Discrete Event Simulation Ch. 1

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• Output Data Analysis - Single System.

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We have a working simulator, generating data.
What do we do with the data? First Possibility
set it up with "typical input conditions" and
"typical parameters" - all of which we more or
less know how to do (by now) - and let it go. At
the end of the run, if the results look good,
recommend the system (or system changes) if the
results look bad, send the system designers back
to the drawing board. This is where some
research papers stop... Second Possibility you
remember that just about all games of chance
favor the house. If we all really believed that
as a deterministic statement, nobody would gamble
- yet, many of us gamble because we believe that,
somehow, the odds favor at least an occasional
"turn of luck", and that we might be there to
benefit. This is another way of saying that a
single run means absolutely nothing...
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We will have to carry out a number of runs, and
we will have to be careful about not making too
many unwarranted assumptions about
independence. Let Y1, Y2, , Ym be the variables
obtained from a single run of length m. These
could be the number of cars arrived at a gasoline
station during the j-th hour, 1 j m the
number of people passed through customs during
the j-th minute the numebr of breadloaves baked
during the j-th hour the Geiger counter clicks
during the j-th second, etc One of the problems
with these variables is that they are neither
independent nor identically distributed - we
should expect them to be highly correlated (a
busy hour is usually followed or preceded by a
busy hour a high reading on the Geiger counter
is usually followed and preceded by a high
reading etc.). This means that any of our
previous results on confidence intervals for the
mean (or any other statistic) cannot be applied
because they required independence.
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We need some methods that will give us IID
samples run the simulation n times, using
different random seeds of the input variable
generators, but with the same initial conditions.
We obtain a table of values y11, y12, , y1i,
, y1m y21, y22, , y2i, , y2m
yn1, yn2, , yni, , ynm where uji will
denote the i-th random number in the j-th run.
Although the problems with the row values are the
same as before, the i-th column values should be
IID observations of the random variable Yi. We
will use this idea of independence across runs to
derive some data analysis methods for example
is an unbiased estimate of E(Yi).
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Example
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One of the first problems we have to deal with is
that of determining the difference between
transient behavior and steady-state
(time-independent) behavior. Since the system
starts "empty", it will take some time before it
reaches any kind of "normal" (read steady-state)
behavior. During this period of time our results
will be, in some sense at least,
"unrepresentative". Let I denote the initial
conditions of the system.. For the output
stochastic process, let Fi(yI) P(Yi yI),
for i 1, 2, , the probability that the event
Yi y occurs given the initial condition I.
Fi(yI) is called the transient distribution of
the output process at time i for the initial
conditions I. Fi(yI) will, in general, be
different for each i and for each set of initial
conditions I. The next slide shows some possible
distributions for several values of i.
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If Fi(yI) -gt F(y) as i -gt 8, for all i and any
initial conditions I, the function F(y) is called
the steady-state distribution of the output
process Y1, Y2, This steady-state distribution
is attained only in the limit, but we can usually
identify an integer k such that the distribution
functions of Yk, Yk1, are (and remain)
sufficiently close so that, for our purposes,
they can be replaced by a single distribution,
the limit one. Furthermore the random variables
will not be independent but will approximately
constitute a covariance-stationary stochastic
process. Example
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Example
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From the examples (and theoretical
considerations) the steady-state distribution
does not depend on the initial conditions,
although the rate of convergence of the
distributions of the Yi's does.
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Types of Simulations with Regard to Output
Analysis.