in MODULE
D:
WAITINGLINE
MODELS
s such diverse settings as service systems, maintenance
activities, and shop-floor control.
True (Introduction, easy)
W
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i
t
i
n
g
l
i
n
e
m
o
d
e
l
s
a
r
e
u
s
e
f
u
l
t
o
o
p
e
r
a
t
i
o
n
The two characteristics of the waiting line itself are whether its
length is limited or unlimited and the discipline of the people or
items in it.
True (Characteristics of a waiting-line system, easy)
3.
1.
2.
A waiting-line system has three parts: the size of the arrival
population, the behavior of arrivals, and the statistical distribution
of arrivals.
False (Characteristics of a waiting-line system, easy)
4.
A copy center has five machines that serve many customers
throughout the day; the waiting-line system for copy service has an
infinite population while the waiting-line system for copier
maintenance has a finite population
True (Characteristics of a waiting-line system, easy)
5.
In queuing problems, arrival rates are generally described by the
normal probability distribution.
False (Characteristics of a waiting-line system, moderate)
6.
Balk and renege are elements of queue discipline.
False (Characteristics of a waiting-line problem, easy)
7.
A hospital emergency room always follows a first-in, first-served
queue discipline in the interest of fairness.
False (Characteristics of a waiting-line system, moderate)
8.
In queuing problems, the term renege refers to the fact that
some customers leave the queue before service is completed.
True (Characteristics of a waiting-line system, moderate)
9.
A waiting-line system with one waiting line and three sequential
processing stages is a multi- channel single-phase system.
False (Characteristics of a waiting-line problem, easy)
10.
If the service time within a queuing system is constant, the service rate
can be easily described by a negative exponential distribution.
False (Characteristics of a waiting-line system, moderate)
11.
The cost of waiting decreases as the service level increases.
True (Queuing costs, moderate)
12.
LIFS (last-in, first-served) is a common queue discipline, most
often seen where people, not objects, form the waiting line.
False (Characteristics of a waiting-line problem, moderate)
1 3.
A bank office with five tellers, each with a separate line of customers, exhibits the
characteristics of a multi-phase queuing system.
False (Characteristics of a waiting-line system, moderate)
14.
In the analysis of queuing models, the Poisson distribution often describes arrival rates and
service times are often described by the negative exponential distribution.
True (Characteristics of a waiting-line system, moderate)
15.
The study of waiting lines calculates the cost of providing good service but does not value
the cost of customers' waiting time.
False (Queuing costs, moderate)
16.
As the average service rate grows larger, the slope of the distribution of service time
probabilities grows larger and larger, eventually becoming positive.
False (The variety of queuing models, easy)
17.
Four of the most widely used waiting line modelsM/M/1 or A, M/M/S or B, M/D/1
or C, and Limited population or Dall share three characteristics: Poisson arrivals,
FIFO discipline, and exponential service times.
False (The variety of queuing models, moderate)
18.
In the M/M/1 waiting line model with an arrival rate of 2 per hour and a service rate of 6
per hour, the utilization factor for the system is approximately 0.333.
True (The variety of queuing models, easy) {AACSB: Analytic Skills}
19.
The greater the margin by which the arrival rate exceeds the service rate, the better the
performance of the waiting line.
False (The variety of queuing models, easy)
20.
An M/M/1 model and an M/D/1 model each have an arrival rate of 1 per minute and a
service rate of 3 per minute; the average queue length of the M/M/1 will be twice that of
the M/D/1.
True (The variety of queuing models, moderate)
21.
A finite population waiting line model has an average service time T of 100 minutes
and an average time between service requirements U of 400 minutes; the service
factor X is 0.25. False (The variety of queuing models, moderate) {AACSB:
Analytic Skills}
MULTIPLE CHOICE
22.
Study of waiting-line models helps operations managers better
understand a. service systems such as bank teller stations
b. maintenance activities that might repair broken machinery
c. shop-floor control activities
d. service systems such as amusement park
rides e. all of the above
e (Introduction, easy)
545
23.
24.
Which of the following is not a common queuing
situation?
a. grocery shoppers being served by checkout clerks
b. commuters slowing or stopping at toll plazas to pay
highway tolls c.
machinery waiting to be repaired or
maintained
d. parcel delivery truck following its computer-generated route
e. patients in a health clinic waiting to see one of several doctors
d (Introduction, easy)
In queuing problems, which of the following probability distributions is typically used to
describe the number of arrivals per unit of time?
a.
binomial b.
normal
c. Poisson
d.
exponential e.
lognormal
c (Characteristics of a waiting-line system, easy)
25.
In queuing problems, which of the following probability distributions is typically used to
describe the time to perform the service?
a.
binomial b.
normal
c. Poisson
d. negative
exponential e.
lognormal
d (Characteristics of a waiting-line system, easy)
26.
The common measures of a queuing system's performance include
a. probability that the service facility will be idle, average queue length, probability
that the waiting time will exceed a specified duration
b. average time each customer spends in the system, probability that the service system
will be
idle, average time each customer spends in the
queue
c. average queue length, maximum time a customer may spend in the queue, the
utilization factor for the system
d. average time each customer spends in the system, maximum queue length,
probability of a specific number of customers in the system
e. none of the above
b (Characteristics of a waiting-line system, moderate)
27.
