Bài giảng Operations Management - Module D: Waiting-Line Models

Operations ManagementWaiting-Line ModelsModule D1OutlineCHARACTERISTICS OF A WAITING-LINE SYSTEMArrival CharacteristicsWaiting-Line CharacteristicsService Facility CharacteristicsMeasuring the Queue’s PerformanceQueuing Costs2OutlineTHE VARIETY OF QUEUING MODELSModel A: Single-Channel Queuing Model with Poisson Arrivals and Exponential Service TimesModel B: Multiple-Channel Queuing ModelModel C: Constant Service Time ModelModel D: Limited Population ModelOTHER QUEUING APPROACHES3When you complete this chapter, you should be able to Identify or Define:The assumptions of the four basic waiting-line modelsExplain or be able to use:How to apply waiting-line modelsHow to conduct an economic analysis of queuesLearning Objectives4‘The other line always moves faster.’‘If you change lines, the one you left will start to move faster than the one you’re in.’© 1995 Corel Corp.Thank you for holding. Hello...are you there?You’ve Been There Before!5Bank Customers Teller Deposit etc.Doctor’s Patient Doctor TreatmentofficeTraffic Cars Light Controlledintersection passage Assembly line Parts Workers AssemblyTool crib Workers Clerks Check out/in tools Situation Arrivals Servers Service ProcessWaiting Line Examples6First studied by A. K. Erlang in 1913Analyzed telephone facilitiesBody of knowledge called queuing theoryQueue is another name for waiting lineDecision problemBalance cost of providing good service with cost of customers waitingWaiting Lines7Level of serviceCostService costTotal waiting line costWaiting time costOptimal Waiting Line Costs8Queue: Waiting lineArrival: 1 person, machine, part, etc. that arrives and demands serviceQueue discipline: Rules for determining the order that arrivals receive serviceChannel: Number of waiting linesPhase: Number of steps in serviceWaiting Line Terminology9Three Parts of a Queuing System at Dave’s Car-Wash10Service FacilityWaiting LinePopulation Arrival rate distribution Poisson Other Pattern of arrivalsRandomScheduledArrival CharacteristicsCharacteristics of a Waiting Line SystemSize of the source populationLimitedUnlimited Behavior of the arrivalsJoin the queue, and wait until servedBalk; refuse to join the lineRenege; leave the line11Service FacilityWaiting LinePopulationWaiting Line CharacteristicsLength of the queue limited unlimited Service priority FIFO otherCharacteristics of a Waiting Line System - Continued12Service FacilityWaiting LinePopulationService Facility Characteristics Number of channels single multiple Number of phases in service system single multiple Service time distribution negative exponential otherCharacteristics of a Waiting Line System - Continued13Service systemWaitinglineService facilityInput source© 1995 Corel Corp.Waiting Line System14Input Source(Population)SizeInfiniteInput Characteristics15Input CharacteristicsInput Source(Population)Size© 1995 Corel Corp.Fixed number of aircraft to serviceInfiniteFinite16Input Source(Population)SizeArrivalPatternFiniteInfiniteRandomNon-RandomInput Characteristics17Input Source(Population)SizeArrivalPatternFiniteInfiniteRandomNon-RandomPoissonOtherInput Characteristics18Number of events that occur in an interval of timeExample: Number of customers that arrive in 15 min.Mean =  (e.g., 5/hr.)Probability: = 0.5 = 6.0Poisson Distribution19Poisson Distributions for Arrival TimesProbabilityProbability=2=420Input Source(Population)SizeBehaviorArrivalPatternFiniteInfiniteRandomNon-RandomPatientImpatientPoissonOtherInput Characteristics21Input Source(Population)SizeBehaviorArrivalPatternFiniteInfiniteRandomNon-RandomPatientImpatientBalkPoissonOtherInput Characteristics22Input sourceService facilityWaitinglineService system© 1995 Corel Corp.Line was too long!Balking23Input Source(Population)SizeBehaviorArrivalPatternFiniteInfiniteRandomNon-RandomPatientImpatientBalkRenegePoissonOtherInput Characteristics24RenegingInput sourceService facilityWaitinglineService system© 1995 Corel Corp.I give up!25Waiting LineLengthUnlimited© 1995 Corel Corp.Waiting Line Characteristics26Waiting LineLengthLimitedUnlimited© 1995 Corel Corp.© 1995 Corel Corp.Waiting Line Characteristics27Waiting LineLengthQueueDisciplineLimitedUnlimitedFIFO(FCFS)RandomPriorityWaiting Line Characteristics28ServiceFacilityConfigurationMulti-ChannelSingleChannelSinglePhaseService Characteristics29Probability t>x=1=2=3=4Service time, & time between arrivalsExample: Service time is 20 min.Mean service rate = e.g., customers/hr.Mean service time = 1/Equation:Negative Exponential Distribution30Negative Exponential DistributionAverage service time = 1 hourAverage service time = 20 minutes31ArrivalsServed unitsService facilityQueueService systemDockWaiting ship lineShips at seaShip unloading systemEmpty shipsSingle-Channel, Single-Phase System32Cars& foodSingle-Channel, Multi-Phase SystemArrivalsServed unitsService facilityQueueService systemPick-upWaiting carsCars in areaMcDonald’s drive-throughPayService facility33ArrivalsServed unitsService facilityQueueService systemService facilityExample: Bank customers wait in