Serial and within-stage independent parallel model equivalence on the minimum completion time

A set of functional equations is investigated that, when satisfied, yields equivalence on the minimum completion time between serial models and parallel models having the property of within-stage independence. Within these classes of models, it is shown that any parallel model is equivalent to a serial model, but not any serial model is equivalent to a parallel model. The necessary and sufficient conditions for a serial model to have an equivalent parallel counterpart are observed and then two sufficient conditions on the survivor functions that produce this result are exhibited. A number of examples satisfying the various theorems are discussed and a special case leading to an extension of a property of exponential distributions is derived.