The shopper who says to himself, Ive waited too long in this line. I dont really need to
buy this product today, and leaves the store is an illustration of which element of arrival
behavior?
a. random
arrival b.
renege
c. random
departure d. balk
e. none of the above
b (Characteristics of a waiting-line system, moderate)
546
28.
A waiting line, or queuing, system has three parts, which are
a. distribution of arrival times, discipline while waiting, and distribution of
service times b. arrival rate, service rate, and utilization rate
c. arrival discipline, queue discipline, and service
sequencing
d. arrival or inputs, queue discipline or the waiting line itself, and the service
facility e.
sequencing policy, penalty for reneging, and expediting of
arrivals
d (Characteristics of a waiting-line system,
moderate)
29.
The source population is considered to be either
in its
size. a.
finite or
infinite
b.
fixed or
variable
c. known
or
unknown
d. random or
scheduled e. small
or large
a (Characteristics of a waiting-line system,
moderate)
30.
The potential restaurant customer who says to her husband, The line looks too long;
let's eat somewhere else, is an illustration of which element of queue discipline?
a. first-in, firstout b. balk
c.
renege
d. random
departure e. none
of the above
b (Characteristics of a waiting-line system,
moderate)
31.
An airline ticket counter, with several agents for one line of customers, is an
example of a a. single channel, single phase system
b.
single channel, multi-phase
system c.
multi-channel,
single phase system d.
multichannel, multi-phase system e.
none of the above
c (Characteristics of a waiting-line system,
moderate)
32.
A concert hall, employing both ticket takers and ushers to seat patrons, behaves
typically as a a.
multi-channel, single phase system
b.
multi-channel, multi-phase
system
c. single channel, single phase
system d.
single channel, multiphase system e.
none of the
above
b (Characteristics of a waiting-line system,
moderate)
33.
If the food service for the university operates a cafeteria with a single serving line, that
system behaves most like a
a. single channel, single phase
system b. single channel, multiphase system c.
multichannel, single phase system d.
multi-channel, multi-phase system
e. none of the above
b (Characteristics of a waiting-line system,
moderate)
547
34.
The sign at the bank that reads Wait here for the first available teller suggests the use of a
waiting line
system. a.
single phase
b.
multiphase
c. single
channel
d.
multi-channel e.
multiple line
d (Characteristics of a waiting-line system,
moderate)
35.
A small hair styling salon has several operators. While customers do not have
appointments, each is waiting to be served by a specific operator. This scenario provides
an example of a
a. multiple-channel, multi-phase, limited queue
length
b. single-channel, multi-phase, limited queue
length c.
multi-channel, limited queue
length
d. multiple single-channel systems, limited queue
length
e. none of the
above
d (Characteristics of a waiting-line system,
moderate)
36.
A large discount store and supermarket has a hair styling salon on its premises. The salon
has several operators. Salon customers can shop in other parts of the store until their name
is called for salon service, at which time the customer will be served by the next available
stylist. This scenario provides an example of a
a. multiple-channel, multi-phase, unlimited queue
length b. single-channel, multi-phase, limited queue
length
c. multi-channel, unlimited queue
length
d. multiple single-channel systems, limited queue
length e.
none of the above
c (Characteristics of a waiting-line system,
moderate)
37.
A university has only one technician in the repair station to care for the computers in the
student labs. This system is most likely
a. a single channel, limited queue
system
b. a single channel, limited population
system c.
a multi-channel, limited
queue system
d. a multi-channel, limited population
system e.
none of the above
b (Characteristics of a waiting-line system,
moderate)
38.
Women and children first! declares the captain of a sinking ship. His directive employs
which of the following queue disciplines in disembarking passengers?
a. priority
b.
random
c. FIFO
or
FIFS
d.
LIFO or
LIFS
e. none of the
above
a (Characteristics of a waiting-line system,
moderate)
548
3 9.
A university has several technicians in the repair station to care for the computers in the
student labs. This system is most likely
a. single channel, limited queue system
b. single channel, limited population
system c.
multi-channel, limited
queue system
d. multi-channel, limited population system
e. none of the above
d (Characteristics of a waiting-line system, moderate)
40.
A system in which the customer receives service from only one station and then exits the
system is a. a single-phase system
b. a single channel system
c. a multiple-channel
system d. a multiplephase system
e. none of the above
a (Characteristics of a waiting-line system, easy)
41.
In a repetitive focus factory, the number of phases found in the system might
refer to a.
the number of successive operations that have to be performed
on a part
b. the number of machines doing the same necessary
operations c. the number of parts waiting to be processed
d. all of the above depending on the
layout e.
none of the above
a (Characteristics of a waiting-line system, moderate)
42.
Which of the following is a measure of queue
performance?
a. utilization factor
b. average queue length
c. probability of a specific number of customers in the
system d. average waiting time in the line
e. all of the above
e (Characteristics of a waiting-line system, moderate)
43. Which of the following is most likely to be served in a last-in, first-served (LIFS) queue
discipline?
a. customers checking out at a grocery
store b. the in-basket on a manager's
desk
c. patients entering a hospital emergency room
d. patrons waiting to be seated in a casual-dining
restaurant e. all of the above
b (Characteristics of a waiting-line system, easy)
44.
