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McGraw-Hill/Irwin Copyrig ht 2009 by The McGraw -Hill Com panies, Inc. All Rights Reserved.
Chapter 18
Waiting Lines
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Chapter 18: Learning Objectives
You should be able to: Explain why waiting lines form in systems that are underloaded Identify the goal of queuing management List the measures of system performance that are used in
queuing Discuss the assumptions of the basic queuing models presented Solve typical problems
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Queuing Theory
Queuing theory Mathematical approach to the analysis of waiting lines Applicable to many environments
Call centers Banks Post offices Restaurants
Theme parks Telecommunications systems Traffic management
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Simple Queuing System
Calling population
Arrivals Waitingline
ExitService
System
Processing Order
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Queuing Models: Infinite Source
Four basic infinite source models All assume a Poisson arrival rate
1. Single server, exponential service time
2. Single server, constant service time3. Multiple servers, exponential service time4. Multiple priority service, exponential service time
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Infinite-Source Symbols
lineintingnumber waiexpectedmaximumThe
(channels)serversof number The
systemin theunitsof y probabilitThe
systemin theunitszeroof y probabilitThe timeService1
systemin thespendcustomerstimeaverageThe
lineinwaitcustomerstimeaverageThe
nutilizatiosystemTheserved beingcustomersof number averageThe
systemin thecustomer of number averageThe
servicefor waitingcustomersof number averageThe
server per rateServiceratearrivalCustomer
max
0
L
M
n P
P
W
W
r
L
L
n
s
q
s
q
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System Utilization
Average number of customers being served
Basic Relationships
M
r
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Basic Relationships
Littles Law For a stable system the average number of customers
in line or in the system is equal to the averagecustomers arrival rate multiplied by the average timein the line or system
qq
s s
W L
W L
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Basic Relationships
The average number of customers Waiting in line for service:
In the system:
The average t ime customers are Waiting in line for service
In the system
q L
r L L q s
q
q LW
s
q s
LW W
1
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Single Server, Exponential Service Time
M/M/1
n
n
n
n
q
P
P P
P
L
1
1
0
0
2
2
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Single Server, Constant Service Time
M/D/1 If a system can reduce variability, it can shorten waiting lines
noticeably For, example, by making service time constant, the average
number of customers waiting in line can be cut in half
Average time customers spend waiting in line is also cut by half. Similar improvements can be made by smoothing arrival rates
(such as by use of appointments)
)(2
2
q L
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Multiple Servers (M/M/S)
Assumptions: A Poisson arrival rate and exponential service time Servers all work at the same average rate
Customers form a single waiting line (in order tomaintain FCFS processing)
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M/M/S
s
qW
s
M
n
M n
M
q
W
W P
M W
M M
n P
P M M
L
1
1!!
!11
1
00
02
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Maximum Line Length
An issue that often arises in service system design ishow much space should be allocated for waiting lines
The approximate line length, n , that will not be exceededa specified percentage of the time can be determinedusing the following:
1 percentage
specified1
where
lnln
or loglog
q L K
K K n
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Operations Strategy Managers must carefully weigh the costs and benefits of
service system capacity alternatives Options for reducing wait times:
Work to increase processing rates, instead of increasing thenumber of servers
Use new processing equipment and/or methods Reduce processing time variability through standardization Shift demand