Lesson Topics - Pepperdine Universityseaver-faculty.pepperdine.edu/jburke2/ba452/PowerP2/ReviewQB_8.pdfLesson Topics Analytical Formulas for Multiple Channels for operating characteristics

B.8 Waiting Line Economic Analysis Review Questions

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Lesson Topics

Analytical Formulas for Multiple Channels

for operating characteristics have been derived, under the queue discipline first-come, first-served, for several queuing models with multiple channels. M/M/2 Queuing System designates M = Markov

(memoryless) arrival distribution (Poisson), M = Markov service-time distribution (exponential), and 2 service channels, under first-come, first-served.

Economic Analysis (6) of waiting lines maximizes

profit for a firm by maximizing value for customers. Maximizing customer value trades off quick service with low purchase prices (resulting from lower costs). Economic Analysis with Teamwork (7) maximizes firm’s profits and customer’s value by trading off having more service channels with teams providing faster service in each channel.


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Economic Analysis

Question. Two new checkout scanning systems are

under consideration Target. Arrivals to the checkout

stand follow the Poisson distribution with = 2 per

minute. The cost for waiting is $18 per hour. The first

system has an exponential service rate of 5 per minute and costs $10 per

hour to operate. The second system has an exponential service rate of 8

per minute and costs $20 per hour to operate. Which system should be

chosen?

You can use the Management Scientist to compute your answer, but do not

use the Economic Analysis check box. You should explain your answer.


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Answer to Question:

The cost for waiting is $18 per hour, or $18/60 = $0.30 per minute.

First system, k = 1 channel, = 2 per minute, = 5 per minute.

Management Scientist reports .6667 average number of units in the

system. Therefore, the cost per minute is .3x.6667 + 10/60 = .3667.

Second system, k = 1 channel, = 2 per minute, = 8 per minute.


system. Therefore, the cost per minute is .3x.3333 + 20/60 = .4333.

Choose first system.


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Economic Analysis

Question. Two alternative checkout scanning

systems are under consideration by a retail store.

Arrivals to the checkout stand follow the Poisson

distribution with an average of one customer arriving

every 30 minutes. The cost for waiting is $100 per hour. The first system

costs $10 per hour to operate, and it has an exponential service rate, and it

can serve an average of one customer every 15 minutes. The second

system costs $20 per hour to operate, and it has an exponential service

rate, and it can serve an average of one customer every 10 minutes.

Which system should be chosen?

You may use any or all of the following analytical formulas for an M/M/1

system to compute your answer:

The probability of no units in the system: P0 = 1-

The average number of units in the waiting line: Lq =

The average number of units in the system: L = Lq +

The average time a unit spends in the waiting line: Wq = Lq/

The average time a unit spends in the system: W = 1/

The probability that an arriving unit has to wait: Pw =

Probability of n units in the system: Pn = ()nP0


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Answer to Question:

First system: k = 1 channel, = 2 per hour, and = 4 per hour.

The average number of units in the waiting line:

Lq = = 4/(4(2)) = 1/2

The average number of units in the system: L = Lq + = ½ + 2/4 = 1

Cost to operate = $10 + $100*L = $110 per hour, which is $1.83 per

minute.

Second system: k = 1 channel, = 2 per hour, and = 6 per hour.


Lq = = 4/(6(4)) = 1/6

The average number of units in the system: L = Lq + = 1/6 + 2/6 = .5

Cost to operate = $20 + $100*L = $70 per hour, which is $1.17 per minute.

So, choose the second system.


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Economic Analysis

Question. Pepperdine University is considering

whether it should employ two, three, or four

attendants at the One Stop student services windows.

To decide, Pepperdine hires management scientists

to make four measurements. 1) After assuming that arrivals to the

checkout stand follow the Poisson distribution, the scientists measure that

an average of one customer arrives every 30 seconds. 2) After assuming

that service times at the checkout stand follow the exponential distribution,

the scientists measure that it takes an average of 40 seconds to serve each

customer. 3) The scientists ask school administrators and find out each

cashier costs $30 per hour to employ. Finally, considering that median

income for a family in Malibu is $123,293, the scientists figure the cost of

customer waiting is $22 per hour. How many attendants should

Pepperdine employ? (Two? Three? Four?)

Is it possible to employ just one attendant?

How would the optimal number of attendants change for Cal State Chico,

which has all the same numbers above, except considering that median

income for a family in Chico is $43,077, the scientists figure the cost of

customer waiting is only $5 per hour.




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Answer to Question: An average of one customer arrives every 30

seconds means = 120 customers arriving per hour. An average of 40

seconds to serve each customer means = 90 customers served per hour.

