CHAPTER 14
Waiting Lines and Queuing Theory Models
TRUE/FALSE
14.1
The three parts of a queuing system are the arrivals, the queue, and the service facility.
14.2
Two characteristics of arrivals are the line length and queue discipline.
14.3
Queuing theory models can also apply to customers placing telephone calls and being placed on
hold.
14.4
The only objective of queuing theory is to minimize customer dissatisfaction.
14.5
Should a customer leave a queue before being served, it is said that the customer has reneged.
14.6
Balking refers to customers who enter the queue but may become impatient and leave without
completing their transactions.
14.7
Most systems use the queue discipline known as the first-in, first-out rule.
14.8
In a very complex queuing model, if all of the assumptions of the traditional models are not
met, then the problem cannot be handled.
14.9
Before using exponential distributions to build queuing models, the quantitative analyst should
determine if the service time data fit the distribution.
14.10
For practical purposes, queue length is almost always modeled with a finite queue length.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.11
The Greek letter  is used to represent the average service rate at each channel.
14.12
For a single channel model that has Poisson arrivals and exponential service rates, the Greek
letter  is the utilization factor.
14.13
In a multi-channel, single-phase queuing system, the arrival will pass through at least two
different service facilities.
14.14
In a multi-channel model  =  /( M ).
14.15
A goal of many waiting line problems is to help a firm find the ideal level of services to be
offered.
14.16
Any waiting line problem can be investigated using an analytical queuing model.
14.17
One of the difficulties in waiting line analyses is that it is sometimes difficult to place a value
on customer waiting time.
14.18
The goal of most waiting line problems is to identify the service level that minimizes service
cost.
14.19
One of the limitations of analytical waiting line models is that they do not give information on
extreme cases (e.g., maximum waiting time or maximum number in the queue).
14.20
An "infinite calling population" occurs when the likelihood of a new arrival does not depend
upon the number of past arrivals.
14.21
All practical problems can be described by an "infinite" population waiting model.
14.22
On a practical note – if we are using waiting line analysis for a problem studying customers
calling a telephone number for service, balking is probably not an issue.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.23
On a practical note– if we are using waiting line analysis to study cars passing through a single
tollbooth, reneging is probably not an issue.
14.24
On a practical note – if we are studying patrons moving through checkout lines at a grocery
store, and we note that these patrons sometimes move from one line to another, we should
consider balking as an issue.
14.25
On a practical note – if we were to study the waiting lines in a hair salon which had only five
chairs for patrons waiting, we would have to use a finite queue waiting line model.
14.26
All practical waiting line problems can be viewed as having a FIFO queue discipline.
14.27
A hospital emergency room will usually employ a FIFO queue discipline.
14.28
If we wish to study a bank, in which patrons entered the building and then, depending upon the
service desired, chose one of several tellers in front of which to form a line, we would employ a
set of single-channel queuing models.
14.29
On a practical note – we should probably view the checkout counters in a grocery store as a set
of single channel systems.
14.30
A cafeteria, in which cold dishes are separated from hot dishes, is probably best viewed as a
single-channel, single-phase system.
14.31
An emergency room might be viewed as a multi-channel, multi-phase system.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.32
A single highway with multiple tollbooths should be viewed as a single-channel system.
14.33
In a doctor's office, we would expect the arrival rate distribution to be Poisson distributed, and
the service time distribution to be negative exponential.
14.34
The M/M/1 queuing model assumes that the arrival rate does not change over time.
14.35
The analytical queuing models typically provide operating characteristics that are averages
(e.g., average waiting time, average number of customers in the queue).
14.36
The analytical queuing models can be used to tell us how many people are presently waiting in
line.
14.37
The quantity  is the probability that one or more customers are in a single channel system.
14.38
In the multi-channel model (M/M/m), we must assume that the average service time for all
channels is the same.
14.39
If we compare a single-channel system with  = 15, to a multi-channel system (with 3 channels)
with the service rate for the individual channel of  = 5, we will find that the average wait time
is less in the single-channel system.
14.40
If we compare a single-channel system with exponential service rate (=5) to a constant service
time model (=5), we will find that the average wait time in the constant service time model is
less than that in the probabilistic model.
14.41
As a general rule, any time that the number of people in line can be a significant portion of the
total population, we should use a finite population model.
14.42
Whether or not we use the finite population queuing model depends upon the relative arrival
and service rates, not just the size of the population from which the arrivals come.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.43
Whether or not we use the finite population queuing model depends upon the amount of space
we have in which to form the queue.
14.44
If a waiting line problem is particularly complex, we may have to turn to a simulation model.
14.45
If we are using a simulation queuing model, we still have to abide by the assumption of a
Poisson arrival rate, and negative exponential service rate.
14.46
Using a simulation model allows one to ignore the common assumptions required to use
analytical models.
*14.47
If we are studying the arrival of automobiles at a highway toll station, we can assume an
infinite calling population.
*14.48
If we are studying the need for repair of electric motors on a small assembly line, we can
assume an infinite calling population.
*14.49
The difference between balking and reneging is that balking implies that the arrival never
joined the queue, while reneging implies that the arrival joined the queue, but became impatient
and left.
*14.50
When looking at the arrivals at the ticket counter of a movie theater, we can assume an
unlimited queue.
*14.51
When looking at the arrivals at a barbershop, we must assume a finite queue.
*14.52
A bank, in which a single queue is used to move customers to several tellers, is an example of a
single-channel system.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
*14.53
A fast food drive-through system is an example of a multi-channel queuing system.
*14.54
A fast food drive-through system is an example of a multi-phase queuing system.
*14.56
In a single-channel, single-phase system, reducing the service time only reduces the total
amount of time spent in the system, not the time spent in the queue.
