## Calculating Activity limes

Figure 8-13: The complete network from Table 8-1.

figure 8-14: Distribution of all i possible activity times for an activity figure 8-14: Distribution of all i possible activity times for an activity

The expected time, TE, is found by

where a = optimistic time estimate, b = pessimistic time estimate, and m = most likely time estimate, the mode.

Note in Table 8-1 that some activity durations are known with certainty, whi is to say that a, b, and m are the same (see activity b, for instance). Note further tha the most likely time may be the same as the optimistic time (a = m) as in activity e, or that the most likely time may be identical to the pessimistic time (m = b) as:i| activity a. The range about m may be symmetric where m - a = b — m as in activity c, or may be quite asymmetric, as in activities h and j.

The above formula for calculating expected times is usually said to be based q| the beta statistical distribution.* This distribution is used rather than the mor| common normal distribution because it is highly flexible in form and can take in| account such extremes as where a = m or b = m.

TE is an estimate of the mean of the distribution. It is a weighted average ofj m, and b with weights of 1-4-1, respectively. Again, we emphasize that this sai| method can be applied to finding the expected level of resource usage given the appropriate estimate of the modal resource level as well as optimistic and pessimistic estimates.

Recently, Sasieni noted (48| that writers (including himself) have been usig the formula used here to estimate TE. He pointed out that it could not be derive| from the formula for the beta distribution without several assumptions that wejj^ not necessarily reasonable, and he wondered about the original source of the fatt, mula. Fortunately, for two generations of writers on the subject, Littlefield a(K|r Randolph |27) cited a U.S. Navy paper that derives the approximations used her\$ and states the not unreasonable assumptions on which they are based. Gallaghel |16| makes a second derivation of the formula using a slightly different set of af sumptions.

•We remind readers who wish a short refresher on elementary statistics and probability that available in Appendix E at the end of this book.