MS&E 223
Lecture Notes #4
Simulation
Input Distributions
Peter J. Haas
Spring Quarter 2009-10
Input Distributions
Ref
: Chapter 6 in Law and Kelton
To specify a simulation model for a discrete-event system, we need to define the distributions of the
clock-setting “input sequences” to the model.
Examples:
(1)
interarrival sequences
(2)
processing time sequences for a production system
(3)
interest rate sequence for a financial model
As a simplifying assumption, it is often assumed that the input sequences are i.i.d.
Even having made this
(major!) assumption, this leaves us with a couple of questions to be answered:
1.
What type of distribution should we use (e.g. gamma or Weibull)?
2.
Once we’ve settled on the type (or “family”) of probability distribution, what parameter values
should we use in our simulation (e.g. if we decide to use a gamma distribution, what values for
the scale parameter
λ
and shape parameter
α
should we use)?
Often, real-world historical data is available to guide us in making these decisions.
However, knowledge
of some probability theory can also help us here, particularly in answering the first question.
1.
Theoretical Justification for Normal Random Variables
Suppose that the quantity X that we are trying to model can be expressed as a sum of other random
variables as follows:
X
=
Y
1
+ ... + Y
n
.
In great generality, various versions of the central limit theorem assert that, for n large,
X
D
=
N(
μ
,
σ
2
)
where
μ
= E(X) and
σ
2
= var(X).
Note that the Y
i
’s need not be identically distributed, nor do they need
to be independent of one another. (They need to be “not too dependent” and “not too different”.)
Moral of the Story
: If X is the sum of a large number of other random quantities, then X can be
approximately modeled as a normal random variable.
2.
Theoretical Justification for Log-Normal Random Variables
Suppose that the quantity X can be expressed as a product of other random variables as follows:
X
=
Z
1
Z
2
⋅⋅⋅
Z
n
.
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