又稱隨機(jī)服務(wù)系統(tǒng)理論課件

Chapter14.

QueuingModels第十四章.排隊(duì)模型運(yùn)籌學(xué)排隊(duì)模型Chapter14.

QueuingModels第十四現(xiàn)實(shí)生活中的排隊(duì)模型BankAirportHospitalRoadManufacturingHotelRestaurantWC……運(yùn)籌學(xué)排隊(duì)模型現(xiàn)實(shí)生活中的排隊(duì)模型Bank運(yùn)籌學(xué)排隊(duì)模型存在的問題如何以最經(jīng)濟(jì)的方式控制排隊(duì)系統(tǒng)，使其達(dá)到特定的要求？提供過多的服務(wù)能力來控制排隊(duì)系統(tǒng)將會造成過量的成本提供的服務(wù)能力不足將會導(dǎo)致過多的等待，降低顧客滿意度并造成顧客流失，減少收益排隊(duì)模型存在的問題如何以最經(jīng)濟(jì)的方式控制排隊(duì)系統(tǒng)，使其達(dá)到特什么是排隊(duì)論什么是排隊(duì)論?什么是排隊(duì)論什么是什么是排隊(duì)論排隊(duì)論是研究排隊(duì)系統(tǒng)的理論，又稱隨機(jī)服務(wù)系統(tǒng)理論，它提供了很多不同的排隊(duì)模型，通過這些排隊(duì)模型能夠找到服務(wù)成本和服務(wù)水什么是排隊(duì)論排隊(duì)論是研究排隊(duì)系A(chǔ)BasicQueuingSystemABasicQueuingSystem排隊(duì)系統(tǒng)的描述涉及的要素顧客隊(duì)列服務(wù)臺到達(dá)間隔時(shí)間服務(wù)時(shí)間排隊(duì)規(guī)則運(yùn)籌學(xué)排隊(duì)模型排隊(duì)系統(tǒng)的描述涉及的要素運(yùn)籌學(xué)排隊(duì)系統(tǒng)的描述績效測度等待顧客數(shù)顧客等待時(shí)間服務(wù)臺忙期服務(wù)臺閑期服務(wù)臺利用率運(yùn)籌學(xué)排隊(duì)模型排隊(duì)系統(tǒng)的描述績效測度運(yùn)籌學(xué)TableofContents(主要內(nèi)容)ElementsofaQueuingModel(Section14.1)(排隊(duì)模型的要素)SomeExamplesofQueuingSystems(Section14.2)(排隊(duì)系統(tǒng)的一些實(shí)例)MeasuresofPerformanceforQueuingSystems(Section14.3)(排隊(duì)系統(tǒng)的績效測度)TableofContents(主要內(nèi)容)ElemenTableofContents(主要內(nèi)容)ACaseStudy:TheDupitCorp.Problem(Section14.4)(案例研究：杜皮特公司問題)SomeSingle-ServerQueuingModels(Section14.5)(一些單服務(wù)臺排隊(duì)模型)SomeMultiple-ServerQueuingModels(Section14.6)(一些多服務(wù)排隊(duì)模型)TableofContents(主要內(nèi)容)ACaseTableofContents(主要內(nèi)容)PriorityQueuingModels(Section14.7)(有優(yōu)先權(quán)的排隊(duì)模型)SomeInsightsaboutDesigningQueuingSystems(Section14.8)(關(guān)于設(shè)計(jì)排隊(duì)系統(tǒng)的一些啟示)EconomicAnalysisoftheNumberofServerstoProvide(Section14.9)(服務(wù)臺數(shù)量的經(jīng)濟(jì)分析)TableofContents(主要內(nèi)容)PrioriHerrCutter’sBarberShopHerrCutterisaGermanbarberwhorunsaone-manbarbershop.(赫爾卡特先生是一個(gè)德國理發(fā)師，開了一家單人理發(fā)店)HerrCutteropenshisshopat8:00A.M.(赫爾卡特每天早上8點(diǎn)開始工作)Thetableshowshisqueuingsysteminactionoveratypicalmorning.(下表顯示了一個(gè)典型的上午排隊(duì)系統(tǒng)情況)HerrCutter’sBarberShopHerrHerrCutter’sBarberShopHerrCutter’sBarberShopArrivals(到達(dá))Thetimebetweenconsecutivearrivalstoaqueuingsystemarecalledtheinterarrivaltimes.(連續(xù)兩個(gè)顧客到達(dá)排隊(duì)系統(tǒng)的時(shí)間間隔稱為到達(dá)間隔時(shí)間)Theexpectednumberofarrivalsperunittimeisreferredtoasthemeanarrivalrate.(單位時(shí)間到達(dá)的期望數(shù)量稱為平均到達(dá)率)Arrivals(到達(dá))ThetimebetweencArrivals(到達(dá))Thesymbolusedforthemeanarrivalrateis(平均到達(dá)率的符號如下)Themeanoftheprobabilitydistributionofinterarrivaltimesis(間隔時(shí)間的概率分布均值是)Arrivals(到達(dá))ThesymbolusedfoArrivals(到達(dá))Mostqueuingmodelsassumethattheformoftheprobabilitydistributionofinterarrivaltimesisanexponentialdistribution.(大多數(shù)排隊(duì)模型假設(shè)到達(dá)間隔時(shí)間的概率分布形式是指數(shù)分布)Arrivals(到達(dá))MostqueuingmodelEvolutionoftheNumberofCustomersEvolutionoftheNumberofCusTheExponentialDistributionforInterarrivalTimesTheExponentialDistributionfThereisahighlikelihoodofsmallinterarrivaltimes,butasmallchanceofaverylargeinterarrivaltime.Thisischaracteristicofinterarrivaltimesinpractice.(小間隔時(shí)間出現(xiàn)的可能性很大，而大間隔時(shí)間出現(xiàn)的機(jī)會則很小，這個(gè)間隔時(shí)間的特征能在實(shí)踐中觀察到)PropertiesoftheExponentialDistributionThereisahighlikelihoodof指數(shù)分布到達(dá)間隔時(shí)間服從指數(shù)分布則到達(dá)過程服從泊松分布指數(shù)分布到達(dá)間隔時(shí)間服從指數(shù)分布則到達(dá)過程服從泊松分布泊松分布泊松分布Formostqueuingsystems,theservershavenocontroloverwhencustomerswillarrive.Customersgenerallyarriverandomly.