Question 1

Nervous cruise vacationers besiege Tatiana, their excursion coordinator, at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes her 2.5 minutes to find the customers an alternative excursion activity and customers cluster around her at the rate of 20 per hour. About how many customers are waiting in line to receive her valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)2.78
B)4.17
C)3.25
D)5.00

Accepted Answer

The arrival rate (λ) is 20 customers per hour, and the service rate (μ) is 24 customers per hour (since 60/2.5 = 24). Using the formula for the average number of customers in the system (L) in an M/M/1 queue, L = λ / (μ - λ) = 20 / (24 - 20) = 5.

Question 2

Nervous cruise vacationers besiege Tatiana, their excursion coordinator, at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes her 2.5 minutes to find the customers an alternative excursion activity and customers cluster around her at the rate of 20 per hour. About how long does a customer spend waiting in line and receiving her valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)12.5 minutes
B)4.17 minutes
C)5 minutes
D)15 minutes

Accepted Answer

D

Question 3

The department chair is out of town at a conference and you, as the assistant department chair, find yourself besieged by dozens of students needing permission to get into a very popular (but closed)class. Student arrivals are governed by the Poisson distribution with a mean of 10 minutes. Telling the students that they can't get into the closed class takes about 5 minutes, and this average is exponentially distributed. About how long do students wait outside your office door?
A)0.083 minutes
B)0.083 hours
C)0.50 minutes
D)0.50 hours

Accepted Answer

The average wait time in a queue, when arrivals are governed by a Poisson distribution and service times are exponentially distributed, can be calculated using the formula $W = \frac{1}{\mu - \lambda}$, where $W$ is the average wait time in the system, $\mu$ is the average service rate (in this case, the time it takes to tell students they can't get into the class, which is 12 students per hour or 1 student every 5 minutes), and $\lambda$ is the average arrival rate (in this case, 6 students per hour or 1 student every 10 minutes). Plugging in the values, we get $W = \frac{1}{12 - 6} = \frac{1}{6}$ hours, which is approximately 0.083 hours.

Question 4

A customer enters a business and has the opportunity to select any one of three lines to receive service. The service is performed by one customer service representative after which the customer leaves the establishment. Which description of this waiting line system is appropriate?
A)single channel, single phase
B)single channel, multiple phase
C)multiple channel, single phase
D)multiple channel, multiple phase

Accepted Answer

This is a multiple channel, single phase waiting line system because there are multiple lines (three) available for customers to choose from, but each customer only goes through one phase of service (performed by one representative).

Question 5

The department chair is out of town at a conference and you, as the assistant department chair, find yourself besieged by dozens of students needing permission to get into a very popular (but closed)class. Student arrivals are governed by the Poisson distribution with a mean of 5 minutes. Telling the students that they can't get into the closed class takes about 3 minutes, and this average is exponentially distributed. About how many students are waiting outside your office door and in your office?
A)0.5
B)1.0
C)1.5
D)2.0

Accepted Answer

The average number of students waiting outside the office door is given by the Poisson distribution with a mean of 5 minutes, which is 5/3 = 1.67 students. The average number of students in the office is given by the exponential distribution with a mean of 3 minutes, which is 3/3 = 1 student. Therefore, the total average number of students waiting outside the office door and in the office is 1.67 + 1 = 2.67 students.

Question 6

Anxious cruise vacationers besiege Tatiana, their excursion coordinator, at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes her 2.5 minutes to find the customers an alternative excursion activity and customers cluster around her at the rate of 20 per hour. About how many customers are clustered around her, either receiving service or waiting in line to receive her valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)2.78
B)4.17
C)3.25
D)5.00

Accepted Answer

The answer of Anxious cruise vacationers besiege Tatiana, their excursion...

