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Computer Simulation. The Essence of Computer Simulation. A stochastic system is a system that evolves over time according to one or more probability distributions.

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The essence of computer simulation
The Essence of Computer Simulation

  • A stochastic system is a system that evolves over time according to one or more probability distributions.

  • Computer simulation imitates the operation of such a system by using the corresponding probability distributions to randomly generate the various events that occur in the system.

  • Rather than literally operating a physical system, the computer just records the occurrences of the simulated events and the resulting performance of the system.

  • Computer simulation is typically used when the stochastic system ivolved is too complex to be analyzed satisfactorily by analytical models.

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Example 1 a coin flipping game
Example 1: A Coin-Flipping Game

  • Rules of the game:

    • Each play of the game involves repeatedly flipping an unbiased coin until the difference between the number of heads and tails tossed is three.

    • To play the game, you are required to pay $1 for each flip of the coin. You are not allowed to quit during the play of a game.

    • You receive $8 at the end of each play of the game.

  • Examples:

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Computer simulation of coin flipping game
Computer Simulation of Coin-Flipping Game

  • A computer cannot flip coins. Instead it generates a sequence of random numbers.

  • A number is a random number between 0 and 1 if it has been generated in such a way that every possible number within the interval has an equal chance of occurring.

  • An easy way to generate random numbers is to use the RAND() function in Excel.

  • To simulate the flip of a coin, let half the possible random numbers correspond to heads and the other half to tails.

    • 0.0000 to 0.4999 correspond to heads.

    • 0.5000 to 0.9999 correspond to tails.

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Freddie the newsboy
Freddie the Newsboy

  • Freddie runs a newsstand in a prominent downtown location of a major city.

  • Freddie sells a variety of newspapers and magazines. The most expensive of the newspapers is the Financial Journal.

  • Cost data for the Financial Journal:

    • Freddie pays $1.50 per copy delivered.

    • Freddie charges $2.50 per copy.

    • Freddie’s refund is $0.50 per unsold copy.

  • Sales data for the Financial Journal:

    • Freddie sells anywhere between 40 and 70 copies a day.

    • The frequency of the numbers between 40 and 70 are roughly equal.

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Application of crystal ball
Application of Crystal Ball

  • Four steps must be taken to use Crystal Ball on a spreadsheet model:

    • Define the random input cells.

    • Define the output cells to forecast.

    • Set the run preferences.

    • Run the simulation.

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Step 1 define the random input cells
Step 1: Define the Random Input Cells

  • A random input cell is an input cell that has a random value.

  • An assumed probability distribution must be entered into the cell rather than a single number.

  • Crystal Ball refers to each such random input cell as an assumption cell.

  • Procedure to define an assumption cell:

    • Select the cell by clicking on it.

    • If the cell does not already contain a value, enter any number into the cell.

    • Click on the Define Assumption button.

    • Select a probability distribution from the Distribution Gallery.

    • Click OK to bring up the dialogue box for the selected distribution.

    • Use the dialogue box to enter parameters for the distribution (preferably referring to cells on the spreadsheet that contain these parameters).

    • Click on OK.

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Step 2 define the output cells to forecast
Step 2: Define the Output Cells to Forecast

  • Crystal Ball refers to the output of a computer simulation as a forecast, since it is forecasting the underlying probability distribution when it is in operation.

  • Each output cell that is being used to forecast a measure of performance is referred to as a forecast cell.

  • Procedure for defining a forecast cell:

    • Select the cell.

    • Click on the Define Forecast button on the Crystal Ball tab or toolbar, which brings up the Define Forecast dialogue box.

    • This dialogue box can be used to define a name and (optionally) units for the forecast cell.

    • Click on OK.

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Step 3 set the run preferences
Step 3: Set the Run Preferences

  • Setting run preferences refers to such things as choosing the number of trials to run and deciding on other options regarding how to perform the simulation.

  • This step begins by clicking on the Run Preferences button on the Crystal Ball tab or toolbar.

  • The Run Preferences dialogue box has five tabs to set various types of options.

  • The Trials tab allows you to specify the maximum number of trials to run for the computer simulation.

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Step 4 run the simulation
Step #4: Run the Simulation

  • To begin running the simulation, click on the Start Simulation button.

  • Once started, a forecast window displays the results of the computer simulation as it runs.

  • The following can be obtained by choosing the corresponding option under the View menu in the forecast window display:

    • Frequency chart

    • Statistics table

    • Percentiles table

    • Cumulative chart

    • Reverse cumulative chart

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Choosing the right distribution
Choosing the Right Distribution

  • A continuous distribution is used if any values are possible, including both integer and fractional numbers, over the entire range of possible values.

  • A discrete distribution is used if only certain specific values (e.g., only some integer values) are possible.

  • However, if the only possible values are integer numbers over a relatively broad range, a continuous distribution may be used as an approximation by rounding any fractional value to the nearest integer.

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A popular central tendency distribution normal
A Popular Central-Tendency Distribution: Normal

  • Some value most likely (the mean)

  • Values close to mean more likely

  • Symmetric (as likely above as below mean)

  • Extreme values possible, but rare

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The uniform distribution
The Uniform Distribution

  • Fixed minimum and maximum value

  • All values equally likely

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The discrete uniform distribution
The Discrete Uniform Distribution

  • Fixed minimum and maximum value

  • All integer values equally likely

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A distribution for random events exponential
A Distribution for Random Events: Exponential

  • Widely used to describe time between random events (e.g., time between arrivals)

  • Events are independent

  • Rate = average number of events per unit time (e.g., arrivals per hour)

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Yes no distribution
Yes-No Distribution

  • Describes whether an event occurs or not

  • Two possible outcomes: 1 (Yes) or 0 (No)

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Distribution for number of times an event occurs binomial
Distribution for Number of Times an Event Occurs: Binomial

  • Describes number of times an event occurs in a fixed number of trials (e.g., number of heads in 10 flips of a coin)

  • For each trial, only two outcomes are possible

  • Trials independent

  • Probability remains the same for each trial

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