofBS

Severalarticlesaboutsoftcomputingmethodsandmathematicalmodelingforthisproblemwerefoundintheliterature.Wahl,Stabler&Wolfle[21]presenteda2DmodelforthepathlosscalculationofWiMaxnetworkplanninginahybridenvi-ronmentbetweenurbanandindoor.Thepredictionconceptdoesnotrelysolelyonthedirectraylikeempiricalmodelsanddoesnotconsiderhundredsofraysforasingleradiolinklikeraytracing,butfocusesonthemostdominantpathbetweentransmitterandreceiver, allowingthe computation ofthetransitionfroman urban to an indoorscenario andviceversa.Admed,Mughni&Akhtar[1]took anoverviewofsomeavailablepathlossmodelssuchasOkumura&Hata,ECC–33and

SUI.Similarly,theworksofAmaldi,Bosio, Malucelli&Yuan[2],Eisenblatter&Geerd[7],Nawrocki,Dohler&Aghvami[13]andTsourakis&Voudouris[20]introducedseveralmethodsforthemathematicalmodelingofWiMaxnetworks.Regardingthesoftcomputingmethods,Taplinetal.[18] introducedsomeautomatic WiMaxnetwork planningona2D mapsuchasFTP, BTP,HillClimb(HC)andTabuSearch(TS).Thetwo firstalgorithmsbelongtothegreedyapproach.WhileFTPtriestoincrease or decreasethenumber ofsectorsineach BS forthe bestperformance ofthe network,BTPsortsall BSs intheascendingorderandputssectorsonthemoneafteranothertoensurethateachuserisservedbyitsbestBS.Thetwolastonesareneighborhood-basedsearchingalgorithms.Theexperimentsonthreedifferentmapsshowedthat TSachievesthebestresultsamongallalgorithms,butithashighcomputationalcomplexity.Onthecontrary,BTPcangenerateacceptableresultsinareasonable time.Carneiroetal.[4]developeda planningtool thatallowstheWiMaxnetworkplanninginthegraphicalenvironmentofArcGISsincetheobjectiveofthisresearchwastorelatethegeographicaldataofagivenlocation,

to the numberof BaseStations neededto cover that location,in terms ofpower andcapacity.The planning tool, based

on localsearch methods, allows usersto chooseseveralparameters, fromthe numberofbase stationsto thenumber ofusers, makingthenetworkoperateindifferentenvironments, providingsimilaritiesto dailysituations.Zhang[22] offered

a comprehensive explanationon how todesign, plan,andoptimize WiMaxnetworks by heuristicmethods involving thetopology,capacity,congestioncontrol,mediumaccesscontrol,schedulingandQuality-of-Service(QoS).Hu,Chen&Banzhaf[10] usedthe adaptive-population-sizeGenetic Algorithmwithindividual representation andgeneticvariation operationsbeing re-constructed toenhance the search capability ofthe algorithm Simulationresults showedthat thisalgorithm isrobusttodifferentscenariosandhasbettersearchprocessthanaconventionalfixedpopulationsizescheme.Hurley,Allen,Ryan &Taplin[11] introducedamathematical modelfortheautomated designoffixed wirelessaccessnetworksthroughtheautomaticselectionandconfigurationofbasestationsites,andpresentedastochasticoptimizationalgorithmtogener-atethefixedwirelessaccessnetworkinfrastructuredesign.Sebastiaoetal.[15]designedageneticalgorithm-basedWiMaxplanning tool,which provides plannerswithpractical andusefulinformationthrough quickcoverage/capacity basedpro-cedures,andoutputsthenumberandpositionofthebasestationsandanestimationofthetotalcostofimplementation,basedondataprovidedbydifferentequipmentmanufacturers.Itwas appliedforthezoneofCovilhã,Portugal,whereGISare usedforrepresentationofruralandsparseurbanareas.SimilartoCarneiroetal.[4],Sapumohottietal.[14] alsopre-sentedNetworkPlanningCellTool(NPCET),anetworkplanningtoolthatwasdesignedforplanningruralWiMaxnetworks

