Transcript Chapter 7A - Department of Management and Information Systems

Chapter 7A
Waiting Line Analysis
McGraw-Hill/Irwin
©2011 The McGraw-Hill Companies, All Rights Reserved
Learning Objectives
 Waiting line characteristics
 Describe what waiting line (queuing)
analysis is.
 Suggestions for managing queues
 Model some common waiting line
situations and estimate service utilization,
the length of a waiting line, and average
customer wait time.
7A-2
Waiting Line Characteristics
 Waiting is a fact of life
 Americans wait up to 30 minutes daily or
about 37 billion hours in line yearly
 US leisure time has shrunk by more than
35% since 1973
 An average part spends more than 95% of
its time waiting
 Waiting is bad for business and occurs in
every arrival
LO 1
7A-3
Where the Time Goes
In a life time, the average
person will spend:
SIX MONTHS Waiting at stoplights
EIGHT MONTHS Opening junk mail
ONE YEAR Looking for misplaced Objects
TWO YEARS Reading E-mail
FOUR YEARS Doing housework
FIVE YEARS Waiting in line
SIX YEARS Eating
7A-4
Economics of the Waiting
Line Problem
 A central problem in many service
settings is the management of
waiting time

Reducing waiting time costs money
 When people waiting are employees,
it is easy to value their time
 When people waiting are customers,
it is more difficult to value their time
LO 2

Lost sales is one (low) value
7A-5
The Practical View of
Waiting Lines
LO 2
7A-6
More on Waiting Lines
 One important variable is the number of
arrivals over the hours that the service
system is open
 Customers demand varying amounts of
service, often exceeding normal capacity
 We can control arrivals



Short lines
Specific hours for specific customers
Specials
 We can affect service time by using faster
or slower servers
LO 2
7A-7
The Queuing System
 Source population and the way
customers arrive at the system
 The servicing system
 The condition of the customers
exiting the system

Do they go back to source population or
not?
LO 4
7A-8
Components of the
Queuing System Visually
Customers
come in
Customers are
served
Customers
leave
LO 4
7A-9
Suggestions for Managing Queues
 Determine an acceptable waiting time for your
customers
 Try to divert your customer’s attention when waiting
 Inform your customers of what to expect
 Keep employees not serving the customers out of
sight
 Segment customers
 Train your servers to be friendly
 Encourage customers to come during the slack
periods
 Take a long-term perspective toward getting rid of
queues
LO 3
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The General Waiting Framework
 Arrival process could be random

If random:
 We assume Poisson arrival
  = Average rate of arrival
 1 = Inter-arrival time

 Service process could be random or constant

If random



We assume Exponential
 = Average rate of service
 1  = Service time
If constant, service time is same for all
 Service intensity      1
LO 4
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Customer Arrivals
 Finite population: limited-size customer pool
that will use the service and, at times, form a
line (e.g., machine repairs)

When a customer leaves its position as a member for
the population, the size of the user group is reduced
by one
 Infinite population: population large enough
so that the population size caused by
subtractions or additions to the population
does not significantly affect the system
probabilities (e.g., waiting for gasoline)
LO 4
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Distribution of Arrivals
 Arrival rate: the number of units
arriving per period

Constant arrival distribution: periodic,
with exactly the same time between
successive arrivals

Variable (random) arrival distributions:
arrival probabilities described statistically


Exponential distribution
Poisson distribution
LO 4
7A-13
Distributions
 Exponential distribution: when
arrivals at a service facility occur in
a purely random fashion

The probability function is f(t) = λe-λt
 Poisson distribution: where one
is interested in the number of
arrivals during some time period T

LO 4
The probability function is
n  T


T e
PT n 
n!
7A-14
Customer Arrivals in
Queues
LO 4
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Other Arrival
Characteristics
 Arrival patterns
 Size of arrival units
 Degree of patience



Balking
Reneging
Jockeying
LO 4
7A-16
The Queuing System
 Length


Infinite potential length
Limited line capacity
 Number of lines
 Queue discipline: a
priority rule or set of
rules for determining
the order of service to
customers in a waiting
line
LO 4
7A-17
Service Time Distribution
 Constant

Service is provided by automation
 Variable


Service provided by humans
Described using exponential distribution
LO 4
7A-18
Line Structure
Channel
Phase
LO 4
7A-19
Exiting the Queuing
System
LO 4
7A-20
Properties of Some Specific
Waiting Line Models
LO 4
7A-21
Notation for Equations
LO 4
7A-22
Equations for Solving
Three Model Problems
Model 1
Model 2
 
pn  1     n
Model 3
Model 4
LO 4
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Example 1:
Model 1
Assume a drive-up window at a fast food restaurant.
Customers arrive at the rate of 25 per hour. The
employee can serve one customer every 2 minutes.
Assume Poisson arrival and exponential service rates.
Determine:
A. What is the average utilization of the employee?
B. What is the average number of customers in line?
C. What is the average number of customers in the system?
D. What is the average waiting time in line?
E. What is the average waiting time in the system?
F. What is the probability that exactly two cars will be in the
system?
7A-24
Example 1: Model 1
A. What is the average utilization of the employee?
 = 25 cust /hr
1 cust omer
=
= 30 cust /hr
2 mins (1hr/60mins)
 25 cust /hr
= =
= 0.8333
 30 cust /hr
7A-25
Example 1: Model 1
B. What is the average number of customers in line?

