The exact overall time distribution of a project with uncertain task durations

This paper presents a new technique for computing the exact overall duration of a project, when task durations have independent distributions. A project is represented as an Activity-on-Arc (AoA) graph, where a task begins as soon as all its predecessor tasks have finished. Task durations use a probability density function (p.d.f.) which combines piecewise polynomial segments and Dirac delta functions, defined over a finite interval. A semi-analytical procedure is proposed to compute the cumulative distribution function (c.d.f.) directly by integrating a linear transformation of the p.d.f. of the task durations. Graph reduction techniques by Hopcroft and Tarjan and by Valdes allow the problem to be broken into a series of smaller subproblems, improving computational efficiency. Examples are presented to illustrate the proposed method.