Question 1

Many complex waiting line situations cannot be analyzed through neatly derived formulas.In these cases,________ is the best method of analysis.

Accepted Answer

The answer of Many complex waiting line situations cannot be...

Question 2

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 answer of Frenzied cruise vacationers besiege Tatiana,their excursion coordinator,and...

Question 3

Barber College is the best deal in town! For only $5,you can get a shampoo and cut by any one of four untrained stylists completed in 7.5 minutes on average.Perhaps this is why customers show up on average two minutes apart (both arrival times and service times are exponentially distributed). How long does a customer spend waiting for a haircut?
A) 17 minutes
B) 20 minutes
C) 23 minutes
D) 26 minutes

Accepted Answer

The answer of Barber College is the best deal in...

Question 4

Customers pass by any one of forty ticket takers on their way in the arena for a WWE event.This is a single ________ system.

Accepted Answer

The answer of Customers pass by any one of forty...

Question 5

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 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 6

The only way to enter Jellystone Park is past Ranger Smith,who checks your camping gear and warns you about the bears in the park.The next step of the entrance process is to pay your site fee to Ranger Dwayne.This is a single ________ system.

Accepted Answer

The answer of The only way to enter Jellystone Park...

Question 7

Hopeful 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 long do customers spend waiting in line and finding out the unhappy news on average?
A) 35 seconds
B) 30 seconds
C) 25 seconds
D) 20 seconds

Accepted Answer

The answer of Hopeful patrons line up to spin the...

Question 8

Barber College is the best deal in town! For only $5,you can get a shampoo and cut by any one of four untrained stylists completed in 7.5 minutes on average.Perhaps this is why customers show up on average two minutes apart (both arrival times and service times are exponentially distributed). On average,how many customers are waiting in line?
A) 9
B) 11
C) 13
D) 15

Accepted Answer

We can model the arrivals as a Poisson process with a rate of λ = 0.5 customers per minute (since customers arrive on average 2 minutes apart). We can also model the service times as exponential distributions with a mean of 7.5 minutes. Using Little's Law, we know that the expected number of customers in the system (in line plus being served) is equal to λ * W, where W is the expected time spent in the system. The expected time spent in the system is equal to Wq + 1/μ, where Wq is the expected time spent waiting in line and μ is the service rate (equal to 1/7.5). Plugging in the values, we have:

λ = 0.5
μ = 1/7.5
Wq = (λ^2)/(μ(μ-λ)) = (0.5^2)/(1/7.5*(1/7.5-0.5)) = 0.6944
W = Wq + 1/μ = 0.6944 + 1/(1/7.5) = 6.4444

Finally, we can calculate the expected number of customers in the system:

λ * W = 0.5 * 6.4444 = 3.2222

Since customers are discrete, the actual number of customers in line can vary from 0 to infinity, but on average we expect 3.2222 customers in line. The closest option is C, 13, which is twice the expected number of customers in line (since the service time is about half the time between arrivals).

Question 9

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

We can use the M/M/1 queueing model to solve this problem, where arrivals follow a Poisson distribution and service times follow an exponential distribution.

The utilization factor for the system is (14 assistants + 1 coordinator)/(200 customers/hour) = 0.075.

Using the M/M/1 queueing model, we can calculate the average time a customer spends waiting in line and receiving service (also known as the residence time) as:
R = (1/μ) / (1 - ρ)
where μ is the service rate (the reciprocal of the mean service time) and ρ is the utilization factor.

Since the arrivals are at a rate of 200 per hour, the interarrival time follows an exponential distribution with parameter λ = 200. The mean interarrival time is therefore 1/λ = 0.005 hours.

We can then find the mean service time by using the fact that the arrival rate and the service rate should be equal in the steady state:
ρ = λ/μ
μ = λ/ρ = 200/0.075 = 266.67 customers/hour
The mean service time is then 1/μ = 0.00375 hours.

Putting it all together:
R = (1/μ) / (1 - ρ) = (1/266.67) / (1 - 0.075) = 0.00557 hours or 20.05 seconds.

