数据、模型与决策（第12版）课件 第12章 排队论

QueuingAnalysisChapter13Copyright©2016PearsonEducation,Inc.

ElementsofWaitingLineAnalysisTheSingle-ServerWaitingLineSystemUndefinedandConstantServiceTimesFiniteQueueLengthFiniteCallingPopulationTheMultiple-ServerWaitingLineAdditionalTypesofQueuingSystemsChapterTopicsCopyright©2016PearsonEducation,Inc.

Asignificantamountoftimeisspentinwaitinglinesbypeople,products,etc.Providingquickserviceisanimportantaspectofqualitycustomerservice.Thebasisofwaitinglineanalysisisthetrade-offbetweenthecostofimprovingserviceandthecostsassociatedwithmakingcustomerswait.Queuinganalysisisaprobabilisticformofanalysis.Theresultsarereferredtoasoperatingcharacteristics.Resultsareusedbymanagersofqueuingoperationstomakedecisions.OverviewCopyright©2016PearsonEducation,Inc.

Waitinglinesformbecausepeopleorthingsarriveataservicefasterthantheycanbeserved.Mostoperationshavesufficientservercapacitytohandlecustomersinthelongrun.Customershowever,donotarriveataconstantratenoraretheyservedinanequalamountoftime.ElementsofWaitingLineAnalysis(1of2)Copyright©2016PearsonEducation,Inc.

Waitinglinesarecontinuallyincreasinganddecreasinginlengthandapproachanaveragerateofcustomerarrivalsandanaverageservicetimeinthelongrun.Decisionsconcerningthemanagementofwaitinglinesarebasedontheseaveragesforcustomerarrivalsandservicetimes.Averagesareusedinformulastocomputeoperatingcharacteristicsofthesystemwhichinturnformthebasisofdecisionmaking.ElementsofWaitingLineAnalysis(2of2)Copyright©2016PearsonEducation,Inc.

Componentsofawaitinglinesysteminclude

arrivals(customers),servers,(cashregister/operator),customersinlineformawaitingline.Factorstoconsiderinanalysis: Thequeuediscipline. Thenatureofthecallingpopulation Thearrivalrate Theservicerate.TheSingle-ServerWaitingLineSystem(1of2)TheSingle-ServerWaitingLineSystem(2of2)Figure13.1TheFastShopMarketwaitinglinesystemCopyright©2016PearsonEducation,Inc.

QueueDiscipline:Theorderinwhichwaitingcustomersareserved.CallingPopulation:Thesourceofcustomers(infiniteorfinite).ArrivalRate:Thefrequencyatwhichcustomersarriveatawaitinglineaccordingtoaprobabilitydistribution(frequentlydescribedbyaPoissondistribution).ServiceRate:

Theaveragenumberofcustomersthatcanbeservedduringatimeperiod(oftendescribedbythenegativeexponentialdistribution).Single-ServerWaitingLineSystemComponentDefinitionsCopyright©2016PearsonEducation,Inc.

Assumptionsofthebasicsingle-servermodel: Aninfinitecallingpopulation Afirst-come,first-servedqueuediscipline Poissonarrivalrate ExponentialservicetimesSymbols:

=thearrivalrate(averagenumberofarrivals/timeperiod)

=theservicerate(averagenumberserved/timeperiod)Customersmustbeservedfasterthantheyarrive(

<

)oraninfinitelylargequeuewillbuildup.Single-ServerWaitingLineSystemSingle-ServerModelCopyright©2016PearsonEducation,Inc.

Probabilitythatnocustomersareinthequeuingsystem:Probabilitythatncustomersareinthesystem:Averagenumberofcustomersinsystem:Averagenumberofcustomerinthewaitingline:Single-ServerWaitingLineSystemBasicSingle-ServerQueuingFormulas(1of2)Copyright©2016PearsonEducation,Inc.

Averagetimecustomerspendswaitingandbeingserved:Averagetimecustomerspendswaitinginthequeue:Probabilitythatserverisbusy(utilizationfactor):Probabilitythatserverisidle:Single-ServerWaitingLineSystemBasicSingle-ServerQueuingFormulas(2of2)Copyright©2016PearsonEducation,Inc.

