Exam 21: Waiting-Line Models

Genco, Inc., a small manufacturer of diesel-generator sets has four shearing machines. Because of the age of these machines, they need minor repairs after 30 hours of use. Analysis of previous breakdowns indicates that breakdowns follow a Poisson distribution. The facility employs one repairman specifically to repair these machines. Average repair time is two hours following an exponential distribution. a. What is the service factor for this system? b. What is the average number of these machines in service? c. What is the impact of adding a second repairman?

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Question 1

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(a) X = 2/(2 + 30) = .0625; (b) 4 - .2987 = 3.7123 machines; (c) 4 - .2514 = 3.7486, there is a slight improvement in availability of these machines. The table below summarizes the software results from ExcelOM.
$\begin{array}{|l|l|l|l|l}\hline\text { One server } & & \text { Two servers } & \\\hline \text { Average server utilization (r) } & 0.246753 & \text { Average server utilization ( } 1 \text { ) } & 0.124954 \\\hline \text { Average number of customers } & & \text { Average number of } & \\\text { in the queue }(\mathrm{Lq}) & 0.051957 & \begin{array}{l}\text { Avstomers in the queue }(\mathrm{L})\end{array} & 0.001464\\\hline\text { Average number of customers } & & \text { Average number of } & \\\text { in the system }(\mathrm{L}) & 0.29871 & \begin{array}{l}\text { customers in the system }(\mathrm{L})\end{array} & 0.251373 \\\hline \text { Average waiting time in the } &0.421129 & \text { Average waiting time in the } &0.011717 \\\text { queue }\left(\mathrm{W}_{\mathrm{q}}\right)\\\hline \begin{array}{l}\text { Average time in the system } \\\text { (W) }\end{array} & 2.421129 & \begin{array}{l}\text { Average time in the system } \\(\mathrm{W})\end{array} & 2.011717 \\\hline\text { Probability (\% of time) system } & & \text { Probability (\% of time) } & \\\text { is empty }\left(\mathrm{P}_{0}\right) & 0.753247 & \text { System is empty }\left(\mathrm{P}_{0}\right) & 0.772099 \\\hline \text { Effective arrival rate } & 0.123376 & \text { Effective arrival rate } & 0.124954\\\hline\end{array}$

Suppose that a fast food restaurant wants the average line to be 4 customers and that 80 customers arrive each hours. How many minutes will the average customer be forced to wait in line?

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Question 2

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Little's Law 4 customers = 80 customers/hour ∗ X hour wait
X = .05 hours = 3 minute average wait

The sign at the bank that reads "Wait here for the first available teller" suggests the use of a ________ waiting line system.

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Question 3

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D

A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per hour. What is the probability that the waiting line is empty?

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Question 4

Waiting-line models are useful to operations in such diverse settings as service systems, maintenance activities, and shop-floor control.

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Question 5

The ________ probability distribution is a continuous probability distribution often used to describe the service time in a queuing system.

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Question 6

A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ = 4. For this system, the probability of more than two units in the system is approximately

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Question 7

A copy centre has five machines that serve many customers throughout the day; the waiting-line system for copy service has an infinite population while the waiting-line system for copier maintenance has a finite population.

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Question 8

The cost of waiting decreases as the service level increases.

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Question 9

In the basic queuing model (M/M/1), arrival rates are distributed by

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Question 10

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

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Question 11

A dental clinic at which only one dentist works is open only two days a week. During those two days, the traffic arrivals follow a Poisson distribution with patients arriving at the rate of three per hour. The doctor serves patients at the rate of one every 15 minutes. a. What is the probability that the clinic is empty (except for the dentist)? b. What is the probability that there are one or more patients in the system? c. What is the probability that there are four patients in the system? d. What is the probability that there are four or more patients in the system?

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Question 12

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

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Question 13

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. How many students, on the average, will be waiting in line at any one time?

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Question 14

A concert hall, employing both ticket takers and ushers to seat patrons, behaves typically as a

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Question 15

A bank office with five tellers, each with a separate line of customers, exhibits the characteristics of a multiphase queuing system.

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Question 16

Describe the difference between FIFO and LIFO queue disciplines.

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Question 17

The ________ of a waiting line and the probability that the queue is empty add to one.

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Question 18

The potential restaurant customer who says to her husband, "The line looks too long; let's eat somewhere else," is an illustration of which element of queue discipline?

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Question 19

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

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Question 20

Exam 21: Waiting-Line Models

Suppose that a fast food restaurant wants the average line to be 4 customers and that 80 customers arrive each hours. How many minutes will the average customer be forced to wait in line?

The sign at the bank that reads "Wait here for the first available teller" suggests the use of a ________ waiting line system.

A waiting line meeting the M/M/1 assumptions has an arrival rate of 4 per hour and a service rate of 12 per hour. What is the probability that the waiting line is empty?

Waiting-line models are useful to operations in such diverse settings as service systems, maintenance activities, and shop-floor control.

The ________ probability distribution is a continuous probability distribution often used to describe the service time in a queuing system.

A waiting-line system that meets the assumptions of M/M/1 has λ = 1, μ = 4. For this system, the probability of more than two units in the system is approximately

A copy centre has five machines that serve many customers throughout the day; the waiting-line system for copy service has an infinite population while the waiting-line system for copier maintenance has a finite population.

The cost of waiting decreases as the service level increases.

In the basic queuing model (M/M/1), arrival rates are distributed by

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

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

A concert hall, employing both ticket takers and ushers to seat patrons, behaves typically as a

A bank office with five tellers, each with a separate line of customers, exhibits the characteristics of a multiphase queuing system.

Describe the difference between FIFO and LIFO queue disciplines.

The ________ of a waiting line and the probability that the queue is empty add to one.

The potential restaurant customer who says to her husband, "The line looks too long; let's eat somewhere else," is an illustration of which element of queue discipline?

A queuing model which 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 ________.

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