(對于大多數(shù)排隊(duì)系統(tǒng)，服務(wù)臺無法控制顧客何時(shí)到達(dá)，顧客的到達(dá)一般是隨機(jī)的)PropertiesoftheExponentialDistributionFormostqueuingsystems,theHavingrandomarrivalsmeansthatinterarrivaltimesarecompletelyunpredictable,inthesensethatthechanceofanarrivalinthenextminuteisalwaysjustthesame.(隨機(jī)到達(dá)意味著到達(dá)時(shí)間完全是不可預(yù)測的，下一分鐘有顧客到達(dá)的概率和其他分鐘相同)PropertiesoftheExponentialDistributionHavingrandomarrivalsmeanstTheonlyprobabilitydistributionwiththispropertyofrandomarrivalsistheexponentialdistribution.(唯一符合隨機(jī)到達(dá)的到達(dá)間隔時(shí)間分布是指數(shù)分布)Thefactthattheprobabilityofanarrivalinthenextminuteiscompletelyuninfluencedbywhenthelastarrivaloccurrediscalledthelack-of-memoryproperty.(下一分鐘到達(dá)的概率完全不受上一次到達(dá)的影響的事實(shí)稱為無記憶性)PropertiesoftheExponentialDistributionTheonlyprobabilitydistributThenumberofcustomersinthequeue(orqueuesize)isthenumberofcustomerswaitingforservicetobegin.(隊(duì)列中的顧客數(shù)也稱隊(duì)列大小或隊(duì)列長度，是等候服務(wù)的顧客數(shù)量)Thenumberofcustomersinthesystemisthenumberinthequeueplusthenumbercurrentlybeingserved.(系統(tǒng)中的顧客數(shù)是隊(duì)列中的顧客數(shù)加上正在接受服務(wù)的顧客數(shù)量)TheQueue(隊(duì)列)ThenumberofcustomersintheThequeuecapacityisthemaximumnumberofcustomersthatcanbeheldinthequeue.(隊(duì)列容量是隊(duì)列所能容納的最大顧客數(shù)量)Aninfinitequeueisoneinwhich,forallpracticalpurposes,anunlimitednumberofcustomerscanbeheldthere.(無限隊(duì)列是為了應(yīng)用方便，可以容納無限顧客數(shù)量的隊(duì)列)TheQueue(隊(duì)列)ThequeuecapacityisthemaxiWhenthecapacityissmallenoughthatitneedstobetakenintoaccount,thenthequeueiscalledafinitequeue.(當(dāng)容量很小，需要將其考慮在內(nèi)時(shí)，這個(gè)隊(duì)列就稱為有限隊(duì)列)TheQueue(隊(duì)列)WhenthecapacityissmallenoThequeuediscipline

referstotheorderinwhichmembersofthequeueareselectedtobeginservice.(排隊(duì)規(guī)則指選擇隊(duì)列中的成員接受服務(wù)的順序)Themostcommonisfirst-come,first-served(FCFS).(最普通的排隊(duì)規(guī)則是先到先服務(wù))Otherpossibilitiesincluderandomselection,somepriorityprocedure,orevenlast-come,first-served.(其他的可能規(guī)則包括隨機(jī)選擇、有優(yōu)先權(quán)的服務(wù)甚至后到先服務(wù))TheQueue(隊(duì)列)Thequeuediscipline

referstoWhenacustomerentersservice,theelapsedtimefromthebeginningtotheendoftheserviceisreferredtoastheservicetime.(當(dāng)一個(gè)顧客接受服務(wù)時(shí)，從服務(wù)開始到服務(wù)結(jié)束經(jīng)過的時(shí)間稱為服務(wù)時(shí)間)Basicqueuingmodelsassumethattheservicetimehasaparticularprobabilitydistribution.(基本的排隊(duì)模型假設(shè)服務(wù)時(shí)間是一個(gè)特定的概率分布)Service(服務(wù))WhenacustomerentersserviceThesymbolusedforthemeanoftheservicetimedistributionis(用于表示服務(wù)時(shí)間分布均值的符號是)Service(服務(wù))Theinterpretationofitselfisthemeanservicerate.(就是平均服務(wù)率) =Expectedservicecompletionsperunittimeforasinglebusyserver(一個(gè)連續(xù)工作的服務(wù)臺單位時(shí)間完成的服務(wù)量的期望值)ThesymbolusedforthemeanoSomeService-TimeDistributionsExponentialDistribution(指數(shù)分布)Themostpopularchoice.(最常用的分布形式)Mucheasiertoanalyzethananyother.(比其它分布更容易分析)Althoughitprovidesagoodfitforinterarrivaltimes,thisismuchlesstrueforservicetimes.(盡管指數(shù)分布很適合描述到達(dá)間隔時(shí)間，但是卻不太符合服務(wù)時(shí)間的特點(diǎn))SomeService-TimeDistributionSomeService-TimeDistributionsProvidesabetterfitwhentheserviceprovidedisrandomthanifitinvolvesafixedsetoftasks.(服務(wù)是隨機(jī)時(shí)，指數(shù)分布比較適合；服務(wù)包含一系列固定任務(wù)時(shí)，指數(shù)分布不太適合)Standarddeviation:s=Mean(標(biāo)準(zhǔn)差等于均值)ConstantServiceTimes(固定服務(wù)時(shí)間)Abetterfitforsystemsthatinvolveafixedsetoftasks.