*14.57
The wait time for a single-channel system is more than twice that for a two channel system
using two servers working at the same rate as the single server.
MULTIPLE CHOICE
14.58
The expected cost to the firm of having customers or objects waiting in line to be serviced is
termed the
(a)
(b)
(c)
(d)
(e)
14.59
expected service cost.
expected waiting cost.
total expected cost.
expected balking cost.
expected reneging cost.
Which of the following is not an assumption in common queuing mathematical models?
(a)
(b)
(c)
(d)
(e)
Arrivals come from an infinite, or very large, population.
Arrivals are Poisson distributed.
Arrivals are treated on a first-in, first-out basis and do not balk or renege.
Service times follow the negative exponential distribution.
The average arrival rate is faster than the average service rate.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.60
Which of the following is not a key operating characteristic for a queuing system?
(a)
(b)
(c)
(d)
(e)
14.61
Three parts of a queuing system are
(a)
(b)
(c)
(d)
14.62
the mean number of people served divided by the mean number of arrivals per time period.
the average time a customer spends waiting in a queue.
the proportion of the time the service facilities are in use.
the percent idle time.
none of the above
Which of the following is not a characteristic of the calling population and its behavior?
(a)
(b)
(c)
(d)
(e)
14.65
single-channel, multi-phase system.
single-channel, single-phase system.
multi-channel, multi-phase system.
multi-channel, single-phase system.
none of the above
The utilization factor  for a system is defined as
(a)
(b)
(c)
(d)
(e)
14.64
the inputs, the queue, and the service facility.
the calling population, the queue, and the service facility.
the calling population, the waiting line, and the service facility.
All of the above are appropriate labels for the three parts of a queuing system.
Upon arriving at a convention, if a person must line up to first register at a table, then proceed
to a table to gather some additional information, and then pay at another single table, this is an
example of a
(a)
(b)
(c)
(d)
(e)
14.63
utilization rate
percent idle time
average time spent waiting in the system and in the queue
average number of customers in the system and in the queue
none of the above
Size is considered to be limited or unlimited.
Queue discipline.
A customer is usually patient.
Customers can arrive randomly.
none of the above
In queuing theory, the objective is to
(a) maximize productivity.
(b) minimize customer dissatisfaction as measured in balking and reneging.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
(c) minimize the sum of the costs of waiting time and providing service.
(d) minimize the percent of idle time.
(e) minimize queue length.
14.66
In queuing problems, the size of the calling population is important because
(a) it is usually easier to deal with the mathematics if the calling population is considered
infinite.
(b) it is usually easier to deal with the mathematics if the calling population is considered
finite.
(c) it is impossible to deal with the mathematics (except through monte carlo simulation) if
the calling population is infinite.
(d) it is impossible to deal with the mathematics (except through monte carlo simulation) if
the calling population is finite.
(e) none of the above
14.67
An arrival in a queue that reneges is one who
(a)
(b)
(c)
(d)
(e)
14.68
after joining the queue, becomes impatient and leaves.
refuses to join the queue because it is too long.
goes through the queue, but never returns.
jumps from one queue to another, trying to get through as quickly as possible.
none of the above
A balk is an arrival in a queue who
(a)
(b)
(c)
(d)
refuses to join the queue because it is too long.
after joining the queue, becomes impatient and leaves.
goes through the queue, but never returns.
jumps from one queue to another, trying to get through as quickly as possible.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.69
Queue discipline may be
(a)
(b)
(c)
(d)
(e)
14.70
If the arrival rate and service times are kept constant and the system is changed from a
single-channel to a two-channel system, then the average time an arrival will spend in the
waiting line or being serviced (W) is
(a)
(b)
(c)
(d)
(e)
14.71
increased by 50 percent.
reduced by 50 percent.
exactly doubled.
the same.
none of the above
If everything else remains constant, including the mean arrival rate and service rate, except that
the service time becomes constant instead of exponential,
(a)
(b)
(c)
(d)
(d)
14.72
FIFO (first-in, first-out).
FIFS (first-in, first-served).
LIFS (last-in, first-served).
by assigned priority.
all of the above
the average queue length will be halved.
the average waiting time will be doubled.
the average queue length will increase.
the average queue length will double and the average waiting time will double.
none of the above
If a queuing situation becomes extremely complex,
(a)
(b)
(c)
(d)
(e)
there is always a mathematical model to solve it.
the only alternative is to study the real situation.
there are tables available for any combination of complexities.
computer simulation is an alternative.
you should make simplifying assumptions and use the mathematical procedure which most
closely approximates the system to be studied.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.73 Customers enter the waiting line at a cafeteria on a first come, first served basis. The
arrival rate follows a Poisson distribution, while service times follow an exponential
distribution. If the average number of arrivals is six per minute and the average service rate of
a single server is eight per minute, what is the average number of customers in the system?
(a)
(b)
(c)
(d)
(e)
14.74
Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival
rate follows a Poisson distribution, while service times follow an exponential distribution. If
the average number of arrivals is six per minute and the average service rate of a single server
is eight per minute, what is the average number of customers waiting in line behind the person
being served?
(a)
(b)
(c)
(d)
(e)
14.75
0.50
0.75
2.25
3.00
none of the above
0.50
0.75
2.25
3.00
none of the above
Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first
served basis. The arrival rate follows a Poisson distribution, while service times follow an
exponential distribution. If the average number of arrivals is six per minute and the average
service rate of a single server is eight per minute, what proportion of the time is the server
busy?
(a)
(b)
(c)
(d)
(e)
0.25
0.50
0.75
2.25
3.00
452
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.76
Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first
served basis. The arrival rate follows a Poisson distribution, while service times follow an
exponential distribution. If the average number of arrivals is six per minute and the average
service rate of a single server is eight per minute, on average, how much time will elapse from
the time a customer enters the line until he/she leaves the cafeteria?
