Question 1

A single-phase waiting-line system meets the assumptions of constant service time or M/D/1.Units arrive at this system every 12 minutes on average.Service takes a constant 8 minutes.What is the average number in the system Ls in units?

Accepted Answer

The answer of A single-phase waiting-line system meets the assumptions...

Question 2

A waiting line model meeting the assumptions of M/M/1 has an arrival rate of 2 per hour and a service rate of 6 per hour.What is the utilization factor for the system?

Accepted Answer

The answer of A waiting line model meeting the assumptions...

Question 3

Which one of the following is NOT a characteristic of a Model A or M/M/1 system?

Accepted Answer

A Model A or M/M/1 system assumes an exponential service time pattern, a single number of servers, a Poisson arrival rate pattern, and a single number of phases. However, it does not assume a finite population size.

Question 4

A queuing model that follows the M/M/1 assumptions has λ = 3 and μ = 2.What is the average number of units in the system?

Accepted Answer

In an M/M/1 queue, the system is stable only if λ < μ, meaning the arrival rate is less than the service rate. If λ is greater than μ, as in this case (λ = 3, μ = 2), the queue will grow indefinitely, leading to an unstable system.

Question 5

A single-phase waiting-line system meets the assumptions of constant service time or M/D/1.Units arrive at this system every 10 minutes on average.Service takes a constant 4 minutes.What is the average length of the queue Lq in units?

Accepted Answer

The M/D/1 system can be analyzed using Little's Formula which states L = λW, where L is the average number of units in the system, λ is the arrival rate, and W is the average time a unit spends in the system.
Since we only want to find Lq, we can use the formula Lq = λWq, where Wq is the average waiting time in the queue.
To find λ, we can use the fact that units arrive every 10 minutes on average, which gives us an arrival rate of 6 units per hour (since there are 60 minutes in an hour).
To find W, we need to consider both the service time and the waiting time in the queue. Since the system is in steady-state, the average waiting time in the queue is equal to the average time spent in the system minus the service time. Thus, W = Wq + 4 minutes.
Let's use Little's Formula to find W: 
L = λW
L = 6 * W
We know that L = Lq + ρ, where ρ is the average number of units being served (in this case, it's 1 since there's only one server). Since the system is in steady-state, Lq is the same as L - ρ. Thus,
L - ρ = λW
Lq = λW - ρ
Lq = 6 * (Wq + 4) - 1
Lq = 6Wq + 23
Now we need to find Wq. Since the system is M/D/1, we can use the formula Wq = (ρ^2) / (1 - ρ) * (1 / μ), where μ is the service rate (in this case, it's 1/4 units per minute).
ρ = λ / μ = 6 / (1/4) = 24
Wq = (24^2) / (1 - 24) * (1/4) = 0.133333...
Therefore, Lq = 6Wq + 23 = 23.8 or around 0.133 units on average are waiting in the queue. 
The best choice is B.

Question 6

A queuing model that follows the M/M/1 assumptions has λ = 10 and μ = 12.What is the average number of units in the system?

Accepted Answer

The answer of A queuing model that follows the M/M/1...

Question 7

Students arrive randomly at the help desk of the computer lab.There is only one service agent,and the time required for inquiry varies from student to student.Arrival rates have been found to follow the Poisson distribution,and the service times follow the negative exponential distribution.The average arrival rate is 12 students per hour,and the average service rate is 20 students per hour.On average,how long does it take to service each student?

Accepted Answer

The answer of Students arrive randomly at the help desk...

Question 8

In the basic queuing model (M/M/1),what probability distribution describes arrival rates?

Accepted Answer

The basic queuing model (M/M/1) assumes a Poisson distribution for arrival rates, which is a common assumption in queuing theory.

Question 9

Which one of the following is NOT a characteristic of a Model B or M/M/S system?

Accepted Answer

Model B or M/M/S systems are characterized by having multiple servers (S > 1), which contradicts option B that states it has a single server.

Question 10

In the basic queuing model (M/M/1),what probability distribution describes service times?

