Lesson II'5 The Simplest Waiting Line Model

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From Chapter 11 (Sections 1,2,3,5)
Waiting Line Models

• Lesson II.5 The Simplest Waiting Line Model
• Lesson II.6 Economic Analysis
• Tool Summary
• Analytical Formulae
• Example 1 M/M/2 Queuing System
• Example 2 Economic Analysis
• Example 3 Another Economic Analysis
• Review Problems

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Tool Summary

• Tool Summary
• Use formulae or Management Scientist to compute
  performance
• Probability that no units are in the system P0
• Average number of units in waiting line Lq
• Average number of units in system L Lq l/m
• Average time a unit spends in waiting line Wq
  Lq/l
• Average time a unit spends in the system W Wq
  1/m
• Probability that an arriving unit has to wait for
  service Pw
• Probability of n units in the system Pn
• Compute total hourly cost for units in the system
  
• ( waiting cost per hour) x (Average number of
  units in system)
• Note average number of units in system is the
  only right choice above average time in waiting
  line does not count the number of units.

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Analytical Formulae
Formulae for M/M/k under FCFS

• Required assumptions for formulae
• Multiple channels (with one central waiting line)
• Poisson arrival-rate distribution
• Exponential service-time distribution
• Unlimited maximum queue length
• Infinite calling population
• Examples
• Four-teller transaction counter in bank
• Two-clerk returns counter in retail store

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Analytical Formulae
Formulae for M/M/k under FCFS

• Probability that no units are in the system
• Average number of units in waiting line
• Average number of units in system L Lq l/m
• Average time a unit spends in waiting line Wq
  Lq/l
• Average time a unit spends in the system W Wq
  1/m

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Analytical Formulae
Formulae for M/M/k under FCFS

• 6) Probability that an arriving unit has to
  wait for service Pw (1/k!) (l/m)k
  (km/(km-l)) P0
• Probability of n units in the system
• (l/m)n /n! P0
  for n lt k
• Pn
• (l/m)n /(k! k(n-k)) P0
  for n gt k

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Example 1 M/M/2 Queuing System
Verbal description

• M/M/2 Queuing System
• Smith, Jones, Johnson, and Thomas, Inc. has
  begun a major advertising campaign which it
  believes will increase its business 50. To
  handle the increased volume, the company has
  hired an additional floor trader, Fred Hanson,
  who works at the same speed as Joe Ferris.
• Note that the new arrival rate of orders, l, is
  50 higher than that of Example 1 in Lesson 2.5.
  Thus, l 1.5(20) 30 per hour.

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Example 1 M/M/2 Queuing System
Utilization factor
M/M/1 P0 1-l/m Lq l2/(m(m-l)) L Lq
l/m Wq Lq/l W 1/(m-l) Pw l/m Pn (l/m)nP0

• Sufficient Service Rate
• Will Joe Ferris alone be able to handle the
  increase in orders?
• Answer Since Joe Ferris processes orders at a
  mean rate of µ 30 per hour, then ? µ 30
  and the average time a unit spends in the system
  is W 1/(m-l) 1/0 infinity.
• That implies the queue of orders will grow
  infinitely large. Hence, Joe alone cannot handle
  that increase in demand.

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Example 1 M/M/2 Queuing System
Probability of n units in system

• Probability of n Units in System
• What is the probability that neither Joe nor Fred
  will be working on an order at any point in time?
• Answer This is an M/M/k queue with ? 30 per
  hour, ? 30 per hour, and k 2. The
  probability that neither Joe nor Fred will be
  working the probability of no units in the
  system. Analytical Formula 1 says that is
• 1/(1 (1/1!)(30/30)1
  (1/2!)(1)22(30)/(2(30)-30)
• 1/(1 1 1) 1/3 .333

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Example 1 M/M/2 Queuing System
Average time in system

• Average Time in System
• What is the average turnaround time for an order
  with both Joe and Fred working?
• Answer The average turnaround time the average
  time a unit spends in the system, W. Analytical
  Formula 2 and 3 say
• and L Lq (? /µ) 1/3 (30/30) 4/3.
  Finally,
• W L/?????(4/3)/30 4/90 hr. 2.67
  min.

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Example 1 M/M/2 Queuing System
Average length of queue

• Average length of queue
• What is the average number of orders waiting to
  be filled with both Joe and Fred working?
• Answer The average number of orders waiting to
  be filled the average number of units in the
  waiting line, Lq. That was calculated earlier as
  1/3.

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Example 1 M/M/2 Queuing System
Waiting line module input
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Example 1 M/M/2 Queuing System
Waiting line module output
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Example 2 Economic Analysis
Verbal description

• The advertising campaign of Smith,
  Jones, Johnson and Thomas, Inc. (Example 1 in
  Lesson 2.5 and Example 1 in Lesson 2.6) was so
  successful that business doubled. The mean rate
  of stock orders arriving at the exchange is now
  40 per hour and the company must decide how many
  floor traders to employ. Each floor trader hired
  can process an order in an average time of 2
  minutes. (So far, ? 40/hr. and m 30/hr.)

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Example 2 Economic Analysis
Problem formulation

• The brokerage firm has determined the average
  waiting cost per minute for an order to be .50.
  (So, you can charge .50 more per order if you
  can process it an average of 1 minute faster.)
  Floor traders hired will earn 20 per hour in
  wages and benefits. Hence, compare the total
  hourly cost of hiring 2 traders with that of
  hiring 3 traders.
• Answer Total hourly cost
• (Total salary cost per hour)
• (Total hourly cost for orders in the
  system)
• (20 per trader per hour) x (Number of
  traders)
• (30 waiting cost per hour) x (Average
  number of orders in system)
• 20k 30L.