(更適合于包含一系列固定任務(wù)的系統(tǒng))Standarddeviation(標(biāo)準(zhǔn)差):s=0.SomeService-TimeDistributionSomeService-TimeDistributionsErlangDistribution(愛爾朗分布)Fillsthemiddlegroundbetweentheexponentialdistributionandconstant.(服務(wù)時(shí)間的波動量介于指數(shù)分布和常量之間)Hasashapeparameter,kthatdeterminesthestandarddeviation.(有一個(gè)形狀參數(shù)k決定其標(biāo)準(zhǔn)差)SomeService-TimeDistribution愛爾郎分布愛爾郎分布ErlangDistribution(愛爾朗分布)ErlangDistribution(愛爾朗分布)StandardDeviationandMeanforDistributionsStandardDeviationandMeanfoLabelsforQueuingModelsToidentifywhichprobabilitydistributionisbeingassumedforservicetimes(andforinterarrivaltimes),aqueuingmodelconventionallyislabeledasfollows:(為了表示服務(wù)時(shí)間[以及到達(dá)間隔時(shí)間]服從什么概率分布，基本排隊(duì)系統(tǒng)的排隊(duì)模型通常用如下的符號表示)LabelsforQueuingModelsToidLabelsforQueuingModelsThesymbolsusedforthepossibledistributionsare(用于表示可能的分布的符號是)M=Exponentialdistribution(Markovian)指數(shù)分布(馬爾科夫)D=Degeneratedistribution(constanttimes)退化分布(固定時(shí)間)運(yùn)籌學(xué)排隊(duì)模型LabelsforQueuingModelsThesLabelsforQueuingModelsEk=Erlangdistribution(shapeparameter=k)(愛爾朗分布[形態(tài)參數(shù)=k])GI=Generalindependentinterarrival-timedistribution(anydistribution)(獨(dú)立的到達(dá)間隔時(shí)間的一般分布[允許任意類型的分布])G=Generalservice-timedistribution(anyarbitrarydistribution)(服務(wù)時(shí)間的一般分布[允許任意類型的分布])LabelsforQueuingModelsEk=SummaryofUsualModelAssumptionsInterarrivaltimesareindependentandidenticallydistributedaccordingtoaspecifiedprobabilitydistribution.(到達(dá)間隔時(shí)間獨(dú)立同分布)Allarrivingcustomersenterthequeuingsystemandremainthereuntilservicehasbeencompleted.(所有到達(dá)的顧客都進(jìn)入排隊(duì)系統(tǒng)，并待在那里直到服務(wù)結(jié)束)SummaryofUsualModelAssumptSummaryofUsualModelAssumptionsThequeuingsystemhasasingleinfinitequeue,sothatthequeuewillholdanunlimitednumberofcustomers(forallpracticalpurposes).(排隊(duì)系統(tǒng)有一個(gè)無限隊(duì)列，因此隊(duì)列將可以容納無限量的顧客[出于應(yīng)用的原因])Thequeuedisciplineisfirst-come,first-served.(排隊(duì)規(guī)則是先到先服務(wù))SummaryofUsualModelAssumptSummaryofUsualModelAssumptionsThequeuingsystemhasaspecifiednumberofservers,whereeachserveriscapableofservinganyofthecustomers.(排隊(duì)系統(tǒng)擁有特定數(shù)量的服務(wù)臺，每一個(gè)服務(wù)臺能夠?yàn)槿我庖晃活櫩吞峁┓?wù))SummaryofUsualModelAssumptSummaryofUsualModelAssumptionsEachcustomerisservedindividuallybyanyoneoftheservers.(每一位顧客由一個(gè)服務(wù)臺單獨(dú)提供服務(wù))Servicetimesareindependentandidenticallydistributedaccordingtoaspecifiedprobabilitydistribution.(服務(wù)時(shí)間是獨(dú)立的，服從特定的概率分布)SummaryofUsualModelAssumptExamplesofCommercialServiceSystemsthatAreQueuingSystemsExamplesofCommercialServiceExamplesofInternalServiceSystemsThatAreQueuingSystemsExamplesofInternalServiceSExamplesofTransportationServiceSystemsThatAreQueuingSystemsExamplesofTransportationSerChoosingaMeasureofPerformanceManagerswhooverseequeuingsystemsaremainlyconcernedwithtwomeasuresofperformance:(檢查排隊(duì)系統(tǒng)的管理人員主要考慮兩種績效測度)Howmanycustomerstypicallyarewaitinginthequeuingsystem?(排隊(duì)系統(tǒng)中有多少顧客在等待)Howlongdothesecustomerstypicallyhavetowait?(這些顧客要等待多少時(shí)間)ChoosingaMeasureofPerformaChoosingaMeasureofPerformanceWhencustomersareinternaltotheorganization,thefirstmeasuretendstobemoreimportant.(當(dāng)顧客是提供服務(wù)的組織[內(nèi)部服務(wù)系統(tǒng)]的內(nèi)部顧客時(shí)，第一個(gè)測度比較重要)Havingsuchcustomerswaitcauseslostproductivity.(讓內(nèi)部服務(wù)系統(tǒng)的顧客等待會導(dǎo)致?lián)p失生產(chǎn)力)ChoosingaMeasureofPerformaChoosingaMeasureofPerformanceCommercialservicesystemstendtoplacegreaterimportanceonthesecondmeasure.