Spreadsheet Modeling - PowerPoint PPT Presentation

Loading...

PPT – Spreadsheet Modeling PowerPoint presentation | free to download - id: 499e8e-NDZlN



Loading


The Adobe Flash plugin is needed to view this content

Get the plugin now

View by Category
About This Presentation
Title:

Spreadsheet Modeling

Description:

Spreadsheet Modeling & Decision Analysis A Practical Introduction to Management Science 4th edition Cliff T. Ragsdale Queuing Theory Introduction to Queuing Theory It ... – PowerPoint PPT presentation

Number of Views:377
Avg rating:3.0/5.0
Slides: 35
Provided by: Clif66
Learn more at: http://clt.astate.edu
Category:

less

Write a Comment
User Comments (0)
Transcript and Presenter's Notes

Title: Spreadsheet Modeling


1
Spreadsheet Modeling Decision Analysis
  • A Practical Introduction to Management Science
  • 4th edition
  • Cliff T. Ragsdale

2
Queuing Theory
Chapter 13
3
Introduction to Queuing Theory
  • It is estimated that Americans spend a total of
    37 billion hours a year waiting in lines.
  • Places we wait in line...
  • ? stores ? hotels ? post offices
  • ? banks ? traffic lights ? restaurants
  • ? airports ? theme parks ? on the phone
  • Waiting lines do not always contain people...
  • ? returned videos
  • ? subassemblies in a manufacturing plant
  • ? electronic message on the Internet
  • Queuing theory deals with the analysis and
    management of waiting lines.

4
The Purpose of Queuing Models
  • Queuing models are used to
  • describe the behavior of queuing systems
  • determine the level of service to provide
  • evaluate alternate configurations for providing
    service

5
Queuing Costs

Total Cost
Cost of providing service
Cost of customer dissatisfaction
Service Level
6
Common Queuing System Configurations
7
Characteristics of Queuing SystemsThe Arrival
Process
  • Arrival rate - the manner in which customers
    arrive at the system for service.
  • where l is the arrival rate (e.g., calls arrive
    at a rate of l5 per hour)
  • See file Fig13-3.xls

8
Characteristics of Queuing SystemsThe Service
Process
  • Service time - the amount of time a customer
    spends receiving service (not including time in
    the queue).
  • where m is the service rate (e.g., calls can be
    serviced at a rate of m7 per hour)
  • The average service time is 1/m.
  • See file Fig13-4.xls

9
Comments
  • If arrivals follow a Poisson distribution with
    mean l, interarrival times follow an Exponential
    distribution with mean 1/l.
  • Example
  • Assume calls arrive according to a Poisson
    distribution with mean l5 per hour.
  • Interarrivals follow an exponential distribution
    with mean 1/5 0.2 per hour.
  • On average, calls arrive every 0.2 hours or every
    12 minutes.
  • The exponential distribution exhibits the
    Markovian (memoryless) property.

10
Kendall Notation
  • Queuing systems are described by 3 parameters
  • 1/2/3
  • Parameter 1
  • M Markovian interarrival times
  • D Deterministic interarrival times
  • Parameter 2
  • M Markovian service times
  • G General service times
  • D Deterministic service times
  • Parameter 3
  • A number Indicating the number of servers.
  • Examples,
  • M/M/3 D/G/4 M/G/2

11
Operating Characteristics
  • Typical operating characteristics of interest
    include
  • U - Utilization factor, of time that all
    servers are busy.
  • P0 - Prob. that there are no zero units in the
    system.
  • Lq - Avg number of units in line waiting for
    service.
  • L - Avg number of units in the system (in line
    being served).
  • Wq - Avg time a unit spends in line waiting for
    service.
  • W - Avg time a unit spends in the system (in
    line being served).
  • Pw - Prob. that an arriving unit has to wait
    for service.
  • Pn - Prob. of n units in the system.

12
Key Operating Characteristics of the M/M/1 Model
13
The Q.xls Queuing Template
  • Formulas for the operating characteristics of a
    number of queuing models have been derived
    analytically.
  • An Excel template called Q.xls implements the
    formulas for several common types of models.
  • Q.xls was created by Professor David Ashley of
    the Univ. of Missouri at Kansas City.

