Models of Hierarchal Replacement

Replacement problems can take a wide variety of forms. We discuss techniques which can be applied to a rather broad class of problems, including as special cases many practical situations. The models incorporate allowances for both-i) replacement from aging stocks, ii) hierarchal structures for replacement elements of different kinds. For the most part, each level of elements in the hierarchy has its own (common) life distribution (whether in stock or in service); it is shown how to modify the distribution in a straightforward way for effects of aging in stock. Of the very large range of possible replacement procedures, we select for special study the two extreme cases of continually increasing specialization (many for one) and continually increasing generality (one for many). A convenient and flexible notation is presented, and methods of using standard probabilistic techniques in deriving life distributions of the systems are described.