(商業(yè)服務(wù)系統(tǒng)[外部顧客接受商業(yè)組織的服務(wù)]會認(rèn)為第二個(gè)測度更加重要)Outsidecustomersaretypicallymoreconcernedwithhowlongtheyhavetowaitthanwithhowmanycustomersarethere.(比起已經(jīng)有多少顧客在等待，顧客更關(guān)心他們會等待多久)ChoosingaMeasureofPerformaDefiningtheMeasuresofPerformanceL=Expectednumberofcustomersinthesystem,includingthosebeingserved(thesymbolLcomesfromLineLength).DefiningtheMeasuresofPerfoDefiningtheMeasuresofPerformanceLq=Expectednumberofcustomersinthequeue,whichexcludescustomersbeingserved.DefiningtheMeasuresofPerfoDefiningtheMeasuresofPerformanceW=Expectedwaitingtimeinthesystem(includingservicetime)foranindividualcustomer(thesymbolWcomesfromWaitingtime).DefiningtheMeasuresofPerfoDefiningtheMeasuresofPerformanceWq=Expectedwaitingtimeinthequeue(excludesservicetime)foranindividualcustomer.DefiningtheMeasuresofPerfoDefiningtheMeasuresofPerformanceThesedefinitionsassumethatthequeuingsystemisinasteady-statecondition.(這些定義假設(shè)排隊(duì)系統(tǒng)處于平穩(wěn)條件中)運(yùn)籌學(xué)排隊(duì)模型DefiningtheMeasuresofPerfoRelationshipbetweenL,W,LqandWq并且知其一即知其四RelationshipbetweenL,W,LqUsingProbabilitiesasMeasuresofPerformanceInadditiontoknowingwhathappensontheaverage,wemayalsobeinterestedinworst-casescenarios.(除了知道系統(tǒng)性能指標(biāo)的各種平均值之外，我們還可能對最壞情況感興趣)Whatwillbethemaximumnumberofcustomersinthesystem?(Exceedednomorethan,say,5%ofthetime.)(在95%的時(shí)間里，系統(tǒng)的最大顧客數(shù)不會超過某個(gè)值?)UsingProbabilitiesasMeasureUsingProbabilitiesasMeasuresofPerformanceWhatwillbethemaximumwaitingtimeofcustomersinthesystem?(Exceedednomorethan,say,5%ofthetime.)(在95%的時(shí)間里，系統(tǒng)的顧客最大等待時(shí)間不會超過某個(gè)值?)Statisticsthatarehelpfultoanswerthesetypesofquestionsareavailableforsomequeuingsystems(統(tǒng)計(jì)手段在回答這類問題時(shí)非常有用，因此它也可用于一些排隊(duì)系統(tǒng)中):UsingProbabilitiesasMeasureUsingProbabilitiesasMeasuresofPerformancePn=Steady-stateprobabilityofhavingexactlyncustomersinthesystem.(在系統(tǒng)中有n個(gè)顧客的平穩(wěn)概率)P(W≤t)=Probabilitythetimespentinthesystemwillbenomorethant.(在系統(tǒng)中的時(shí)間消耗不超過t的概率)P(Wq≤t)=Probabilitythewaittimewillbenomorethant.(等待時(shí)間不超過t的概率)UsingProbabilitiesasMeasureUsingProbabilitiesasMeasuresofPerformanceExamplesofcommongoals(常用目標(biāo)的一些例子):Nomorethanthreecustomers95%ofthetime:P0+P1+P2+P3≥0.95(在95%的時(shí)間里系統(tǒng)中的顧客數(shù)不超過3個(gè))Nomorethan5%ofcustomerswaitmorethan2hours:P(W≤2hours)≥0.95(等待時(shí)間超過2小時(shí)的顧客數(shù)不超過總顧客數(shù)的5%)UsingProbabilitiesasMeasureTheDupitCorp.ProblemTheDupitCorporationisalongtimeleaderintheofficephotocopiermarketplace.Dupit’sservicedivisionisresponsibleforprovidingsupporttothecustomersbypromptlyrepairingthemachineswhenneeded.Thisisdonebythecompany’sservicetechnicalrepresentatives,ortechreps.TheDupitCorp.ProblemTheDupTheDupitCorp.Problem杜皮特公司在辦公復(fù)印機(jī)市場上長期處于領(lǐng)導(dǎo)地位杜皮特公司的服務(wù)部門負(fù)責(zé)為公司的顧客提供高質(zhì)量的服務(wù)支持，在需要的時(shí)候維修公司的設(shè)備。這項(xiàng)工作由公司的技術(shù)服務(wù)代表完成TheDupitCorp.Problem杜皮特公司在辦TheDupitCorp.ProblemCurrentpolicy:Eachtechrep’sterritoryisassignedenoughmachinessothatthetechrepwillbeactiverepairingmachines(ortravelingtothesite)75%ofthetime.Arepaircallaverages2hours,sothiscorrespondsto3repaircallsperday.Machinesaverage50workdaysbetweenrepairs,soassign150machinesperrep.TheDupitCorp.ProblemCurrentTheDupitCorp.Problem目前的政策：每一位技術(shù)服務(wù)代表的地域應(yīng)當(dāng)有足夠多的機(jī)器設(shè)備，使得技術(shù)服務(wù)代表在75%的時(shí)間里都處于維修工作狀態(tài)(或在前往維修地點(diǎn)的路上)每臺設(shè)備的平均維修時(shí)間是2小時(shí)，一天平均要接到3個(gè)維修電話機(jī)器平均50個(gè)工作日需要維修一次，因此為每位技術(shù)服務(wù)代表分配150臺機(jī)器TheDupitCorp.Problem目前的政策：每目前存在的問題剛買了一款新式彩色打印-復(fù)印機(jī)，現(xiàn)在出了問題，等了這么久還沒有人來維修，服務(wù)水平太差了，我要投訴?。