(25)
Lq =
=
= 4.167
 ( -  ) 30(30- 25)
2
2
C. What is the average number of customers in the
system?

25
Ls =
=
=5
 -  (30- 25)
7A-26
Example 1: Model 1
D. What is the average waiting time in line?
Wq =
Lq

= 0.1667 hrs = 10 mins
E. What is the average waiting time in the system?
Ws =
Ls

= 0.2 hrs = 12 mins
7A-27
Example 1: Model 1
F. What is the probability that exactly two cars will be in
the system (one being served and the other waiting
in line)?
pn
 
= (1 - )( )
 
Since  
n
 

n


P

1



, then, n

25 25 2
p 2 = (1 )( ) = 0.1157
30 30
7A-28
Example 2: Model 2
An automated pizza vending machine heats and
dispenses a slice of pizza in 4 minutes.
Customers arrive at a rate of one every 6 minutes with
the arrival rate exhibiting a Poisson distribution.
Determine:
A. The average number of customers in line.
B. The average total waiting time in the system.
7A-29
Example 2: Model 2
A. The average number of customers in line.
2
(10)2
Lq =
=
= 0.6667
2 (  -  ) (2)(15)(15- 10)
B. The average total waiting time in the system.

10
Wq =
=
= 0.06667hrs = 4 mins
2 ( -  ) 2(15)(15 - 10)
1
1
Ws = Wq + = .06667hrs +
= 0.1333hrs = 8 mins

15/hr
7A-30
Problem 1: Page 262— Model 1
a) Percent Idle:

4
1  60 min 
1  60 min 

  6 / hr 1    1   0.33

  4 / hr  
10 min  1hr 
15min  1hr 
6
b) Time waiting in line:
1.33
2
42

W

 0.33hrs  20 min
q
Wq 
 Lq 

 1.33
4 / hr

     66  4
Lq
c) Length of line:
2
42
Lq 

 1.33
     66  4
d) Probability of waiting:
P1waiting  Pn2
P2  P3  P4  ... P  1  Pn1
 1  P0  P1 
 
Queue
Server
System
0
1

4  4 
4  4  



 1  1     1      1  0.3333 0.2222
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6  6 
6  6  



  0.4444


Problem 10: Page 263— Model 1
a) Average number in system

20
1  60 min 
1  60 min 
L


 2 patients
s


  20 / hr  

  30 / hr
   30  20
3 min  1hr 
2 min  1hr 
b) Average time in system:
L
2
2
60  min  6 min
Ws  s  hrs. Ws 

20
20
 
c) Probability:
P3system  Pn3
Queue
P3  P4  P5  ... P  1  Pn2
Server
System
 1  P0  P1  P2   1  0.3333 0.2222 0.1481  0.2963
d) Utilization:




20
 0.6667
30
7A-32
Problem 10: Page 263
Suppose we wish, instead, to determine the
probability of three or more patients in line:

1  60 min 
1  60 min 

  20 / hr  

  30 / hr
3 min  1hr 
2 min  1hr 
 
Queue
Server
System
P3line  Pn4
P4  P5  P6  ... P  1  Pn3
 1  P0  P1  P2  P3   1  0.3333 0.2222 0.1481 0.0988
 0.1976
7A-33
Example 3: Page 249
Customers in Line
 Utilization of the teller
 Average number in line
 Average number in system
 Average waiting time in line
 Average waiting time in system,
including service
LO 4
7A-34
Example 3: Solution
 15
 
 0.75  75 percent
 20


15
Lq 

 2.25 cust omers
     2020  15

15
Ls 

 3 cust omers
   20  15
2
2
Lq
LO 4
2.25
Wq 

 0.15 hours or 9 minutes

15
Ls 3
Ws 
  0.2 hour or 12 minutes
 15
7A-35
Example 3: No More Than
Three Cars
LO 4
7A-36
Determining s and  for a
Given Service Level
In a Model 1 car wash facility   10 and   12 . Find the
number of parking spaces needed to guarantee a service
level of 98%. (Let s=number in the system). Then, we
want:
p0  p1  p2  ... ps  0.98
 
n


p

1



But, n
 1   0  1  1  1   2  ... 1   s  0.98
1       2   2   3  ...   s   s1  0.98 1   s 1  0.98
 s 1  0.02
s  1ln   ln0.02
s 1 
ln0.02
ln 
 ln1  SL 
  1
s  
 ln  
 ln1  SL 
 

 s  1  7A-37
Inv .
And
Example 3: 95% SL
 ln1  SL 

 s 1 
Inv .
 
 ln1  .95 
 

 3 1 
Inv .
  2.99573

4


Inv .
 
   0.74893Inv.
  0.47287
LO 4
7A-38