(a)
(b)
(c)
(d)
(e)
14.77
A post office has a single line for customers to use while waiting for the next available postal
clerk. There are two postal clerks who work at the same rate. The arrival rate of customers
follows a Poisson distribution, while the service time follows an exponential distribution. The
average arrival rate is three per minute and the average service rate is two per minute for each
of the two clerks. What is the average length of the line?
(a)
(b)
(c)
(d)
(e)
14.78
0.25
0.50
0.75
2.25
3.00
3.429
1.929
1.143
0.643
none of the above
A post office has a single line for customers to use while waiting for the next available postal
clerk. There are two postal clerks who work at the same rate. The arrival rate of customers
follows a Poisson distribution, while the service time follows an exponential distribution. The
average arrival rate is three per minute and the average service rate is two per minute for each
of the two clerks. How long does the average person spend waiting for a clerk to become
available?
(a)
(b)
(c)
(d)
(e)
3.429
1.929
1.143
0.643
none of the above
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.79
A post office has a single line for customers to use while waiting for the next available postal
clerk. There are two postal clerks who work at the same rate. The arrival rate of customers
follows a Poisson distribution, while the service time follows an exponential distribution. The
average arrival rate is three per minute and the average service rate is two per minute for each
of the two clerks. What proportion of the time are both clerks idle?
(a)
(b)
(c)
(d)
(e)
14.80
A finite population model differs from the other models because, with a finite population,
(a)
(b)
(c)
(d)
14.81
the queue line is never empty.
there is a dependent relationship between the length of the queue and the arrival rate.
the service rate will be less than the arrival rate.
the average number in the system is the same as the average number in the queue.
At an automatic car wash, cars arrive randomly at a rate of 9 cars every 20 minutes. The car
wash takes exactly 2 minutes (this is constant). On average, what would the length of the line
be?
(a)
(b)
(c)
(d)
(e)
14.82
0.643
0.250
0.750
0.143
none of the above
8.1
4.05
9
1
none of the above
At an automatic car wash, cars arrive randomly at a rate of 9 cars every 20 minutes. The car
wash takes exactly 2 minutes (this is constant). On average, how long would each car spend at
the car wash?
(a)
(b)
(c)
(d)
(e)
0.9 minutes
0.45 minutes
9 minutes
18 minutes
none of the above
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.83
According to Table 14-1, which provides a queuing problem solution, what proportion of the
time is the system totally empty?
(a)
(b)
(c)
(d)
(e)
14.84
According to Table 14-1, which provides a queuing problem solution, on average, how long
does each customer spend waiting in line?
(a)
(b)
(c)
(d)
(e)
14.85
0.111
0.333
0.889
0.667
none of the above
0.333 minutes
0.889 minutes
0.222 minutes
0.722 minutes
0.111 minutes
According to Table 14-1, which provides a queuing problem solution, what is the utilization
rate of the service facility?
(a)
(b)
(c)
(d)
(e)
0.111
0.889
0.222
0.722
0.667
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.86
According to Table 14-2, which provides a queuing problem solution, on average, how many
units are in the line?
(a)
(b)
(c)
(d)
(e)
14.87
According to Table 14-2, which provides a queuing problem solution, what proportion of the
time is at least one server busy?
(a)
(b)
(c)
(d)
(e)
14.88
5.455
3.788
1.091
0.758
0.833
0.833
0.758
0.091
0.909
none of the above
According to Table 14-2, which provides a queuing problem solution, there are two servers in
this system. Counting each person being served and the people in line, on average, how many
people would be in this system?
(a)
(b)
(c)
(d)
(e)
5.455
3.788
9.243
10.900
none of the above
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.89
According to Table 14-3, which presents a queuing problem solution for a queuing problem
with a constant service rate, on average, how much time is spent waiting in line?
(a)
(b)
(c)
(d)
(e)
14.90
According to Table 14-3, which presents a queuing problem solution for a queuing problem
with a constant service rate, on average, how many customers are in the system?
(a)
(b)
(c)
(d)
(e)
14.91
1.875 minutes
1.125 minutes
0.625 minutes
0.375 minutes
none of the above
1.875
1.125
0.625
0.375
none of the above
According to Table 14-3, which presents a queuing problem solution for a queuing problem
with a constant service rate, on average, how many customers arrive per time period?
(a) 3
(b) 4
(c) 1.875
(d) 1.125
(e) none of the above
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.92 According to Table 14-3, which presents a queuing problem with a constant service
rate, on average, how many minutes does a customer spend in the service facility?
(a)
(b)
(c)
(d)
(e)
14.93
The most appropriate cost to be considered in making a waiting line decision is the
(a)
(b)
(c)
(d)
(e)
14.94
percent of time the system is idle.
average percent of time the customers wait in line.
average time the service system is open.
percent of time that a single customer is in the system.
none of the above
Which of the following is usually the most difficult cost to determine?
(a)
(b)
(c)
(d)
(e)
14.97
Arrivals come from an infinite, or very large, population.
Arrivals are Poisson distributed.
Arrivals are treated on a first-in, first-out basis and do not balk or renege.
Service rates follow the normal distribution.
The average service rate is faster than the average arrival rate.
The utilization factor is defined as the
(a)
(b)
(c)
(d)
(e)
14.96
expected service cost.
expected waiting cost.
total expected cost.
expected balking cost.
expected reneging cost.
Which of the following is not an assumption in common queuing mathematical models?
(a)
(b)
(c)
(d)
(e)
14.95
0.375 minutes
4 minutes
0.625 minutes
0.25 minutes
none of the above
service cost
facility cost
calling cost
waiting cost
none of the above
Lines at banks where customers wait to go to a teller window are usually representative of a
(a) single-channel, multi-phase system.