∧壳按嬖诘膯栴}剛買了一款新式彩色TheDupitCorp.ProblemProposedNewServiceStandard:Theaveragewaitingtimebeforeatechrepbeginsthetriptothecustomersiteshouldnotexceedtwohours.(建議的新服務(wù)標(biāo)準(zhǔn)：在技術(shù)服務(wù)代表開始前往顧客所在地修理設(shè)備之前，顧客的平均等待時(shí)間不應(yīng)超過兩個(gè)小時(shí))TheDupitCorp.ProblemProposeAlternativeApproachestotheProblemApproachSuggestedbyJohnPhixitt:Modifythecurrentpolicybydecreasingthepercentageoftimethattechrepsareexpectedtoberepairingmachines.(約翰建議的做法：修改目前的政策，降低期望技術(shù)服務(wù)代表進(jìn)行維修工作的時(shí)間百分比。減少分配數(shù)量，增加服務(wù)代表)AlternativeApproachestotheAlternativeApproachestotheProblemApproachSuggestedbytheVicePresidentforEngineering:Providenewequipmenttotechrepsthatwouldreducethetimerequiredforrepairs.(工程副總裁建議的做法：為技術(shù)服務(wù)代表提供新的裝備，以減少設(shè)備修理時(shí)間)AlternativeApproachestotheAlternativeApproachestotheProblemApproachSuggestedbytheChiefFinancialOfficer:Replacethecurrentone-persontechrepterritoriesbylargerterritoriesservedbymultipletechreps.(首席財(cái)務(wù)總監(jiān)建議的做法：將現(xiàn)在的單個(gè)技術(shù)服務(wù)代表負(fù)責(zé)區(qū)域轉(zhuǎn)變?yōu)檩^大的區(qū)域，由多個(gè)技術(shù)服務(wù)代表提供服務(wù))AlternativeApproachestotheAlternativeApproachestotheProblemApproachSuggestedbytheVicePresidentforMarketing:Giveownersofthenewprinter-copierpriorityforreceivingrepairsoverthecompany’sothercustomers.(營銷副總裁建議的做法：授予這種新的打印-復(fù)印機(jī)的所有者比公司的其他顧客優(yōu)先接受維修服務(wù)的權(quán)利)AlternativeApproachestotheTheQueuingSystemforEachTechRepThecustomers:Themachinesneedingrepair.(顧客：需要修理的設(shè)備)Customerarrivals:Thecallstothetechreprequestingrepairs.(顧客到達(dá)：打給每一個(gè)技術(shù)服務(wù)代表要求修理的電話)Thequeue:Themachineswaitingforrepairtobeginattheirsites.(隊(duì)列：顧客所在地等待修理的所有設(shè)備)Theserver:Thetechrep.(服務(wù)者：技術(shù)服務(wù)代表)TheQueuingSystemforEachTeTheQueuingSystemforEachTechRepServicetime:Thetotaltimethetechrepistiedupwithamachine,eithertravelingtothemachinesiteorrepairingthemachine.(Thus,amachineisviewedasleavingthequeueandenteringservicewhenthetechrepbeginsthetriptothemachinesite.)TheQueuingSystemforEachTeTheQueuingSystemforEachTechRep服務(wù)時(shí)間：技術(shù)服務(wù)代表花在一臺設(shè)備上的總時(shí)間，包括到待修設(shè)備所在地的行進(jìn)時(shí)間和維修時(shí)間(因此當(dāng)技術(shù)服務(wù)代表開始前往設(shè)備所在地時(shí)，這臺設(shè)備就可以視作已經(jīng)離開隊(duì)列進(jìn)入服務(wù)系統(tǒng)了)TheQueuingSystemforEachTeNotationforSingle-ServerQueuingModelsNotationforSingle-ServerQueNotationforSingle-ServerQueuingModels運(yùn)籌學(xué)排隊(duì)模型NotationforSingle-ServerQueTheM/M/1ModelAssumptions(假設(shè))Interarrivaltimeshaveanexponentialdistributionwithameanof.(到達(dá)間隔時(shí)間服從均值為的指數(shù)分布)Servicetimeshaveanexponentialdistributionwithameanof1/m.(服務(wù)時(shí)間服從均值為1/m的指數(shù)分布)Thequeuingsystemhasoneserver.(排隊(duì)系統(tǒng)有一個(gè)服務(wù)臺)TheM/M/1ModelAssumptions(假設(shè))TheM/M/1Model運(yùn)籌學(xué)排隊(duì)模型TheM/M/1Model運(yùn)籌學(xué)TheM/M/1Model運(yùn)籌學(xué)排隊(duì)模型TheM/M/1Model運(yùn)籌學(xué)TheM/M/1ModelThe

probabilityofhavingexactlyncustomersinthesystemis(系統(tǒng)中剛好有n個(gè)顧客的概率為)Theprobabilitythatthewaitingtimeinthesystemexceedstis(在系統(tǒng)中的等待時(shí)間超過t的概率是)TheM/M/1ModelTheprobabilityTheM/M/1ModelTheprobabilitythatthewaitingtimeinthequeueexceedstis(在隊(duì)列中的等待時(shí)間超過t的概率是)運(yùn)籌學(xué)排隊(duì)模型TheM/M/1ModelTheprobabilityM/M/1QueuingModelfortheDupit’sCurrentPolicyM/M/1QueuingModelfortheDuJohnPhixitt’sApproach(ReduceMachines/Rep)Theproposednewservicestandardisthattheaveragewaitingtimebeforeservicebeginsbetwohours(i.e.,Wq

