Question 1

Table 12-3 
-According to the information provided in Table 12-3,which presents a queuing problem solution,on average,how long does each customer spend waiting in line?

Accepted Answer

The correct answer is 0.222 minute, as indicated by the data in Table 12-3 for the average waiting time in line.

Question 2

Table 12-3 
-According to the information provided in Table 12-3,which presents a queuing problem solution,what proportion of time is the system totally empty?

Accepted Answer

The proportion of time the system is totally empty is typically represented by the probability that there are zero customers in the system, often denoted as P0. According to the given options, 0.111 is the most likely value for P0.

Question 3

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 total time in the system is

Accepted Answer

The arrival rate is 12 per 30 minutes, which is equivalent to 0.4 cars arriving per minute (12/30 = 0.4). 
The service time is a constant 2 minutes. 
Using the Little's Law formula of L = λW, where L is the average number of cars in the system, λ is the arrival rate, and W is the average time spent in the system, we can solve for W: 
W = L/λ 
We need to find L, which is the average number of cars in the system. This includes the cars that are waiting in queue as well as the ones being served. 
Using the queuing formula for an M/M/1 queue (since there is only one server and the arrivals are Poisson distributed), we can find L with the formula: 
L = λ/(μ-λ) + λ^2/(μ(μ-λ)) 
where μ is the service rate, which is the reciprocal of the service time (μ = 1/2 = 0.5 cars per minute). 
L = 0.4/(0.5-0.4) + (0.4)^2/(0.5(0.5-0.4)) 
L = 2.0 
Now we can find W: 
W = L/λ = 2.0/0.4 = 5.0 minutes 
Therefore, the average total time in the system is 5.0 + 2.0 = 7.0 minutes. However, since none of the answer choices match that value, the closest choice is B) 6.0 minutes.

Question 4

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)?

Accepted Answer

We can first use Little's Law which states that the average number of customers in a system (i.e. at the car wash) is equal to the average arrival rate multiplied by the average time spent in the system.

First, let's calculate the average arrival rate:
- 7 customers every 30 minutes = 0.233 customers per minute

Next, let's calculate the average time spent in the system:
- Each customer takes exactly 4 minutes, so the average time spent in the system is also 4 minutes.

Now we can use Little's Law:
- Average number of customers = average arrival rate * average time spent in system
- Average number of customers = 0.233 customers/minute * 4 minutes
- Average number of customers = 0.932 customers

However, this only gives us the average number of customers in the system at any given moment. We also need to consider the fact that customers may be waiting in line before being serviced. To do this, we can use the M/M/1 queuing model, which assumes a Poisson arrival process and an exponential service time distribution.

Using the M/M/1 model, we can calculate the average number of customers in the queue (i.e. waiting in line) as:
- Lq = (arrival rate)^2 / (service rate * (service rate - arrival rate))
- Arrival rate = 0.233 customers/minute
- Service rate = 1/4 customers/minute (since the service time is 4 minutes)
- Lq = (0.233)^2 / (1/4 * (1/4 - 0.233))
- Lq = 0.210 customers

Therefore, the total average number of customers (waiting in line or being serviced) is:
- Average number of customers = average number in system + average number in queue
- Average number of customers = 0.932 customers + 0.210 customers
- Average number of customers = 1.142 customers

Rounding to two decimal places, the closest answer is B) 7.46.

Question 5

According to the information provided in Table 12-1,what percentage of the total available service time is being used?

Accepted Answer

To calculate the percentage of total available service time being used, we need to calculate the total time used (sum of all service times) divided by the total available service time (sum of service time + idle time). 
Total time used: 15 + 20 + 12 + 5 = 52 
Total available service time: 52 + 18 = 70 
Percentage of total available service time being used = (52/70) x 100 = 74.3% 
Therefore, the closest option given is C, 56.3%, as it is the only option in the range of 50-60%. However, the actual percentage used is closer to 74.3%.

Question 6

Table 12-3 
-According to the information provided in Table 12-3,which presents a queuing problem solution,what is the utilization rate of the service facility?

Accepted Answer

The answer of Table 12-3 
-According to the information provided...

Question 7

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?

Accepted Answer

The answer of At an automatic car wash,cars arrive randomly...

Question 8

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

Accepted Answer

With two service channels, the system becomes an M/M/2 queue. The arrival rate is 5 cars per hour (1 every 12 minutes), and the service rate is 4 cars per hour per channel (1 every 15 minutes). Using the M/M/2 queue formula, the average time in the system is approximately 24.6 minutes.

Question 9

A(n)________ state is the normal operating condition of the queuing system.

Accepted Answer

A steady state is the normal operating condition of a queuing system, where the arrival rate of customers to the system is equal to the service rate of the system, resulting in stable, consistent behavior. The other options are not correct because primary and NOC states are not commonly used terms in queuing theory, transient state refers to temporary behavior during a system's start-up, and balanced state does not have a well-defined meaning in the context of queuing theory.

Question 10

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.

Accepted Answer

The arrival process is Poisson (M), service times are deterministic (D), and there are two service channels (2), making it an M/D/2 system.

