排队论的matlab仿真(包括仿真代码)Word文档下载推荐.docx
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排队论的matlab仿真(包括仿真代码)Word文档下载推荐.docx
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nandρ=AnifA<
n.Thequantityρindicatestheserverutilization.TheErlangCformula(1.3)canbeeasilycalculatedbythefollowingiterativescheme
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<
n<
10,0.1<
A<
10.
2.when10<
20,3<
20.
3.when30<
100,13<
120.
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=[30:
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
length(n_2)
length(A_2)
p_2(j,i)=erlang_b(A_2(j),n_2(i));
length(n_3)
length(A_3)
p_3(j,i)=erlang_b(A_3(j),n_3(i));
%useloglogtofiguretheresultwithinlogarithmcoordinate.
loglog(A_1,p_1,'
k-'
A_2,p_2,'
A_3,p_3,'
);
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
%erlang_b(A,n)isthefunctionofstep1.
function[Pc]=erlang_c(A,n)
Pc=1/((A/n)+(n-A)/(n*erlang_b(A,n)));
Thentofigureoutthetableinthelogarithmcoordinateaswhatshowninthestep1.
%P_cistheblockingprobabilityoferlangCmodel.
%foreachpart,calltheerlang_c()function.
p_1_c(j,i)=erlang_c(A_1(j),n_1(i));
%µ
÷
Ó
Ã
º
¯
Ê
ý
erlang_c
p_2_c(j,i)=erlang_c(A_2(j),n_2(i));
p_3_c(j,i)=erlang_c(A_3(j),n_3(i));
loglog(A_1,p_1_c,'
g*-'
A_2,p_2_c,'
A_3,p_3_c,'
text(6,.115,'
text(14,.115,'
text(20,.115,'
text(30,.115,'
40'
)
text(39,.115,'
text(47,.115,'
60'
text(55,.115,'
70'
text(65,.115,'
80'
text(75,.115,'
90'
text(85,.115,'
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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