This research develops exact methods to calculate project duration distributions and to calculate Van Slyke's (1963) criticality for arrows, the probability that an arrow is on a critical path, assuming nonnegative integer duration distributions. These calculations for project duration distributions correct estimates made by the Program Evaluation and Review Technique (PERT), and the Van Slyke criticality calculations extend the arrow criticality analysis by the Critical Path Method (CPM) into the probabilistic realm;Exact methods for calculating project duration distributions and Van Slyke's criticality are demonstrated on series networks, parallel networks, parallel-series networks, and the Wheatstone network. The Van Slyke criticality equation for parallel networks is in a form that appears to improve upon one proposed by Dodin & Elmaghraby (1985). The present form is generalized to, in principle, include all networks;The exact methods are enhanced by developing a procedure to limit the number of calculations needed to analyze large networks. The procedure identifies paths through a large network, calculates the minimum and maximum path durations, and ranks the paths by duration. A smaller skeletal network is constructed from the arrows of the longest paths and is analyzed by exact methods. The procedure emphasizes accuracy for the longer project durations, of greatest concern to project managers and schedulers, while limiting the number of necessary calculations;The procedure for large networks is illustrated on the 40-arrow Kleindorfer (1971) network. Of the 51 Kleindorfer paths, the procedure selected 6 paths to construct a skeletal network. Analysis of the skeletal network yields a project duration distribution that is correct in its range and in the duration probabilities for the upper 5% of the distribution. Analysis results are compared with SLAM II and FORTRAN simulations. No arrow criticality appears to be seriously miscalculated. The project duration distribution is calculated to be bimodal, in keeping with the simulation;Conditions under which the just mentioned bimodality can occur are determined for parallel, normally-distributed paths. The large-network procedure warns when these oddly shaped distributions are possible.