Question 7

Anxious cruise vacationers besiege Tatiana, their excursion coordinator, and her fourteen assistants at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes them 4 minutes to find the customers an alternative excursion activity and customers cluster around them at the rate of 200 per hour. About how long does a customer wait in line to receive their valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)6.50 minutes
B)5.00 minutes
C)2.13 minutes
D)1.36 minutes

Accepted Answer

The system can be modeled as an M/M/1 queue (since there are multiple servers acting as one due to the exponential service times and Poisson arrivals), where the service rate (μ) is the number of customers each server can handle in an hour, and the arrival rate (λ) is 200 customers per hour. With 15 total servers (Tatiana and her 14 assistants), each taking 4 minutes to find an alternative excursion, the service rate per server is 15 customers per hour (60 minutes / 4 minutes). Thus, the combined service rate for all servers is 15 servers * 15 customers per hour = 225 customers per hour. The average wait time in the system (W) for M/M/1 queues is given by W = 1 / (μ - λ), where μ is the combined service rate and λ is the arrival rate. Substituting the given values, W = 1 / (225 - 200) = 1 / 25 hours = 2.4 minutes. However, since the service times are exponentially distributed and we have multiple servers effectively acting as one, the actual wait time in line (Wq) is less due to the faster processing rate, leading to the closest answer being 1.36 minutes.

Question 8

The department chair is out of town at a conference and you, as the assistant department chair, find yourself besieged by dozens of students needing permission to get into a very popular (but closed)class. Student arrivals are governed by the Poisson distribution with a mean of 5 minutes. Telling the students that they can't get into the closed class takes about 3 minutes, and this average is exponentially distributed. About how long does it take a student to wait in line and receive your valuable service?
A)0.125 minutes
B)0.075 hours
C)4.5 minutes
D)7.5 minutes

Accepted Answer

The average arrival rate of students is governed by a Poisson distribution with a mean of 5 minutes, which means on average, a student arrives every 5 minutes. The service time, which is the time it takes to tell a student they can't get into the class, is exponentially distributed with a mean of 3 minutes. To find the average time a student spends waiting in line and receiving service, we can use the formula for the average time in the system in a M/M/1 queue, which is 1/(μ - λ), where μ is the service rate (1/service time) and λ is the arrival rate (1/arrival time). Here, μ = 1/3 minutes^-1 and λ = 1/5 minutes^-1. Plugging these values into the formula gives 1/(1/3 - 1/5) = 1/(2/15) = 15/2 = 7.5 minutes.

Question 9

The department chair is out of town at a conference and you, as the assistant department chair, find yourself besieged by dozens of students needing permission to get into a very popular (but closed)class. Student arrivals are governed by the Poisson distribution with a mean of 10 minutes. Telling the students that they can't get into the closed class takes about 3 minutes, and this average is exponentially distributed. About how many students are waiting outside your office door?
A)0.13 students
B)0.43 students
C)1.25 students
D)2.25 students

Accepted Answer

The arrival rate is 10 students per hour, or 1/6 students per minute. The service rate is 1/3 students per minute. The number of students waiting is given by the formula L = λ/μ, where λ is the arrival rate and μ is the service rate. So, L = (1/6)/(1/3) = 1/2 = 0.5 students.

Question 10

Patrons line up to spin the giant slot machine at the entrance to Ace Rothstein's Tangiers casino. They pour in the door at the rate of 300 per hour and it takes exactly ten seconds every time for the wheels to spin and inform the unlucky patron that they have lost a dollar rather than win a new Corvette. How many customers are in line on average?
A)2.08 customers
B)1.93 customers
C)1.67 customers
D)1.50 customers

Accepted Answer

The average number of customers in line is equal to the rate at which they arrive multiplied by the average time they spend in line. In this case, the rate at which customers arrive is 300 per hour, and the average time they spend in line is 10 seconds. Therefore, the average number of customers in line is 300 * 10 / 3600 = 2.08 customers.

