## Merging Dependent Paths

So far the discussion has been about the merging of independent paths. Setting the condition of independence certainly simplifies matters greatly. We know from our study of multiple random variables that if the random variables are not independent, then there is a covariance between them and a degree of correlation. We also know that if one path outcome is conditioned on the other path's outcome, then we must invoke Bayes' Theorem to handle the conditions.

p(A on time AND B on time) = p(A on time given B is on time) * p(B is on time)

We see right away on the right side of the equation that the "probability of A on time given B on time" is made smaller by the multiplication of the probability of "B on time." From what we have already discussed, making "probability of A on time given B on time" smaller is a shift right of the schedule. The exact amount is more difficult to estimate because of having to estimate "probability of A on time given B on time," but the heuristic is untouched: the milestone join of two paths, whether independent or not, will shift right. Only the degree of shift depends on the conditions between the paths.

Thinking of the milestone as an "outcome milestone," we can also approach this examination from the point of view of risk as represented by the variance of the outcome distribution. We may not know this distribution outright, although by the discussion that follows we might assume that it is somewhat Normal. In any event, if the two paths are not independent, what can we say about the variance of the outcome distribution at this point? Will it be larger or smaller?

We reason as follows: The outcome performance of the milestone is a combined effect of all the paths joining. If we look at the expected value of the two paths joining, and the paths are independent, then we know that:

E(A and B) = E(A) * E(B), paths independent But if the paths are not independent, then the covariance between them comes into play:

E(A and B) = E(A) * E(B) + Cov(A and B) where paths A and B are not independent.

The equation for the paths not independent tells us that the expected value may be larger

(longer) or smaller (shorter) depending on the covariance. The covariance will be positive — that is, the outcome will be larger (the expected value of the outcome at the milestone is longer or shifted right) — if both paths move in the same direction together. This is often the case in projects because of the causes within the project that tend to create dependence between paths. The chief cause is shared resources, whether the shared resource is key staff, special tools or environments, or other unique and scarce resources. So the equation we are discussing is valuable heuristically even if we often do not have the information to evaluate it numerically. The heuristic most interesting to project managers is:

Parallel paths that become correlated by sharing resources stretch the schedule!

The observation that sharing resources will stretch the schedule is not news to most project managers. Either through experience or exposure to the various rules of thumb of project management, such a phenomenon is generally known. What we have done is given the rule of thumb a statistical foundation and given the project manager the opportunity, by means of the formula, to figure out the actual numerical impact. However, perhaps the most important way to make use of this discussion is by interpreting results from a Monte Carlo simulation. When running the Monte Carlo simulation, the effects of merge points and shift right will be very apparent. This discussion provides the framework to understand and interpret the results provided.