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The theory of practice indicates there should be
QSM's SLIM-Estimate tool (see www.qsm.com) employs a reciprocal fourth-power formula that relates effort and duration of a project. That is, the relationship is of the form:
Effort ~ Time-4
This generates a curve which looks like this:
At very short time cost doesn't just go up, it goes up a lot. While if we have the latitude to take our time on a project we can get the same system built for a lot less. I developed a qualitative explanation for this in the article Real Work, Necessary Friction, Optional Chaos.
Limits of Theory
Other researchers (particularly Barry Boehm in his COCOMO I and somewhat in his COCOMO II models) have indicated that the ends of the curve, rather than extending out forever, "curl over." Anecdotally, Fred Brooks said this in The Mythical Man Month all those years ago: that adding people to a project (ie., increasing the effort and cost) can make the project take longer.
I posit a couple of other functions that operate alongside the Time-4 curve but which only appear when time is very small or very large. The composite curve would look like this:
The Other Theory
In this article, I hypothesize that the "curled" function is the sum of three discrete functions (though they may not really be "discrete" as in operate independently). They would look like this:
I furthermore suggested that the Compression Curve might have the form:
Effort ~ log(Time)
And that the Relaxation Curve might have the form:
Effort ~ Time * exp(Time)
Almost as the mirror image of the Compression Curve.
Testing the Hypothesis
We at QSM are working to find data points in our 10,000 project database toward these extremes and to find out if these fomulae approximate what happens. Watch this space.