排隊論的matlab仿真(包括仿真代碼)

1、WirelessNetworkExperimentThree:QueuingTheoryABSTRACTThisexperimentisdesignedtolearnthefundamentalsofthequeuingtheory.MainlyabouttheM/M/SandM/M/n/nqueuingmodels.KEYWORDS:queuingtheory,M/M/s,M/M/n/n,ErlangB,ErlangC.INTRODUCTIONAqueueisawaitinglineandqueueingtheoryisthemathematicaltheoryofwaitinglines.

2、Moregenerally,queueingtheoryisconcernedwiththemathematicalmodelingandanalysisofsystemsthatprovideservicetorandomdemands.Incommunicationnetworks,queuesareencounteredeverywhere.Forexample,theincomingdatapacketsarerandomlyarrivedandbuffered,waitingfortheroutertodeliver.Suchsituationisconsideredasaqueue

3、.Aqueueingmodelisanabstractdescriptionofsuchasystem.Typically,aqueueingmodelrepresents(1)thesystemsphysicalconfiguration,byspecifyingthenumberandarrangementoftheservers,and(2)thestochasticnatureofthedemands,byspecifyingthevariabilityinthearrivalprocessandintheserviceprocess.Theessenceofqueueingtheor

4、yisthatittakesintoaccounttherandomnessofthearrivalprocessandtherandomnessoftheserviceprocess.ThemostcommonassumptionaboutthearrivalprocessisthatthecustomerarrivalsfollowaPoissonprocess,wherethetimesbetweenarrivalsareexponentiallydistributed.Theprobabilityoftheexponentialdistributionfunctionisf(t)=入e

5、入t.ErlangBmodelOneofthemostimportantqueueingmodelsistheErlangBmodel(i.e.,M/M/n/n).ItassumesthatthearrivalsfollowaPoissonprocessandhaveafinitenservers.InErlangBmodel,itassumesthatthearrivalcustomersareblockedandclearedwhenalltheserversarebusy.TheblockedprobabilityofaErlangBmodelisgivenbythefamousErla

6、ngBformula,(I)PE(n3=,蛙、2k=i)kTwherenisthenumberofserversandA=A/istheofferedloadinErlangs,入isthearrivalrateand1/p.istheaverageservicetime.Formula(1.1)ishardtocalculatedirectlyfromitsrightsidewhennandAarelarge.However,itiseasytocalculateitusingthefollowingiterativescheme:ErlangCmodelTheErlangdelaymode

7、l(M/M/n)issimilartoErlangBmodel,exceptthatnowitassumesthatthearrivalcustomersarewaitinginaqueueforaservertobecomeavailablewithoutconsideringthelengthofthequeue.Theprobabilityofblocking(alltheserversarebusy)isgivenbytheErlangCformula,Wherep=1ifAnandp=AifAn.Thequantitypindicatestheserverutilization.nT

8、heErlangCformula(1.3)canbeeasilycalculatedbythefollowingiterativeschemewhereP(n,A)isdefinedinEq.(1.1).BDESCRIPTIONOFTHEEXPERIMENTSUsingtheformula(1.2),calculatetheblockingprobabilityoftheErlangBmodel.DrawtherelationshipoftheblockingprobabilityPB(n,A)andofferedtrafficAwithn=1,2,10,20,30,40,50,60,70,8

9、0,90,100.Compareitwiththetableinthetextbook(P.281,table10.3).Fromtheintroduction,weknowthatwhenthenandAarelarge,itiseasytocalculatetheblockingprobabilityusingtheformula1.2asfollows.P(n,A)=APB(n_Bm+APB(n-1,A)itusethetheoryofrecursionforthecalculation.Butthedenominatorandthenumeratoroftheformulabothne

10、edtorecurs(PB(n-1,A)whendoingthematlabcalculation,itwastetimeandreducethematlabcalculationefficient.Sowechangetheformulatobe:PB(n,A)=APPB(n,A)=APB(n1,A)nAPb(n1,A)1=1/(12nAPB(n_1AAPB(nAPB(n-1,A)1,A)Thenthecalculationonlyneedrecursoncetimeandismoreefficient.Thematlabcodefortheformulais:erlang_b.m%*%Fi

11、le:erlanb_b.m%A=offeredtrafficinErlangs.%n=numberoftrunckedchannels.%Pbistheresultblockingprobability.%*functionPb=erlang_b(A,n)ifn=0Pb=1;%P(0,A)=1elsePb=1/(1+n/(A*erlang_b(A,n-1);%userecursionerlang(A,n-1)endendAswecanseefromthetableonthetextbooks,itusesthelogarithmcoordinate,sowealsousethelogarith

12、mcoordinatetoplottheresult.Wedividethenumberofservers(n)intothreeparts,foreachpartwecandefineaintervalofthetrafficintensity(A)basedonthefigureonthetextbooks:1.when0n10,0.1A10.when10n20,3A20.when30n100,13A*rofTrunkdClia-Hntla(C13I4?tAnM-MiLBSM10.0血0.01-TrafficlEueniiEyisBrito*Nwal*rofTrunkdClia-Hntla