Therefore, on average, a customer spends 20.05 seconds waiting in line and receiving their valuable service. Since there are 200 customers per hour, this is equivalent to (20.05/3600) * 200 = 1.11 minutes per customer. However, the question asks for the combined waiting and service time, so we need to double this value:
2 * 1.11 = 2.22 minutes per customer.

Therefore, the closest answer choice is B) 5.36 minutes.

Question 10

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 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

The ATM provides a constant service time, which reduces variability compared to the exponentially distributed service time of the teller. This results in a more efficient queue, reducing the average number of customers waiting in line by half.

Question 11

One assumption in most waiting line analyses is that the number of customers is ________.

Accepted Answer

The answer of One assumption in most waiting line analyses...

Question 12

Customers that leave the system after waiting in line for a while are ________ but customers that don't even join the line because it looks too long are ________.

Accepted Answer

The answer of Customers that leave the system after waiting...

Question 13

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 coordinator,and...

Question 14

If 5 minutes passes between customer arrivals then the ________ is 12/hour.

Accepted Answer

The answer of If 5 minutes passes between customer arrivals...

Question 15

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 answer of Patrons line up to spin the giant...

Question 16

A single channel single server system requires that the ________ rate be larger than the ________ rate.

Accepted Answer

The answer of A single channel single server system requires...

Question 17

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 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 utilization remains the same because the service rate of the ATM is the same as the average service rate of the teller, both being 30 seconds per customer.

Question 18

Regulars 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 long do customers wait in line on average?
A) 10 seconds
B) 15 seconds
C) 20 seconds
D) 25 seconds

Accepted Answer

The answer of Regulars line up to spin the giant...

Question 19

Barber College is the best deal in town! For only $5,you can get a shampoo and cut by any one of four untrained stylists completed in 7.5 minutes on average.Perhaps this is why customers show up on average two minutes apart (both arrival times and service times are exponentially distributed). Management is concerned about the waiting time for customers and decides to train one more student.If the arrival and service rates remain the same as in the original scenario,what happens to the number of customers in the system?
A) There are about half as many as in the original scenario.
B) There are about a third as many as in the original scenario.
C) There are about a fourth as many as in the original scenario.
D) There are about a fifth as many as in the original scenario.

Accepted Answer

The answer of Barber College is the best deal in...

Question 20

Barber College is the best deal in town! For only $5,you can get a shampoo and cut by any one of four untrained stylists completed in 7.5 minutes on average.Perhaps this is why customers show up on average two minutes apart (both arrival times and service times are exponentially distributed). What percentage of the time are the four stylists working with customers?
A) 94%
B) 90%
C) 84%
D) 80%

Accepted Answer

We can use the M/M/c queuing model to solve this problem. Here, c = 4 (number of servers) and ρ is the utilization of the servers, which is equal to λ/μc, where λ is the arrival rate and μ is the service rate of each server.
From the problem, we know that the arrival rate is 0.5 customers per minute (i.e. 2 minutes per customer) and the average service time is 7.5 minutes per customer. Therefore, μ = 1/7.5 = 0.1333 customers per minute.
Using the formula ρ = λ/μc, we get ρ = (0.5/0.1333*4) = 0.938.
Now, we can use the formula P(c busy) = ((c*ρ)^c / c!) / (Σ((c*ρ)^k / k!), k=0 to c), which gives us the probability that all servers are busy. P(c busy) equals the probability that the four stylists are working with customers.
Substituting the values, we get P(4 busy) = ((4*0.938)^4 / 4!) / (Σ((4*0.938)^k / k!), k=0 to 4) = 0.94 or 94%.
Therefore, the answer is (A) 94%.

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

Many complex waiting line situations cannot be analyzed through neatly derived formulas.In these cases,________ is the best method of analysis.

Customers pass by any one of forty ticket takers on their way in the arena for a WWE event.This is a single ________ system.

The only way to enter Jellystone Park is past Ranger Smith,who checks your camping gear and warns you about the bears in the park.The next step of the entrance process is to pay your site fee to Ranger Dwayne.This is a single ________ system.

One assumption in most waiting line analyses is that the number of customers is ________.

Customers that leave the system after waiting in line for a while are but customers that don't even join the line because it looks too long are .

If 5 minutes passes between customer arrivals then the ________ is 12/hour.

A single channel single server system requires that the rate be larger than the rate.