inMalaysiainvolvingwirelesspropagation,GISandmaps,networkplanningandprogramming

Even though the relevantsoft computingmethods and mathematicalmodels were available, they contain some tations that shouldbe improvedfurther Firstly, mostexisting mathematical modelsdidnot count forrestrictedlocations

limi-of BSson aterrain so that aBS could be put onunsuitableregions such aslakes andrivers.Evenif restrictedlocationsare calculated,themodels wereconstructedsolely onthebasis offlatplane (two-dimensions)butnot on3D sothatthecalculation fromapathlossmodelisnotaccurate.Assuch,theavailablesoftcomputingmethodsworkingonthosemod-els resulted inlessaccuracyandineffectiveness Secondly, some softcomputingalgorithms limitedthe objectivefunction

to either the performance of the network or the investment costs so that the planning is just semi-automatic Thus, itmakes sensetodesignageneralized3DmathematicalmodelthattakesintoaccounttherestrictedlocationsofBSsandanautomaticmulti-objectiveWiMaxnetworkplanningmethodon3DGIS

OurmajorideasinthenewalgorithmthatwecallWNPA-3DTaresummarizedbelow.Firstly,wepresentapre-processingprocedureusingthelatestresultsofparallelRandomForestclassificationalgorithm(Thong,Son& Hoa,[19])todeterminevalidpositionsofBSsonaterrainof3DGIS Secondly,baseduponthosepositions,we designageneralizedmathematicalmodelfortheconsidered problem,takingintoaccount3D obstacles inthepathlosscalculationprocess Thirdly,a hybridalgorithmbetweenthegreedyBTPandimprovedParticleSwarm Optimization(Gongetal.,[9])incorporatedwithparallelcomputingispresentedinordertogenerateoptimalsolutionsofthemodel.Experimentalvalidationoftheproposedmethod

incomparisonwithotherrelevantonesisintensivelyperformed.Thoseideasareallourcontributionsinthisarticle.Therestsofthispaperareorganizedasfollows.Section2introducesthenovelmethod– WNPA-3DT.Section3validatesthe proposedapproachthrough variousdatasetsandrelevantworks.Finally,wegive someconclusions andoutlinefutureworksinthelastsection

2 The proposed algorithm

Inthissection,wedescribetheproposedalgorithm,namedasMulti-Objective WiMax Network Planning Algorithm on 3D rains (WNPA-3DT). ItincludesthedeterminationofvalidpositionsofBSsontheterrain,themathematicalmodelingofourconsideredproblem,thedescriptionofthepathlosscalculationusedinthemodelandthedetailsofthehybridalgorithm,whichisusedtospecifytheoptimalplanningsolutions.Thosepartsarepresentedinsomesub-sectionsbelow,respectively

Ter-2.1 Terrain classification

Givena terrainintheDigital ElevationModel(DEM) formatrepresentedby amatrixofheight values.Theaimofthissub-sectionistodeterminesomebasicobjectsinthatterrainsuchasmountains,plateaus,hills,flatlands,riversandlakes.From thisclassification,thevalidpositionsofBSsontheterrainare totallyspecified.ThisproblembelongstotheclassofImageClassificationwhichdividesthe collectionofterrain dataintotwogroups:theTraining andtheTestingsets.Whilethe Trainingsetisusedtoconstructaclassification model,theTesting isdesignedfortheverificationoftheperformance

ofthemodel.Inthispart,we usetheparallelRandomForest (Thong,Son& Hoa,[19]) forboth theconstructionandtheverificationofthemodel.