14
The M/M/s Model
  • Assumptions
  • There are s servers.
  • Arrivals follow a Poisson distribution and occur
    at an average rate of l per time period.
  • Each server provides service at an average rate
    of m per time period, and actual service times
    follow an exponential distribution.
  • Arrivals wait in a single FIFO queue and are
    serviced by the first available server.
  • llt sm.

15
An M/M/s Example Bitway Computers
  • The customer support hotline for Bitway Computers
    is currently staffed by a single technician.
  • Calls arrive randomly at a rate of 5 per hour and
    follow a Poisson distribution.
  • The technician services calls at an average rate
    of 7 per hour, but the actual time required to
    handle a call follows an exponential
    distribution.
  • Bitways president, Rod Taylor, has received
    numerous complaints from customers about the
    length of time they must wait on hold for
    service when calling the hotline.
  • Continued

16
Bitway Computers (continued)
  • Rod wants to determine the average length of time
    customers currently wait before the technician
    answers their calls.
  • If the average waiting time is more than 5
    minutes, he wants to determine how many
    technicians would be required to reduce the
    average waiting time to 2 minutes or less.

17
Implementing the Model
  • See file Q.xls

18
Summary of Results Bitway Computers
  • Arrival rate 5 5
  • Service rate 7 7
  • Number of servers 1 2
  • Utilization 71.43 35.71
  • P(0), probability that the system is empty 0.2857
    0.4737
  • Lq, expected queue length 1.7857 0.1044
  • L, expected number in system 2.5000 0.8187
  • Wq, expected time in queue 0.3571 0.0209
  • W, expected total time in system 0.5000 0.1637
  • Probability that a customer waits 0.7143 0.1880

19
The M/M/s Model With Finite Queue Length
  • In some problems, the amount of waiting area is
    limited.
  • Example,
  • Suppose Bitways telephone system can keep a
    maximum of 5 calls on hold at any point in time.
  • If a new call is made to the hotline when five
    calls are already in the queue, the new call
    receives a busy signal.
  • One way to reduce the number of calls
    encountering busy signals is to increase the
    number of calls that can be put on hold.
  • If a call is answered only to be put on hold for
    a long time, the caller might find this more
    annoying than receiving a busy signal.
  • Rod wants to investigate what effect adding a
    second technician to answer hotline calls has on
  • the number of calls receiving busy signals
  • the average time callers must wait before
    receiving service.

20
Implementing the Model
  • See file Q.xls

21
Summary of ResultsBitway Computers With Finite
Queue
  • Arrival rate 5 5
  • Service rate 7 7
  • Number of servers 1 2
  • Maximum queue length 5 5
  • Utilization 68.43 35.69
  • P(0), probability that the system is
    empty 0.3157 0.4739
  • Lq, expected queue length 1.0820 0.1019
  • L, expected number in system 1.7664 0.8157
  • Wq, expected time in queue 0.2259 0.0204
  • W, expected total time in system 0.3687 0.1633
  • Probability that a customer waits 0.6843 0.1877
  • Probability that a customer balks 0.0419 0.0007

22
The M/M/s Model With Finite Population
  • Assumptions
  • There are s servers.
  • There are N potential customers in the arrival
    population.
  • The arrival pattern of each customer follows a
    Poisson distribution with a mean arrival rate of
    l per time period.
  • Each server provides service at an average rate
    of m per time period, and actual service times
    follow an exponential distribution.
  • Arrivals wait in a single FIFO queue and are
    serviced by the first available server.

23
M/M/s With Finite Population Example The Miller
Manufacturing Company
  • Miller Manufacturing owns 10 identical machines
    that produce colored nylon thread for the
    textile industry.
  • Machine breakdowns follow a Poisson distribution
    with an average of 0.01 breakdowns per operating
    hour per machine.
  • The company loses 100 each hour a machine is
    down.
  • The company employs one technician to fix these
    machines.
  • Service times to repair the machines are
    exponentially distributed with an avg of 8 hours
    per repair. (So service is performed at a rate of
    1/8 machines per hour.)
  • Management wants to analyze the impact of adding
    another service technician on the average time
    to fix a machine.
  • Service technicians are paid 20 per hour.