Employing one cashier means the waiting line will grow without limit, so

that is not optimal.

Employing two attendants means the average number of customers in the

system is 2.4, which means the hourly cost is 2x$30 + 2.4x$22 = $112.8.

Employing three attendants means the average number of customers in the

system is 1.478, which means the hourly cost is 3x$30 + 1.478x$22 =

$122.52.

Employing four attendants means the average number of customers in the

system is 1.3592, which means the hourly cost is 4x$30 + 1.3592x$22 =

$149.90.

Hence, Pepperdine University should employ two attendants, to achieve

the minimum hourly cost of $112.80.

Cal State Chico faces hourly costs of 2x$30 + 2.4x$5 = $72 for two

attendants, 3x$30 + 1.478x$5 = $97.39 for three attendants, and 4x$30 +

1.3592x$5 = $126.80 for four attendants.

Hence, Cal State Chico should also employ two attendants, to achieve the

minimum hourly cost of $52.


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Economic Analysis

Question. A Florida coastal community experiences

a population increase during the winter, with seasonal

residents arriving from nothern states and Canada.

Staffing at a local post office is often in a state of

change due to the relatively low volumne of customers in the summer.

Consider a typical winter month. Assume Poisson arrival times and

exponential service times. Customers arrive at a rate of 55 per hour. The

average service time is 1 minute per customer. Customer waiting time is

valued at $20 per hour. From that data, determine the recommended

number of tellers at the post office, if each teller costs $20 per hour in

wages and benefits.

Now, consider a typical summer month. Assume Poisson arrival times and

exponential service times. Customers arrive at a rate of 24 per hour. The

average service time is 1 minute per customer. Customer waiting time is

valued at $20 per hour. From that data, determine the recommended

number of tellers at the post office, if each teller costs $20 per hour in

wages and benefits.

For both answers, you may use computer software to compute your

answer, but be sure to define the computer input and explain how the

computer output solves the problem. Do not use the Economic Analysis

checkbox in the Management Scientist.


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Answer to Question: The cost of waiting is $20 per hour, and the cost of

each channel (teller) is $20.

First, consider a typical winter month: = 55 per hour arrive, = 60 per

hour served.

For k = 1 channel, Management Scientist reports 11 average number of

units in the system. Therefore, total hourly cost is $20x11 + 20 = $240.

For k = 2 channels, Management Scientist reports 1.16 average number of

units in the system. Therefore, total hourly cost is $20x1.16 + 40 = $63.20.

For k = 3 channels, Management Scientist reports 0.9489 average number

of units in the system. Therefore, total hourly cost is $20x0.9489+ 60 =

$78.98.

And for k > 4 channels, the channel cost alone is at least $80.

Choose 2 channels (that is, 2 tellers).

Finally, consider a typical summer: = 24 per hour arrive, = 60 per hour

served.

For k = 1 channel, Management Scientist reports 0.6667 average number

of units in the system. Therefore, total hourly cost is $20x0.6667 + 20 =

$33.3333.

And for k > 2 channels, the channel cost alone is at least $40.

Choose 1 channel (that is, 1 teller).


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Economic Analysis

Question. Hinck Electrical Contractor, Inc. is

considering opening a second service counter to

better serve the electrical contractor customers. The

arrival rate is 10 per hour. The service rate is 14 per

hour. Assume Poisson arrival and exponential service-time distributions. If

the cost of waiting is $30 and the cost of each service counter is $22 per

hour, then should the second counter be opened?




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Answer to Question:

One counter, k = 1 channel, = 10 per hour, = 14 per hour.

Management Scientist reports 2.5 average number of units in the system.

Therefore, the cost per hour 30x2.5 + 22 = 97.

Two counters, k = 2 channels, = 10 per hour, = 14 per hour.


system. Therefore, the cost per hour 30x.8187 + 44 = 68.561.

Open the second counter.


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Economic Analysis with Teamwork

Question. Food on Foot is a 501(c)(3) nonprofit

organization dedicated to providing the poor and

homeless of Los Angeles with nutritious meals,

clothing, and assistance in the transition to

employment and life off the streets. Food on Foot operates a weekly meal

program every Sunday in Hollywood. They are considering two alternative

distribution systems.

Arrivals to the distribution stand follow the Poisson distribution with an average of one homeless person arriving every 4 minutes. The cost for a homeless person waiting is $1 per hour. The first system involves one volunteer and costs $10 per hour to operate (the opportunity cost of the volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 2 minutes. The second system involves two volunteers working as a team (one talks to the homeless and the other serves the meals) and costs $20 per hour to operate (the opportunity cost of the 2 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 1 minute. Which system should be chosen?