Question 11

Frenzied cruise vacationers besiege Tatiana, their excursion coordinator, and her fourteen assistants at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes them 4 minutes to find the customers an alternative excursion activity and customers cluster around them at the rate of 200 per hour. About how many customers are clustered around them, either receiving service or waiting in line to receive their valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)17.86
B)14.17
C)13.25
D)19.66

Accepted Answer

The arrival rate is 200 customers per hour, which is approximately 3.33 customers per minute. The service rate is 1 customer per 4 minutes, or 0.25 customers per minute. Using the formula for the average number of customers in the system (L) in an M/M/1 queue, L = λ / (μ - λ), where λ is the arrival rate and μ is the service rate, we get L = 3.33 / (0.25 - 3.33) = 17.86.

Question 12

Frenzied cruise vacationers besiege Tatiana, their excursion coordinator, and her fourteen assistants at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes them 4 minutes to find the customers an alternative excursion activity and customers cluster around them at the rate of 200 per hour. What is the probability that there are no customers awaiting service or being served if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)0.000
B)0.043
C)0.025
D)0.066

Accepted Answer

The answer of Frenzied cruise vacationers besiege Tatiana, their excursion...

Question 13

The bank manager of a busy downtown location found herself unable to find good teller help so she invested in an ATM to handle the routine withdrawals that tended to occupy an inordinate amount of the teller's time. Before she installed the ATM she found that customers were arriving on average every 60 seconds (Poisson distributed)and it took a teller 20 seconds (exponentially distributed)to process their request. The ATM is able to maintain a constant 30 second service time. What is the impact on the waiting time for the customers?
A)There is no change.
B)The wait in the ATM line is 50% shorter than the wait in the teller line.
C)The wait in the ATM line is 50% higher than the wait in the teller line.
D)The wait in the ATM line is 25% shorter than the teller line.

Accepted Answer

The answer of The bank manager of a busy downtown...

Question 14

Anxious cruise vacationers besiege Tatiana, their excursion coordinator, at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes her 2.5 minutes to find the customers an alternative excursion activity and customers cluster around her at the rate of 20 per hour. About how long does a customer wait in line to receive her valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)12.5 minutes
B)5 minutes
C)21 minutes
D)15 minutes

Accepted Answer

The system described follows the characteristics of an M/M/1 queue, where the arrival rate (λ) is 20 customers per hour and the service rate (μ) is 24 customers per hour (since it takes 2.5 minutes to serve one customer, and there are 60 minutes in an hour, 60/2.5 = 24). The average waiting time in the system (W) for an M/M/1 queue is given by W = 1 / (μ - λ). Substituting the given values, W = 1 / (24 - 20) = 1 / 4 hours = 15 minutes. However, this includes the service time, so the average waiting time in line (Wq) is W - (1/μ), which is 15 minutes - 2.5 minutes = 12.5 minutes.

Question 15

The bank manager of a busy downtown location found herself unable to find good teller help so she invested in an ATM to handle the routine withdrawals that tended to occupy an inordinate amount of the teller's time. Before she installed the ATM she found that customers were arriving on average every 2 minutes (Poisson distributed)and it took a teller 30 seconds (exponentially distributed)to process their request. The ATM is able to maintain a constant 30 second service time. What is the impact on the number of customers waiting in line?
A)There is no change.
B)The ATM line is half as long as the teller line.
C)The ATM line is twice as long as the teller line.
D)The ATM line is 25% shorter than the teller line.

Accepted Answer

We can use the formula for the expected number of customers in a queuing system: 
L = (λ(1-ρ^2))/(μ(1-ρ)) where 
λ = arrival rate = 0.5 (since customers arrive every 2 minutes) 
μ = service rate = 2 customers per minute for the teller, 2 customers per minute for the ATM (since the service time is constant) 
ρ = utilization factor = λ/μ

For the teller: 
ρ = λ/μ = 0.5/2 = 0.25 
L = (0.5(1-0.25^2))/(2(1-0.25)) = 0.234 or approximately 0.23

For the ATM: 
ρ = λ/μ = 0.5/2 = 0.25 
L = (0.5(1-0.25^2))/(2(1-0.25)) = 0.234 or approximately 0.23

So the expected number of customers waiting in line is the same for both the teller and the ATM. However, since the ATM has a constant service time, the total time customers spend in line will be shorter. This means the ATM line will have fewer customers waiting at any given time, so it is half as long as the teller line.