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Example 2 Economic Analysis
Cost of two servers

• This is an M/M/2 queue with ? 40 per hour
  and ? 30 per hour. Analytical Formulae
  1, 2, 3
• P0 1 / 1(1/1!)(40/30)(1/2!)(40/30)2(6
  0/(60-40))
• 1 / 1 (4/3) (8/3) 1/5
• say the average time an order waits in the system
  is
• L Lq (? /µ) 16/15 4/3 2.40
• Hence, total cost (20)(2) 30(2.40)
  112.00 per hour

? 40/hr. m 30/hr. Cost 20k 30L
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Example 2 Economic Analysis
Cost of three servers

• This is an M/M/3 queue with ? 40 per hour
  and ? 30 per hour. Analytical Formulae
  1, 2, 3
• P0 1/1(1/1!)(40/30)(1/2!)(40/30)2
• (1/3!)(40/30)3(90/(90-40))
• 1 / 1 4/3 8/9 32/45 15/59
• say the average time an order waits in the system
  is
• L .1446 40/30 1.4780
• Hence, total cost (20)(3) 30(1.4780)
  104.35 per hour

? 40/hr. m 30/hr. Cost 20k 30L
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Example 2 Economic Analysis
System cost comparison

• System cost comparison
• Wage Waiting Total
• Cost/Hr Cost/Hr Cost/Hr
• 2 Traders 40.00 82.00 112.00
• 3 Traders 60.00 44.35 104.35
• Thus, the total cost of having 3 traders is
  less than that of 2 traders.

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Example 3 Another Economic Analysis
Verbal description

• A fast-food franchise is considering adding a
  drive-up window to a particular location.
  
• Assume customer arrivals follow a Poisson
  probability distribution, with an arrival rate of
  l 24 cars per hour.
• Assume customer service times follow an
  exponential distribution.
• Arriving customers place orders at an intercom
  station at the back of the parking lot and then
  drive to the service window to pay for and
  receive their orders.
• The following three service alternatives are
  being considered.

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Example 3 Another Economic Analysis
Verbal description

• System A. One employee fills the order and
  takes the money from the customer. The
  average service time for this alternative is 2
  minutes (30 customers per hour). All together,
  (k, l, m) (1, 24, 30).
• System B. One employee fills the order while a
  second employee takes the money from the
  customer. The average service time for this
  alternative is 1.25 minutes (48 customers per
  hour). All together,
  (k, l, m) (1, 24, 48).
• System C. Two service windows, each with an
  employee that fills the order and takes the money
  from the customer. The average service time for
  this alternative is 2 minutes (30 customers per
  hour) for each channel. All together, (k, l, m)
  (2, 24, 30).

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Example 3 Another Economic Analysis
Verbal description

• Customer waiting time is valued at 25 per hour.
• So, you can charge 25/60 more per order if you
  can process orders an average of 1 minute (1/60
  hour) faster.
• The cost of each employee is 6.50 per hour.
• Each channel costs 20 for equipment and space.
• Which system is most profitable?

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Example 3 Another Economic Analysis
Verbal description

• The labor plus equipment-space cost for each
  channel of each system is

• The waiting plus channel cost for each each
  system is
• System B is thus most profitable (it costs the
  minimum, 58.00). That is, one employee fills the
  order while a second employee takes the money
  from the customer.

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Example 3 Another Economic Analysis
Verbal description

• If, instead, the drive-up window were for a
  location in a poor part of town, where
  customer waiting time is valued at 2 per hour,
  which system is most profitable?
• Answer Change the waiting plus channel cost for
  each system from
  to reduce time value from 25 to 2.
  System A is now most profitable.
• Finally, if the drive-up window were for outside
  the colony in Malibu, where customer waiting time
  is valued at 200 per hour, which system is most
  profitable?
• Answer System B is most profitable, but if the
  value of time were high enough, then System C
  would be most profitable.

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Review Problems

• Questions with Economic Analysis of Waiting Lines
• Exam 2 Version C, Question 3
• http//faculty.pepperdine.edu/jburke2/ba452/Exam2/
  Exam2cAnswers.pdf
• Exam 2 Version D, Question 3
• http//faculty.pepperdine.edu/jburke2/ba452/Exam2/
  Exam2dAnswers.pdf
• Exam 2 Version E, Question 3
• http//faculty.pepperdine.edu/jburke2/ba452/Exam2/
  Exam2eAnswers.pdf
• Exam 2 Version F, Question 3 (like Version E
  above)
• http//faculty.pepperdine.edu/jburke2/ba452/Exam2/
  Exam2fAnswers.pdf
• Exam 2 Version G, Question 3 (applied to
  Homeless)
• http//faculty.pepperdine.edu/jburke2/ba452/Exam2/
  Exam2gAnswers.pdf

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Review Problems

• Questions with Economic Analysis of Waiting Lines
• Final Exam Version B, Question 3
• http//faculty.pepperdine.edu/jburke2/ba452/FinalE
  xam/FinalbAnswers.pdf
• Final Exam Version C, Question 3
• http//faculty.pepperdine.edu/jburke2/ba452/FinalE
  xam/FinalcAnswers.pdf

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End of Lesson II.6
BA 452

Quantitative Analysis