Calculating the theoretical project completion time of large networks in polynomial time

One of the most important theoretical problems in project management is to obtain the distribution of the total completion time in PERT networks. For practical and managerial purposes what matters is the criticality of each activity within a PERT network, which can be assessed using a sound approach to calculate the completion time. Critical activities are activities that if delayed would delay the entire project. A sequence of critical activities throughout the network is called a critical path. The critical path is the longest path in the network and it is possible to have more than one critical path at once. But unlike CPM, in stochastic activity networks the duration time of individual activities varies and so activities are critical for some combinations of duration times and may not be critical for other combinations. Therefore, activities have a given probability of being critical (i.e. being part of the longest path). We call this probability criticality. The focus of this paper is to describe an analytical method for calculating the theoretical expectation of the project completion time as well as the criticality index of each activity