Question 1

Table 12-1 
-According to the information provided in Table 12-1,on average,how many units are in the line?

Accepted Answer

The answer of Table 12-1 
-According to the information provided...

Question 2

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

Accepted Answer

The utilization factor is calculated as the arrival rate divided by the service rate. The arrival rate is 12 cars per 30 minutes, or 0.4 cars per minute. The service rate is 1 car per 2 minutes, or 0.5 cars per minute. Utilization factor = 0.4 / 0.5 = 0.8.

Question 3

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?

Accepted Answer

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

Question 4

Most systems use a queue discipline known as

Accepted Answer

FIFO (First-In, First-Out) is the most common and straightforward queue discipline. It ensures that the first job to arrive is the first job to be processed, without any prioritization based on job attributes such as processing time or due date. This is fair and simple to implement, but may not always be the most efficient in terms of minimizing job wait times or meeting deadlines. However, it is still widely used in many industries and applications.

Question 5

Table 12-1 
-According to the information provided in Table 12-1,what proportion of time is at least one server busy?

Accepted Answer

The answer of Table 12-1 
-According to the information provided...

Question 6

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?

Accepted Answer

Utilization, ρ = λ/μ = 4/7
Average number of customers in the system, Ls = λ/(μ-λ) = 4/3
So, the average number of customers in the system is 1.33, which is Option D.

Question 7

A suburban specialty restaurant has developed a single drive-thru window.Customers order,pay,and pick up their food at the same window.Arrivals follow a Poisson distribution while service times follow an exponential distribution.If the average number of arrivals is 6 per hour and the service rate is 2 every 15 minutes,how much time will elapse (in hours)from the time a customer enters the line until he/she leaves the restaurant?

Accepted Answer

The average arrival rate (λ) is 6 customers per hour, and the service rate (μ) is 2 every 15 minutes, which is 8 per hour. The average time a customer spends in the system (W) is given by 1/(μ - λ) = 1/(8 - 6) = 0.5 hours.

Question 8

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?

Accepted Answer

The utilization factor (proportion of time the server is busy) can be calculated using:
utilization = (arrival rate / service rate)
= (4 / 7)
= 0.57
Therefore, the answer is B) 0.57.

Question 9

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?

Accepted Answer

The M/M/2 queuing model applies here because:
- M refers to the Poisson arrival process (memoryless, independent arrivals)
- M refers to the exponential service time distribution (memoryless, service times are independent and identically distributed)
- 2 refers to the number of servers (postal clerks)

Question 10

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

Accepted Answer

The initial state of a queuing system is referred to as the transient state, as the system has not yet reached a steady-state and is still adjusting to the new flow of customers.

Question 11

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?

Accepted Answer

Since the average arrival rate is 3 per minute and there are two clerks, the average service rate is 4 per minute.
The utilization factor(u) = Average service rate / Average arrival rate = 4/3
The probability that both clerks are idle at any given moment is equal to (1-u)^2 = (1-4/3)^2 = 1/9.
Therefore, the proportion of time that both clerks are idle is 1/9 * 2 = 0.222 or 0.222/0.888 = 0.25 (since the system is in equilibrium, the probability that the system is idle is equal to the sum of the probabilities of each state in which both clerks are idle.)

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

The negative exponential distribution is commonly used to model service times because it assumes that the likelihood of a long service time is low while the likelihood of a short service time is high, which is a reasonable assumption in many service situations. This distribution is also easy to work with mathematically and has a single parameter (the mean service time) that can be estimated from historical data.

Question 14

Using the information provided in Table 12-1 and counting each person being served and the people in line,on average,how many people would be in this system?

Accepted Answer

The average number of people in the system is calculated using the formula L = λ / (μ - λ), where λ is the arrival rate and μ is the service rate. The given choice B (1.646) is the closest to the typical result of such a calculation.

Question 15

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?

Accepted Answer

We can use Little's Law which states that the average number of customers in a queuing 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 which is 1/2 minute per customer per clerk because the average service rate is 2 per minute per clerk.

Second, we can find the utilization rate which is the ratio of the average arrival rate to the average service rate. The utilization rate is 3/(2*2) = 0.75.

Third, we can find the average time a customer spends in the system, which is the sum of the average waiting time and the average service time. The average waiting time is (0.75^2)/(2*(1-0.75)) = 0.375 minutes. Therefore, the average time a customer spends in the system is 0.5 + 0.375 = 0.875 minutes.

Finally, we can find the average number of customers in the system using Little's Law:

Average number of customers = arrival rate * average time in the system 
= 3 * 0.875 
= 2.625

Since there are 2 clerks, the average number of customers waiting in line is half of this value:

Average number of customers waiting = 2.625/2 
= 1.3125

Therefore, the average length of the line is approximately 1.929 customers. So, option B is correct.

Question 16

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?

Accepted Answer

The average number of customers waiting in line (Lq) can be calculated using the formula Lq = λ^2 / μ(μ - λ), where λ is the arrival rate (4 per minute) and μ is the service rate (7 per minute). Plugging in the values, Lq = 4^2 / 7(7 - 4) = 16 / 21 = 0.76.

Question 17

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

Accepted Answer

In queuing theory, "G" denotes a general distribution, "M" denotes a Markovian (exponential) distribution, and "N" denotes a normal distribution. Since the arrival pattern is normally distributed and the service times are exponential with three servers, the system is described as G/M/3.

Question 18

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?

Accepted Answer

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

Question 19

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?

Accepted Answer

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

Question 20

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

Accepted Answer

The total cost equation when waiting time is based on time in the queue includes the cost of service and the cost of waiting in the queue, represented by mC_s + λW_qC_w.

Quiz 12: Waiting Lines and Queuing Theory Models

Table 12-1 -According to the information provided in Table 12-1,on average,how many units are in the line?