13、(C13I4?tAnM-MiLBSM10.0血0.01-TrafficlEueniiEyisBrito*Wecanseefromthetwopicturesthat,theyareexactlythesamewitheachotherexceptthattheresultoftheexperimenthavenotconsideredthesituationwithn=3,4,5,.,12,14,16,18.2.Usingtheformula(1.4),calculatetheblockingprobabilityoftheErlangCmodel.Drawtherelationshipoft

14、heblockingprobabilityPC(n,A)andofferedtrafficAwithn=1,2,10,20,30,40,50,60,70,80,90,100.Fromtheintroduction,weknowthattheformula1.4is:PCPC(n,A)=nPB(n,A)n-A(1-PB(n，A)SinceeachtimewecalculatetheP(n,A),weneedtorecursntimes,sotheformulaisnotBefficient.Wechangeittobe:PC(n,A)=nPB(n,A)n-A(1-pB(n，A)=1/n_A(1_

15、PB(n，A)PC(n,A)=nPB(n,A)n-A(1-pB(n，A)Thenweonlyneedrecursonce.PB(n,A)iscalculatedbythe“erlang_b”functionasstep1.Thematlabcodefortheformulais:erlang_c.m%*%File:erlanb_b.m%A=offeredtrafficinErlangs.%n=numberoftrunckedchannels.%Pbistheresultblockingprobability.%erlang_b(A,n)isthefunctionofstep1.%*functi

16、onPc=erlang_c(A,n)Pc=1/(A/n)+(n-A)/(n*erlang_b(A,n);endThentofigureoutthetableinthelogarithmcoordinateaswhatshowninthestep1.Thematlabcodeis:%*%forthethreeparts.%nisthenumberservers.%Aisthetrafficindensity.%P_cistheblockingprobabilityoferlangCmodel.%*n_1=1:2;A_1=linspace(0.1,10,50);%50pointsbetween0.

17、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);%pOA_Eyerlang_cendendfori=1:length(n_2)forj=1:length(A_2)p_2_c(j,i)=erlang_c(A_2(j),n_2(i);endendfori=1:len

18、gth(n_3)forj=1:length(A_3)p_3_c(j,i)=erlang_c(A_3(j),n_3(i);endend%*%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(

19、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.EBu住星口右xi-KFqaEBu住星口右xi-KFqa在ThenweputthetableoferlangBanderlangCintheonefigure,tocomparetheirc

20、haracteristic.E盂口in苫JTE盂口in苫JTiH-gqEd1D1io1Tr-sffic-ndensrlyinErbrig!爐iIfl3Thematlabcodeis:Thematlabcodeis:mms_function.mid1id1a1icrTraficindensfcyinErl-angsThelinewith*istheerlangCmodel,thelinewithout*istheerlangBmodel.Wecanseefromthepicturethat,foraconstanttrafficintensity(A),theerlangCmodelhasahi

21、gherblockingprobabilitythanerlangBmodel.Theblockingprobabilityisincreasingwithtrafficintensity.Thesystemperformsbetterwhenhasalargern.ADDITIONALBONUSWriteaprogramtosimulateaM/M/kqueuesystemwithinputparametersoflamda,mu,k.Inthispart,wewillfirstlysimulatetheM/M/kqueuesystemusematlabtogetthefigureofthe

22、performanceofthesystemsuchastheleavetimeofeachcustomerandthequeuelengthofthesystem.Aboutthesimulation,wefirstlycalculatethearrivetimeandtheleavetimeforeachcustomer.Thenanalysisoutthequeuelengthandthewaittimeforeachcustomeruse“for”loops.Thenwelettheinputtobelamda=3,mu=1andS=3,andanalysisperformanceof

23、thesystemforthefirst10customersindetail.Finally,wewilldotwotesttocomparedtheperformanceofthesystemwithinputlamda=1,mu=1andS=3andtheinputlamda=4,mu=1andS=3.functionblock_rate,use_rate=MMS_function(mean_arr,mean_serv,peo_num,server_num)%firstpart:computethearrivingtimeinterval,servicetime%interval,wai

24、tingtime,leavingtimeduringthewholeserviceinterval%state=zeros(5,peo_num);%representthestateofeachcustomerby%usinga5*peo_nummatrix%themeaningofeachlineis:arrivingtimeinterval,servicetime%interval,waitingtime,queuelengthwhenNO.ncustomer%arrive,leavingtimestate(1,:)=exprnd(1/mean_arr,1,peo_num);%arrivi

25、ngtimeintervalbetweeneach%customerfollowsexponetialdistributionstate(2,:)=exprnd(1/mean_serv,1,peo_num);%servicetimeofeachcustomerfollowsexponetialdistributionfori=1:server_numstate(3,1:server_num)=0;endarr_time=cumsum(state(1,:);%accumulatearrivingtimeintervaltocompute%arrivingtimeofeachcustomersta

26、te(1,:)=arr_time;state(5,1:server_num)=sum(state(:,1:server_num);%computelivingtimeoffirstNO.server_num%customerbyusingfomulararrivingtime+servicetimeserv_desk=state(5,1:server_num);%createavectortostoreleavingtimeofcustomerswhichisinservicefori=(server_num+1):peo_numifarr_time(i)min(serv_desk)state