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Answer to Question:



Lq = = 15*15/(30(15)) = 1/2

The average number of units in the system: L = Lq + = ½ + 15/30 = 1

Cost to operate = $10 + $1*L = $11 per hour.



Lq = = 15*15/(60(45)) = 1/(4*3) = 1/12

The average number of units in the system: L = Lq + = 1/12 + 15/60 =

4/12 = .33

Cost to operate = $20 + $1*L = $20.33.

Choose the first system. (Going beyond the basic answer, ask one of your

two volunteers to put in extra time at work and donate their income. The

homeless people will benefit more from the extra money than from the

faster service.)


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Question. Food on Foot is a 501(c)(3) nonprofit

organization dedicated to providing the poor and

homeless of Los Angeles with nutritious meals,

clothing, and assistance in the transition to

employment and life off the streets. Food on Foot operates a weekly meal

program every Sunday in Hollywood. They are considering three

alternative distribution systems.

Arrivals to the distribution stand follow the Poisson distribution with an average of one homeless person arriving every 6 minutes. The cost for a homeless person waiting is $3 per hour. The first system involves one volunteer (who talks to the homeless and serves food and serves drinks) and costs $1 per hour to operate (the opportunity cost of the volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 3 minutes. The second system involves two volunteers working as a team (one talks to the homeless and serves food, and the other serves drinks) and costs $2 per hour to operate (the opportunity cost of the 2 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 2 minutes. The third system involves three volunteers working as a team (one talks to the homeless, one serves food, and the third serves drinks) and costs $3 per hour to operate (the opportunity cost of the 3 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 1.5 minutes. Which system should be chosen?






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Answer to Question:



Lq = = 10*10/(20(10)) = 1/2

The average number of units in the system: L = Lq + = 1/2 + 1/2 = 1




Lq = = 10*10/(30(20)) = 1/6

The average number of units in the system: L = Lq + = 1/6 + 1/3 = 1/2 =

0.50

Cost to operate = $2 + $3*L = $3.50 per hour.

Third system: k = 1 channel, = 10 per hour, and = 40 per hour.


Lq = = 10*10/(40(30)) = 1/12

The average number of units in the system: L = Lq + = 1/12 + 1/4 = 1/3

= 0.33


Choose the second system. (Going beyond the basic answer, ask any

other volunteers beyond two to put in extra time at work and donate their

income. The homeless people will benefit more from the extra money than

from the faster service.)


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Question. Food on Foot operates a weekly meal

program every Sunday in Hollywood. They are

considering three alternative distribution systems.

Arrivals to the distribution stand follow the Poisson distribution with an average of one homeless person arriving every 12 minutes. The cost for a homeless person waiting is $2 per hour. The first system involves one volunteer (who talks to the homeless and serves food and serves drinks) and costs $10 per hour to operate (the opportunity cost of the volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 6 minutes. The second system involves two volunteers working as a team (one talks to the homeless and serves food, and the other serves drinks) and costs $20 per hour to operate (the opportunity cost of the 2 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 3 minutes. The third system involves three volunteers working as a team (one talks to the homeless, one serves food, and the third serves drinks) and costs $30 per hour to operate (the opportunity cost of the 3 volunteer’s time), and it has an exponential service rate, and it can serve an average of one homeless person every 1 minute. Which system should be chosen? Re-compute your answer if the customers value their time at $20 per hour, rather than $2 per hour. You may use any or all of the following analytical formulas for an M/M/1










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Answer to Question:



Lq = = 5*5/(10(5)) = 1/2

The average number of units in the system: L = Lq + = 1/2 + 1/2 = 1




Lq = = 5*5/(20(15)) = 1/12

The average number of units in the system: L = Lq + = 1/12 + 1/4 = 1/3

= 0.3333


Third system: k = 1 channel, = 5 per hour, and = 60 per hour.


Lq = = 5*5/(60(55)) = 1/132

The average number of units in the system: L = Lq + = 1/132 + 1/12 =

0.0909


Choose the first system. (Going beyond the basic answer, ask any other

volunteers beyond two to put in extra time at work and donate their income.

The homeless people will benefit more from the extra money than from the

faster service.)

At $20 per hour for customers,

First system cost to operate = $10 + $20*(1) = $30 per hour.

Second system cost to operate = $20 + $20*(0.333) = $26.66 per hour.

Third system cost to operate = $30 + $20*(0.090) = $31.80 per hour.

Choose the second system.


