Total variate distributions for feedback and recycling networks

Abstract

In the first part, a general closed activity network formula is inverted by complex integration methods to obtain exact total variate distributions assuming independent additivity of individual activity variates such as time or cost. Inversions are made assuming networks with independent gamma distributed branch activities with equal second parameters. The generalized W-function, a weighted characteristic function, as introduced by users of GERT, Graphical Evaluation and Review Technique, is the network function used in the characteristic function inversion to obtain the total variate distribution. Four cases of gamma branch W-functions are considered and exact general distribution formulas are derived under certain restrictions. Specific examples are presented for each case, and resultant cumulative total variate densities are tabulated as a function of distribution parameters. Specific examples for the first part are also solved for the total variate distribution by using a conditional total variate distribution argument solved by ordinary differential equation techniques whenever possible. Approximate total time distributions for percentage points in one of the basic examples are given assuming normal and then gamma distributed network activities. Approximate cost per unit time distributions are derived assuming χ²--distributed variates for approximate percentage point calculations. In the last part, total time distribution for single and multiple recycling purification processes are determined for given initial and desired purities. Examples are presented for χ² and normally distributed recrystallization and liquid-liquid extraction situations. Discrete distributions of fractional purity for single recycling systems are derived for given stopping times and initial and desired purities. Gamma and normally distributed activities are assumed in the derivation of fractional purity density formulas.