Stock-Flow diagrams

[gallaher@teleport.com (Ed Gallah]

Paul Kucera presented a simple stock-flow diagram.

Here is an expanded version:



> Stock-Flow the representation of system levels (states or accumulations) as
> boxes with the pipes containing valves used to show the things
> that change the levels.
> ___________
> | |
> Source ----X----->| Level |-----X-----> Sink
> inflow |___________| outflow
> ^ ^
| |
/ r ________/ Drain



(This only works in a fixed pitch font).
Needless to say the pictures can be made to look more elegant.


Consider water flowing in and out of a barrel.

The source is considered "exogenous" to the model. That is, when we turn
on the faucet, water will alway come out. If this is a factor in the
function of the model, then the model would need to be expanded to take
this into account.

The sink is similar. After the water leaves the barrel, it is irrelevant
to the model. This would not be the case if the barrel were in your
THIS IS SPAM and the water was running out onto the floor.

The information link from Level to Outflow indicates that Level (i.e.
height of water, or pressure head) affects the rate of outlow. Outflow in
turn affects the level of water, creating a "causal loop", or "feedback"
loop, or "circular causal structure".

In addition, this model indicates that outflow will also be a function of
the size of the drain hole.

A very powerful feature of this approach is that the model shown above be
applied almost directly to (a) intravenous drug infusion, drug metabolism,
and therapeutic levels, (b) radioactive decay, (c) a cooling cup of coffee,
(d) mRNA and protein synthesis, decay, and steady state levels, (e) other.

Once you have developed this model and played with it, conducted
sensitivity analyses with each of the parameters, and so on, you will know
a whole lot about -all- these dynamics systems. You also find that when
you see data which describe the performance of some system over time (e.g.
cooling cup of coffee), you can begin to "guess" what the underlying
structure might be. It starts becoming "intuitive".

This reminds me of a friend who was complimented on his guitar playing at a
party. After thanking the person, Pete said under his breath to me, "Yeah,
Ive been playing 2 hours a day for 20 years, and it gets attributed to
talent!"

The point here is that dynamic behavior begins to become intuitive after
many, many hours of running simple models and critically -thinking- about
why the system behaves as it does. I have found this to be the hardest
point to get across to beginners. They build a model and run a few
simulations, and think they understand it.

In the above model, one can alter inputs from very low (100 ml/min) to very
high (20 l/min). Also, inputs could be pulses (buckets; small or large;
widely spaced, or more frequent). Or inputs could include both a water
faucet -and- buckets. Or the water faucet could be turned on low for a
while, then high, then low, then high.

If we add a bucket, the water declines exponentially until the barrel is
empty (more correctly approaches an asymptote). If we add a second
bucket before the first is gone, we see an accumulation. How might the
water accumulate over time? Does it increase indefinitely? Or does it
eventually reach a stable oscillation? How high is the peak and how low
the trough? How long does it take to reach this stable oscillation?

Some of these questions are not too relevant to water barrels, but they are
very relevant to drug administration regimes.

The output could be linear (minor modification of the structure above),
simulating an output pump. Or the drain could be large, or small. Or the
drain could be large and gradually become plugged with debris. (Or the
liver could begin to fail so that drug elimination gradually declines).

This will be trivial to the experienced, of course, but not so trivial to
newcomers.

Ed Gallaher
gallaher@teleport.com