In a repetitive focus factory, the number of channels available for the processing of a
certain part would likely refer to
a. the number of successive operations that have to be performed on
that part b. the number of machines doing the same necessary
operations
c. the number of parts waiting to be
processed d. all of the above depending
on the layout
e. none of the above
b (Characteristics of a waiting-line system, moderate)
549
4 5.
A waiting line meeting the assumptions of M/M/1 has average time between arrivals of 20
minutes and services items in an average of 10 minutes each; the utilization factor is
approximately
a. 0.25
b.
0.33 c.
0.50
d.
0.67 e.
3.00
c (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
46.
A waiting line model meeting the assumptions of M/M/1 has an arrival rate of 2 per
hour and a service rate of 6 per hour; the utilization factor for the system is
approximately
a. 0.25
b.
0.33 c.
0.50
d.
0.67 e.
3.00
b (The variety of queuing models, easy) {AACSB: Analytic
Skills}
47.
As the average service rate increases, the shape of the negative exponential distribution
of service times
a. grows steadily steeper without
limit
b. has an ever steeper slope that eventually turns
positive
c. becomes less gently curved as it moves ever closer to the graph
origin d. takes on a more uniform slope over a wide range of
service times
e. changes in appearance from convex to
concave
c (The variety of queuing models,
easy)
48.
Which one of the following is not a characteristic of a Model A or M/M/1
system?
a. exponential service time
pattern b. single number of
channels
c. single number of
phases
d. Poisson arrival rate
pattern e.
limited
population size
e (The variety of queuing models,
moderate)
49.
Which one of the following is not a characteristic of a Model B or M/M/S
system?
a. unlimited population
size b. single channel
c. single
queue
d.
single
phase
e. Poisson arrival rate
pattern
b (The variety of queuing models,
moderate)
550
5 0.
Which one of the following is not a characteristic of a Model C or M/D/1 system?
a. single
channel b.
single phase
c. Poisson arrival rate
pattern
d. exponential service time
pattern e.
unlimited
population size
d (The variety of queuing models,
moderate)
51.
In the basic queuing model (M/M/1), service times are
described by a. continuous probability distributions
b. negative exponential probability
distributions c. Poisson probability
distributions
d. normal probability
distributions e. lognormal
distributions
b (Characteristics of a waiting-line system,
moderate)
52.
In the basic queuing model (M/M/1), arrival rates are
distributed by a.
continuous probability distributions
b.
normal probability
distributions
c. negative exponential probability
distributions d. Poisson distributions
e. lognormal
distributions
d (The variety of queuing models,
moderate)
53.
A single-phase waiting-line system meets the assumptions of constant service time or
M/D/1. Units arrive at this system every 10 minutes on average. Service takes a constant 4
minutes. The average length of the queue Lq is
a. 0.4
b.
0.133 c.
4.167
d.
4.583 e.
6
b (Characteristics of a waiting-line system, moderate) {AACSB: Analytic
Skills}
54.
Which of the following is not an assumption of the M/M/1 model?
a. The first customers to arrive are the first customers
served.
b. Each arrival comes independently of the arrival immediately before and after that
arrival. c.
The population from which the arrivals come is very large or infinite
in size.
d.
Customers do not
renege.
e. Service times occur according to a normal
curve.
e (The variety of queuing models,
moderate)
551
5 5.
A single-phase waiting-line system meets the assumptions of constant service time or
M/D/1. Units arrive at this system every 12 minutes on average. Service takes a constant 8
minutes. The average length of the queue Lq is approximately
a. 0.67
b. 2.5
c. 4.5
d. 5.0
e. 7.5
a (Characteristics of a waiting-line system, moderate) {AACSB: Analytic Skills}
56.
A single-phase waiting-line system meets the assumptions of constant service time or
M/D/1. Units arrive at this system every 12 minutes on average. Service takes a constant 8
minutes. The average number in the system Ls is approximately
a. 2.25
b. 2.5
c. 3.0
d.
1.33 e.
5.0
d (Characteristics of a waiting-line system, moderate) {AACSB: Analytic Skills}
57.
A queuing model which follows the M/M/1 assumptions has = 2 and = 3. The average
number in the system is
a. 2/3
b. 1
c. 1.5
d. 2
e. 6
d (The variety of queuing models, moderate) {AACSB: Analytic Skills}
58.
A queuing model which follows the M/M/1 assumptions has = 3 and = 2. The average
number in the system is
a. -3
b. 3
c. 0.667
d. 150 percent
e. growing without limit, since is larger than .
e (The variety of queuing models, moderate) {AACSB: Analytic Skills}
59.
Students arrive randomly at the help desk of the computer lab. There is only one service
agent, and the time required for inquiry varies from student to student. Arrival rates have
been found to follow the Poisson distribution, and the service times follow the negative
exponential distribution. The average arrival rate is 12 students per hour, and the average
service rate is 20 students per hour. What is the average service time for this problem?
a. 1 minute
b. 2 minutes
c. 3 minutes
d. 5 minutes
e. 20
minutes
c (The variety of queuing models, moderate) {AACSB: Analytic Skills}
552
6 0.
A queuing model which follows the M/M/1 assumptions has = 10 and = 12. The
average number in the system is
a. 0.83
b.
2
c. 2.5
d. 5
e. 6
d (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
61.
A queuing model which follows the M/M/1 assumptions has = 2 and = 8. The average
number in the system Ls is
and the utilization of the
system is
.
a. 3;
100
percent
b. 0.33; 25
percent c.
4;
33 percent
d. 6; 25
percent e.
4; 25
percent
b (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
62. Four of the most widely used waiting line modelsM/M/1 or A, M/M/S or B, M/D/1 or C,
and
Limited population or Dall share three characteristics,
which are a. normal arrivals, FIFO discipline, and normal
service times
b. Poisson arrivals, FIFO discipline, and a single-service
phase
c. Poisson arrivals, FIFO discipline, and exponential service
times
d. Poisson arrivals, no queue discipline, and exponential service
times e. none of these
b (The variety of queuing models,
moderate)
63.
A queuing model which follows the M/M/1 assumptions has = 2 and = 3. The average
waiting time in the system is
a. 2/3
b. 1
c. 1.5
d. 2
e. 6
b (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
6 4.
Students arrive randomly at the help desk of the computer lab. There is only one service
agent, and the time required for inquiry varies from student to student. Arrival rates have
been found to follow the Poisson distribution, and the service times follow the negative
exponential distribution. The average arrival rate is 12 students per hour, and the average
service rate is 20 students per hour. What is the utilization factor?
a. 20%
b.
30% c.
40%
d.
50% e.
60%
e (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
553
6 5.
A finite population waiting line model has an average service time T of 100 minutes
and an average time between service requirements U of 400 minutes; the service
factor X is
a. 0.20
b.
0.25 c.
4
d. 5
e. 300 minutes
a (The variety of queuing models, easy) {AACSB: Analytic Skills}
66.
A finite population waiting line model has an average service time T of 200 minutes
and an average time between service requirements U of 300 minutes; the service
factor X is
a. 0.20
b.
0.40 c.
0.60
d.
0.67 e.
2.5
b (The variety of queuing models, easy) {AACSB: Analytic Skills}
67.
Students arrive randomly at the help desk of the computer lab. There is only one service
agent, and the time required for inquiry varies from student to student. Arrival rates have
been found to follow the Poisson distribution, and the service times follow the negative
exponential distribution. The average arrival rate is 12 students per hour, and the average
service rate is 20 students per hour. A student has just entered the system. How long is she
expected to stay in the system?
a. 0.125 minute
b. 0.9 minute
c. 1.5
minutes d.
7.5 minutes e.
0.075 hour
d (The variety of queuing models, moderate) {AACSB: Analytic Skills}
68.
Students arrive randomly at the help desk of the computer lab. There is only one service
agent, and the time required for inquiry varies from student to student. Arrival rates have
been found to follow the Poisson distribution, and the service times follow the negative
exponential distribution. The average arrival rate is 12 students per hour, and the average
service rate is 20 students per hour. How many students, on the average, will be waiting in
line at any one time?
a. 0.9 students
b. 1.5
students c.
3 students
d. 4 students
e. 36 students
a (The variety of queuing models, moderate) {AACSB: Analytic Skills}
554
6 9.
A waiting-line system that meets the assumptions of M/M/S has = 5, = 4, and M = 2.
For these values, Po is approximately 0.23077 and Ls is approximately 2.05128. The
average time a unit spends waiting in this system
a. is
approximately
0.1603 b.
is
approximately 0.2083 c.
is
approximately
0.4103
d.
is
approximately 0.8013
e. cannot be calculated because is larger
than
a (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
70. A waiting-line system that meets the assumptions of M/M/1 has = 1, = 4. For this
system, Po is
and utilization is
.
a. 0.75;
0.25 b.
0.80; .20 c.
-3; -4
d.
3;
4
e. none
of
these
a (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
71.
A waiting-line system that meets the assumptions of M/M/S has = 5, = 4, and M = 2.
For these values, Po is approximately 0.23077 and Ls is approximately 2.05128. The
average number of units waiting in the queue
a. is
approximately
0.1603 b.
is
approximately 0.4103 c.
is
approximately
0.8013
d.
is
approximately 1.0417
e. cannot be calculated because is larger
than
c (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
72.
A waiting-line system that meets the assumptions of M/M/1 has = 1, = 4. For this
system, the probability of more than two units in the system is approximately
a. zero
b.
0.015625 c.
0.0625
d.
0.25
e.
0.9375
b (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
73.
A waiting-line system that meets the assumptions of M/M/1 has = 1, = 4. For this
system, the probability of fewer than two units in the system is approximately
a. 0.0625
b. 0.25
c. 0.75
d.
0.9375 e.
certain
d (The variety of queuing models, difficult) {AACSB: Analytic
Skills}
555
FILL-IN-THE-BLANK
74.
A waiting line or
is where items or people are in a line awaiting service;
is a body of knowledge about waiting lines.
queue, queuing theory (Introduction, easy)
75.
A(n)
is a discrete probability distribution that often describes the arrival
rate in queuing theory.
Poisson distribution (Characteristics of a waiting-line system, moderate)
76. The
of a waiting line and the probability that the queue is empty add to
one.
utilization (Characteristics of a waiting-line system, moderate)
77.
A(n)
occurs when an arrival refuses to enter a waiting line; a(n)
occurs when an arrival joins a waiting line, then leaves it.
balk, renege (Characteristics of a waiting-line system, moderate)
78.
A waiting line has a(n)
population if, as arrivals take place, the
likelihood of additional arrivals decreased.
finite is or limited (Characteristics of a waiting-line system, moderate)
79.
Of the three types of queue discipline, only
is assumed by the four
primary waiting line models.
FIFO or FIFS or first-in, first served (Characteristics of a waiting-line system,
moderate)
80.
A(n)
queuing system has one line and one server.
single-channel (Characteristics of a waiting-line system, moderate)
81.
A(n)
82.
A(n)
queuing system is one in which the customer receives service
from only one station and then exits the system.
single-phase (Characteristics of a waiting-line system, moderate)
83.
The
probability distribution is a continuous probability distribution
often used to describe the service time in a queuing system.
negative exponential (Characteristics of a waiting-line system, moderate)
84.
In a finite or limited population waiting line, the
is calculated from the
average service time and average time between service requirements before the problem
can be completed. service factor (Characteristics of a waiting-line system, moderate)
queuing system has one waiting line, but several servers; a(n)
queuing system is one in which the customer receives services
from several stations before exiting the system
multiple-channel, multiple-phase (Characteristics of a waiting-line system, moderate)
556
SHORT ANSWERS
85.
Provide an example of a limited or finite population for a queue.
Answers will vary, but the textbook provides copiers at a copying shop in
terms of breakdowns and repair. (Characteristics of a waiting-line system,
moderate)
86.
What does it mean to have a patient customer in a waiting line?
A patient customer is one who waits in the queue until he or she has been served,
and exits the system normally. A patient customer does not balk or renege.
(Characteristics of a waiting-line system, moderate)
87.
What is queue discipline? State three rules for queue discipline. Which of these rules
appears most frequently in the four main models?
Queue discipline is the rule that determines the sequence in which arrivals will
receive service. Choices include (1) FIFO, FIFS, or first-in, first-served, (2) LIFO,
LIFS, or last-in, first-served, and (3) priority, such as the triage area of a hospital
emergency room. FIFO is the only one assumed by the four models. (Characteristics
of a waiting-line system, moderate)
88.
Describe the difference between FIFO and LIFO queue disciplines.
FIFO (first-in, first-out) serves first the customer who entered earliest and who has
been in line longest; LIFO serves first the customer who entered most recently. FIFO
exhibits a fairness to people waiting; LIFO is something like an in-basket on a desk,
where the top piece of paper was the last entered, but the first to be serviced.
(Characteristics of a waiting-line system, moderate)
89.
Customers take a number as they join the waiting line of the customer service counter at a
discount store. There are two customer service agents. Provide the most likely
characteristics of this system. a. name of model
b. number of
channels c. number
of phases
d. arrival rate
distribution e. service
time distribution f.
population size
g. queue discipline
Multi-channel system (M/M/S); two; single; Poisson; exponential; unlimited;
and FIFO. (Characteristics of a waiting-line system, moderate)
90.
Students arrive randomly at the help desk of a computer lab. There is only one service
agent, and the service time varies from one student to the other. Provide the most likely
characteristics for this system.
a. name of model
b. number of
channels c. number
of phases
d. arrival rate
distribution e. service
time distribution f.
population size
g. queue discipline
Single channel system (M/M/1); single; single; Poisson; exponential; unlimited;
and FIFO. (Characteristics of a waiting-line system, moderate)
557
9 1.
Why does it matter whether a population of arrivals is limited or unlimited? Compose
your answer in a well-organized, convincing paragraph.
If an arrival population is unlimited, then one arrival has no effect on the likelihood
of the
next. All arrivals will be determined by the underlying probability distribution. If,
however, the population is limited, then one arrival, by leaving the outside population
and entering the waiting line, means that there are fewer potential arrivals. In turn,
that decreases the probability of the next arrival, and requires techniques beyond the
typical probability distributions. In an extreme case, if all members of the population
are in the waiting line,
then there can no further arrivals at all!(Characteristics of a waiting-line system,
moderate)
92.
What are Ls and Lq, as used in waiting line terminology? Which is larger, Ls or Lq?
Explain.
Ls is the average number of units in the system. Lq is the average number of units
in the waiting line. Ls includes the average number of units in line as well as the
average number of units being served. Consequently, Ls is larger than Lq by the
average number of units actually being serviced. (Characteristics of a waiting-line
system, moderate)
93.
There is only one bay and one type of service at an automatic car wash. Provide the
most likely characteristics of this system.
a. name of model
b. number of
channels c. number
of phases
d. arrival rate
distribution e. service
time distribution f.
population size
g. queue discipline
Constant service (M/D/1); single; single; Poisson; constant; unlimited; and
FIFO. (Characteristics of a waiting-line system, moderate)
94.
What is the waiting-line problem? Why is it important to operations?
A waiting-line problem concerns the question of finding the appropriate level of
service where customers waiting for service form a line (or queue). Analysis of
queues helps operations managers understand service systems, maintenance
activities, and shop-floor control activities. (Characteristics of a waiting-line
system, easy)
95.
What are the components in a waiting-line system?
The components of a waiting-line system are the arrivals, or inputs, the waiting
line itself, and the service facility. (Characteristics of a waiting-line system, easy)
96.
Describe the important operating characteristics of a queuing system.
The operating characteristics usually considered of theoretical importance are:
The average time each customer or object spends in the queue
The average length of the queue
The average time each customer spends in the system (waiting time plus service
time)
The average number of customers in the system
The probability that the service facility will be idle
The utilization factor
The probability of a specific number of customers in the system
(Characteristics of a waiting-line system, moderate)
558
9 7.
In an earlier chapter, you were introduced to a "traditional" view of costs and a "full cost"
view of costs, with respect to a certain type of analysis. How might that lesson apply
here? In particular, might operations managers pay more attention to some kinds of costs
than others?
This is a critical-thinking exercise for students. Service costs are probably easy for
operations managers to identify and to calculate, and may well be direct company
expenses. On the
other hand, waiting costs may be more subjective, hard to quantify, or perhaps are
borne by
customers, not the firm. In this case, operations managers might pay more attention
to what costs they bear and experience, and less attention to those that are borne by
others. The
result of this bias leads to a reduction in the level of service, because the "savings" in
reduced
wait costs are not fully recognized. (Queuing costs, easy) {AACSB: Reflective
Thinking}
98. What costs are present in waiting line analysis? How do these costs vary with the level of
service?
The costs are the cost of providing service, and the cost of waiting time. The cost of
providing service rises in a linear fashion with increases in the service level. The cost
of waiting time decreases in a nonlinear fashion as the service level increases.
(Queuing costs, easy)
99.
What are the assumptions underlying the M/M/1 waiting line model? Which of these also
hold for the M/D/1 model?
The assumptions underlying the standard waiting line or queuing model are:
1. Arrivals are served on a first come, first served (FCFS, or FIFO) basis;
and every arrival waits to be served regardless of the length of the line or
queue.
2. All arrivals are independent of preceding arrivals, and the average number of
arrivals per unit time (arrival rate) does not change over time.
3. Arrival rates are described by a Poisson probability distribution, and arrivals
come from
an infinite or very large source.
4. Service times vary from one customer to another and are independent of one
another, but their average rate is known.
5. Service times are described by a negative exponential probability distribution.
6. The effective service rate is faster than the arrival
rate. Assumptions 4 and 5 do not apply to M/D/1.
(The variety of queuing models, moderate)
100. You have seen that, in an M/D/1 problem, the average queue length is exactly one-half the
average queue length of an otherwise identical M/M/1 problem. Are all other performance
statistics one- half as large also? Explain.
Only the average queue length and average waiting time are halved. Utilization is
exactly the
same. Ls and Ws are smaller in M/D/1 than in M/M/1, but not by exactly onehalf. (The variety of queuing models, moderate)
101. Why must the service rate be greater than the arrival rate in a single-channel system?
If the service rate is not greater than the arrival rate, the line will increase in
length indefinitely. (The variety of queuing models, moderate)
102. Most banks have changed from having a line in front of each teller to a system where one
line feeds all tellers. Which system is better? Why?
It is inherently more efficient to have a queuing system whereby one line feeds all
tellers. (The variety of queuing models, moderate)
559
PROBLEMS
103. A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a
service rate of 12 per hour. What is the probability that the waiting line is empty?
Po = 1 - / = 1 - 4/12 = 8/12 or 0.667. (The variety of queuing models, easy)
{AACSB: Analytic Skills}
104. A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a
service rate of 12 per hour. What is the average time a unit spends in the system and the
average time a unit spends waiting?
Ws = 1 / ( - ) = 1 / (12 4) = 1/8 or 0.125; Wq = / (*(-)) = 4 / (12*8) = 1/24 or
0.0417. (The variety of queuing models, easy) {AACSB: Analytic Skills}
105. A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a
service rate of 12 per hour. What is the average time a unit spends in the system and the
average time a unit spends waiting?
Ws = 1 / ( - ) = 1 / (12 10) = 1/2 or 0.5; Wq = / (*(-)) = 10 / (12*2) = 10 / 24 or
0.4167. (The variety of queuing models, easy) {AACSB: Analytic Skills}
106. A waiting line meeting the M/M/1 assumptions has an arrival rate of 10 per hour and a
service rate of 12 per hour. What is the probability that the waiting line is empty?
Po = 1 - / = 1 - 10/12 = 2/12 or 0.1667. (The variety of queuing models, easy)
{AACSB: Analytic Skills}
107. A crew of mechanics at the Highway Department garage repair vehicles that break down
at an average of = 7.5 vehicles per day (approximately Poisson in nature). The
mechanic crew can service an average of = 10 vehicles per day with a repair time
distribution that approximates an exponential distribution.
a. What is the utilization rate for this service system?
b. What is the average time before the facility can return a breakdown to service?
c. How much of that time is spent waiting for service?
d. How many vehicles are likely to be in the system at any one time?
(a) Utilization is = 7.5 / 10 = .75 or 75 percent; (b) Ws = 1 / (10 7.5) = 1 / 2.5 =
0.4 days; (c) Wq = 7.5 / 10*(10-7.5) = 0.3 days; (d) Ls = 7.5 / (10-7.5) = 7.5 / 2.5 = 3
units.
(The variety of queuing models, easy) {AACSB: Analytic Skills}
108. A crew of mechanics at the Highway Department garage repair vehicles that break down
at an average of = 7 vehicles per day (approximately Poisson in nature). The mechanic
crew can service an average of = 11 vehicles per day with a repair time distribution
that approximates an exponential distribution.
a. What is the utilization rate for this service system?
b. What is the average time before the facility can return a breakdown to service?
c. How much of that time is spent waiting for service?
d. How many vehicles are likely to be waiting for service at any one time?