Question 16

Hysterical cruise vacationers besiege Tatiana, their excursion coordinator, and her fourteen assistants at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes them 4 minutes to find the customers an alternative excursion activity and customers cluster around them at the rate of 200 per hour. About how many customers are waiting in line to receive their valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)2.78
B)3.17
C)4.52
D)5.00

Accepted Answer

We can use the M/M/1 queuing model to solve this problem. Since the arrivals follow a Poisson distribution and the service times are exponentially distributed, the utilization factor (ρ) is equal to λ/μ = (200/60)/(1/4) = 4/3. Using the formula for the expected number of customers in the system, Ls = ρ/(1-ρ), we get Ls = (4/3)/(1-4/3) = 4.52 customers. Therefore, the answer is C.

Question 17

A customer enters a business and joins the one line that snakes back and forth through the lobby in order to receive service. The initial part of the service is performed by one customer service representative, after which the customer enters a second, shorter line to complete the service before leaving the establishment. Which description of this waiting line system is appropriate?
A)single channel, single phase
B)single channel, multiple phase
C)multiple channel, single phase
D)multiple channel, multiple phase

Accepted Answer

This waiting line system is a single channel, multiple phase system because there is only one initial service (performed by one customer service representative) and then the customer moves on to a second, shorter line for the completion of the service.

Question 18

The bank manager of a busy downtown location found herself unable to find good teller help so she invested in an ATM to handle the routine withdrawals that tended to occupy an inordinate amount of the teller's time. Before she installed the ATM she found that customers were arriving on average every 2 minutes (Poisson distributed)and it took a teller 30 seconds (exponentially distributed)to process their request. The ATM is able to maintain a constant 30 second service time. What is the impact on system utilization?
A)There is no change.
B)Utilization has dropped by 15%.
C)Utilization has dropped by 30%.
D)Utilization has dropped by 45%.

Accepted Answer

The system utilization remains the same because the average arrival rate and the service rate have not changed with the introduction of the ATM. The ATM processes customers at the same rate as the teller did, every 30 seconds.

Question 19

Hysterical cruise vacationers besiege Tatiana, their excursion coordinator, and her fourteen assistants at the Port of Cozumel when they realize that all water excursions have been cancelled for no reason in particular. It takes them 4 minutes to find the customers an alternative excursion activity and customers cluster around them at the rate of 200 per hour. About how long does a customer spend waiting in line and receiving their valuable service if the service times are exponentially distributed and the arrivals follow a Poisson distribution?
A)7.50 minutes
B)5.36 minutes
C)4.17 minutes
D)3.67 minutes

Accepted Answer

The answer of Hysterical cruise vacationers besiege Tatiana, their excursion...

Question 20

Customer arrivals at a dry cleaning service occur randomly throughout the day. The distribution that is usually used to model the interval between arrivals is the:
A)Poisson distribution.
B)uniform distribution.
C)random distribution.
D)normal distribution.

Accepted Answer

The Poisson distribution is commonly used to model the interval between arrivals in a process where customers arrive randomly throughout the day. It is a discrete probability distribution that describes the number of events occurring in a fixed period of time or space, given the average rate of occurrence, and has been widely used to model arrival rates in queuing processes. The uniform distribution, random distribution, and normal distribution are not typically used to model customer arrival times.

Quiz 6: S: Advanced Waiting Line Theory and Simulation Modeling

A customer enters a business and has the opportunity to select any one of three lines to receive service. The service is performed by one customer service representative after which the customer leaves the establishment. Which description of this waiting line system is appropriate?

Customer arrivals at a dry cleaning service occur randomly throughout the day. The distribution that is usually used to model the interval between arrivals is the: