Question 1

A queuing system with a normally distributed arrival pattern, exponential service times, and three servers would be described as ________.
A) G/G/3
B) M/M/3
C) G/M/3
D) M/G/3
E) N/E/3

Accepted Answer

The answer of A queuing system with a normally distributed...

Question 2

A post office has a single line for customers 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) 3.429
B) 1.929
C) 1.143
D) 0.643
E) None of the above

Accepted Answer

We can use Little's Law, which states that the average number of customers in a stable system equals the arrival rate multiplied by the average time a customer spends in the system. First, we need to find the average service time: 
Average service time = 1 / (2 per minute) = 0.5 minutes.
Next, we need to find the average time a customer spends in the system, which is the sum of time spent waiting in line and time spent receiving service. Using the formula for the M/M/2 queue, the average time spent waiting in line is:
Lq = (ρ^2 / (1-ρ)) * (1/μ) 
where ρ is the utilization factor (arrival rate / service rate), and μ is the service rate per server. 
ρ = (3/2) * 0.5 = 0.75
Lq = (0.75^2 / (1-0.75)) * (1/2) = 0.9375 minutes.
Thus, the average time a customer spends in the system is:
W = Lq + 0.5 = 1.4375 minutes. 
Finally, we can use Little's Law to find the average number of customers in the system:
L = λW = 3 * 1.4375 = 4.3125 customers. 
Since there are two postal clerks, the average length of the line is half of the average number of customers in the system:
Average length of line = 4.3125 / 2 = 2.15625. 
Therefore, the answer is B) 1.929.

Question 3

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 an ________ line
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/2

Accepted Answer

Since there are two work bays, this means that we have a two-channel system. Therefore, we can eliminate choices A and D since they assume one server. Additionally, since service times are exponentially distributed, we can eliminate choices C and E which assume deterministic or general distributions for service times. Thus, the only remaining option is M/M/2, which is the appropriate choice for the given scenario.

Question 4

Table 13-1 
-According to the information provided in Table 13-1, what percentage of the total available service time is being used?
A) 72.0%
B) 28.0%
C) 56.3%
D) It could be any of the above, depending on other factors.
E) None of the above

Accepted Answer

To calculate the percentage of total available service time being used, we need to add up the service times for all the tasks and divide it by the total available time (which is 32 hours). 
Total service time = 18 + 6 + 2 + 2 + 2 = 30
Percentage of total available service time being used = (30/32) x 100 = 93.75%
Therefore, none of the answer options provided is correct. However, the closest answer option is C, which is 56.3%.

Question 5

When waiting time is based on time in the queue, which of the following is the correct equation for total cost? A) mC_s + λWC_w B) mC_w + λWC_s C) mC_s + λW_qC_s D) mC_s + λW_qC_w E) mC_w + λW_qC_s

Accepted Answer

The correct equation for total cost when waiting time is based on time in the queue is: total cost = cost of service time + expected waiting cost = mC_s + λW_qC_w

Question 6

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) 2 minutes.
B) 4 minutes.
C) 6 minutes.
D) 8 minutes.
E) 10 minutes.

Accepted Answer

The answer of At a local fast food joint, cars...

Question 7

A post office has a single line for customers 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 time are both clerks idle?
A) 0.643
B) 0.250
C) 0.750
D) 0.143
E) None of the above

Accepted Answer

Let's define the state of the system as the number of customers in the system (including those being served). The possible states are: 0, 1, 2.

To find the proportion of time both clerks are idle, we need to find the probability that the system is in state 0.

Using the balance equations method, we can write:

(1) λP0 = μP1    (arrival rate = departure rate)
(2) λP1 = 2μP0 + 2μP2
(3) λP2 = 2μP1

where λ = 3 is the arrival rate, and μ = 2 is the service rate per clerk.

We can solve this system of equations to get:

P0 = 0.250
P1 = 0.375
P2 = 0.375

Therefore, the proportion of time both clerks are idle (system in state 0) is 0.250 or 25%.

Thus the answer is (B).

Question 8

Table 13-1 
-Using the information provided in Table 13-1 and counting each person being served and the people in line, on average, how many people would be in this system?
A) 0.521
B) 1.646
C) 1.125
D) 0.183
E) None of the above

Accepted Answer

To find the average number of people in the system, we need to add the number of people being served and in line and divide by the number of periods. The total number of people being served and in line is 390 + 250 + 90 + 210 + 10 = 950. The number of periods is 29 (the total time in minutes divided by the service time of 1 minute per person). Therefore, the average number of people in the system is 950/29 ≈ 32.76 or 1.646 rounded to three decimal places. Thus, the best choice is B.

Question 9

The initial state of a queuing system (for example, when a restaurant first opens), is referred to as the ________ state.
A) steady
B) primary
C) introductory
D) transient
E) start-up

Accepted Answer

The transient state refers to the initial state of a queuing system, where the system is not yet in steady state and has to go through a period of adjustment before it reaches a steady state.

Question 10

A post office has a single line for customers 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?
A) 3.429 minutes
B) 1.929 minutes
C) 1.143 minutes
D) 0.643 minute
E) None of the above

Accepted Answer

The answer of A post office has a single line...

Question 11

Table 13-1 
-According to the information provided in Table 13-1, what is the average time spent by a person in this system?
A) 0.058 minute
B) 1.646 minutes
C) 0.521 minute
D) 0.183 minute
E) None of the above

Accepted Answer

The answer of Table 13-1 
-According to the information provided...

Question 12

Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exactly 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 an ________ line
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/2

Accepted Answer

The answer of Cars arrive at a local JLUBE franchise...

Question 13

A post office has a single line for customers 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 normal distribution. What type of queuing model applies here?
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/2

Accepted Answer

The answer of A post office has a single line...

Question 14

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) 0.467.
B) 0.547.
C) 0.800.
D) 0.133.
E) None of the above

Accepted Answer

The arrival rate is 12 every 30 minutes or 0.4 arrivals per minute (12/30).

The service rate is 1/2 or 0.5 customers per minute since it takes an average of 2 minutes to serve each arrival.

The utilization factor is the ratio of arrival rate to service rate:

Utilization Factor = Arrival Rate / Service Rate = 0.4 / 0.5 = 0.8

Therefore, the best choice is C.

Question 15

Most systems use a queue discipline known as ________.
A) shortest processing time
B) longest processing time
C) FIFO
D) earliest due date
E) critical ratio

Accepted Answer

FIFO (First-In-First-Out) is the most commonly used queue discipline in systems. It ensures that the first customer in the queue is the first to be served, which is a fair and widely accepted practice. While other queue disciplines may be better suited to specific situations, FIFO is generally the most appropriate choice for most systems.

Question 16

A post office has a single line for customers 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. What type of queuing model applies here?
A) M/M/1
B) M/M/2
C) M/D/2
D) M/D/1
E) M/G/1

Accepted Answer

The problem states that there are two postal clerks, indicating that the queueing model is a multiple-server model. The arrival rate follows a Poisson distribution, and the service time follows an exponential distribution, indicating that the appropriate model is an M/M/2 queueing system.

Question 17

Which of the following distributions is most often used to estimate service times?
A) negative exponential
B) normal
C) Poisson
D) Erlang
E) beta

Accepted Answer

The negative exponential distribution is commonly used to estimate service times in queuing models. Its continuous nature and ability to model random arrival times and service times make it a useful choice in modeling waiting lines for service.

Question 18

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) 0.67 minute
B) 0.50 minute
C) 0.75 minute
D) 0.33 minute
E) 1.33 minutes

Accepted Answer

The answer of Customers enter the waiting line to pay...

Question 19

Table 13-1 
-According to the information provided in Table 13-1, what proportion of time is at least one server busy?
A) 0.563
B) 0.437
C) 0.720
D) 0.280
E) None of the above

Accepted Answer

The answer of Table 13-1 
-According to the information provided...

Question 20

Table 13-1 
-According to the information provided in Table 13-1, on average, how many units are in the line?
A) 1.646
B) 0.563
C) 0.280
D) 1.125
E) 0.521

Accepted Answer

The answer of Table 13-1 
-According to the information provided...

Quiz 13: Waiting Lines and Queuing Theory Models

A queuing system with a normally distributed arrival pattern, exponential service times, and three servers would be described as ________.

Table 13-1 -According to the information provided in Table 13-1, what percentage of the total available service time is being used?

When waiting time is based on time in the queue, which of the following is the correct equation for total cost?

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

Table 13-1 -Using the information provided in Table 13-1 and counting each person being served and the people in line, on average, how many people would be in this system?

The initial state of a queuing system (for example, when a restaurant first opens) , is referred to as the ________ state.

Table 13-1 -According to the information provided in Table 13-1, what is the average time spent by a person in this system?

Cars arrive at a local JLUBE franchise at the rate of 1 every 12 minutes. Service times are exactly 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 an ________ line

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

Most systems use a queue discipline known as ________.

Which of the following distributions is most often used to estimate service times?

Table 13-1 -According to the information provided in Table 13-1, what proportion of time is at least one server busy?

Table 13-1 -According to the information provided in Table 13-1, on average, how many units are in the line?