=24customersperhourarriveatcheckoutcounter

=30customersperhourcanbecheckedout

Single-ServerWaitingLineSystemOperatingCharacteristics:FastShopMarket(1of2)Copyright©2016PearsonEducation,Inc.

Single-ServerWaitingLineSystemOperatingCharacteristicsforFastShopMarket(2of2)Copyright©2016PearsonEducation,Inc.

Single-ServerWaitingLineSystemSteady-StateOperatingCharacteristicsBecauseofthesteady-statenatureofoperatingcharacteristics: Utilizationfactor,U,mustbelessthanone:U<1,or

/

<1and

<

.Theratioofthearrivalratetotheserviceratemustbelessthanone.Inotherwords,theserviceratemustbegreaterthanthearrivalrate. Theservermustbeabletoservecustomersfasterthan thearrivalrateinthelongrun,orwaitinglinewillgrow toinfinitesize.Copyright©2016PearsonEducation,Inc.

Amanagerwishestotestseveralalternativesforreducingcustomerwaitingtime:AdditionofanotheremployeetopackuppurchasesAdditionofanothercheckoutcounter.Alternative1:Additionofanemployee (raisesserviceratefrom

=30to

=40customersperhour).Cost$150perweek,avoidslossof$75perweekforeachminuteofreducedcustomerwaitingtime.Systemoperatingcharacteristicswithnewparameters: Po=.40probabilityofnocustomersinthesystem L=1.5customersonaverageinthequeuingsystem

Single-ServerWaitingLineSystemEffectofOperatingCharacteristics(1of6)Copyright©2016PearsonEducation,Inc.

Systemoperatingcharacteristicswithnewparameters (continued): Lq=0.90customerontheaverageinthewaitingline W=0.063houraveragetimeinthesystempercustomer Wq=0.038houraveragetimeinthewaitinglinepercustomer U=.60probabilitythatserverisbusyandcustomermustwait I=.40probabilitythatserverisavailable Averagecustomerwaitingtimereducedfrom8to2.25minutesworth$431.25perweek.

Single-ServerWaitingLineSystemEffectofOperatingCharacteristics(2of6)Copyright©2016PearsonEducation,Inc.

Alternative2:Additionofanewcheckoutcounter($6,000plus$200perweekforadditionalcashier).

=24/2=12customersperhourpercheckoutcounter

=30customersperhourateachcounterSystemoperatingcharacteristicswithnewparameters: Po=.60probabilityofnocustomersinthesystem L=0.67customerinthequeuingsystem Lq=0.27customerinthewaitingline W=0.055hourpercustomerinthesystem Wq=0.022hourpercustomerinthewaitingline U=.40probabilitythatacustomermustwait I=.60probabilitythatserverisidle Single-ServerWaitingLineSystemEffectofOperatingCharacteristics(3of6)Copyright©2016PearsonEducation,Inc.

Savingsfromthereducedwaitingtimeworth: $500perweek-$200=$300netsavingsperweek.After$6,000isrecovered,alternative2wouldprovide: $300-281.25=$18.75moresavingsperweek.

Single-ServerWaitingLineSystemEffectofOperatingCharacteristics(4of6)Copyright©2016PearsonEducation,Inc.

Table13.1OperatingcharacteristicsforeachalternativesystemSingle-ServerWaitingLineSystemEffectofOperatingCharacteristics(5of6)Copyright©2016PearsonEducation,Inc.

Figure13.2Costtrade-offforservicelevelsSingle-ServerWaitingLineSystemEffectofOperatingCharacteristics(6of6)Copyright©2016PearsonEducation,Inc.

Exhibit13.1Single-ServerWaitingLineSystemSolutionwithExcelandExcelQM(1of2)FormulaforLq,averagenumberinqueue=(1/(D4-D3))*60=(D3/(D4*(D4-D3)))*60Copyright©2016PearsonEducation,Inc.

Exhibit13.2Single-ServerWaitingLineSystemSolutionwithExcelandExcelQM(2of2)Clickon“ExcelQM”toaccessthe“Chapter”menuCopyright©2016PearsonEducation,Inc.