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Question. Pepperdine University student volunteers

are considering how to best serve food to the

homeless. To decide, Pepperdine hires management

scientists to make four measurements. 1) After

assuming that arrivals of homeless people follow the Poisson distribution,

the scientists measure that an average of one homeless person arrives

every 2 minutes. 2) The scientists ask social workers and find out each

homeless person values their time at $2 per hour. 3) After polling the

student volunteers, the scientists find out each volunteer values their

volunteer time at $10 per hour. (That is, each volunteer would be

indifferent between donating either 1 hour of their time or $10.) Finally, the

scientists consider 5 alternative methods for serving food:

Method A: There is one line, and 4 volunteers work along the line. It takes

30 seconds on average to serve a homeless person under this method.

Method B: There is one line, and 8 volunteers work along the line. It takes


Method C: There are 2 lines, and 4 volunteers work along each line. For

each line, it takes 30 seconds on average to serve a homeless person

under this method.

Method D: There are 4 lines, and 2 volunteers work along each line. For


under this method.

Method E: There is 1 line, and 1 volunteer works on the line. It takes 300

seconds on average to serve a homeless person under this method.

Which method is best? In your answer, be sure to show your computations

for each of the 5 methods.


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You may use computer software to compute your answer, but be sure to

define the computer input and explain how the computer output solves the

problem. Do not use the Economic Analysis checkbox in the Management

Scientist.


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Answer to Question: The cost of a volunteer is $10 per hour, and the

cost of waiting is $2 per hour.

Method A, V = 4 volunteers, k = 1 channel, = 30 per hour arrive rate, =

120 per hour served. Management Scientist reports L = 0.3333 average

number of units in the system. Therefore, the hourly operating cost is

$10x4 + $2x.333 = $40.67.

Method B, V = 8, k = 1, = 30, = 240, L = 0.1429. Hourly cost is

$10x8 + $2x.1429 = $80.29.

Method C, V = 8, k = 2, = 30, = 120, L = 0.2540. Hourly cost is

$10x8 + $2x.2540 = $80.51.

Method D, V = 8, k = 4, = 30, = 60, L = 0.5003. Hourly cost is

$10x8 + $2x.5003 = $81.00.

Method E, V = 1, k = 1, = 30, = 12, L = infinity. Hourly cost is infinity.

Choose Method A.


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under this method.



under this method.






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Scientist.


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Answer to Question: The cost of a volunteer is $20 per hour, and the

cost of unit waiting is $1 per hour.




$20x4 + $1x.3333 = $80.33.


$20x8 + $1x.1429 = $160.14.


$20x8 + $1x.2540 = $160.25.


$20x8 + $1x.5003 = $161.50.


Choose Method A.


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under this method.



under this method.






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How would your answer change if the volunteers, rather than serving the

homeless, are serving middle managers, who each value their time at $30

per hour.




Scientist.


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Answer to Question: The cost of a volunteer is $10 per hour.

First, answer when the cost of waiting is $3 per hour.




$15x4 + $3x.2857 = $60.86


$15x8 + $3x.125 = $120.38


$15x8 + $3x.1678 = $120.50


$15x8 + $3x.2500 = $81.75


Choose Method A.

Second, answer when the cost of waiting is $30 per hour.




$15x4 + $30x.2857 = $68.6


$15x8 + $30x.125 = $123.8


$15x8 + $30x.1678 = $125.0


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$15x8 + $30x.2500 = $88.5


Choose Method A.


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2 minutes on average to serve a homeless person under this method.


1 minute on average to serve a homeless person under this method.



under this method.



under this method.






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How would your answer change if the volunteers, rather than serving the

homeless, are serving lawyers, who each value their time at $100 per hour.




Scientist.


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Answer to Question: The cost of a volunteer is $10 per hour.

First, answer when the cost of waiting is $1 per hour.




$10x4 + $1x1.0000 = $41


$10x8 + $1x.333 = $80.33


$10x8 + $1x.1255 = $80.1255


$10x8 + $1x.1875 = $80.1875


Choose Method A, at $41 per hour.

Second, answer when the cost of waiting is $100 per hour.




$10x4 + $100x1.0000 = $140


$10x8 + $100x.333 = $113.33


$10x8 + $100x.1255 = $92.55


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$10x8 + $100x.1875 = $98.75


Choose Method C, at $92.55 per hour.

Documents

Lesson Topics - Pepperdine Universityseaver-faculty.pepperdine.edu/jburke2/ba452/PowerP2/ReviewQB_8.pdfLesson Topics Analytical Formulas for Multiple Channels for operating characteristics