(a) Utilization is = 7 / 11 = .636 or 64 percent; (b) Ws = 1 / (11-7) = 1/4 = 0.25 days;
(c) Wq = 7 / 11*(11-7) = 7 / 44 = 0.16 days; (d) Lq = 7*7 / 11*(11-7) = 49 / 44 = 1.114
units
(The variety of queuing models, easy) {AACSB: Analytic Skills}
560
109. A crew of mechanics at the Highway Department garage repair vehicles which break
down at an average of = 5 vehicles per day (approximately Poisson in nature). The
mechanic crew can service an average of = 10 vehicles per day with a repair time
distribution that approximates an exponential distribution.
a. What is the probability that the system is empty?
b. What is the probability that there is precisely one vehicle in the
system? c. What is the probability that there is more than one vehicle
in the system? d. What is the probability of 5 or more vehicles in the
system?
(a)P0 = 1 5/10 = 0.50; (b) Pn>1 =(5/10)2 = 0.25; the probability of exactly one is .50
-.25 = .25; (c) 0.25 as previously calculated; (d) the probability of five or more is Pn>4
= (5/10)5 = 0.0313. (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
110. A crew of mechanics at the Highway Department garage repair vehicles that break down
at an average of = 8 vehicles per day (approximately Poisson in nature). The mechanic
crew can service an average of = 11 vehicles per day with a repair time distribution
that approximates an exponential distribution. The crew cost is approximately $300 per
day. The cost associated with lost productivity from the breakdown is estimated at $150
per vehicle per day (or any fraction thereof). What is the expected cost of this system?
The number of vehicles out of service is Ls = 8 / (11-8) = 8/3 = 2.667. The cost of
waiting is
$150 x Ls = $150 x 2.667 = $400. Server cost is $300 per day for a total
of $700. (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
111. A crew of mechanics at the Highway Department garage repair vehicles that break down
at an average of = 8 vehicles per day (approximately Poisson in nature). The mechanic
crew can service an average of = 10 vehicles per day with a repair time distribution
that approximates an exponential distribution.
a. What is the probability that the system is empty?
b. What is the probability that there is precisely one vehicle in the
system? c. What is the probability that there is more than one vehicle
in the system? d. What is the probability of 5 or more vehicles in the
system?
(a) P0 = 1 8/10 = 0.20; (b) Pn>1 =(8/10)2 = 0.64; the probability of exactly one is .36
-.20 = .16; (c) 0.64 as previously calculated; (d) Pn>4 = (8/10)5 = 0.32768. (The variety
of queuing models, moderate) {ACSB: Analytic Skills}
112. A crew of mechanics at the Highway Department garage repair vehicles that break down
at an average of = 8 vehicles per day (approximately Poisson in nature). The mechanic
crew can service an average of = 11 vehicles per day with a repair time distribution
that approximates an exponential distribution. The crew cost is approximately $300 per
day. The cost associated with lost productivity from the breakdown is estimated at $150
per vehicle per day (or any fraction
thereof). Which is cheaper, the existing system with one service crew, or a revised system
with two service crews?
Ls for the single server is 8 / (11-8) = 8/3 = 2.667. The single-server system server cost
is $300 per day; wait cost is $150 x 2.667 = $400, for a total of $700. For the two-
server system, Ls =
0.8381. The two-server system will double the server cost to $600, but reduce the wait
cost to
$150 x .8381 = $125.72, for a total of $725.72. The single-server system is
cheaper. (The variety of queuing models, difficult) {AACSB: Analytic
Skills}
561
113. A dental clinic at which only one dentist works is open only two days a week. During
those two days, the traffic is uniformly busy with patients arriving at the rate of three
per hour. The doctor serves patients at the rate of one every 15 minutes.
a. What is the probability that the clinic is empty (except for the dentist)?
b. What percentage of the time is the dentist busy?
c. What is the average number of patients in the waiting room?
d. What is the average time a patient spends in the office (wait plus service)?
e. What is the average time a patient waits for service?
(a) Po = 1 3/4 = 0.25; (b) The dentist is busy when the clinic is not empty, or 1 - .25 =
0.75 or
75 percent of the time; (c) Lq = 3*3 / 4*(4-3) = 2.25; (d) Ws = 1 / (4-3) = 1 hour;
(e) Wq = 3 / 4*(4-3) = 0.75 hours. (The variety of queuing models, easy) {AACSB:
Analytic
Skills}
114. A dental clinic at which only one dentist works is open only two days a week. During
those two days, the traffic arrivals follow a Poisson distribution with patients arriving at
the rate of three per hour. The doctor serves patients at the rate of one every 15 minutes.
a. What is the probability that the clinic is empty (except for the dentist)?
b. What is the probability that there are one or more patients in the system?
c. What is the probability that there are four patients in the system?
d. What is the probability that there are four or more patients in the system?
(a) Po = 1 3/4 = 0.25; (b) The probability that there are one or more patients is Pn>0 =
3/4 or
.75; (c) The probability of exactly four patients is Pn>3 Pn>4=.3164 .2373 = .0791; (d)
.3164
as previously calculated. (The variety of queuing models, moderate) {AACSB: Analytic
Skills}
115. At the order fulfillment center of a major mail-order firm, customer orders, already
packaged for shipment, arrive at the sorting machines to be sorted for loading onto the
appropriate truck for the parcel's address. The arrival rate at the sorting machines is at the
rate of 100 per hour following a Poisson distribution. The machine sorts at the constant
rate of 150 per hour.
a. What is the utilization rate of the system?
b. What is the average number of packages waiting to be
sorted? c. What is the average number of packages in the
sorting system? d. How long must the average package wait
until it gets sorted?
e. What would Lq and Wq be if the service rate were exponential, not constant?