≤1/4day).(建議的新服務(wù)標(biāo)準(zhǔn)是將每一個(gè)服務(wù)開始前的平均等待時(shí)間降為2小時(shí))JohnPhixitt’sApproach(ReducJohnPhixitt’sApproach(ReduceMachines/Rep)JohnPhixitt’ssuggestedapproachistolowerthetechrep’sutilizationfactorsufficientlytomeetthenewservicerequirement.(約翰建議的方法是降低技術(shù)服務(wù)代表的有效因子以適應(yīng)新的服務(wù)要求)JohnPhixitt’sApproach(ReducJohnPhixitt’sApproach(ReduceMachines/Rep)運(yùn)籌學(xué)排隊(duì)模型JohnPhixitt’sApproach(ReducM/M/1ModelforJohnPhixitt’sSuggestedApproach(ReduceMachines/Rep)M/M/1ModelforJohnPhixitt’s方案1的結(jié)果指定給每一位技術(shù)服務(wù)代表的設(shè)備數(shù)從150下降到100現(xiàn)有的技術(shù)服務(wù)代表人數(shù)10000人，工資約6億美元/年公司大約要增加5000位技術(shù)服務(wù)代表，增加的費(fèi)用大約為3億美元/年方案1的結(jié)果指定給每一位技術(shù)服務(wù)代表的設(shè)備數(shù)從150下降到1TheM/G/1ModelAssumptions(假設(shè))：Interarrivaltimeshaveanexponentialdistributionwithameanof1/l.(到達(dá)間隔時(shí)間服從均值為1/l的指數(shù)分布)Servicetimescanhaveanyprobabilitydistribution.Youonlyneedthemean(1/m)andstandarddeviation(s).(服務(wù)時(shí)間可以為任何概率分布，僅需要知道均值和標(biāo)準(zhǔn)差)Thequeuingsystemhasoneserver.(排隊(duì)系統(tǒng)有一個(gè)服務(wù)臺)TheM/G/1ModelAssumptions(假設(shè))Theprobabilityofzerocustomersinthesystemis(系統(tǒng)中沒有顧客的概率是)Theexpectednumberofcustomersinthequeueis(隊(duì)列中的顧客期望數(shù)是)TheM/G/1ModelTheprobabilityofzerocustomTheM/G/1ModelTheexpectednumberofcustomersinthesystemis(系統(tǒng)中的顧客期望數(shù)是)Theexpectedwaitingtimeinthequeueis(隊(duì)列中的期望等待時(shí)間是)TheM/G/1ModelTheexpectednuTheexpectedwaitingtimeinthesystemis(在系統(tǒng)中的期望等待時(shí)間是)TheM/G/1Model運(yùn)籌學(xué)排隊(duì)模型Theexpectedwaitingtimeint運(yùn)籌學(xué)排隊(duì)模型運(yùn)籌學(xué)VPforEngineeringApproach(NewEquipment)Theproposednewservicestandardisthattheaveragewaitingtimebeforeservicebeginsbetwohours(i.e.,Wq

≤1/4day).(建議的新服務(wù)標(biāo)準(zhǔn)是將每一個(gè)服務(wù)開始前的平均等待時(shí)間降為2小時(shí))VPforEngineeringApproach(NVPforEngineeringApproach(NewEquipment)TheVicePresidentforEngineeringhassuggestedprovidingtechrepswithnewstate-of-the-artequipmentthatwouldreducethetimerequiredforthelongerrepairs.(工程副總裁建議為技術(shù)服務(wù)代表提供新式的設(shè)備以大大降低需要較長時(shí)間的修理工作)VPforEngineeringApproach(NVPforEngineeringApproach(NewEquipment)Aftergatheringmoreinformation,theyestimatethenewequipmentwouldhavethefollowingeffectontheservice-timedistribution:(收集了更多的信息之后，他們估計(jì)新設(shè)備對服務(wù)時(shí)間分布將產(chǎn)生如下影響)Decreasethemeanfrom1/4dayto1/5day.(均值減少)Decreasethestandarddeviationfrom1/4dayto1/10day.(標(biāo)準(zhǔn)差減少)VPforEngineeringApproach(NM/G/1ModelfortheVPofEngineeringApproach(NewEquipment)方案2的結(jié)果：可以滿足要求，增加的費(fèi)用大約5億美元(一次性投入)M/G/1ModelfortheVPofEngiTheM/M/sModelAssumptions(假設(shè))：Interarrivaltimeshaveanexponentialdistributionwithameanof1/l.(到達(dá)間隔時(shí)間服從均值為的指數(shù)分布)Servicetimeshaveanexponentialdistributionwithameanof1/m.(服務(wù)時(shí)間服從均值為的指數(shù)分布)Anynumberofservers(denotedbys).(任意數(shù)量的服務(wù)臺，用s表示)TheM/M/sModelAssumptions(假設(shè)Withmultipleservers,theformulafortheutilizationfactorbecomes(對于多服務(wù)臺系統(tǒng)，有效性因子的公式變?yōu)?TheM/M/sModelbutstillrepresentsthataveragefractionoftimethatindividualserversarebusy.(仍然代表服務(wù)臺處于繁忙狀態(tài)下的時(shí)間比例)Withmultipleservers,theforValuesofLfortheM/M/sModelforVariousValuesofsValuesofLfortheM/M/sModeCFOSuggestedApproach(CombineIntoTeams)Theproposednewservicestandardisthattheaveragewaitingtimebeforeservicebeginsbetwohours(i.e.,Wq

