Probabilistic PERT

A solution is offered to the problem of determining a probability distribution for the length of the longest path from source (start) to sink (finish) in an arbitrary PERT network (directed acyclic graph), as well as determining associated probabilities that the various paths are critical (“bottleneck probabilities”). It is assumed that the durations of delays encountered at a node are random variables having known but arbitrary probability distributions with finite expected values. The solution offered is, in a certain sense, a worst-case bound over all possible joint distributions of delays for given marginal distributions for delays. This research was motivated by the engineering problem of the timing analysis of computer hardware logic block graphs where randomness in circuit delay is associated with manufacturing variations. The probability distribution of the critical pathlength turns out to be a solution of an unconstrained minimization problem, which can be recast as a convex programming problem with linear constraints. The probability that a given path is critical turns out to be the Lagrange multiplier associated with the constraint determined by the path. The discrete version of the problem can be solved numerically by means of various parametric linear programming formulations, in particular by one which is efficiently solved by Fulkerson's network flow algorithm for project cost curves.