(b) single-channel, single-phase system.
(c) multi-channel, multi-phase system.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
(d) multi-channel, single-phase system.
(e) none of the above
14.98
A restaurant in which one must go to the maitre d' in order to be seated in one of three dining
rooms is an example of a
(a)
(b)
(c)
(d)
(e)
14.99
A gasoline station which has a single pump and where the customer must enter the building to
pay is an example of a
(a)
(b)
(c)
(d)
(e)
14.100
single-channel, multi-phase system.
single-channel, single-phase system.
multi-channel, multi-phase system.
multi-channel, single-phase system.
none of the above
single-channel, multi-phase system.
single-channel, single-phase system.
multi-channel, multi-phase system.
multi-channel, single-phase system.
none of the above
A vendor selling newspapers on a street corner is an example of a
(a)
(b)
(c)
(d)
(e)
single-channel, multi-phase system.
single-channel, single-phase system.
multi-channel, multi-phase system.
multi-channel, single-phase system.
none of the above
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.101
The utilization factor  for a system tells one the
(a)
(b)
(c)
(d)
(e)
14.102
Which of the following is not an assumption about the queue in the waiting line models
presented in this chapter?
(a)
(b)
(c)
(d)
(e)
14.103
Queue length is considered to be unlimited.
Queue discipline is assumed to be FIFO.
A customer in the queue is usually patient.
Customers arrive to enter the queue in a random fashion.
none of the above
Assume that we are using a waiting line model to analyze the number of service technicians
required to maintain machines in a factory. Our goal should be to
(a)
(b)
(c)
(d)
(e)
14.104
mean number of people served divided by the mean number of arrivals per time period.
average time a customer spends waiting in a queue.
proportion of the time the service facilities are in use.
percent idle time.
none of the above
maximize productivity of the technicians.
minimize the number of machines needing repair.
minimize the downtime for individual machines.
minimize the percent of idle time of the technicians.
minimize the total cost (cost of maintenance plus cost of downtime).
In queuing problems, the size of the calling population is important because
(a) we have models only for problems with infinite calling populations.
(b) we have models only for problems with finite calling populations.
(c) the size of the calling population determines whether or not the arrival of one customer
influences the probability of arrival of the next customer.
(d) we will have to consider the amount of space available for the queue.
(e) none of the above
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.105
The behavior of jumping from one queue to another trying to get through as quickly as possible
is called:
(a)
(b)
(c)
(d)
(e)
14.106
The customer who arrives at a bank, sees that there is a long line, and leaves to return at
another time is
(a)
(b)
(c)
(d)
(e)
14.107
balking.
cropping.
reneging.
blithering.
none of the above
The term queue discipline describes the
(a)
(b)
(c)
(d)
(e)
14.108
balking.
reneging.
cropping.
blithering.
none of the above
degree to which members of the queue renege.
sequence in which members of the queue arrived.
degree to which members of the queue are orderly and quiet.
sequence in which members of the queue are serviced.
all of the above
If the arrival rate and service times are kept constant and the system is changed from a twochannel system to a single-channel system, then the average time an arrival will spend in the
waiting line is
(a)
(b)
(c)
(d)
(e)
decreased.
increased.
exactly doubled.
the same as before.
could be any of the above depending on other parameters of the problem.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
14.109
If everything else remains constant, including the mean arrival rate and service rate, except that
the service time becomes constant instead of exponential, the
(a)
(b)
(c)
(d)
(e)
14.110
Assume that we wish to study the performance of checkout stations in a large grocery store.
Some of the checkouts are reserved for those customers with fewer than twelve items; other
checkouts are reserved for those customers from the bakery, deli, or pharmacy; and still other
checkouts are open to all customers. We should employ
(a)
(b)
(c)
(d)
(e)
14.111
a multi-channel, multi-phase queuing model.
a number of single-channel, single-phase models.
two separate multi-channel, single-phase models.
simplifying assumptions to make the problem fit one or another of the analytical models.
a simulation model.
If we want to know the maximum number of customers who will be waiting to buy tickets to a
movie in a theater where there are three servers selling tickets, we should employ a
(a)
(b)
(c)
(d)
(e)
14.112
average waiting time will be decreased.
average queue length will be increased.
average number of customers in the system will be increased.
none of the above
(a), (b), & (c)
single-channel, single-phase model.
multi-channel, single-phase model.
multi-channel, multi-phase model.
single-channel, multi-phase model.
none of the above
The most likely queue discipline to be followed in a hospital emergency room is
(a)
(b)
(c)
(d)
(e)
FIFO (first in, first out)
LIFO (last in, first out)
FILO (first in, last out)
WCF (worst case first)
none of the above
462
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.113
Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival
rate follows a Poisson distribution, while service times follow an exponential distribution. If
the average number of arrivals is four per minute and the average service rate of a single server
is seven per minute, what is the average number of customers in the system?
(a)
(b)
(c)
(d)
(e)
14.114
Customers enter the waiting line at a cafeteria on a first come, first served basis. The arrival
rate follows a Poisson distribution, while service times follow an exponential distribution. If
the average number of arrivals is four per minute and the average service rate of a single server
is seven per minute, what is the average number of customers waiting in line behind the person
being served?
(a)
(b)
(c)
(d)
(e)
14.115
0.43
1.67
0.57
1.33
none of the above
0.76
0.19
1.33
1.67
none of the above
Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first
served basis. The arrival rate follows a Poisson distribution, while service times follow an
exponential distribution. If the average number of arrivals is four per minute and the average
service rate of a single server is seven per minute, what proportion of the time is the server
busy?