≤1/4day).(建議的新服務(wù)標(biāo)準(zhǔn)是將每一個(gè)服務(wù)開始前的平均等待時(shí)間降為2小時(shí))CFOSuggestedApproach(CombinCFOSuggestedApproach(CombineIntoTeams)TheChiefFinancialOfficerhassuggestedcombiningthecurrentone-persontechrepterritoriesintolargerterritoriesthatwouldbeservedjointlybymultipletechreps.(首席財(cái)務(wù)官建議將目前的單人技術(shù)服務(wù)代表區(qū)域合并為一些較大的區(qū)域，由多位技術(shù)服務(wù)代表一起提供服務(wù))CFOSuggestedApproach(CombinCFOSuggestedApproach(CombineIntoTeams)CFOSuggestedApproach(CombinM/M/sModelfortheCFO’sSuggestedApproach(CombineIntoTeamsofTwo)M/M/sModelfortheCFO’sSuggCFOSuggestedApproach(TeamsofThree)CFOSuggestedApproach(TeamsM/M/sModelfortheCFO’sSuggestedApproach(CombineIntoTeamsofThree)M/M/sModelfortheCFO’sSuggComparisonofWqwithTerritoriesofDifferentSizesComparisonofWqwithTerritor方案3的結(jié)果如果將三個(gè)連續(xù)的單人技術(shù)服務(wù)代表區(qū)域合并為一個(gè)大區(qū)域，由三個(gè)相同的技術(shù)服務(wù)代表提供服務(wù)，能夠滿足要求沒有直接增加額外更多的成本缺點(diǎn)增加技術(shù)代表前往設(shè)備所在地的時(shí)間團(tuán)隊(duì)作業(yè)的協(xié)調(diào)困難個(gè)性化服務(wù)減弱個(gè)體自豪感下降團(tuán)隊(duì)成員的摩擦問題方案3的結(jié)果如果將三個(gè)連續(xù)的單人技術(shù)服務(wù)代表區(qū)域合并為一個(gè)大ValuesofLfortheM/D/sModelforVariousValuesofsValuesofLfortheM/D/sModeValuesofLfortheM/Ek/2ModelforVariousValuesofkValuesofLfortheM/Ek/2ModPriorityQueuingModelsGeneralAssumptions:Therearetwoormorecategoriesofcustomers.Eachcategoryisassignedtoapriorityclass.Customersinpriorityclass1aregivenpriorityovercustomersinpriorityclass2.Priorityclass2haspriorityoverpriorityclass3,etc.PriorityQueuingModelsGeneralPriorityQueuingModels一般假設(shè)：有兩類或更多類的顧客。每一類都具有一個(gè)優(yōu)先級。優(yōu)先級1的顧客將比優(yōu)先級2的顧客優(yōu)先接受服務(wù)。如果優(yōu)先級多于兩個(gè)，優(yōu)先級2的顧客將具有優(yōu)先于優(yōu)先級3的顧客的權(quán)利PriorityQueuingModels一般假設(shè)：PriorityQueuingModelsGeneralAssumptions(一般假設(shè)):Afterdeferringtohigherprioritycustomers,thecustomerswithineachpriorityclassareservedonafirst-come-fist-servedbasis.(除了在較高優(yōu)先級的顧客后面接受服務(wù)外，同一優(yōu)先級內(nèi)的顧客服從先到先服務(wù)的規(guī)則)PriorityQueuingModelsGeneralPriorityQueuingModelsTwotypesofprioritiesNonpreemptivepriorities:Onceaserverhasbegunservingacustomer,theservicemustbecompleted(evenifahigherprioritycustomerarrives).However,onceserviceiscompleted,prioritiesareappliedtoselectthenextonetobeginservice.PriorityQueuingModelsTwotypPriorityQueuingModels兩類優(yōu)先權(quán)無強(qiáng)制的優(yōu)先規(guī)則：當(dāng)一個(gè)服務(wù)臺開始為一個(gè)顧客服務(wù)時(shí)，服務(wù)必須在不被打斷的條件下完成，即使在服務(wù)過程中有更高優(yōu)先級的顧客到來。然而，這個(gè)服務(wù)一完成，如果隊(duì)列中有顧客，將根據(jù)優(yōu)先權(quán)從中選出一個(gè)顧客進(jìn)行服務(wù)。被選中的顧客是在具有最高優(yōu)先級的顧客中等待時(shí)間最長的一個(gè)PriorityQueuingModels兩類優(yōu)先權(quán)PriorityQueuingModelsTwotypesofprioritiesPreemptivepriorities:Thelowestprioritycustomerbeingservedispreempted(ejectedbackintothequeue)wheneverahigherprioritycustomerentersthequeuingsystem.PriorityQueuingModelsTwotypPriorityQueuingModels兩類優(yōu)先權(quán)強(qiáng)制優(yōu)先規(guī)則：當(dāng)一個(gè)具有更高優(yōu)先權(quán)的顧客進(jìn)入排隊(duì)系統(tǒng)時(shí)，正在接受服務(wù)的最低優(yōu)先級的顧客被擠出(退回到隊(duì)列中)。當(dāng)一個(gè)服務(wù)臺成功完成了一個(gè)服務(wù)工作后，下一個(gè)開始接受服務(wù)的顧客將以上面無強(qiáng)制優(yōu)先規(guī)則描述的相同規(guī)則選出PriorityQueuingModels兩類優(yōu)先權(quán)PreemptivePrioritiesQueuingModelAdditionalAssumptions(附加假設(shè))：Preemptiveprioritiesareusedaspreviouslydescribed.(強(qiáng)制優(yōu)先級如上所述)Forpriorityclassi(i=1,2,…,n),theinterarrivaltimesofthecustomersinthatclasshaveanexponentialdistributionwithameanof1/li.