Statistics for the Maximum of Several Positive Random Variables with Application to Networking

摘要

This report addresses a common problem in activity networks, popularly known as Program Evaluation and Review Technique (PERT) networks. This is the problem of calculating statistics associated with the maximum of several parallel, random activity times. Generally, one desires the mean, standard deviations, and quantiles of the distribution of the maximum of a set of positive, continuous, and independent random variables. An accurate recursive numerical algorithm for calculating these statistics is presented here. This method is somewhat more efficient than more straightforward numerical techniques. The method developed and used here is not restricted by having all of the random variables belong to the same distribution or even by having the same functional form. Pertinent, general formulas are derived as well as some closed-form results for a special case. Numerical examples are presented to assess the method's computational error and to illustrate certain quantitative generalizations. Although not limited to activity network applications, these results and the enclosed computer programs can be used to assess the estimation errors associated with deterministic networking methods such as PERT and CPM.