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Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,

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Presentation on theme: "Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies,"— Presentation transcript:

1 Stevenson and Ozgur First Edition Introduction to Management Science with Spreadsheets McGraw-Hill/Irwin Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 13 Waiting-Line Models Part 3 Probabilistic Decision Models

2 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–2 Learning Objectives 1.Explain why waiting lines can occur in service systems. 2.Identify typical goals for designing of service systems with respect to waiting. 3.Read the description of the queuing problem and identify the appropriate queuing model needed to solve the problem. 4.Manually solve typical problems using the formulas and tables provided in this chapter. 5.Use Excel to solve typical queuing problems associated with this chapter. After completing this chapter, you should be able to:

3 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–3 Learning Objectives (contd) 6.Use Excel and perform sensitivity analysis and what-if analysis with the results of various queuing models. 7.Outline the psychological aspects of waiting lines. 8.Explain the value of studying waiting-line models to those who are concerned with service systems. After completing this chapter, you should be able to:

4 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–4 Figure 13–1The Total Cost Curve Is 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 waiting time to under five minutes

5 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–5 Figure 13–2Major Elements of Waiting-Line Systems Waiting lines are commonly found in a wide range of production and service systems that encounter variable arrival rates and service times. First come, first served (FCFS) Priority Classification

6 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–6 Figure 13–3A Poisson Distribution Is Usually Used to Describe the Variability in Arrival Rate

7 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–7 Assumptions for using the Poisson Distribution 1.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. 2.The expected number of occurrences of an event in an interval is proportional to the size of the interval. 3.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.

8 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–8 Figure 13–4If the Arrival Rate Is Poisson, the Interarrival Time Is a Negative Exponential

9 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–9 Exhibit 13-1Selection of a Specified Function from the Function Wizard

10 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–10 Exhibit 13-2Calculation of a Probability Using the Poisson Distribution

11 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–11 Exhibit 13–3Calculation of a Cumulative Probability Using the Poisson Distribution

12 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–12 Figure 13–5Comparison of Single- and Multiple-Channel Queuing System

13 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–13 Figure 13–6An Exponential Service-Time Distribution

14 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–14 Figure 13–7Graphical Depiction of Probabilities Using the Exponential Distribution

15 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–15 Exhibit 13–4Calculation of a Probability Using the Exponential Distribution

16 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–16 Operating Characteristics Lq=the average number waiting for service L= the average number in the system (i.e., waiting for service or being served) P 0 =the probability of zero units in the system r=the system utilization (percentage of time servers are busy serving customers) W a= the average time customers must wait for service W=the average time customers spend in the system (i.e., waiting for service and service time) M=the expected maximum number waiting for service for a given level of confidence

17 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–17 Table 13–1 Line and Service Symbols for Average Number Waiting, and Average Waiting and Service Times

18 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–18 Basic Single-Channel (M/M/1) Model A single-channel model is appropriate when these conditions exist: –One server or channel. –A Poisson arrival rate. –A negative exponential service time. –First-come, first-served processing order. –An infinite calling population. –No limit on queue length.

19 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–19 Table 13–2 Formulas for Basic Single Server Model

20 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–20 Table 13–2 Formulas for Basic Single Server Model (contd)

21 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–21 Exhibit 13–5Basic Single-Channel Model with Poisson Arrival and Exponential Service Rate (M/M/1 Model)

22 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–22 Table 13–3

23 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–23 Figure 13–8As Utilization Approaches 100 percent, L q and W q Rapidly Increase

24 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–24 Multiple-Channel Model The multiple-channel model is appropriate when these conditions exist: 1.A Poisson arrival rate. 2.A negative exponential service time. 3.First-Come, first-served processing order. 4.More than one server. 5.An infinite calling population. 6.No upper limit on queue length. 7.The same mean service rate for all servers.

25 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–25 Table 13–4Multiple-Channel Formulas

26 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–26 Table 13–4Multiple-Channel Formulas (contd)

27 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–27 Table 13–5Infinite Source Values for L q and P 0 given λ μ and s

28 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–28 Exhibit 13–6Multiple-Channel Model with Poisson Arrival and Exponential Service Rate (M/M/S Model)

29 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–29 Table 13–6

30 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–30 Table 13–7Formulas for Poisson Arrivals, Any Service Distribution

31 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–31 Exhibit 13–7Single-Channel Model with Poisson Arrival and Any Service Distribution (M/G/1 Model)

32 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–32 Exhibit 13–8Single-Channel Model with Poisson Arrival and Constant Service Distribution (M/D/1 Model)

33 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–33 Table 13–8Single-Server, Finite Queue Length Formulas

34 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–34 A Model with a Finite Queue 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.

35 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–35 Exhibit 13–9Single-Channel Model That Involves a Finite Queue Length with Poisson Arrival and Exponential Service Distribution

36 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–36 Table 13–9Finite Calling Population Formulas

37 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–37 Exhibit 13–10Single-Channel Model That Involves a Finite Calling Population with Poisson Arrival and Exponential Service Distribution

38 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–38 Table 13–10Multiple-Server, Priority Service Model

39 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–39 Exhibit 13–11Goal Seek Input Window

40 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–40 Exhibit 13–12Goal Seek Output Window

41 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–41 Exhibit 13–13Worksheet Showing the Results of Goal Seek for Example 13-3 (Car Wash Problem)

42 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–42 Table 13–11Summary of Queuing Models Described in This Chapter

43 Copyright © 2007 The McGraw-Hill Companies. All rights reserved. McGraw-Hill/Irwin 13–43 The Value of Queuing 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.


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