(a)
(b)
(c)
(d)
(e)
0.43
0.57
0.75
0.25
0.33
463
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.116
Customers enter the waiting line to pay for food as they leave a cafeteria on a first come, first
served basis. The arrival rate follows a Poisson distribution, while service times follow an
exponential distribution. If the average number of arrivals is four per minute and the average
service rate of a single server is seven per minute, on average, how much time will elapse from
the time a customer enters the line until he/she leaves the cafeteria?
(a)
(b)
(c)
(d)
(e)
14.117
A post office has a single line for customers to use while waiting for the next available postal
clerk. There are two postal clerks who each work at the same rate. The arrival rate of
customers follows a Poisson distribution, while the service time follows an exponential
distribution. The average arrival rate is seven per minute and the average service rate is four per
minute for each of the two clerks. What is the average length of the line?
(a)
(b)
(c)
(d)
(e)
14.118
0.67 minutes
0.50 minutes
0.75 minutes
0.33 minutes
1.33minutes
3.429
4.932
5.717
7.467
none of the above
A post office has a single line for customers to use while waiting for the next available postal
clerk. There are two postal clerks who each work at the same rate. The arrival rate of
customers follows a Poisson distribution, while the service time follows an exponential
distribution. The average arrival rate is seven per minute and the average service rate is four per
minute for each of the two clerks. How long does the average person spend waiting for a clerk
to become available?
(a)
(b)
(c)
(d)
(e)
0.067
0.817
1.067
0.875
none of the above
464
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.119
A post office has a single line for customers to use while waiting for the next available postal
clerk. There are two postal clerks who each work at the same rate. The arrival rate of
customers follows a Poisson distribution, while the service time follows an exponential
distribution. The average arrival rate is seven per minute and the average service rate is four per
minute for each of the two clerks. What proportion of the time are both clerks idle?
(a)
(b)
(c)
(d)
(e)
14.120
0.875
0.067
0.750
0.817
none of the above
At an automatic car wash, cars arrive randomly at a rate of 7 cars every 30 minutes. The car
wash takes exactly 4 minutes (this is constant). On average, what would the length of the line
be?
(a)
(b)
(c)
(d)
(e)
8.171
7.467
6.533
0.467
none of the above
14.121 At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car
wash takes exactly 4 minutes (this is constant). On average, how long would each car spend at
the car wash?
(a)
(b)
(c)
(d)
(e)
14.122
At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash
takes exactly 4 minutes (this is constant). On average, how long would each driver have to wait
before receiving service?
(a)
(b)
(c)
(d)
(e)
14.123
28 minutes
32 minutes
17 minutes
24 minutes
none of the above
28 minutes
32 minutes
17 minutes
24 minutes
none of the above
At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash
takes exactly 4 minutes (this is constant). On average, how many customers would be at the car
wash (waiting in line or being serviced)?
(a) 8.171
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Waiting Lines and Queuing Theory Models  CHAPTER 14
(b)
(c)
(d)
(e)
14.124
At an automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash
takes exactly 4 minutes (this is constant). The utilization factor for this system is
(a)
(b)
(c)
(d)
(e)
14.125
0.643
2.191
2.307
0.217
0.905
According to the information provided in Table 14-4, what proportion of the time is at least one
server busy?
(a)
(b)
(c)
(d)
(e)
14.128
adding additional parking spaces.
reducing the price you charge for washing the car.
modifying the system to speed up service and reduce waiting time.
adding additional, optional service features.
none of the above
According to the information provided in Table 14-4, on average, how many units are in the
line?
(a)
(b)
(c)
(d)
(e)
14.127
0.467
0.533
1.000
0.933
none of the above
At your automatic car wash, cars arrive randomly at a rate of 7 every 30 minutes. The car wash
takes exactly 4 minutes (this is constant). At the moment, you have space for 7 cars in the
waiting area. You should consider
(a)
(b)
(c)
(d)
(e)
14.126
7.467
6.533
0.467
none of the above
0.643
0.905
0.783
0.091
none of the above
Using the information provided in Table 14-4: Counting each person being served and the
people in line, on average, how many people would be in this system?
466
Waiting Lines and Queuing Theory Models  CHAPTER 14
(a)
(b)
(c)
(d)
(e)
14.129
0.905
2.191
6.037
14.609
none of the above
According to the information provided in Table 14-4, what is the average time spent by a
person in this system?
(a)
(b)
(c)
(d)
(e)
0.905 minutes
2.191 minutes
6.037 minutes
14.609 minutes
none of the above
467
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.130
According to the information provided in Table 14-4, what percentage of the total available
service time is being used?
(a)
(b)
(c)
(d)
(e)
14.131
According to Table 14-5, which presents the solution for a queuing problem with a constant
service rate, on the average, how much time is spent waiting in line?
(a)
(b)
(c)
(d)
(e)
14.132
90.5%
21.7%
64.3%
could be any of the above; depends upon other factors.
none of the above
1.607 minutes
0.714 minutes
0.179 minutes
0.893 minutes
none of the above
According to Table 14-5, which presents the solution for a queuing problem with a constant
service rate, on the average, how many customers are in the system?
(a)
(b)
(c)
(d)
(e)
0.893
0.714
1.607
0.375
none of the above
468
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.133 According to Table 14-5, which presents a queuing problem solution for a queuing
problem with a constant service rate, on average, how many customers arrive per time period?
(a) 5
(b) 7
(c) 1.607
(d) 0.893
(e) none of the above
14.134
According to Table 14-5, which presents the solution for a queuing problem with a constant
service rate, on average, how many minutes does a customer spend in the system?
(a)
(b)
(c)
(d)
(e)
14.135
According to Table 14-5, which presents the solution for a queuing problem with a constant
service rate, what percentage of available service time is actually used?
