Operations management by stevenson 9th student slides chapter 18

Chapter 18
Waiting Lines

McGraw-Hill/Irwin

Copyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved.


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

System

Processing Order

Calling
population

Arrivals

Waiting
line

Service

Exit

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Queuing Models: Infinite Source
• Four basic infinite source models
– All assume a Poisson arrival rate
1.
2.
3.
4.

Single server, exponential service time
Single server, constant service time
Multiple servers, exponential service time
Multiple priority service, exponential service time

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Infinite-Source Symbols
 Customer arrival rate
 Service rate per server
Lq The average number of customers waiting for service
Ls The average number of customer in the system
r The average number of customers being served

 The system utilization
Wq The average time customers wait in line
Ws The average time customers spend in the system
1  Service time

P0 The probability of zero units in the system
Pn The probability of n units in the system
M The number of servers (channels)
Lmax The maximum expected number waiting in line

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Basic Relationships
System Utilization



M
Average number of customers being served


r


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Basic Relationships
• Little’s Law
– For a stable system the average number of customers
in line or in the system is equal to the average
customers arrival rate multiplied by the average time
in the line or system

Ls Ws
Lq Wq
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Basic Relationships
• The average number of customers
– Waiting in line for service:
– In the system:

Lq
Ls Lq  r

• The average time customers are
– Waiting in line for service

– In the system

Wq 

Lq



1 Ls
Ws Wq  
 

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Single Server, Exponential Service Time
• M/M/1
2
Lq 
2
   

P0 1 


Pn P0  


n


Pn 1   


n

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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
2


Lq 
2 (    )
– 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)

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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 to
maintain FCFS processing)

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M/M/S
M


  

Lq 

P
2 0
 M  1! M   
n

M

 


 
 M  1  







P0  

 n 0 n!

 
M !1 
 

 M  

1

Ws 
M  
Wq
PW 
Ws

1

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Maximum Line Length
• An issue that often arises in service system design is
how much space should be allocated for waiting lines
• The approximate line length, n, that will not be exceeded
a specified percentage of the time can be determined
using the following:

log K
ln K
n
or
log 
ln 
where
specified
1
percentage
K
Lq 1   

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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 the
number of servers
– Use new processing equipment and/or methods
– Reduce processing time variability through standardization
– Shift demand

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