David Petersons comments are right on. Here are a few supplemental thoughts:
One of Forresters experiments with delay order is in Appendix H of
Industrial Dynamics (available from Productivity Press, and a must-read).
It compares the effect of 1st-, 3rd-, and infinite-order (pipeline) delays
in the full ID model. Appendix E (Smoothing of Information) and Appendix F
(Noise) are also somewhat relevant to the question.
I find myself wanting to ask, "why normal?" The distribution cant be truly
normal, because that would imply some of the material would have a negative
arrival time, since the normal distribution is unbounded. Thus the left
tail of the arrival distribution must be truncated. Perhaps the data
actually fit an exponential or other distribution equally well anyway.
It might help to explore more deeply why the process behaves as it does.
For example, the observed behavior might be the result of two sequential
steps in a process, one of long duration and zero variance, and one of
short duration that adds all the variance.
Then you could justify tacking a short-duration Nth order delay onto the
end of a long-duration pipeline delay.
Another possibility that comes to mind - the opposite extreme of using a
3rd order delay as an approximation - is to use an infinite order delay
(pipeline, conveyor in ithink, DELAY_FIXED or DELAY MATERIAL in Vensim,
etc.). Since the variance is so narrow, this might be good enough.
There are also several tricky ways to proceed. If your time step is really
1, you could model the flow by separating it into several discrete
components with different (infinite-order) time constants, i.e.:
Outflow(t) = a*Inflow(t-tau-3)+b*Inflow(t-tau-2)
+c*Inflow(t-tau-1)+d*Inflow(t-tau)
+ ...
a, b, c, d, etc. are weights describing the shape of your distribution. You
could approximate the normal distribution with relatively few equations
this way, but it wont work if you change your time step to be smaller than
1, because the outflow given a pulse inflow will then look like a series of
spikes rather than a smooth curve. There are further tricks to avoid this
problem, but they get complicated.
One point about using pipeline delays is that while they make the equations
look simple, they are still computationally complex. A pipeline delay is
really a string of first order delays of duration equal to the simulation
time step. Thus if you have a delay of length 20 and a time step of .25,
you actually are using 80 internal levels.
The most general approach would be to write an external function that
behaves exactly as desired. This is fairly easy, but again hides some
computational complexity that might not be needed anyway.
Regards,
Tom
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Thomas Fiddaman, Ph.D.
Ventana Systems
http://www.vensim.com
34025 Mann Road Tel (360) 793-0903
Sultan, WA 98294 Fax (360) 793-2911
Tom@Vensim.com http://home.earthlink.net/~tomfid/
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