排队论的matlab仿真(包括仿真代码).docx
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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=AnifAwherePB（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.when02.when103.when30Foreachpart,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=[30:
10:
100];
A_3=linspace（13,120,50）;
%*****************************************
%foreachpart,calltheerlang_b（）function.
%*****************************************
fori=1:
length（n_1）
forj=1:
length（A_1）
p_1（j,i）=erlang_b（A_1（j）,n_1（i））;
end
end
fori=1:
length（n_2）
forj=1:
length（A_2）
p_2（j,i）=erlang_b（A_2（j）,n_2（i））;
end
end
fori=1:
length（n_3）
forj=1:
length（A_3）
p_3（j,i）=erlang_b（A_3（j）,n_3（i））;
end
end
%*****************************************
%useloglogtofiguretheresultwithinlogarithmcoordinate.
%*****************************************
loglog（A_1,p_1,'k-',A_2,p_2,'k-',A_3,p_3,'k-'）;
xlabel（'TrafficindensityinErlangs（A）'）
ylabel（'ProbabilityofBlocking（P）'）
axis（[0.11200.0010.1]）
text（.115,.115,'n=1'）
text（.6,.115,'n=2'）
text（7,.115,'10'）
text（17,.115,'20'）
text（27,.115,'30'）
text（45,.115,'50'）
text（100,.115,'100'）
Thefigureonthetextbooksisasfollow:
Wecanseefromthetwopicturesthat,theyareexactlythesamewitheachotherexceptthattheresultoftheexperimenthavenotconsideredthesituationwithn=3,4,5,…,12,14,16,18.
2.Usingtheformula（1.4）,calculatetheblockingprobabilityoftheErlangCmodel.DrawtherelationshipoftheblockingprobabilityPC（n,A）andofferedtrafficAwithn=1,2,10,20,30,40,50,60,70,80,90,100.
Fromtheintroduction,weknowthattheformula1.4is:
PCn,A=nPB（n,A）n-A（1-PB（n,A））
SinceeachtimewecalculatethePBn,A,weneedtorecursntimes,sotheformulaisnotefficient.Wechangeittobe:
PCn,A=nPB（n,A）n-A（1-PB（n,A））=1n-A（1-PB（n,A））nPB（n,A）=1（An+n-AnPBn,A）
Thenweonlyneedrecursonce.PBn,Aiscalculatedbythe“erlang_b”functionasstep1.
Thematlabcodefortheformulais:
erlang_c.m
%**************************************
%File:
erlanb_b.m
%A=offeredtrafficinErlangs.
%n=numberoftrunckedchannels.
%Pbistheresultblockingprobability.
%erlang_b（A,n）isthefunctionofstep1.
%**************************************
function[Pc]=erlang_c（A,n）
Pc=1/（（A/n）+（n-A）/（n*erlang_b（A,n）））;
end
Thentofigureoutthetableinthelogarithmcoordinateaswhatshowninthestep1.
Thematlabcodeis:
%*****************************************
%forthethreeparts.
%nisthenumberservers.
%Aisthetrafficindensity.
%P_cistheblockingprobabilityoferlangCmodel.
%*****************************************
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=[30:
10:
100];
A_3=linspace（13,120,50）;
%*****************************************
%foreachpart,calltheerlang_c（）function.
%*****************************************
fori=1:
length（n_1）
forj=1:
length（A_1）
p_1_c（j,i）=erlang_c（A_1（j）,n_1（i））;
%µ÷Óú¯Êýerlang_c
end
end
fori=1:
length（n_2）
forj=1:
length（A_2）
p_2_c（j,i）=erlang_c（A_2（j）,n_2（i））;
end
end
fori=1:
length（n_3）
forj=1:
length（A_3）
p_3_c（j,i）=erlang_c（A_3（j）,n_3（i））;
end
end
%*****************************************
%useloglogtofiguretheresultwithinlogarithmcoordinate.
%*****************************************
loglog（A_1,p_1_c,'g*-',A_2,p_2_c,'g*-',A_3,p_3_c,'g*-'）;
xlabel（'TrafficindensityinErlangs（A）'）
ylabel（'ProbabilityofBlocking（P）'）
axis（[0.11200.0010.1]）
text（.115,.115,'n=1'）
text（.6,.115,'n=2'）
text（6,.115,'10'）
text（14,.115,'20'）
text（20,.115,'30'）
text（30,.115,'40'）
text（39,.115,'50'）
text（47,.115,'60'）
text（55,.115,'70'）
text（65,.115,'80'）
text（75,.115,'90'）
text（85,.115,'100'）
TheresultofblockingprobabilitytableoferlangCmodel.
ThenweputthetableoferlangBanderlangCintheonefigure,tocomparetheircharacteristic.
100
10-1
Thelinewith‘*’istheerlangCmodel,thelinewithout‘*’istheerlangBmodel.Wecanseefromthepicturethat,foraconstanttrafficintensity（A）,theerlangCmodelhasahigherblockingprobabilitythanerlangBmodel.Theblockingprobabilityisincreasingwithtrafficintensity.Thesystemperformsbetterwhenhasalargern.
ADDITIONALBONUS
WriteaprogramtosimulateaM/M/kqueuesystemwithinputparametersoflamda,mu,k.
Inthispart,wewillfirstlysimulatetheM/M/kqueuesystemusematlabtogetthefigureoftheperformanceofthesystemsuchastheleavetimeofeachcustomerandthequeuelengthofthesystem.
Aboutthesimulation,wefirstlycalculatethearrivetimeandtheleavetimeforeachcustomer.Thenanalysisoutthequeuelengthandthewaittimeforeachcustomeruse“for”loops.
Thenwelettheinputtobelamda=3,mu=1
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