数据模型与决策(运筹学)课后习题和案例答案014s

CD14s-1CDSUPPLEMENTTOCHAPTER14THEFINITEQUEUEANDFINITECALLINGPOPULATIONVARIATIONSOFTHEM/M/SMODELReviewQuestions14s-1WhenthesystemisfullwithKcustomers,anynewarrivingcustomersleavewithoutenteringthesystem.14s-2PKistheprobabilitythatthesystemisfullandthereforetheprobabilitythatanarrivingcustomeristurnedaway.14s-3Averagefractionoftimethatserversarebusy=(1Pk)s.14s-4Thecallingpopulationforaqueueingsystemisthepopulationofallpotentialcustomerswhomightneedtocometothesystemforservice.Afinitecallingpopulationisonethatissmallenoughthatthenumberofcustomersalreadyinthequeueingsystemaffectsthemeanarrivalrateofnewcustomers.14s-5Uponcompletingservice,thetimeuntilamemberofthecallingpopulationneedsserviceagainhasanexponentialdistributionwithameanofN/.14s-6NnNPnn0N.14s-7Utilizationofservers=s.14s-8No,thisapproximationshouldnotbeusedwhenisnearly1unlessKorNishuge.CD14s-2Problems14s.1a)With0spaces,42.9%ofcustomerswillbelost.1234567891011121314ABCDEFGTemplateforM/M/sFiniteQueueModelDataResults0.25(meanarrivalrate)L=0.42857140.333(meanservicerate)Lq=0s=1(#servers)K=1(maxcustomers)W=3Wq=00.75nPn00.571428610.4285714b)With2spaces,15.4%ofcustomerswillbelost.12345678910111213141516ABCDEFGTemplateforM/M/sFiniteQueueModelDataResults0.25(meanarrivalrate)L=1.14857140.333(meanservicerate)Lq=0.5142857s=1(#servers)K=3(maxcustomers)W=5.4324324Wq=2.43243240.75nPn00.365714310.274285720.205714330.1542857CD14s-3c)With4spaces,7.2%ofcustomerswillbelost.123456789101112131415161718ABCDEFGTemplateforM/M/sFiniteQueueModelDataResults0.25(meanarrivalrate)L=1.70092070.333(meanservicerate)Lq=1.005049s=1(#servers)K=5(maxcustomers)W=7.3329065Wq=4.33290650.75nPn00.304128310.228096220.171072230.128304140.096228150.072171114s.2a&b)Withspacefor2cars,21.1%ofcustomerswillbelost,L=0.74cars,andW=2.8minutes.123456789101112131415ABCDEFGHTemplateforM/M/sFiniteQueueModelDataResults20(meanarrivalrate)L=0.736842130(meanservicerate)Lq=0.2105263s=1(#servers)minutesK=2(maxcustomers)W=0.04666672.8Wq=0.01333330.80.6666667nPn00.473684210.315789520.2105263CD14s-4Withspacefor3cars,12.3%ofcustomerswillbelost,L=1.02cars,andW=3.47minutes.12345678910111213141516ABCDEFGHTemplateforM/M/sFiniteQueueModelDataResults20(meanarrivalrate)L=1.015384630(meanservicerate)Lq=0.4307692s=1(#servers)minutesK=3(maxcustomers)W=0.05789473.47Wq=0.02456141.470.6666667nPn00.415384610.276923120.184615430.1230769Withspacefor4cars,7.6%ofcustomerswillbelost,L=1.24cars,andW=4.03minutes.1234567891011121314151617ABCDEFGHTemplateforM/M/sFiniteQueueModelDataResults20(meanarrivalrate)L=1.241706230(meanservicerate)Lq=0.6255924s=1(#servers)minutesK=4(maxcustomers)W=0.06717954.03Wq=0.03384622.030.6666667nPn00.383886310.255924220.170616130.113744140.0758294CD14s-5Withspacefor5cars,4.8%ofcustomerswillbelost,L=1.42cars,andW=4.48minutes.123456789101112131415161718ABCDEFGHTemplateforM/M/sFiniteQueueModelDataResults20(meanarrivalrate)L=1.422556430(meanservicerate)Lq=0.7879699s=1(#servers)minutesK=5(maxcustomers)W=0.07472354.48Wq=0.04139022.480.6666667nPn00.365413510.24360920.16240630.108270740.072180550.0481203c)spacesratecustomersarelost(Pk)changeinPkprofit/hour($4)()(1–Pk)changeinprofit/hour20.21$63.2030.120.09$70.40$7.2040.080.04$73.60$3.2050.050.03$76.00$2.40d)Sinceitcost$200permonthpercarlengthrented,eachadditionalspacemustbringatleast$200permonth(or$1perhour)inadditionalprofit.Fivespacesstillbringmorethanthatso5shouldbeprovided.14s.312345678910111213141516ABCDEFGHTemplateforM/M/sFiniteQueueModelDataResults15(meanarrivalrate)L=115(meanservicerate)Lq=0.0909091s=2(#servers)minutesK=3(maxcustomers)W=0.07333334.40Wq=0.00666670.400.5nPn00.363636410.363636420.181818230.0909091CD14s-6a)Acallwillbeansweredimmediatelyifthereareoneorfewercustomersinthesystem:P0+P1=72.7%.b)Acallwillbeputonholdwhenthereareexactlytwocustomersinthesystem:P2=18.2%.c)Acallerwillgetabusysignalwhenthereare3custoemrsinthesystem:P3=9.1%.14s.4a)TheM/M/smodelwithafinitecallingpopulationfitsthisqueueingsystem.b)1234567891011121314151617ABCDEFGTemplateforM/M/sFiniteCallingPopulationModelDataResults0.333(maxarrivalrate)L=0.7180527380.5(meanservicerate)Lq=0.210953347s=1(#servers)N=3(sizeofpopulation)W=2.832Wq=0.8320.666666667-bar=0.2535496960nPn100.492900609210.328600406320.146044625430.032454361Theprobabilitiesthatthereare0,1,2,or3machinesnotrunningareP0,P1,P2,andP3respectivelyasshowninthespreadsheetabove.ThemeanofthisdistributionisL=0.718asshownabove.c)WL0.7180.253=2.832hours.d)Theexpectedfractionoftimethattherepairtechnicianwillbebusyisthesystemutilization,whichis0.667.CD14s-7e)M/M/smodel:12345678910111213ABCDEGHTemplatefortheM/M/sQueueingModelDataResults0.33333333(meanarrivalrate)L=20.5(meanservicerate)Lq=1.333333333s=1(#servers)W=6Pr(Wt)=0.84648172Wq=4whent=10.666666667Prob(Wqt)=0.56432115whent=1nPn00.333333333FinitequeuevariationoftheM/M/smodelwithK=3:12345678910111213141516ABCDEFGHTemplateforM/M/sFiniteQueueModelDataResults0.33333333(