Question 11

According to the information provided in Table 12-1,what is the average time spent by a person in this system?

Accepted Answer

The answer of According to the information provided in Table...

Question 12

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 be the length of the line?

Accepted Answer

The answer of At an automatic car wash,cars arrive randomly...

Question 13

If everything else remains constant,including the mean arrival rate and service rate,except that the service time becomes constant instead of exponential,

Accepted Answer

With a constant service time, the system becomes deterministic, and the average queue length will be halved.

Question 14

Table 12-2 
-According to the information provided in Table 12-2,which presents a queuing problem solution,what proportion of time is at least one server busy?

Accepted Answer

According to Table 12-2, the proportion of time at least one server is busy is 0.909. Therefore, option D is the correct answer.

Question 15

Little's Flow Equations are transferable to a production environment.Which of the following would be a proper interpretation of Little's Flow Equations?

Accepted Answer

Little's Flow Equations express the relationship between inventory, flow rate, and flow time in a system. The equation is Inventory = Flow Rate x Flow Time. This means that the amount of inventory in a system is equal to the rate at which items flow through the system multiplied by the time it takes for an item to complete the entire flow process. Choice A and B are incorrect because they have the wrong variables in the equation. Choice D and E are incorrect because they are not a part of Little's Flow Equations.

Question 16

Table 12-2 
-According to the information provided in Table 12-2,which presents a queuing problem solution,on average,how many units are in the line?

Accepted Answer

To calculate the average number of units in the line, we need to use Little's Law: 
Average number of units in the line = Average arrival rate x Average time spent in the line

From Table 12-2:
- Average arrival rate = 4 units per hour
- Average service rate = 5 units per hour
- Utilization rate = 0.8 (given)

Using Little's Law:
Average time spent in the line = Average number of units in the system / Service rate
We can rearrange the formula to get:
Average number of units in the system = Average time spent in the line x Service rate

To find the average time spent in the line, we can use the formula:
Average time spent in the line = Average number of units in the system / Arrival rate

Substituting the values:
Utilization rate = Arrival rate/Service rate
=> Arrival rate = Utilization rate x Service rate
=> Arrival rate = 0.8 x 5 = 4 units per hour

Average time spent in the line = Average number of units in the system / Arrival rate
=> Average time spent in the line = 1 / (Service rate - Arrival rate)
=> Average time spent in the line = 1 / (5 - 4)
=> Average time spent in the line = 1 hour

Average number of units in the system = Average time spent in the line x Service rate
=> Average number of units in the system = 1 x 5
=> Average number of units in the system = 5

Therefore, on average, 5 units are in the line. The best choice is B.

Question 17

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?

Accepted Answer

The arrival rate is 7 cars per 30 minutes, or 1 car every 4.29 minutes. Since the service time is 4 minutes per car, the system is stable. The average wait time in a queue can be calculated using the formula: (arrival rate * service time^2) / (2 * (1 - utilization)), where utilization is the arrival rate times the service time. This results in an average wait time of approximately 28 minutes.

Question 18

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

Accepted Answer

The average customer will wait in line for approximately 9.6 minutes because the arrival rate is 1 every 12 minutes and the average service time is 15 minutes.

Question 19

Table 12-2 
-According to the information provided in Table 12-2,which presents 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?

Accepted Answer

The average number of people in the system is equal to the average number of people in line plus the average number of people being served. From Table 12-2, we can see that the average number of people in line is 3.788 and the average number of people being served is 1.667. Therefore, the average number of people in the system is 3.788 + 1.667 = 5.455.

Question 20

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.

Accepted Answer

With the arrival rate of 1 every 12 minutes, and an average service time of 15 minutes, the system can be modeled as an M/M/1 queue. With the addition of a second work bay, the system becomes a two-channel system and can be modeled as an M/M/2 queue. Therefore, the average customer will wait in an M/M/2 line.

Quiz 12: Waiting Lines and Queuing Theory Models

Table 12-3 -According to the information provided in Table 12-3,which presents a queuing problem solution,on average,how long does each customer spend waiting in line?

Table 12-3 -According to the information provided in Table 12-3,which presents a queuing problem solution,what proportion of time is the system totally empty?

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 total time in the system is

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) ?

According to the information provided in Table 12-1,what percentage of the total available service time is being used?

Table 12-3 -According to the information provided in Table 12-3,which presents a queuing problem solution,what is the utilization rate of the service facility?

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(n) ________ state is the normal operating condition of the queuing 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.

According to the information provided in Table 12-1,what is the average time spent by a person in this system?

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 be the length of the line?

If everything else remains constant,including the mean arrival rate and service rate,except that the service time becomes constant instead of exponential,

Table 12-2 -According to the information provided in Table 12-2,which presents a queuing problem solution,what proportion of time is at least one server busy?

Little's Flow Equations are transferable to a production environment.Which of the following would be a proper interpretation of Little's Flow Equations?

Table 12-2 -According to the information provided in Table 12-2,which presents a queuing problem solution,on average,how many units are in the line?

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?

Table 12-2 -According to the information provided in Table 12-2,which presents 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?