(a)
(b)
(c)
(d)
(e)
14.136
0.893 minutes
0.321 minutes
0.714 minutes
1.607 minutes
none of the above
0.217
0.643
0.321
0.179
none of the above
According to Table 14-5, which presents the solution for a queuing problem with a constant
service rate, the probability that the server is idle is
(a)
(b)
(c)
(d)
(e)
0.217
0.643
0.286
0.714
none of the above
469
Waiting Lines and Queuing Theory Models  CHAPTER 14
*14.137 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food
joint takes an average of 2 minutes to serve each arrival. The utilization factor for this system
is
(a)
(b)
(c)
(d)
(e)
0.467
0.547
0.800
0.133
none of the above
*14.138 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food
joint has restructured their serving system so that service takes exactly 2 minutes (this is
constant) per arrival. The utilization factor for this system is
(a)
(b)
(c)
(d)
(e)
0.800
0.723
1.000
0.854
none of the above
*14.139 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. The fast food
joint takes exactly 2 minutes (this is constant). The average wait time for arrivals is
(a)
(b)
(c)
(d)
(e)
5.4 minutes
6.0 minutes
8.0 minutes
2.5 minutes
none of the above
*14.140 At a local fast food joint, cars arrive randomly at a rate of 12 every 30 minutes. Service times
are random (exponential) and average 2 minutes per arrival. The average time in the queue for
each arrival is
(a)
(b)
(c)
(d)
(e)
2 minutes
4 minutes
6 minutes
8 minutes
10 minutes
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Waiting Lines and Queuing Theory Models  CHAPTER 14
*14.141 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are
exponentially distributed with an average of 15 minutes. The average customer waits in line
(a)
(b)
(c)
(d)
(e)
3.0 minutes
2.5 minutes
3.5 minutes
4.0 minutes
none of the above
*14.142 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are
exponentially distributed with an average of 15 minutes. Jack Burns, the JLUBE owner, has
decided to open a second work bay, i.e., make the shop into a two-channel system. Under this
new scheme, the average customer will wait in line
(a)
(b)
(c)
(d)
(e)
 9.6 minutes
 2.5 minutes
 24.6 minutes
 2.1 minutes
none of the above
*14.143 Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are
exponentially distributed with an average of 15 minutes. Jack Burns, the JLUBE owner, has
decided to open a second work bay, i.e., make the shop into a two-channel system. Under this
new scheme, the total time an average customer spends in the system will be
(a)
(b)
(c)
(d)
 37 minutes
 2.1 minutes
 9.6 minutes
 24.6 minutes
*14.144 Cars approach a set of toll booths at the rate of 75 cars per hour. There are five toll booths,
with an average service time of 0.5 minutes. For the two miles before a car reaches a toll
booth, the highway is five lanes wide. Under heavy traffic conditions, a car commits to a
specific toll lane nearly a mile before it reaches the toll booth. On average, how long is the line
in front of a specific toll booth?
(a)
(b)
(c)
(d)
(e)
0.0125 cars
0.0270 cars
0.0176 cars
0.0179 cars
0.0714 cars
*14.145 Cars approach a set of toll booths at the rate of 75 cars per hour. There are five toll booths,
with an average service time of 0.5 minutes. For the two miles before a car reaches a toll
booth, the highway is five lanes wide. Under light traffic conditions, a car does not have to
471
Waiting Lines and Queuing Theory Models  CHAPTER 14
commit to a specific toll lane until actually approaching the toll booths. On average, how long
is the line in front of a specific toll booth?
(a)
(b)
(c)
(d)
(e)
0.0179 cars
0.0001 cars
0.0100 cars
0.5000 cars
none of the above
*14.146 Cars appear to approach a local Burger Basket Restaurant at the rate of 20 per hour. Average
service rate averages 22 per hour. Gale Johnson, the owner, has become concerned about the
waiting time under the current configuration. She is considering enlarging the facility by
constructing a second takeout window. How does the waiting time with two windows change
from that with only a single window?
(a)
(b)
(c)
(d)
(e)
Waiting time drops from  27.3 minutes with one window to  0.7 minutes with two.
Waiting time drops from  32.1 minutes with one window to  21.3 minutes with two.
There is no change.
Waiting time drops from  20.3 minutes with one window to  6.5 minutes with two.
none of the above
*14.147 Customers arrive at the local PharmCal gas station at the rate of 40 per hour. What service rate
is necessary to keep the average wait time less than 5 minutes?
(a)
(b)
(c)
(d)
(e)
 45 per hour
 47 per hour
 49 per hour
 50 per hour
none of the above
472
Waiting Lines and Queuing Theory Models  CHAPTER 14
PROBLEMS
14.148
A new shopping mall is considering setting up an information desk manned by one employee.
Based upon information obtained from similar information desks, it is believed that people will
arrive at the desk at the rate of 15 per hour. It takes an average of two minutes to answer a
question. It is assumed that arrivals are Poisson and answer times are exponentially distributed.
(a)
(b)
(c)
(d)
(e)
(f)
14.149
Find the probability that the employee is idle.
Find the proportion of the time that the employee is busy.
Find the average number of people receiving and waiting to receive some information.
Find the average number of people waiting in line to get some information.
Find the average time a person seeking information spends at the desk.
Find the expected time a person spends just waiting in line to have his question answered.
A new shopping mall is considering setting up an information desk manned by two employees.
Based upon information obtained from similar information desks, it is believed that people will
arrive at the desk at the rate of 20 per hour. It takes an average of four minutes to answer a
question. It is assumed that arrivals are Poisson and answer times are exponentially distributed.
(a) Find the proportion of the time that the employees are busy.
(b) Find the average number of people waiting in line to get some information.
