Chapter 7A - Department of Management and Information Systems
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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
7A-10
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
7A-11
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
7A-12
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
7A-15
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
7A-23
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
66 4
Lq
c) Length of line:
2
42
Lq
1.33
66 4
d) Probability of waiting:
P1waiting Pn2
P2 P3 P4 ... P 1 Pn1
1 P0 P1
Queue
Server
System
0
1
4 4
4 4
1 1 1 1 0.3333 0.2222
7A-31
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:
P3system Pn3
Queue
P3 P4 P5 ... P 1 Pn2
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
P3line Pn4
P4 P5 P6 ... P 1 Pn3
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
2020 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 s1 0.98 1 s 1 0.98
s 1 0.02
s 1ln ln0.02
s 1
ln0.02
ln
ln1 SL
1
s
ln
ln1 SL
s 1 7A-37
Inv .
And
Example 3: 95% SL
ln1 SL
s 1
Inv .
ln1 .95
3 1
Inv .
2.99573
4
Inv .
0.74893Inv.
0.47287
LO 4
7A-38