single line for one of several tellers.Multi-Channel, Single Phase System34Service facilityArrivalsServed unitsService facilityQueueService systemService facilityExample: At a laundromat, customers use one of several washers, then one of several dryers.Service facilityMulti-Channel, Multi-Phase System35Two Examples of the Negative Exponential DistributionAverage Service Rate () = 3 customers per hourAverage Service Time = 20 minutes per customerAverage Service Rate () = 1 customer per hourProbability that Service Time is greater than t=e-t, for t > 0Time (t) in HoursProbability that Service Time  t36Deciding on the Optimum Level of ServiceTotal expected costCost of waiting timeCostLow level of serviceOptimal service levelHigh level of serviceMinimum total costCost of providing service37Average queue time, WqAverage queue length, LqAverage time in system, WsAverage number in system, LsProbability of idle service facility, P0System utilization, Probability of k units in system, Pn > kWaiting-Line Performance Measures38Assumptions of the Basic Simple Queuing ModelArrivals are served on a first come, first served basisArrivals are independent of preceding arrivalsArrival rates are described by the Poisson probability distribution, and customers come from a very large populationService times vary from one customer to another, and are independent of one and other; the average service time is knownService times are described by the negative exponential probability distributionThe service rate is greater than the arrival rate39Simple (M/M/1)Example: Information booth at mallMulti-channel (M/M/S)Example: Airline ticket counterConstant Service (M/D/1)Example: Automated car washLimited PopulationExample: Department with only 7 drillsTypes of Queuing Models40Type: Single-channel, single-phase systemInput source: Infinite; no balks, no renegingArrival distribution: PoissonQueue: Unlimited; single lineQueue discipline: FIFO (FCFS)Service distribution: Negative exponentialRelationship: Independent service & arrivalService rate > arrival rateSimple (M/M/1) Model Characteristics41Simple (M/M/1) Model EquationsAverage number of units in queue Average time in system Average number of units in queue Average time in queue System utilizationLWLWssqq==== - 12 -  ( -  ) ( -  ) = 42Simple (M/M/1) Probability EquationsProbability of 0 units in system, i.e., system idle:Probability of more than k units in system:Where n is the number of units in the system Pk+1011=-=-=Pn>kl( )43Type: Multichannel systemInput source: Infinite; no balks, no renegingArrival distribution: PoissonQueue: Unlimited; multiple linesQueue discipline: FIFO (FCFS)Service distribution: Negative exponentialRelationship: Independent service & arrival Service rates > arrival rateMultichannel (M/M/S) Model Characteristics44Model B (M/M/S) EquationsProbability of zero people or units in the system:Average number of people or units in the system:Average time a unit spends in the system:45Model B (M/M/S) EquationsAverage number of people or units waiting for service:Average time a person or unit spends in the queue46Type: Single-channel, single-phase systemInput source: Infinite; no balks, no renegingArrival distribution: PoissonQueue: Unlimited; single lineQueue discipline: FIFO (FCFS)Service distribution: Negative exponentialRelationship: Independent service & arrival Service rates > arrival rateConstant Service Rate (M/D/1) Model Characteristics47Model C (M/D/1) EquationsAverage number of people or units in the system:Average time a unit spends in the system:Average number of people or units waiting for service:Average time a person or unit spends in the queue48Type: Single-channel, single-phase systemInput source: Limited; no balks, no renegingArrival distribution: PoissonQueue: Limited; single lineQueue discipline: FIFO (FCFS)Service distribution: Negative exponentialRelationship: Independent service & arrivalService rate > arrival rateLimited Population Model (D) Characteristics49Model D (Limited Population) EquationsService Factor:Average number of people or units waiting for service:Average time a person or unit spends in the queue50Model D (Limited Population) Equations - ContinuedAverage number runningAverage number being served:Number in the population:51Model D (Limited Population) Equations - ContinuedWhere:D = probability that a unit will have to wait in the queueF = efficiency factorH = average number of units being servicedJ = average number of units not in the queue or service bay L = average number of units waiting for service52Model D (Limited Population) Equations - ContinuedM = number of service channelsN = number of potential customersT = average service timeU = average time between unit service requirementsW = average time a unit waits in lineX = service factorto be obtained from finite queuing tables53 = Mean number of arrivals per time periode.g., 3 units/hour = Mean number of people or items served per time periode.g., 4 units/hour1/ = 15 minutes/unitRemember:  &  Are Rates© 1984-1994 T/Maker Co.If average service time is 15 minutes, then μ is 4 customers/hour54