Lecture Introduction to Management Science with Spreadsheets: Chapter 13 - Stevenson, Ozgur

Chapter 13 Waiting-line models, after completing this chapter, you should be able to: Explain why waiting lines can occur in service systems, identify typical goals for designing of service systems with respect to waiting, read the description of the queuing problem and identify the appropriate queuing model needed to solve the problem,...

Introduction to Management Science with Spreadsheets Stevenson and Ozgur First Edition Part Probabilistic Decision Models Chapter 13 Waiting­Line Models McGraw­Hill/Irwin Copyright © 2007 by The McGraw­Hill Companies, Inc. All rights reserved Learning Objectives After completing this chapter, you should be able to: Explain why waiting lines can occur in service systems Identify typical goals for designing of service systems with respect to waiting Read the description of the queuing problem and identify the appropriate queuing model needed to solve the problem Manually solve typical problems using the formulas and tables provided in this chapter Use Excel to solve typical queuing problems associated with this chapter Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  13–2 Learning Objectives (cont’d) After completing this chapter, you should be able to: Use Excel and perform sensitivity analysis and what-if analysis with the results of various queuing models Outline the psychological aspects of waiting lines Explain the value of studying waiting-line models to those who are concerned with service systems Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13–3 Figure Figure13–1 13–1 The TheTotal TotalCost CostCurve CurveIsIsU-Shaped U-Shaped The most common goal of queuing system design is to minimize the combined costs of providing capacity and customer waiting An alternative goal is to design systems that attain specific performance criteria (e.g., keep the average Copyright © 2007 The McGraw­Hill  McGraw­ waiting time to under five minutes Companies. All rights reserved.   Hill/Irwin  13–4 Figure Figure13–2 13–2 Major MajorElements ElementsofofWaiting-Line Waiting-LineSystems Systems First come, first served (FCFS) Priority Classification Waiting lines are commonly found in a wide range of production and service systems that encounter variable arrival rates and service times Copyright © 2007 The McGraw­Hill  McGraw­ Companies. All rights reserved.   Hill/Irwin  13–5 Figure Figure13–3 13–3 AAPoisson PoissonDistribution DistributionIsIsUsually UsuallyUsed Usedto toDescribe Describethe the Variability VariabilityininArrival ArrivalRate Rate Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13–6 Assumptions Assumptions for for using using the the Poisson Poisson Distribution Distribution The probability of occurrence of an event (arrival) in a given interval does not affect the probability of occurrence of an event in another nonoverlapping interval The expected number of occurrences of an event in an interval is proportional to the size of the interval The probability of occurrence of an event in one interval is equal to the probability of occurrence of the event in another equal-size interval Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13–7 Figure Figure13–4 13–4 IfIfthe theArrival ArrivalRate RateIsIsPoisson, Poisson,the theInterarrival InterarrivalTime TimeIsIsaa Negative NegativeExponential Exponential Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13–8 Exhibit Exhibit13-1 13-1 Selection Selectionof ofaaSpecified SpecifiedFunction Functionfrom fromthe theFunction FunctionWizard Wizard Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13–9 Exhibit Exhibit13-2 13-2 Calculation CalculationofofaaProbability ProbabilityUsing Usingthe thePoisson PoissonDistribution Distribution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 10 Table Table13–6 13–6 Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 29 Table Table13–7 13–7 Formulas Formulasfor forPoisson PoissonArrivals, Arrivals,Any AnyService ServiceDistribution Distribution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 30 Exhibit Exhibit13–7 13–7 Single-Channel Single-ChannelModel Modelwith withPoisson PoissonArrival Arrivaland andAny AnyService Service Distribution Distribution(M/G/1 (M/G/1Model) Model) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 31 Exhibit Exhibit13–8 13–8 Single-Channel Single-ChannelModel Modelwith withPoisson PoissonArrival Arrivaland andConstant Constant Service ServiceDistribution Distribution(M/D/1 (M/D/1Model) Model) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 32 Table Table13–8 13–8 Single-Server, Single-Server,Finite FiniteQueue QueueLength LengthFormulas Formulas Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 33 AA Model Model with with aa Finite Finite Queue Queue Length Length • Specific assumptions are presented below: – The arrivals are distributed according to the Poisson distribution and the service time distribution is negative exponential However, the service time distribution assumption can be relaxed to allow any distribution – The system has k channels and the service rate is the same for each channel – The arrival is permitted to enter the system if at least one of the channels is not occupied An arrival that occurs when all the servers are busy is denied service and is not permitted to enter the system.McGraw­ Copyright © 2007 The McGraw­Hill  Hill/Irwin  13– Companies. All rights reserved.   34 Exhibit Exhibit13–9 13–9 Single-Channel Single-ChannelModel ModelThat ThatInvolves InvolvesaaFinite FiniteQueue QueueLength Length with withPoisson PoissonArrival Arrivaland andExponential ExponentialService ServiceDistribution Distribution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 35 Table Table13–9 13–9 Finite FiniteCalling CallingPopulation PopulationFormulas Formulas Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 36 Exhibit Exhibit13–10 13–10 Single-Channel Single-ChannelModel ModelThat ThatInvolves InvolvesaaFinite FiniteCalling CallingPopulation Populationwith with Poisson PoissonArrival Arrivaland andExponential ExponentialService ServiceDistribution Distribution Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 37 Table Table13–10 13–10 Multiple-Server, Multiple-Server,Priority PriorityService ServiceModel Model Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 38 Exhibit Exhibit13–11 13–11 Goal GoalSeek SeekInput InputWindow Window Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 39 Exhibit Exhibit13–12 13–12 Goal GoalSeek SeekOutput OutputWindow Window Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 40 Exhibit Exhibit13–13 13–13 Worksheet WorksheetShowing Showingthe theResults Resultsof ofGoal GoalSeek Seekfor for Example Example13-3 13-3(Car (CarWash WashProblem) Problem) Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 41 Table Table13–11 13–11 Summary Summaryof ofQueuing QueuingModels ModelsDescribed DescribedininThis ThisChapter Chapter Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 42 The The Value Value of of Queuing Queuing Models Models • Common complaints about queuing analysis – Often, service times are not negative exponential – The system is not in steady-state, but tends to be dynamic – “Service” is difficult to define because service requirements can vary considerably from customer to customer Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin  13– 43 ... waiting-line models to those who are concerned with service systems Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?13? ??3 Figure Figure13–1 13? ??1 The TheTotal TotalCost... Hill/Irwin ? ?13? ?? 24 Table Table13–4 13? ??4 Multiple-Channel Multiple-ChannelFormulas Formulas Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?13? ?? 25 Table Table13–4 13? ??4... Copyright © 2007 The McGraw­Hill  Companies. All rights reserved.   McGraw­ Hill/Irwin ? ?13? ?? 27 Exhibit Exhibit13–6 13? ??6 Multiple-Channel Multiple-ChannelModel Modelwith withPoisson PoissonArrival Arrivaland andExponential Exponential