27、(3,i)=0;elsestate(3,i)=min(serv_desk)-arr_time(i);%whencustomerNO.iarrivesandthe%serverisallbusy,thewaitingtimecanbecomputeby%minusarrivingtimefromtheminimumleavingtimeendstate(5,i)=sum(state(:,i);forj=1:server_numifserv_desk(j)=min(serv_desk)serv_desk(j)=state(5,i);breakend%replacetheminimumleaving

28、timebythefirstwaitingcustomersleavingtimeendend%secondpart:computethequeuelengthduringthewholeserviceinterval%zero_time=0;%zero_timeisusedtoidentifywhichserverisemptyserv_desk(1:server_num)=zero_time;block_num=0;block_line=0;fori=1:peo_numifblock_line=0find_max=0;forj=1:server_numifserv_desk(j)=zero

29、_timefind_max=1;%meansthereisemptyserverbreakelsecontinueendendiffind_max=1%updateserv_deskserv_desk(j)=state(5,i);fork=1:server_numifserv_desk(k)min(serv_desk)%ifacustomerwillleavebeforetheNO.i%customerarrivefork=1:server_numifarr_time(i)serv_desk(k)serv_desk(k)=state(5,i);breakelsecontinueendendfo

30、rk=1:server_numifarr_time(i)serv_desk(k)serv_desk(k)=zero_time;elsecontinueendendelse%ifnocustomerleavebeforetheNO.icustomerarriveblock_num=block_num+1;block_line=block_line+1;endendelse%thesituationthatthequeuelengthisnotzeron=0;%computethenumberofleaingcustomerbeforetheNO.icustomerarrivesfork=1:se

31、rver_numifarr_time(i)serv_desk(k)n=n+1;serv_desk(k)=zero_time;elsecontinueendendfork=1:block_lineifarr_time(i)state(5,i-k)n=n+1;elsecontinueendendifnblock_line+1%narr_time(i)form=1:server_numifserv_desk(m)=zero_timeserv_desk(m)=state(5,i-block_line+k)breakelsecontinueendendelsecontinueendendblock_li

32、ne=block_line-n+1;else%n=block_line+1meansthequeuelengthiszero%updateserv_deskandqueuelengthfork=0:block_lineifarr_time(i)state(5,i-k)form=1:server_numifserv_desk(m)=zero_timeserv_desk(m)=state(5,i-k)breakelsecontinueendendelsecontinueendendblock_line=0;endendstate(4,i)=block_line;endplot(state(1,:)

33、,*-);figureplot(state(2,:),g);figureplot(state(3,:),r*);figureplot(state(4,:),y*);figureplot(state(5,:),*-);SincethesystemisM/M/SwhichmeansthearrivingrateandserviceratefollowsPoissondistributionwhilethenumberofserverisSandthebufferlengthisinfinite,wecancomputeallthearrivingtime,servicetime,waitingti

34、meandleavingtimeofeachcustomer.Wecantestthecodewithinputlamda=3,mu=1andS=3.Figuresarebelow.Arrivingtimeofeachcustomercu-stamarnurnbar3DSD1D0ServicetimeofeachcustomerWaitingtimeofeachcustomer108?oQueuelengthwheneachcustomerarriveAslamda=mu*server_num,theloadofthesystemcouldbeveryhigh.Thenwewillzoomin

35、theresultpicturestoanalysistheperformanceofthesystemforthefirstly10customer.Arrivingtimeoffirst10customer1a=1.6-Thequeuelengthis1forthe7thcustomer.11a=1.6-Thequeuelengthis1forthe7thcustomer.1.衛(wèi)-12345673910customBrrtuniberQueuelengthoffirst10customermELe.e=-bh_-u1234567mELe.e=-bh_-u1234567B510cuetoiT

36、i&rhurriberLeavingtimeoffirst10customerAswehave3serverinthistest,thefirst3customerwillbeservedwithoutanydelay.Thearrivingtimeofcustomer4isabout1.4andtheminimumleavingtimeofcustomerinserviceisabout1.2.Socustomer4willbeservedimmediatelyandthequeuelengthisstill0.Customer1,4,3isinservice.Thearrivingtime

37、ofcustomer5isabout1.8andtheminimumleavingtimeofcustomerinserviceisabout1.6.Socustomer5willbeservedimmediatelyandthequeuelengthisstill0.Customer1,5isinservice.Thearrivingtimeofcustomer6isabout2.1andthereisaemptyserver.Socustomer6willbeservedimmediatelyandthequeuelengthisstill0.Customer1,5,6isinservic

38、e.Thearrivingtimeofcustomer7isabout2.2andtheminimumleavingtimeofcustomerinserviceisabout2.5.Socustomer7cannotbeservedimmediatelyandthequeuelengthwillbe1.Customer1,5,6isinserviceandcustomer7iswaiting.Thearrivingtimeofcustomer8isabout2.4andtheminimumleavingtimeofcustomerinserviceisabout2.5.Socustomer8cannotbeservedimmediatelyandthe