# Project Management and Scheduling

The expected time for an activity represents the average time it would take if the activity is performed over and over again. It is known empirically that the probability density function of activity duration closely follows a Beta distribution, which defines the following relationship (Heldman &amp. Baca, 267)
Standard Deviation is used for calculating the variability associated with the high degree of uncertainty in estimated time durations.&nbsp.
The time analysis elements such as the forward pass (Early Start time ES and Early Finish time EF) and the Backward Pass (Late Start time LS and Late Finish time LF) are used to find the Critical Path. These times are calculated using the Expected Time (te) for the respective activity relative to the zero date (date when the project clock starts ticking) of the project.

The Backward Pass Late Start (LS) and Late Finish (LF) values are calculated considering that the earliest completion time of the project for the last activity and then working backward towards the predecessors. For all the last activities, the LF will be equal to the respective EF value. Therefore, values of LF for the activities O, N, M, I and C are equal to the respective values of EF.

The Critical Path can be determined by finding the Total Slack for each activity. Total Slack for any activity is the maximum time by which the start of an activity can be delayed without affecting the critical path.

The path with the longest Expected time duration is the critical path. Activities on the critical path have slack as 0 since these activities cannot be delayed at all. Each path from the Start to the Finish node is defined along with the respective time duration. The Expected Completion Time (Et) for each path is equal to the expected duration of the constituent activities. For Example, A-E-H-K-I is calculated as: Et (A-E-H-K-I) = [te (A) + te (E) + te (H) + te (K) + te (I)] = 2.500 + 6.166 +2.833 + 2.000 + 2.500 = 15.999