(a) The utilization rate is = 100/150 = 0.67 or 67 percent; (b) Lq = 100*100 /
(2*150*50) =
.67; (c) Ls = .67 + 100/150 = 1.33; (d) Wq = 100 / (2*150*50) = 0.0067 hours, or 0.4
minutes.
(e) Both values would be exactly doubled from the constant service rate results: Lq =
1.33= .0133. (The variety of queuing models, moderate) {AACSB: Analytic Skills}
W and
q
562
116. At the order fulfillment center of a major mail-order firm, customer orders, already
packaged for shipment, arrive at the sorting machines to be sorted for loading onto the
appropriate truck for the parcel's address. The arrival rate at the sorting machines is at the
rate of 140 per hour following a Poisson distribution. The machine sorts at the constant
rate of 150 per hour.
a. What is the utilization rate of the system?
b. What is the average number of packages waiting to be
sorted? c. What is the average number of packages in the
sorting system? d. How long must the average package wait
until it gets sorted?
(a) The utilization rate is = 140/150 = 0.9333 or 93.33 percent; (b) Lq = 6.53; (c) Ls =
7.47;
(d) Wq = 0.0467 hours, or less than 3 minutes. Parts (b)-(d) are supported by the
excerpt from
ExcelOM results below.
Results
Average server utilization(r)
0.933333
Average number of customers in the queue(Lq) 6.533333
Average number of customers in the system(L) 7.466667
Average waiting time in the queue(Wq)
0.046667
Average time in the system(W)
0.053333
Probability (% of time) system is empty (P0) 0.066667
(The variety of queuing models, moderate) {AACSB: Analytic Skills}
117. A waiting-line system that meets the assumptions of M/M/1 has = 1, = 4. Calculate Po.
Build a table showing the probability of more than 0, 1, 2, 3, 4, 5, 6,and 7 units in the
system. Round to six decimal places in your work
(The variety of queuing models, moderate) {AACSB: Analytic Skills}
563
118. Genco, Inc., a small manufacturer of diesel-generator sets has four shearing machines.
Because of the age of these machines, they need minor repairs after 30 hours of use.
Analysis of previous breakdowns indicates that breakdowns follow a Poisson distribution.
The facility employs one repairman specifically to repair these machines. Average repair
time is two hours following an exponential distribution.
a. What is the service factor for this system?
b. What is the average number of these machines in service?
c. What is the impact of adding a second repairman?
(a) X = 2/(2+30) = .0625; (b) 4 - .2987 = 3.7123 machines; (c) 4 - .2514 = 3.7486,
there is a slight improvement in availability of these machines. The table below
summarizes the software results from ExcelOM.
One server
Two servers
Average server utilization (r) 0.246753
Average number of customers
in the queue (L )
Average server utilization (r) 0.124954
Average number of customers in 01464
the queue (L )
0.0
q
q
0.29871
Average number of customers in
the system (L)
0.251373
Average waiting time in the
queue (Wq)
0.011717
Average time inof time) eystem 2.421129
Probability (% the syst sm (W)
is empty (P )
Average time in the system (W) 2.011717
Probability (% of time) system 0.772099
is empty (P )
0.051957
Average number of customers
in the system (L)
Average waiting time in the
queue (Wq)
0.421129
0
0
0.753247
Effective arrival rate
0.123376
Effective arrival rate
0.124954 (The variety of queuing models, difficult)
{AACSB: Analytic Skills}
119. A finite population waiting line model with a single server has an average service time T
of 200 minutes and an average time between service requirements U of 300 minutes.
Calculate the service factor X. If the population consists of 5 elements, what are the
average number waiting, the average number being serviced, and the average number
running? Refer to Table D.7.
The service factor is X = T / (T + U) = 200 / (200 + 300) = 0.40. For a population of
five, the table factors are D = .952 and F = 0.493. The number waiting is L = N*(1F) = 5*(1-0.493) =
5*.507 = 2.535. The number being serviced is H = F*N*X = .493*5*.4 = 0.986. The
number running is J = N*F*(1-X) = 5*.493*.60 = 1.479. (The variety of queuing
models, moderate)
{AACSB: Analytic Skills}
120. A finite population waiting line model with a single server has an average service time T
of 50 minutes and an average time between service requirements U of 350 minutes.
Calculate the service factor X. If the population consists of 5 elements, what are the
average number waiting, the average number being serviced, and the average number
running? Refer to Table D.7.
The service factor is X = T / (T + U) = 50 / (50 + 350) = 0.125. For a population of five,
th e
table factors are D = .473 and F = 0.920. The number waiting is L = N*(1-F) = 5*(10.920) =
5*.080 =0.400. The number being serviced is H = F*N*X = .920*5*.125 = 0.575. The
number running is J = N*F*(1-X) = 5*.920*.875 = 4.025. (The variety of queuing
models, moderate)
{AACSB: Analytic Skills}
564