24
Implementing the Model
  • See file Q.xls

25
Summary of Results Miller Manufacturing
  • Arrival rate 0.01 0.01 0.01
  • Service rate 0.125 0.125 0.125
  • Number of servers 1 2 3
  • Population size 10 10 10
  • Utilization 67.80 36.76 24.67
  • P(0), probability that the system is
    empty 0.3220 0.4517 0.4623
  • Lq, expected queue length 0.8463 0.0761 0.0074
  • L, expected number in system 1.5244 0.8112
    0.7476
  • Wq, expected time in queue 9.9856 0.8282 0.0799
  • W, expected total time in system 17.986 8.8282
    8.0799
  • Probability that a customer waits 0.6780 0.1869
    0.0347
  • Hourly cost of service technicians 20.00 40.00
    60.00
  • Hourly cost of inoperable machines 152.44
    81.12 74.76
  • Total hourly costs 172.44 121.12 134.76

26
The M/G/1 Model
  • Not all service times can be modeled accurately
    using the Exponential distribution.
  • Examples
  • Changing oil in a car
  • Getting an eye exam
  • Getting a hair cut
  • M/G/1 Model Assumptions
  • Arrivals follow a Poisson distribution with mean
    l.
  • Service times follow any distribution with mean m
    and standard deviation s.
  • There is a single server.

27
An M/G/1 Example Zippy Lube
  • Zippy-Lube is a drive-through automotive oil
    change business that operates 10 hours a day, 6
    days a week.
  • The profit margin on an oil change at Zippy-Lube
    is 15.
  • Cars arrive at the Zippy-Lube oil change center
    following a Poisson distribution at an average
    rate of 3.5 cars per hour.
  • The average service time per car is 15 minutes
    (or 0.25 hours) with a standard deviation of 2
    minutes (or 0.0333 hours).
  • Continued

28
Zippy Lube (continued)
  • A new automated oil dispensing device costs
    5,000.
  • The manufacturer's representative claims this
    device will reduce the average service time by 3
    minutes per car. (Currently, employees manually
    open and pour individual cans of oil.)
  • The owner wants to analyze the impact the new
    automated device would have on his business and
    determine the pay back period for this device.

29
Implementing the Model
  • See file Q.xls

30
Summary of Results Zippy Lube
  • Arrival rate 3.5 3.5 4.371
  • Average service TIME 0.25 0.2 0.2
  • Standard dev. of service time 0.0333 0.0333 0.333
  • Utilization 87.5 70.0 87.41
  • P(0), probability that the system is
    empty 0.1250 0.3000 0.1259
  • Lq, expected queue length 3.1168 0.8393 3.1198
  • L, expected number in system 3.9918 1.5393 3.9939
  • Wq, expected time in queue 0.8905 0.2398 0.7138
  • W, expected total time in system 1.1405 0.4398 0.9
    138

31
Payback Period Calculation
  • Increase in
  • Arrivals per hour 0.871
  • Profit per hour 13.06
  • Profit per day 130.61
  • Profit per week 783.63
  • Cost of Machine 5,000
  • Payback Period 6.381 weeks

32
The M/D/1 Model
  • Service times may not be random in some queuing
    systems.
  • Examples
  • In manufacturing, the time to machine an item
    might be exactly 10 seconds per piece.
  • An automatic car wash might spend exactly the
    same amount of time on each car it services.
  • The M/D/1 model can be used in these types of
    situations where the service times are
    deterministic (not random).
  • The results for an M/D/1 model can be obtained
    using the M/G/1 model by setting the standard
    deviation of the service time to 0 ( s 0).

33
Simulating Queues
  • The queuing formulas used in Q.xls describe the
    steady-state operations of the various queuing
    systems.
  • Simulation is often used to analyze more complex
    queuing systems.
  • See file Fig13-21.xls

34
End of Chapter 13
About PowerShow.com