(c) Find the expected time a person spends just waiting in line to have his question answered.
14.150
Due to a recent increase in business, a secretary in a certain law firm is now having to type
20 letters a day on average. It takes her approximately 20 minutes to type each letter.
Assuming the secretary works 8 hours a day:
(a)
(b)
(c)
(d)
What is the secretary's utilization rate?
What is the average waiting time before the secretary types a letter?
What is the average number of letters waiting to be typed?
What is the probability that the secretary has more than 5 letters to type?
473
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.151
Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can
"shoot" a dog every three minutes. It is estimated that the dogs will arrive independently and
randomly throughout the day at a rate of one dog every six minutes, according to a Poisson
distribution. Also assume that Sam's shooting times are exponentially distributed. Find the:
(a)
(b)
(c)
(d)
(e)
(f)
14.152
Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can
"shoot" a dog every three minutes. It is estimated that the dogs will arrive independently and
randomly throughout the day at a rate of one dog every four minutes, according to a Poisson
distribution. Also assume that Sam's shooting times are exponentially distributed. Find the:
(a)
(b)
(c)
(d)
(e)
(f)
14.153
probability that Sam is idle.
proportion of the time that Sam is busy.
average number of dogs being shot or waiting to be shot.
average number of dogs waiting to be shot.
average time a dog waits before getting shot.
average amount (mean) of time a dog spends between waiting in line and getting shot.
probability that Sam is idle.
proportion of the time that Sam is busy.
average number of dogs being shot or waiting to be shot.
average number of dogs waiting to be shot.
average time a dog waits before getting shot.
average amount (mean) of time a dog spends between waiting in line and getting shot.
Calls arrive at the hotel switchboard at a rate of two per minute. The average time to handle
each of these is 15 seconds. There is only one switchboard operator at the current time. The
Poisson and exponential distribution appear to be relevant in this situation.
(a) What is the probability that the operator is busy?
(b) What is the average time that a call must wait before reaching the operator?
(c) What is the average number of calls waiting to be answered?
14.154
At the start of football season, the ticket office gets very busy the day before the first game.
Customers arrive at the rate of four every ten minutes, and the average time to transact business
is two minutes.
(a) What is the average number of people in line?
(b) What is the average time that a person would spend in the ticket office?
(c) What proportion of time is the server busy?
474
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.155
At the start of football season, the ticket office gets very busy the day before the first game.
Customers arrive at the rate of four every ten minutes, and the average time to transact business
is one minute. There are two servers in the ticket office, each of whom operate at the same rate
of speed.
(a) What is the average number of people in line?
(b) What is the average time that a person would spend in the ticket office?
(c) What proportion of time is the server busy?
14.156
A post office has a single drive-in window for customers to use. The arrival rate of cars
follows a Poisson distribution, while the service time follows an exponential distribution. The
average arrival rate is 20 per hour and the average service time is two minutes.
(a) What is the average number of cars in the line?
(b) What is the average time spent waiting to get to the service window?
(c) What percentage of the time is the postal clerk idle?
14.157
A company has six computers that are used to run an automated manufacturing facility. Each
of these runs an average of 90 minutes without requiring any attention from the technician.
Each time the technician is required to adjust a computer, an average of 15 minutes (following
an exponential distribution) is required to fix the problem.
(a) On average, how many computers are waiting for service?
(b) On average, how long is a computer out of service?
(c) What is the average waiting time in the queue to be serviced?
14.158 The new Providence shopping mall is considering setting up an information desk
manned by one employee. Because of the complex design of the mall, it is expected that people
will arrive at the desk at about twice the rate for most malls. The expected rate is 25 per hour.
It will also take longer to answer their questions – approximately four minutes per person on
average.
(a)
(b)
(c)
(d)
(e)
(f)
Find the probability that the employee is idle.
Find the proportion of the time that the employee is busy.
Find the average number of people receiving and waiting to receive some information.
Find the average number of people waiting in line to get some information.
Find the average time a person seeking information spends waiting and at the desk.
Find the expected time a person spends just waiting in line to have his question answered.
.
14.159
The new Providence shopping mall is considering setting up an information desk manned by
one employee. The layout for this mall is quite complex, leading the mall manager to expect a
higher than normal arrival rate for persons seeking assistance. It appears that a reasonable
expectation is an arrival rate of approximately 25 patrons per hour. Under the original plan, the
manager expected that it would take approximately four minutes for the Information Desk
475
Waiting Lines and Queuing Theory Models  CHAPTER 14
employee to help the average person. By utilizing a new map and special guide signs, the
manager believes that the required service time can be reduced to an average of two minutes
per person. Assuming that he implements the new map and guide signs:
(a)
(b)
(c)
(d)
(e)
(f)
14.160
Find the probability that the Information Desk employee is idle.
Find the proportion of the time that the Information Desk employee is busy.
Find the average number of people receiving and waiting to receive some information.
Find the average number of people waiting in line to get some information.
Find the average time a person seeking information spends waiting and at the desk.
Find the expected time a person spends just waiting in line to have his question answered.
The new Providence shopping mall has been considering setting up an information desk
manned by one employee. The layout for this mall is quite complex, leading the mall manager
to expect a higher than normal arrival rate for persons seeking assistance. It appears that a
reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the
original plan, the manager expected that it would take approximately 4 minutes for the
Information Desk employee to help the average person. By utilizing a new map and special
guide signs, he believes that the required service time can be reduced to an average of two
minutes per patron. The manager has also noticed that the people seeking help at the
information desk may come from one of two groups: (a) mall patrons, and (b) mall staff, or
delivery persons. Therefore, the manager has decided that he wants to consider another option:
establishing two information desks – one desk to help mall patrons, the other to help staff and
delivery persons. The manager believes that he can expect patrons to arrive at the rate of 20
per hour, and mall staff or delivery persons to arrive at the rate of 5 per hour. It is likely to take
an average of two minutes to answer the questions of a patron, and an average of ten minutes to
answer those of a staff or delivery person. Assuming that the two-desk concept is implemented,
(a) find the probability that both Information Desk employees are idle.