Exhibit13.3Single-ServerWaitingLineSystemSolutionwithQMforWindowsCopyright©2016PearsonEducation,Inc.

Constant,ratherthanexponentiallydistributedservicetimes,occurwithmachineryandautomatedequipment.Constantservicetimesareaspecialcaseofthesingle-servermodelwithundefinedservicetimes.Queuingformulasfortheundefinedservicetimemodel:Single-ServerWaitingLineSystemUndefinedandConstantServiceTimesCopyright©2016PearsonEducation,Inc.

Data:Singlefaxmachine;arrivalrateof20usersperhour,Poissondistributed;undefinedservicetimewithmeanof2minutes,standarddeviationof4minutes.Operatingcharacteristics:Single-ServerWaitingLineSystemUndefinedServiceTimesExample(1of2)Copyright©2016PearsonEducation,Inc.

Operatingcharacteristics(continued):Single-ServerWaitingLineSystemUndefinedServiceTimesExample(2of2)Copyright©2016PearsonEducation,Inc.

Intheconstantservicetimemodelthereisnovariabilityinservicetimes;=0.Substituting=0intoequations:Alloftheremainingformulasarethesameasthesingle-serverformulas.Single-ServerWaitingLineSystemConstantServiceTimesFormulasCopyright©2016PearsonEducation,Inc.

Carwashservicingonecaratatime;constantservicetimeof4.5minutes;arrivalrateofcustomersof10perhour(Poissondistributed).Determineaveragelengthofwaitinglineandaveragewaitingtime.

=10carsperhour,

=60/4.5=13.3carsperhourSingle-ServerWaitingLineSystemConstantServiceTimesExampleCopyright©2016PearsonEducation,Inc.

Exhibit13.4UndefinedandConstantServiceTimesSolutionwithExcelAveragenumberinthequeue,Lq=D8+(1/D4)*60=(D6/D3)*60Copyright©2016PearsonEducation,Inc.

Exhibit13.5UndefinedandConstantServiceTimesSolutionwithQMforWindowsCopyright©2016PearsonEducation,Inc.

Inafinitequeue,thelengthofthequeueislimited.Operatingcharacteristics,whereMisthemaximumnumberinthesystem:FiniteQueueLengthCopyright©2016PearsonEducation,Inc.

MetroQuickLubesinglebayservice;spaceforonevehicleinserviceandthreewaitingforservice;meantimebetweenarrivalsofcustomersis3minutes;meanservicetimeis2minutes;bothinter-arrivaltimesandservicetimesareexponentiallydistributed;maximumnumberofvehiclesinthesystemequals4.Operatingcharacteristicsfor

=20,

=30,M=4:FiniteQueueLengthExample(1of2)Copyright©2016PearsonEducation,Inc.

Averagequeuelengthsandwaitingtimes:FiniteQueueLengthExample(2of2)Copyright©2016PearsonEducation,Inc.

Exhibit13.6FiniteQueueModelExampleSolutionwithExcelFormulaforP0incellD7+((D3/D4)/(1-(D3/D4)))-((D5+1)*(D3/D4)^(D5+1))/(1-(D3/D4)^(D5+1))Copyright©2016PearsonEducation,Inc.

Exhibit13.7FiniteQueueModelExampleSolutionwithQMforWindowsCopyright©2016PearsonEducation,Inc.

Inafinitecallingpopulationthereisalimitednumberofpotentialcustomersthatcancallonthesystem.OperatingcharacteristicsforasystemwithPoissonarrivalandexponentialservicetimes:FiniteCallingPopulationCopyright©2016PearsonEducation,Inc.

WheelcoManufacturingCompany;20machines;eachmachineoperatesanaverageof200hoursbeforebreakingdown;averagetimetorepairis3.6hours;breakdownrateisPoissondistributed,servicetimeisexponentiallydistributed.Isrepairstaffsufficient?

=1/200hour=.005perhour

=1/3.6hour=.2778perhour N=20machinesFiniteCallingPopulationExample(1of2)Copyright©2016PearsonEducation,Inc.

…Thesystemiswoefullyinadequate.FiniteCallingPopulationExample(2of2)Copyright©2016PearsonEducation,Inc.

