One Last Graph

There is one other graph that is perhaps of interest. Let's assume that we can "add" the completion forces and the learning forces together. Force is a vector, so the assumption I am making here is that the completion forces and the learning forces are co-linear, or acting in the same direction, a somewhat arbitrary assumption. However, if I make this assumption, then I can sum the forces and get the total force on the project as a function of elapsed time, as Figure 12.10 shows.

Figure 12.10. Total forces on a project (learning + completion).

Let's look at some areas under the curves in the four regions.^[20] I note that all the segments have the same form, so let's look at the peak value of the total force and the length of time the segment is in play; the product of these two will be proportional to the area, as Table 12.2 shows.

^[20] Here I will consider all areas to be positive. That is, I take the absolute value of areas below the axis, so that there are no "negative" areas. If I didn't do this, the total area would integrate to zero.

Table 12.2. Product of Total Force Peak Value and Time Interval Length

Phase	Percent of Elapsed Time	Length of Time Interval	Peak Value of Total Force	Product (Impulse)
Inception	0 10	10	0.8	8
Elaboration	10 40	30	0.4	12
Construction	40 90	50	0.2	10
Transition	90 100	10	1.0	10

For the mathematically and physically inclined, we know that the area is an integral, and the integral of force over time is equal to the change in momentum.^[21] What we see from the table is that the change in momentum is roughly the same over the four phases. The change in momentum (or impulse) is the same during Construction and Transition10 units. The biggest impulse is during Elaboration, 12 units. And the combined impulse of the first and second phases20 unitsis equal to the combined impulse of the third and fourth phases. Half the total impulse is applied during the first 40 percent of the elapsed time; this again expresses the idea of "front loading," which we believe is a good thing.

^[21] More precisely, the integral of the force over time is the impulse, which is equal to the change in momentum.

Note also that the discontinuity in the total force curve at the 40 percent boundary is not too large. That is probably also a good indicator. So the conclusions drawn from using Kruchten's and Royce's numbers for learning and completion percentages in the four phases are consistent with the benefits we believe we see in iterative development projects.

While I have made many assumptions and pushed the model about as far as I am comfortable with, the results do appear to be both internally consistent and consistent with observed behavior.