Algorithms to calculate the distribution of the longest path length of a stochastic activity network with continuous activity durations

Activity networks are used to plan projects by showing precedence relationships between the various activities that constitute such projects [1]. An activity network is a special case of a directed graph in which the nodes (or vertices) represent points in time and the arcs (or edges) represent activities with time values modeling activity durations. Figure 1 shows F1 an example of a stochastic activity network, where the positive random variable Yij denotes the time to complete the activity associated with the arc from node i to node j. The Yijs are assumed to be mutually independent. Common-cause delays, for example, rain delays on an outdoor project, are not considered here. Activity start times are constrained in that no activity emanating from a given node can start until all activities that enter that node have been completed. We consider three techniques for calculating the distribution of the time to complete an activity network: simulation, series-parallel reduction, and conditioning. In some cases we are also able to calculate the probability that a path through the network