(對于優(yōu)先級i，這個(gè)優(yōu)先級顧客的到達(dá)間隔時(shí)間服從均值為的指數(shù)分布)PreemptivePrioritiesQueuingPreemptivePrioritiesQueuingModelAdditionalAssumptions(附加假設(shè))：Allservicetimeshaveanexponentialdistributionwithameanof1/m,regardlessofthepriorityclassinvolved.(不管優(yōu)先級是多少，所有服務(wù)時(shí)間服從均值為的指數(shù)分布)Thequeuingsystemhasasingleserver.(排隊(duì)系統(tǒng)有一個(gè)服務(wù)臺)PreemptivePrioritiesQueuingPreemptivePrioritiesQueuingModelTheutilizationfactorfortheserveris(服務(wù)臺的有效因子是)運(yùn)籌學(xué)排隊(duì)模型PreemptivePrioritiesQueuingNonpreemptivePrioritiesQueuingModelAdditionalAssumptions(附加假設(shè))：Nonpreemptiveprioritiesareusedaspreviouslydescribed.(無強(qiáng)制優(yōu)先級如上所述)Forpriorityclassi(i=1,2,…,n),theinterarrivaltimesofthecustomersinthatclasshaveanexponentialdistributionwithameanof1/li.(對于優(yōu)先級i，這個(gè)優(yōu)先級顧客的到達(dá)間隔時(shí)間服從均值為的指數(shù)分布)NonpreemptivePrioritiesQueuiNonpreemptivePrioritiesQueuingModelAdditionalAssumptions(附加假設(shè))：Allservicetimeshaveanexponentialdistributionwithameanof1/m,regardlessofthepriorityclassinvolved.(不管優(yōu)先級是多少，所有服務(wù)時(shí)間服從均值為的指數(shù)分布)Thequeuingsystemcanhaveanynumberofservers.(排隊(duì)系統(tǒng)可以有任意數(shù)量的服務(wù)臺)NonpreemptivePrioritiesQueuiNonpreemptivePrioritiesQueuingModelTheutilizationfactorfortheserversis(服務(wù)臺的有效因子為)：運(yùn)籌學(xué)排隊(duì)模型NonpreemptivePrioritiesQueuiVPofMarketingApproach(PriorityforNewCopiers)Theproposednewservicestandardisthattheaveragewaitingtimebeforeservicebeginsbetwohours(i.e.,Wq

≤1/4day).TheVicePresidentofMarketinghasproposedgivingtheprinter-copierspriorityoverothermachinesforreceivingservice.Therationaleforthisproposalisthattheprinter-copierperformssomanyvitalfunctionsthatitsownerscannottoleratebeingwithoutitaslongasothermachines.VPofMarketingApproach(PrioVPofMarketingApproach(PriorityforNewCopiers)建議的新服務(wù)標(biāo)準(zhǔn)是將每一個(gè)服務(wù)開始前的平均等待時(shí)間降為2小時(shí)營銷副總裁建議給打印-復(fù)印機(jī)高于其他設(shè)備的接受服務(wù)的優(yōu)先權(quán)。理由是打印-復(fù)印機(jī)有很多的關(guān)鍵功能，如果這臺設(shè)備的停機(jī)等待維修時(shí)間與其他設(shè)備一樣長，其擁有者是無法忍受的VPofMarketingApproach(PrioVPofMarketingApproach(PriorityforNewCopiers)Themeanarrivalratesforthetwoclassesofcopiersare(這兩類復(fù)印機(jī)的平均到達(dá)率分別是)：Theproportionofprinter-copiersisexpectedtoincrease,soinacoupleyears(據(jù)估計(jì)在幾年里這種打印-復(fù)印機(jī)的比例會增加，于是)VPofMarketingApproach(PrioNonpreemptivePrioritiesModelforVPofMarketing’sApproach(CurrentArrivalRates)NonpreemptivePrioritiesModelNonpreemptivePrioritiesModelforVPofMarketing’sApproach(FutureArrivalRates)NonpreemptivePrioritiesModelExpectedWaitingTimeswithNonpreemptivePrioritiesExpectedWaitingTimeswithNo方案4的結(jié)果可以滿足目前的要求將優(yōu)先權(quán)和多服務(wù)臺單隊(duì)列相結(jié)合可以滿足將來的要求無需增加更多額外費(fèi)用缺點(diǎn)降低沒有優(yōu)先權(quán)的顧客的服務(wù)水平和多服務(wù)臺單隊(duì)列模型相同的其它缺點(diǎn)方案4的結(jié)果可以滿足目前的要求TheFourApproachesUnderConsiderationsTheFourApproachesUnderConsTheFourApproachesUnderConsiderationsDecision:Adoptfourthproposal(exceptforsparselypopulatedareaswheresecondproposalshouldbeadopted).(決策：采用第四個(gè)方案[對于人口分散的區(qū)域則采用第二個(gè)方案])TheFourApproachesUnderConsSomeInsightsAboutDesigningQueuingSystemsWhendesigningasingle-serverqueuingsystem,bewarethatgivingarelativelyhighutilizationfactor(workload)totheserverprovidessurprisinglypoorperformanceforthesystem.(當(dāng)設(shè)計(jì)一個(gè)單服務(wù)臺排隊(duì)系統(tǒng)時(shí)，注意相對較高的服務(wù)臺有效因子[工作強(qiáng)度]將使得系統(tǒng)的績效測度令人吃驚地低下)SomeInsightsAboutDesigningSomeInsightsAboutDesigningQueuingSystemsDecreasingthevariabilityofservicetimes(withoutanychangeintheme