Renewal Decisions when Category Life Distributions are of Phase-Type

Abstract
A system must operate for t units of time. A certain component is essential for the operation of the system and must be replaced by a new component whenever it fails. There are n types of replacement categories available (with an infinite supply of each) differing only in price and life distribution. The main problem is to select the proper category for replacement at any time a failure occurs, so as to minimize the total expected cost of running the system. In this paper the problem is studied when category life distributions have a common matrix phase type representation. The generalized Erlang and the hyperexponential distributions, as well as coherent structures of the latter distributions, are some special cases of this representation. Our main result is that in many of these cases, the problem of finding an optimal replacement policy is reduced to that of determining a specified number (at most n − 1) of points on the real time axis. Also provided is a condition for identifying categories which should never be used, thus reducing the problem by eliminating them. This work generalizes previous results obtained when all category life distributions were assumed exponential. Several illustrative examples are provided.

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