(b) determine whether the patron or the staff person is likely to have the longer wait.
14.161
The new Providence shopping mall has been considering setting up an information desk
manned by one employee. The layout for this mall is quite complex, leading the mall manager
to expect a higher than normal arrival rate for persons seeking assistance. It appears that a
reasonable expectation is an arrival rate of approximately 25 patrons per hour. Under the
original plan, the manager expected that it would take approximately 4 minutes for the
Information Desk employee to help the average person. He has now come to realize that
employing only a single person at the information desk would lead to a very lengthy line –
theoretically, an infinite line! He has decided, therefore, to employ two staff members at the
information desk.
(a)
(b)
(c)
(d)
Find the proportion of the time that the employees are busy.
Find the average number of people waiting in line to get some information.
Find the expected time a person spends just waiting in line to have his question answered.
If the manager has a goal that, for the average patron, the time spent having one's question
answered is less than half the time spent waiting, has he met that goal?
476
Waiting Lines and Queuing Theory Models  CHAPTER 14
(e) Assuming that the manager also has the goal that the combined idle time for the two
workers does not exceed 45 minutes in an 8 hour day has he met that goal?
(f) What would the arrival rate have to be for the manager to meet the 45 minute idle time
goal?
14.162
Bank Boston now has a branch at Bryant College. The branch is always busiest at the
beginning of the college year when freshmen and transfer students arrive on campus and open
accounts. This year, freshmen arrived at the office at a rate of 40 per day (8-hour day). On
average, it takes the Bank Boston staff person about ten minutes to process each account
application.
(a)
(b)
(c)
(d)
14.163
What is the staff person's utilization rate?
What is the average time a student has to wait before getting his application processed?
What is the probability that there are more than three students waiting in line?
The office has a total of five chairs for students, four for waiting, and one at the service
desk. Is it likely that any students will have to stand?
Sam the Vet is running a rabies vaccination clinic for cats at the local grade school. Sam can
"shoot" a cat every four minutes. It is estimated that the cats will arrive independently and
randomly throughout the day at a rate of one cat every five minutes, according to a Poisson
distribution. Also assume that Sam's shooting times are exponentially distributed.
(a) What is the probability that a cat will have to wait?
(b) On the average, how many cats will be in the waiting room?
(c) If a cat has to wait more than 20 minutes, it will become obnoxious. Is this likely to
present a serious problem?
477
Waiting Lines and Queuing Theory Models  CHAPTER 14
.
14.164
Sam the Vet is running a rabies vaccination clinic for dogs at the local grade school. Sam can
"shoot" a dog every three minutes. It is estimated that the dogs will arrive independently and
randomly throughout the day at a rate of one dog every six minutes, according to a Poisson
distribution. Also assume that Sam's shooting times are exponentially distributed.
Sam would like to have each waiting dog placed in a holding pen during the waiting period. If
Sam wants to be certain to have enough cages to accommodate all dogs at least 90 percent of
the time, how many cages should he prepare?
14.165
Cars arrive at the entrance to a parking lot at the rate of 20 per hour. The average time to get a
ticket and proceed to a parking space is two minutes. There is only one lot attendant at the
current time. The Poisson and exponential distribution appear to be relevant in this situation.
(a) What is the probability that an approaching auto must wait?
(b) What is the average waiting time?
(c) What is the average number of autos waiting to enter the garage?
14.166
At the start of football season, the ticket office gets very busy the day before the first game.
Customers arrive at the rate of four every ten minutes. Would the customer be better off if the
stadium employed a single ticket seller who could service a customer in two minutes, or two
ticket sellers, each of whom could service a customer in three minutes?
478
Waiting Lines and Queuing Theory Models  CHAPTER 14
14.167 At the start of ballet season, the ticket office gets very busy the day before the first
performance. Customers arrive at the rate of four every fifteen minutes, and the average time to
transact business is 6 minutes. There are two servers in the ticket office, both of whom operate
at the same speed.
(a) What is the average number of people in line?
(b) What is the average time that a person would spend in the ticket office?
(c) What proportion of time is at least one server busy?
14.168
A post office has a single drive-in window for customers to use. The arrival rate of cars
follows a Poisson distribution, while the service time follows an exponential distribution. The
average arrival rate is 20 per hour and the average service time is two minutes.
If the post office wants to accommodate all of the waiting cars at least 95 percent of the time,
how many car-lengths should they make the driveway leading to the window?
14.169
A company has six computers that are used to run an automated manufacturing facility. Each
of these runs an average of 90 minutes without requiring any attention from the technician.
Each time the technician is required to adjust a computer, an average of 12 minutes (following
an exponential distribution) is required to fix the problem.
How many spares should the technician keep on hand if she wishes to be 90 percent certain that
she will have a working machine to swap for a defect before repairing the defective machine?
Assume that it takes only three minutes to swap the machines.
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Waiting Lines and Queuing Theory Models  CHAPTER 14
SHORT ANSWER/ESSAY
14.170
With regard to queue theory, define what is meant by balking.
14.171
With regard to queue theory, define what is meant by reneging.
14.172
How is FIFO used in describing a queuing theory problem?
14.173
List three key operating characteristics of a queuing system.
14.174
What is meant by a single-channel queuing system?
14.175
What is meant by a multi-channel queuing system?
14.176
What is meant by a single-phase system?
14.177
What is meant by a multi-phase system?
14.178
What is represented by ?
480