Random and Exogenous Variation

Random Sources of Variation

The random sources of variation are generated using pseudorandom number generators. The netWorks application provides a set of standard generators. These include

Exponential, with mean $1/\lambda$ and density function
$f(x)=\frac{1}{\lambda}\exp^{-x/\lambda}, { for } 0 \leq x, 0 \leq \lambda$
Nonhomogenous Poisson, with rate parameter $\lambda(t)$ and density function
$f(x)=\frac{1}{\lambda(t)}\exp^{-x/\lambda(t)}, { for } 0 \leq x, 0 \leq \lambda(t)$
where $\lambda(t)$ is cyclic and continuous for all $t \geq 0$ with
$\lambda^*\equiv \sup_{t \geq 0}\lambda(t)$
Gamma, with parameters $\lambda$ and n and density function
$f(x)=\frac{x^{n-1}\exp^{-x/\lambda}}{\lambda^n\Gamma(n)}, { for } 0 \lt x, 0 \lt \lambda, 0 \lt n$
Erlang, with parameters $\lambda$ and n and density function
$f(x)=\frac{x^{n-1}\exp^{-x/\lambda}}{\lambda^n\Gamma(n)}, { for } 0 \lt x, 0 \lt \lambda, n\epsilon Z^+$
Uniform, with parameter U and density function
$f(x)=\frac{1}U, { for } 0 \leq x, 0 \leq U$
IUniform, with parameter U and density function
$f(x)=\frac{1}{\lfloor U\rfloor}, { for } x \epsilon Z, 0 \leq U$
Deterministic, with parameter U and density function
f(x) = 1, for x = U

Each of these generators has a control panel for setting parameters. For example, if you select Exponential from the list box in Figure B.1 and then click the button, the exponential control panel in Figure B.2 is displayed. In this window, you set the inital seed value and the mean $1/\lambda$ .

Figure B.2: The Exponential Control Panel

The seed value for each distribution is initialized from a data set of seed values provided with the application. When a new distribution is created, it retrieves the next seed value from the seed data set. You can explicitly change this value if you want to use another seed for the random variable.

Note also that, by default, the Exponential mean in Figure B.2 is set to 1. You can change the mean with the slider. The slider has a range of 0 to 10 for this parameter. If 10 is not large enough for the mean, you can increase it by pointing the cursor at the slider and clicking the right mouse button. This displays a pop-up menu with the entry Attributes.

Selecting Attributes displays the window in Figure B.3.

Figure B.3: The Slider Attributes Window

In this window you can change various parameters of the slider including the minimum and maximum values. To change the maximum value that an Exponential mean can have, enter a maximum value for the Exponential mean slider and click the button. Now the slider can be moved to any value up to that maximum.

You can also change a parameter value by positioning the cursor over the field displaying the parameter value and clicking the left mouse button. You then enter the new value and press the Return/Enter key, and the parameter value is updated.

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