## The Central Limit Theorem Applied to Networks

Take notice that the critical path through the network always connects the beginning node or milestone and the ending node or milestone. The ending milestone can be thought of as the output milestone, and all the tasks in between are input to the final output milestone. Furthermore, if the project manager has used three-point estimates for the task durations, then the duration of any single task is a random variable best represented by the expected value of the task. [2] The total duration of the critical path from the input or beginning milestone to the output milestone, itself a 0-duration event, or the date assigned to the output milestone, represents the length of the overall schedule. The length of the overall schedule is a summation of random variables and is itself a random variable, L, of length:

L = S Di = (Di + D2 + Di ...) where Di are the durations of the tasks along the critical path.

We know from our discussion of the Central Limit Theorem that for a "large" number of durations in the sum the distribution of L will tend to be Normal regardless of the distributions of the individual tasks. This statement is precisely the case if all the distributions are the same for each task, but even if some are not, then the statement is so close to approximately true that it matters little to the project manager that L may not be exactly Normal distributed. Figure 7-8 illustrates this point.

EETA Dislrbuiian

EETA Dislrbuiian

I OiilrithiliCfl iOr MikS Of IHG '.VCS

I OiilrithiliCfl iOr MikS Of IHG '.VCS

Figure 7-8: The Output Milestone Distribution.