Inthe construction step,some criteriaforthe classificationofsix basicobjectsabove are defined Eachterrain intheTraining setisscannedwiththosecriteriatodetectobjects Thoseobjectsare storedina databaseaccordingto theDEMMarkup Language(DML)standardserving fortheconstruction ofthe classificationmodel.Descriptionsof thecriteriaareshownbelow

• Boundary:acollectionofboundarypointsofanobject.Basically,thenumberofthosepointsshouldbeasmanyassibleforthebestprediction.Nonetheless,inordertoreducethecomputationalcomplexity,wekeepacertainnumber

pos-ofpoints andusetheadaptiveBandWidthalgorithm(Dagher,[5]) toapproximatethemintotheboundarylinesoftheobjectwithagivenerrorthresholdθ

• Inner values:

◦ Themaximalandminimalheightsoftheobject.Notethatan objectisacollectionofheightvalues(a.k.a.z values)

sinceweareworkingwiththeDEMstandard

◦ Theanglebetweenthelineconnectingthehighestandthelowestpointsoftheobjectanditsprojection

◦ Thecoordinate(x,y)ofthehighestpointoftheobject

◦ Thenumberofheightvaluesoftheobject

• Neighborhood:informationaboutfourneighboredobjectsinthedirections:South,North,EastandWest.Thisisticreflectsthespatialrelationshipsinaterrain,e.g.plateausaremorelikelytobeneartoflatlandsthanrivers

character-• Reference parameters:somegeographicvaluessuchastheprojection,thecodeoftheterraininWGS84ReferenceSystem,etc are usedto transformall above characteristics into a unique standard.Due to thistransformation,all objects ofterrainsintheTrainingsetcanbesynchronizedintheDMLstandard

Inorder to constructtheclassification model fromthedatabase, the parallel RandomForest algorithm (Thong,Son &Hoa,[19]) isused.RandomForest, originatedbyBreiman [3],isamachine learningalgorithm basedonthe decisiontreeapproachfortheclassification ofsatellite andremote sensing images.Theadvantages ofRandomForestare theindepen-denceoftheselectionofTrainingsets,highaccuracy,robustto outliersandsupported byseveralusefultoolssuchasthecalculationsof variablesimportance and classificationerrors Inessence, Random Forest includes a collectionofdecisiontrees, constructedfrom a randomsubset ofthe original dataset,andthe final classification resultdepends onwhich re-sultisthemostappearinginalltrees.ThetreesconstructionprocessinRandomForestdoesnot requirethetree-pruning,whichismostlyusedinthetraditionaldecisiontreealgorithmsuchasID3.EventhoughtheclassificationaccuracyofRan-domForestisbetterthanthoseofthetraditionaldecisiontreealgorithms,itscomputationaltimeisstillamajorweakness(Breiman,[3]).TheparallelRandomForestalgorithm(Thong,Son&Hoa,[19])wasdesignedtoacceleratethewholecompu-tationalprocessusingparallelcomputingwithmultipleprocessors.Eachprocessorisresponsibletogeneratesomedecisiontreesfromsmall,randomsubsetsoftheTrainingsetandtolookforthemostappearingclassinallitstreesforadatasetintheTestingset,whichissentfromtheMasterprocessor.Thus,thecomputationaltimeofboththeconstructionandthever-ificationstepsaresignificantlyreduced.Thepseudo-codebelowshowssomestepstogeneratedecisiontreesinaprocessor.Thanks tothedesignoftheparallel RandomForest,theverificationstepisparalleledperformedinallprocessors TheMasterprocessor synthesizesall resultsfromotherprocessors incharge andfinds themostappearingclassamongthem.UsingtheparallelRandomForest,wecandeterminethevalidpositionsofBSsontheterrain.Thosepositionsareincluded

inthemodelforourproblem,whichwillbepresentedinsub-section2.2

2.2 The mathematical modeling

TheWiMaxNetworkPlanningon3DGISproblemisdescribedasfollows.Assumethatwehaveaterrainincludingsomevalidpositions of BSsand fixed positions ofusers EachBS can be attached by some sectorsin differentdirections,andtheperformanceofaBS tousersdependsonthepositionofBS andtheanglesofsectors.Determine anoptimalplanningsolutionthatsatisfiesthefollowingconstraints:

• Thecoveragetoallusersislargerthanathreshold α

• TheQoStoallusersislargerthanathresholdβ

• Maximaloverloadcapacityofasectoris γ

• ThenumbersofBSsandsectorsareminimal

Weformulatetheproblemasfollows

Input:

• Terrain & users data: a terrain T whose sizes are Ncols×Nrows and the cell’s distances are (height,width) ValidpositionsofBSs,thenumberandpositionsofusersaredeterminedfromsub-section2.1

Input: DML data –X ( N , whereN isthe number of records andr isthe dimension of data;N treesis the number of decision trees;

Output: N treesdecision trees

Random Forest Construction Algorithm:

3: No_trees=No_trees+1

4: N child=rand (2, N );r child=rand (2, r

5: ChooseN childrandom data andr childrandom attributes from the Training set

6: Assume that the current table isS.Split each attribute and its class inS intosmall tablesS iwherei isan attribute ofS

7: Calculate Entropy value ofS:

Entropy ( S )=

j

−p jlog2p j ,

wherep jis the number of datasets in relation with classj in S

8: DivideS i into subsets S j in terms of domain of values and calculateEntropy ( S j )as in Step 7 If the domain is discrete thenS jrefers

to possible values of attributei.Otherwise, choose a random numberk anddivide the domain intok equalparts whose average values representing forS j; j=1, k

9: CalculateGain ( S , i )wherei isan attribute ofS bythe formula below:

Gain ( S , A )=Entropy ( S )− 

v∈V alues ( A )

|S v|

|S|Entropy ( S v ),

whereV alues ( A )is the set of possible values of attributeA,andS vis a subset ofS containingdata whose attribute isA andvalue isv

10: Choose attributei havingmaximal value ofGain ( S , i )as the root node

11: DivideS intosub-tablesD jin terms of domain of values according to the attributei

12: IfEntropy ( D j )=0 then the class of all data inD jis the leaf node

13: Remove attributei from S andperform the similar steps from Step 6 to Step 12 for the new table until the Entropy values of all sub-table

are equal to zero

14: Until No_trees=N trees

•Network parameters:The height ofBS andthethresholds (α , β, γ).Assume that theheightsofBSsare equal,andallsectors areof thesame types M A X S E R V isthe maximal emitted powerof asector and M A X Q o S is themaximalsignalstrengththatausercanreceivefromasector

Output:

•TheperformanceofthenetworkincludingCoverageandQoS

•TheinvestmentcostsconsistingofthenumbersofBSsandsectors

•TheconfigurationofsectorsonBSssuchastheanglesandthedirections

isthepositionofuseru iontheterrain

• S= {s1,s2, ,s Nsec}isasetofsectors

s i= {P i,b j, θi, ϕi,f eq i,Prec( i,u k,T),G i,LFadei },

◦ Nsec isthenumberofsectors

◦ P i isthebroadcastcapacityofsectors i,measuredindB

◦b jistheBScontainingsectors i

◦ θi istheangleofantenna,whosevaluefallsinto(60,90,120,180)degrees

◦ ϕ isthedirectionofantenna,whosevalueisfrom0to360degrees

◦ f eq i isthefrequencyofsectors i,measuredinMHz.

◦ G i istheantennagainofsectors i,measuredindB

◦ LFadei istheattenuationcoefficientofsectors i,measuredindB

◦ Prec( i,u k,T)isthesignalstrengthofsectors itouseru konterrainT ,measuredindBandcalculatedbytheequationbelow(Hurley,Allen,Ryan&Taplin,[11]):

where D B S isthe minimaldistancebetweenBSsonthe terrain.Inthemodels(1)–(14),we definedthe setsofusers–U

andBSs– B withsectors– S and thendetermined 4single objectives: F1 (measuresthecoverage qualityfrom B to U ),

F2(measurestheQoSvaluefromB to U ), F3 (measuresthenumberofBSs)andF4 (measuresthenumberofsectors)withthe equivalent formulae.The multi-objectiveoptimizationproblemis then givenin equations(9)–(14)whoseconstraintsrelatetonetworkparameters(10)–(13)and3Dterrain(14).Somespecialcasesoftheproblem(9)–(14)are:

• Ifconstraints(10)–(12),(14)arenotprovidedandS doesnotexistand y i=0 inb i and y j=0 inu jthenwe getthemodelofTaplinetal.[18].Additionally,ifequation(9)isnotincludedthenwereceivethemodelofCarneiroetal.[4]

• If constraints(10)–(12), (14) are not provided and equation (9) is not included and S does not exist and y i=0 in

b i thenwe getthe modelsofAdmed, Mughni&Akhtar [1]andSapumohottietal.[14] Additionally,iftheobjectivechangestoF=F1+F2−→Max inequation(9)thenwereceivethemodelsofSebastiaoetal.[15]

• Withoutconstraint(14)andS does notexistandy i=0 inb iand y j=0 inu jthenwe getthemodelofHu,Chen&Banzhaf[10]

2.3 Path loss calculation

In sub-section2.2, we havealreadyknown that q( i,u k,T)is the attenuationfromsector s i touser u k on terrain T

But how can we calculate thisquantity? Thispart mentions a modelfor the path losscalculation, which is the answerfor that question Since we are workingon a terrain, the numberand positions ofobstacles between BS b i (∀i∈[1,N])

containing sectors i anduseru j (∀j∈[1,M])mustbepre-determined.Baseduponthoseobstacles,themodelforthepathlosscalculationisspecified

Firstly, theproblem ofdetermining how manyobstacles exist betweena BS anda userrefers to theLine-of-Sight orVisibilityproblem(Ghosh,[8]).OurpreviousworkinSon[16]presentedamethodforthisproblembydividingthelinecon-nectingtheBSandtheuserbymultiplesplittingpoints.Thelengthoftwoconsecutivesplittingpointsis(height+width) /2where(height,width)arethecell’sdistancesofterrainT Thosepointsareverifiedwhethertheboundingrectanglesofter-rain T consisting of them are the obstacles or not If so, the positions and the number of obstacles are marked Thisverificationis performedby parallelcomputing, andthe coordinatesofthe boundingrectangleconsistingofthe splittingpoint(x,z areshownbelow

L f reespace=32.4+20∗log(R i) +20∗log(f eq i). (21)

R i istheEuclideandistancefromBSb i touseru j,measuredinkilometers,and f eq iisthefrequencyofsectors i,measured

inMHz

Inordertocalculate L ke,letusexaminetheone-obstaclemodelinFig 1.Inthisfigure,h1,h2,h3 aretheheightsofBS

b i,theobstacleanduseru j,respectively.d1,d2 aretheEuclideandistancesfromb i totheobstacleandfromtheobstacle

d

1+d2

λ ∗d

1∗d2

whereλisthewavelength,andv istheFresnelreflectioncoefficientoftheobstacle

Fig 1 The one-obstacle model.

Incaseofmany-obstaclesmodel,thestrategybelowisappliedtocalculateL ke

• Determinethemainobstaclethathashighestheightamongall

• CalculateL main ke oftheone-obstaclemodelincludingBSb i,themainobstacleanduseru j

• Determinethesecondmainobstacle,whichhashighestheightamongall,betweenBSb i andthemainobstacle

• CalculateL le f t ke oftheone-obstaclemodelincludingBSb i,thesecondmainobstacleandthemainobstacle

• PerformthesimilarstepsandreceiveL right ke

L ke incaseofmany-obstaclesmodeliscalculatedbelow

Usingtheserules,we cancalculatetheattenuationfroma sectortoauserona3D terrain.Thus,allparametersinthemodelofsub-section2.2aretotallyspecified

2.4 The hybrid algorithm for the determination of Optimal Planning Solutions

Thissub-sectionpresentsa hybridmethodtodeterminetheoptimalplanningsolutionsforourproblem.Ourideasareusingtheimproved ParticleSwarm Optimization (Gong etal., [9]),integratedwithBTPalgorithm(Taplin etal., [18]) andparallelcomputing.ParticleSwarmOptimization(PSO),introducedbyKennedy&Eberhart[12],isastochastic,swarm-basedoptimizationalgorithmthatsimulatesthefood-lookingbehaviorsofbirds.PSOwassuccessfullyappliedtomanyoptimiza-tion problemssuch as GraphColoring, Traveling Salesman, etc Nonetheless, three mainproblems ofPSO that can affectthe performance ofthealgorithm are theinitialization, the convergence to local optima and the computational time.Gong et

al.[9]pointedout thatbadinitializationofparticlesmayresultinthequalityofthesolutions,andPSOtendstoconverge

to localoptima The large computational time ofPSOis another problemifthe numberof iterations increases.Thus, animprovementofPSOthatcanhandlethoselimitationsisrequiredforourproblem

Inthissub-section,weconsider theusesoftheparallel BTPalgorithmfortheinitializationproblem Beingintroduced

inSection1,BTPcangenerateacceptableresultsbasedonthegreedyapproachinareasonabletime.Aparallelversion ofthisalgorithmisintroducedbothtoacceleratethecomputingprocess,especiallyincasesofverylargeterraindata,andtoachieve theinitialsolutionsofPSO Inordertotacklewiththe convergencetolocaloptimaproblem,we usetheideasofGongetal.[9]aboutthemutationoperator.AnimprovementoftraditionalPSOalgorithmincorporatingwiththemutationoperatorandsomeadditionaltechniquessuchasparametersupdatingandparallelcomputingschemeisdesignedtohandlethetwolastproblemsofPSO.Themechanismoftheproposedapproach–WNPA-3DTisdescribedinFig 2

Letusmakea deeperanalysisaboutthismechanism Firstly,the parallelBTPalgorithm isusedto generatetheinitialsolutionsofPSO.Thisalgorithmdividestheterrainintosmallgridswhosesizesare η1× η2 (height≤ η1;width≤ η2)andputs BSstoall grids’ nodes.EachBS isequippedwithfour sectorsinall directionssuch asSouth,North,EastandWest.ThealgorithmchecksthoseBSsandsectorsandtriestoremovesomeofthemuntiltheconstraints(10)–(14)donothold.Outputs of thisalgorithm are the numbers andpositions of BSs andsectors aswell as their configurations such asthedirectionofantennas.Thepseudo-codeoftheparallelBTPalgorithmisshownbelow

AfterwereceivetheoutputsoftheparallelBTPalgorithmthenusetheimprovedPSOtogeneratethefinalsolutions.IntheInitprocedure,PSOalgorithmisinitializedby N pop particles–P= p1,p2, ,p N pop whosefirstoneisinheritedfromtheparallelBTPandtherestare randomlyinitialized withthemaximalnumberofBSsbeingthenumberofBSsfromtheparallelBTP(M A X_B S).Each p i (i=1,N pop)isencodedasfollows

• X d i j(V i j d)istheposition(velocity)ofb j inp i ( j=1,M A X_B S, i=1,N pop,d=1,2)

• on k i j isthecheckingvariableforthepossibilityofputtingsectors k onBSb j ( j=1,M A X_B S, i=1,N pop,k=1,Nsec).Itsdomainofvaluesis{0,1}

Fig 2 The mechanism of the proposed approach (WNPA-3DT).

• θk

i j isthe updatingvelocity ofdirectionofsector s k onBS b j ( j=1,M A X_B S, i=1,N pop,k=1,Nsec).Its domainofvaluesis[0,360]

• ϕk

i j isthedirectionofsectors konBSb j ( j=1,M A X_B S, i=1,N pop,k=1,Nsec).Itsdomainofvaluesis[0,360]

• pbest d i j isthebestpositionofb j inp i ( j=1,M A X_B S, i=1,N pop,d=1,2)

•on k pbest_i j is the checkingvariable forthe possibility of puttingsector s k on BS b j ( j=1,M A X_B S, i=1,N pop, k=

1,Nsec)inrelationwithpbest d

i j

• ϕk

pbest_i j isthedirectionofsectors konBSb j ( j=1,M A X_B S, i=1,N pop,k=1,Nsec)inrelationwithpbest d

i j

• f pbest i isthebestfitnessvalueofp i (i=1,N pop)

