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Lesson II'5 The Simplest Waiting Line Model

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Title: Lesson II'5 The Simplest Waiting Line Model


1
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

2
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.

3
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

4
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

5
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

6
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.

7
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.

8
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

9
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.

10
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.

11
Example 1 M/M/2 Queuing System
Waiting line module input
12
Example 1 M/M/2 Queuing System
Waiting line module output
13
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.)

14
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.

15
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
16
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
17
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.

18
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.

19
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).

20
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?

21
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.

22
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.

23
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

24
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

25
End of Lesson II.6
BA 452

Quantitative Analysis
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