Accepted Answer

In the basic queuing model (M/M/1), service times are assumed to follow a negative exponential distribution. This means that the likelihood of a customer completing service within a certain time follows an exponential decay curve, where shorter times are more likely than longer times. This assumption is common in queuing models because it allows for straightforward calculations of key performance metrics, such as the expected waiting time for customers in the queue.

Question 11

A queuing model that follows the M/M/1 assumptions has λ = 2 and μ = 3.What is the average number of units in the system?

Accepted Answer

The answer of A queuing model that follows the M/M/1...

Question 12

A single-phase waiting-line system meets the assumptions of constant service time or M/D/1.Units arrive at this system every 12 minutes on average.Service takes a constant 8 minutes.What is the average length of the queue Lq in units?

Accepted Answer

In an M/D/1 queue, the average length of the queue Lq can be calculated using the formula Lq = (λ^2 * σ^2) / (2 * (1 - ρ)), where λ is the arrival rate, σ is the service time, and ρ is the utilization factor (λ * σ). Here, λ = 1/12 per minute, σ = 8 minutes, and ρ = (1/12) * 8 = 2/3. Plugging these values into the formula gives Lq = (1/12)^2 * 8^2 / (2 * (1 - 2/3)) = 0.67.

Question 13

A finite population waiting line model is also called M/M/1 with finite source.

Accepted Answer

A finite population waiting line model is indeed referred to as M/M/1 with finite source. The M/M/1 model stands for Markovian/Markovian/1 server and is used to analyze waiting times in a queuing system with a single server and a Poisson arrival process. The "finite source" aspect refers to the fact that there is a fixed number of customers in the population that the system serves.

Question 14

Four of the most widely used waiting line models-M/M/1 or A,M/M/S or B,M/D/1 or C,and Finite population or D-all share which of the following three characteristics?

Accepted Answer

The answer of Four of the most widely used waiting...

Question 15

Which of the following is NOT an assumption of the M/M/1 model?

Accepted Answer

The M/M/1 model assumes that service times occur according to an exponential probability distribution, not a normal probability distribution.

Question 16

Which one of the following is NOT a characteristic of a Model C or M/D/1 system?

Accepted Answer

The answer of Which one of the following is NOT...

Question 17

An M/M/1 model and an M/D/1 model each have an arrival rate of 1 per minute and a service rate of 3 per minute;the average queue length of the M/M/1 will be twice that of the M/D/1.

Accepted Answer

The average queue length of an M/M/1 model can be calculated using the formula Lq = (ρ^2)/(1-ρ), where ρ is the traffic intensity, which is equal to λ/μ in this case (arrival rate/ service rate). Thus, ρ = 1/3, and Lq = (1/9)/(2/3) = 1/6. 
For the M/D/1 model, the queue length will be zero if the service time is less than the inter-arrival time, and will be equal to the arrival rate minus the service rate otherwise. Since the service rate (3) is greater than the arrival rate (1), the queue length for the M/D/1 model will be 2, which is twice as small as the queue length for the M/M/1 model. Therefore, the statement is true.

Question 18

A queuing model that follows the M/M/1 assumptions has λ = 2 and μ = 8.The average number in the system Ls is ________ and the utilization of the system is ________.

Accepted Answer

The average number in the system Ls for an M/M/1 queue is given by Ls = λ / (μ - λ). Here, Ls = 2 / (8 - 2) = 0.33. The utilization of the system is given by ρ = λ / μ = 2 / 8 = 0.25 or 25 percent.

Question 19

In the finite population waiting line model,the average length of the queue is calculated from the probability the system is empty.

Accepted Answer

In the finite population waiting line model, the average length of the queue can indeed be calculated based on the probability that the system is empty, as this probability affects the overall utilization and throughput of the system, which in turn influences the average queue length.

Question 20

A waiting line meeting the assumptions of M/M/1 has an average time between arrivals of 20 minutes,and it services items in an average of 10 minutes each.What is the utilization factor?

Accepted Answer

The utilization factor (ρ) is calculated as the arrival rate (λ) divided by the service rate (μ). Here, λ = 1/20 per minute and μ = 1/10 per minute, so ρ = (1/20) / (1/10) = 0.50.

Quiz 25: Waiting-Line Models