Example 1.Supposethatwehaveasolution: M A X_B S=5,Nmax _ seci =4 (i=1,5)andtheconfigurationofputtingsectors

onBSsasinequation(28)

We clearly recognize that the numberofBSs used forthissolution is: N i=4 since there isno sector that isput on

BS b5.Themaximalnumberofsectorsis20,andthetotalnumberofusedsectorsis:N isec=8

Input: Input in sub-section 2.2 andη1 ,η2 parameters

Output: The number of BSs(M A X_B S),the positions of BSs( x i , y i , z i ) (i=1, M A X_B S),the number of sectors(numSec)and the direction of

6: Doublex, y, z, ϕ // Position of BS and the direction of antenna

7: nc = ( width×Ncols )÷η2 nr = ( height×Nrows )÷η1

8: Fori=1 tonr // Step 8 to 22 are performed parallel in all processors

max _ sec 17: s numSec add Arc ( ϕ k )

18: b m addSec ( numSec )

22: CalculateMcoverage of all sectors by equations (4) – (6) where the attenuation values from sectors to users are paralleled computed from the

path loss model

23: Sort the coverages in the ascending order

24: Add BSs and sectors to the current solution

• M A X_B S isthemaximalnumberofBSsinp i

• N isec isthenumberofsectorsinp i

• Nmax _ secj isthemaximalnumberofsectorsthatcanbeputonBSb j (∀j∈[1,N]).

• Mcoverageisthecoveragevalue

• M Q o S istheQoSvalue

Based upon the fitnessvalues ofparticles, thebest values ofparticles (pBest) andtheswarm (gBest)are determinedaccordingly.Somenotionsbelowshowthesebestvalues:

gbest_ j isthedirectionofsectors konBSb j ( j=1,M A X_B S, k=1,Nsec)inrelationwithgbest d j

• f gbest isthebestfitnessvalueofallparticles

TheupdatingprocessesofpBestandgBestareshowninequations(30)and(31),respectively

If f pbest i >f ithen

f i = f i; pbest d =X d;

If f gbest>f pbest i then

f gbest= f pbest i ; gbest d j=pbest d i j;

on k gbest_ j=on k pbest_i j; ϕk gbest_ j= ϕk pbest_i j;

End If

In Fig 2, we use k processors in a parallel computing systemfor the calculations of fitness and pBest values of allparticles.Those pBestvaluesaresynchronizedattheMasterprocessorforthecalculationofgBest.Thisvalueisthensent

toallprocessorsfortheupdatingofotherparameters,whichisdescribedasfollows

•Update the possibilityof puttingsector s k on BS b j ( j=1,M A X_B S, k=1,Nsec) Denote rands(0/1) asthe randomfunctionwhoseoutputis0or1

If rands(0 1) +on k i j+on k pbest_i j+on k gbest_ j

on k i j=1

Else

on k i j=0

•Updatethedirectionandthevelocityofdirectionofsector s k onBSb j ( j=1,M A X_B S, k=1,Nsec)

If on k i j=1 then

θi j k= θk

i j+c1



ϕk pbest_i j− ϕk i j

 +c2

2 arethecoefficientsofpbest d i j andgbest d j,respectively

Beingmentionedinsomefirstlinesofthissub-section,themutationoperatorisappliedtoourproposedalgorithmafterall parameters have beenupdated Itchanges the positionandthe velocity ofb j in p i,the directionandthevelocity ofdirectionofsectors konb j (i=1,N pop, j=1,M A X_B S, k=1,Nsec)inacertainextent

Beingmentionedinsomefirstlinesofthissub-section,themutationoperatorisappliedtoourproposedalgorithmafterall parameters have beenupdated Itchanges the positionandthe velocity ofb...

onBSsasinequation(28)

We clearly recognize that the numberofBSs used forthissolution is: N i=4 since there isno sector that isput on

BS b5.Themaximalnumberofsectorsis20,andthetotalnumberofusedsectorsis:N