Exhibit13.8FiniteCallingPopulationExampleSolutionwithExcelandExcelQM(1of2)P0=1/G26Summationcomponentforn=1incellG6ArraywithsummationcomponentsforP0formulaCopyright©2016PearsonEducation,Inc.

Exhibit13.9FiniteCallingPopulationExampleSolutionwithExcelandExcelQM(2of2)Clickon“ExcelQM”toaccessthemacroforthefinitepopulationmodelCopyright©2016PearsonEducation,Inc.

Exhibit13.10FiniteCallingPopulationExampleSolutionwithQMforWindowsCopyright©2016PearsonEducation,Inc.

Multiple-ServerWaitingLine(1of3)Figure13.3Amultiple-serverwaitinglineCopyright©2016PearsonEducation,Inc.

Inmultiple-servermodels,twoormoreindependentserversinparallelserveasinglewaitingline.BiggsDepartmentStoreservicedepartment;first-come,first-servedbasis.Multiple-ServerWaitingLine(2of3)Copyright©2016PearsonEducation,Inc.

Multiple-ServerWaitingLineQueuingFormulas(1of3)Assumptions: First-comefirst-servedqueuediscipline Poissonarrivals,exponentialservicetimes Infinitecallingpopulation.Parameterdefinitions:

=arrivalrate(averagenumberofarrivalspertimeperiod)

=theservicerate(averagenumberservedpertimeperiod)perserver(channel)c=numberofserversc

=meaneffectiveservicerateforthesystem(must exceedarrivalrate)Copyright©2016PearsonEducation,Inc.

Multiple-ServerWaitingLineQueuingFormulas(2of3)Copyright©2016PearsonEducation,Inc.

Multiple-ServerWaitingLineQueuingFormulas(3of3)Copyright©2016PearsonEducation,Inc.

Multiple-ServerWaitingLineBiggsDepartmentStoreExample(1of2)

=10,

=4,c=3Copyright©2016PearsonEducation,Inc.

Multiple-ServerWaitingLineBiggsDepartmentStoreExample(2of2)Copyright©2016PearsonEducation,Inc.

Exhibit13.11Multiple-ServerWaitingLineSolutionwithExcelFormulaforP0=((((D3)*(D4)*((D3/D4)^D5)*(D7))/(FACT(D5-1)*(((D5*D4)-D3)^2))))+(D3/D4)=(1/FACT(D5))*((D3/D4)^D5)*((D5*D4*D7)/((D5)*(D4)-(D3)))Copyright©2016PearsonEducation,Inc.

Exhibit13.12Multiple-ServerWaitingLineSolutionwithExcelQMCopyright©2016PearsonEducation,Inc.

Exhibit13.13Multiple-ServerWaitingLineSolutionwithQMforWindowsCopyright©2016PearsonEducation,Inc.

Figure13.4Singlequeueswithsingle&multipleserversinsequenceAdditionalTypesofQueuingSystems(1of2)Copyright©2016PearsonEducation,Inc.

Otheritemscontributingtoqueuingsystems:

Systemsinwhichcustomersbalk

fromenteringsystem,orleavetheline(renege).

Serverswhoprovideserviceinotherthanafirst-come, first-servedmannerServicetimesthatarenotexponentiallydistributedorareundefinedorconstantArrivalratesthatarenotPoissondistributed

Jockeying(i.e.,movingbetweenqueues)AdditionalTypesofQueuingSystems(2of2)Copyright©2016PearsonEducation,Inc.

ProblemStatement:CitizensNorthernSavingsBankloanofficercustomerinterviews. Customerarrivalrateoffourperhour,Poissondistributed;officerinterviewservicetimeof12minutespercustomer.

Determineoperatingcharacteristicsforthissystem.Addanadditionalofficercreatingamultiple-serverqueuingsystemwithtwochannels.Determineoperatingcharacteristicsforthissystem.ExampleProblemSolution(1of7)Copyright©2016PearsonEducation,Inc.

Solution: Step1:DetermineOperatingCharacteristicsfortheSingle-ServerSystem

=4customersperhourarrive,

=5customersper hourareserved Po=(1-

/

)=(1–4/5)=.20probabilityofno