排队论的matlab仿真(包括仿真代码).docx
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排队论的matlab仿真(包括仿真代码).docx
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WirelessNetwork
ExperimentThree:
QueuingTheory
ABSTRACT
Thisexperimentisdesignedtolearnthefundamentalsofthequeuingtheory.MainlyabouttheM/M/SandM/M/n/nqueuingmodels.
KEYWORDS:
queuingtheory,M/M/s,M/M/n/n,ErlangB,ErlangC.
INTRODUCTION
Aqueueisawaitinglineandqueueingtheoryisthemathematicaltheoryofwaitinglines.Moregenerally,queueingtheoryisconcernedwiththemathematicalmodelingandanalysisofsystemsthatprovideservicetorandomdemands.Incommunicationnetworks,queuesareencounteredeverywhere.Forexample,theincomingdatapacketsarerandomlyarrivedandbuffered,waitingfortheroutertodeliver.Suchsituationisconsideredasaqueue.Aqueueingmodelisanabstractdescriptionofsuchasystem.Typically,aqueueingmodelrepresents
(1)thesystem'sphysicalconfiguration,byspecifyingthenumberandarrangementoftheservers,and
(2)thestochasticnatureofthedemands,byspecifyingthevariabilityinthearrivalprocessandintheserviceprocess.
Theessenceofqueueingtheoryisthatittakesintoaccounttherandomnessofthearrivalprocessandtherandomnessoftheserviceprocess.ThemostcommonassumptionaboutthearrivalprocessisthatthecustomerarrivalsfollowaPoissonprocess,wherethetimesbetweenarrivalsareexponentiallydistributed.Theprobabilityoftheexponentialdistributionfunctionisft=λe-λt.
lErlangBmodel
OneofthemostimportantqueueingmodelsistheErlangBmodel(i.e.,M/M/n/n).ItassumesthatthearrivalsfollowaPoissonprocessandhaveafinitenservers.InErlangBmodel,itassumesthatthearrivalcustomersareblockedandclearedwhenalltheserversarebusy.TheblockedprobabilityofaErlangBmodelisgivenbythefamousErlangBformula,
wherenisthenumberofserversandA=λ/μistheofferedloadinErlangs,λisthearrivalrateand1/μistheaverageservicetime.Formula(1.1)ishardtocalculatedirectlyfromitsrightsidewhennandAarelarge.However,itiseasytocalculateitusingthefollowingiterativescheme:
lErlangCmodel
TheErlangdelaymodel(M/M/n)issimilartoErlangBmodel,exceptthatnowitassumesthatthearrivalcustomersarewaitinginaqueueforaservertobecomeavailablewithoutconsideringthelengthofthequeue.Theprobabilityofblocking(alltheserversarebusy)isgivenbytheErlangCformula,
Whereρ=1ifA>nandρ=AnifA wherePB(n,A)isdefinedinEq.(1.1). DESCRIPTIONOFTHEEXPERIMENTS 1.Usingtheformula(1.2),calculatetheblockingprobabilityoftheErlangBmodel.DrawtherelationshipoftheblockingprobabilityPB(n,A)andofferedtrafficAwithn=1,2,10,20,30,40,50,60,70,80,90,100.Compareitwiththetableinthetextbook(P.281,table10.3). Fromtheintroduction,weknowthatwhenthenandAarelarge,itiseasytocalculatetheblockingprobabilityusingtheformula1.2asfollows. PBn,A=APB(n-1,A)m+APB(n-1,A) itusethetheoryofrecursionforthecalculation.Butthedenominatorandthenumeratoroftheformulabothneedtorecurs(PBn-1,A)whendoingthematlabcalculation,itwastetimeandreducethematlabcalculationefficient.Sowechangetheformulatobe: PBn,A=APB(n-1,A)n+APB(n-1,A)=1n+APBn-1,AAPBn-1,A=1(1+nAPBn-1,A) Thenthecalculationonlyneedrecursoncetimeandismoreefficient. Thematlabcodefortheformulais: erlang_b.m %************************************** %File: erlanb_b.m %A=offeredtrafficinErlangs. %n=numberoftrunckedchannels. %Pbistheresultblockingprobability. %************************************** function[Pb]=erlang_b(A,n) ifn==0 Pb=1;%P(0,A)=1 else Pb=1/(1+n/(A*erlang_b(A,n-1)));%userecursion"erlang(A,n-1)" end end Aswecanseefromthetableonthetextbooks,itusesthelogarithmcoordinate,sowealsousethelogarithmcoordinatetoplottheresult.Wedividethenumberofservers(n)intothreeparts,foreachpartwecandefineaintervalofthetrafficintensity(A)basedonthefigureonthetextbooks: 1.when0 2.when10 3.when30 Foreachpart,usethe“erlang_b”functiontocalculateandthenuse“loglog”functiontofigurethelogarithmcoordinate. Thematlabcodeis: %***************************************** %forthethreeparts. %nisthenumberservers. %Aisthetrafficindensity. %Pistheblockingprobability. %***************************************** n_1=[1: 2]; A_1=linspace(0.1,10,50);%50pointsbetween0.1and10. n_2=[10: 10: 20]; A_2=linspace(3,20,50); n_3=[
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