Introduction to
Management Science
with Spreadsheets
Stevenson and Ozgur
First Edition
Part 3 Probabilistic Decision Models
Chapter 13
Waiting-Line Models
McGraw-Hill/Irwin
Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved.
Learning Objectives
After completing this chapter, you should be able to:
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.
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Learning Objectives (cont’d)
After completing this chapter, you should be able to:
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.
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Figure 13–1
The 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
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Figure 13–2
Major Elements of Waiting-Line 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.
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Figure 13–3
A Poisson Distribution Is Usually Used to Describe the
Variability in Arrival Rate
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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.
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Figure 13–4
If the Arrival Rate Is Poisson, the Interarrival Time Is a
Negative Exponential
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Exhibit 13-1
Selection of a Specified Function from the Function Wizard
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Exhibit 13-2
Calculation of a Probability Using the Poisson Distribution
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Exhibit 13–3
Calculation of a Cumulative Probability Using the Poisson
Distribution
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Figure 13–5
Comparison of Single- and Multiple-Channel Queuing System
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Figure 13–6
An Exponential Service-Time Distribution
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Figure 13–7
Graphical Depiction of Probabilities Using the Exponential Distribution
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Exhibit 13–4
Calculation of a Probability Using the Exponential Distribution
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Operating Characteristics
Lq = the average number waiting for service
L
= the average number in the system (i.e.,
waiting for service or being served)
P0 = the probability of zero units in the system
r
Wa
= the system utilization (percentage of time servers
are busy serving customers)
=
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
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Table 13–1
Line and Service Symbols for Average Number Waiting, and
Average Waiting and Service Times
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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.
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Table 13–2
Formulas for Basic Single Server Model
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Table 13–2
Formulas for Basic Single Server Model (cont’d)
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Exhibit 13–5
Basic Single-Channel Model with Poisson Arrival and
Exponential Service Rate (M/M/1 Model)
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Table 13–3
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Figure 13–8
As Utilization Approaches 100 percent, Lq and Wq Rapidly
Increase
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Multiple-Channel Model
• The multiple-channel model is appropriate when
these conditions exist:
1.
2.
3.
4.
5.
6.
7.
A Poisson arrival rate.
A negative exponential service time.
First-Come, first-served processing order.
More than one server.
An infinite calling population.
No upper limit on queue length.
The same mean service rate for all servers.
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Table 13–4
Multiple-Channel Formulas
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Table 13–4
Multiple-Channel Formulas (cont’d)
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Table 13–5
Infinite Source Values for Lq and P0 given λ ⁄ μ and s
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Exhibit 13–6
Multiple-Channel Model with Poisson Arrival and Exponential
Service Rate (M/M/S Model)
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Table 13–6
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Table 13–7
Formulas for Poisson Arrivals, Any Service Distribution
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Exhibit 13–7
Single-Channel Model with Poisson Arrival and Any Service
Distribution (M/G/1 Model)
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Exhibit 13–8
Single-Channel Model with Poisson Arrival and Constant
Service Distribution (M/D/1 Model)
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Table 13–8
Single-Server, Finite Queue Length Formulas
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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.
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Exhibit 13–9
Single-Channel Model That Involves a Finite Queue Length
with Poisson Arrival and Exponential Service Distribution
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Table 13–9
Finite Calling Population Formulas
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Exhibit 13–10
Single-Channel Model That Involves a Finite Calling Population with
Poisson Arrival and Exponential Service Distribution
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Table 13–10
Multiple-Server, Priority Service Model
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Exhibit 13–11
Goal Seek Input Window
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Exhibit 13–12 Goal Seek Output Window
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Exhibit 13–13 Worksheet Showing the Results of Goal Seek for
Example 13-3 (Car Wash Problem)
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Table 13–11
Summary of Queuing Models Described in This Chapter
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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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