Diffusion approximations and nearly optimal maintenance policies for system breakdown and repair problems

Abstract

We consider the problem of service system deterioration and maintenance when there are only small statistical differences in the quality (or time required) of each produced item when the system is in the different states of deterioration, but where those marginal differences are economically important. This is somewhat analogous to the situation in the modeling of queues in heavy traffic, where the main effects which are dealt with might also be considered to be “marginal” ones. In our case the production of each item takes a random length of time and the deterioration during any production or sampling period can have a fairly general (and state dependent) statistical relation with this time and with the quality of the production. Due to this generality, there are several continuous parameter interpolations (of the sequence of conditional probabilities of the system states, given the observed data) which are appropriate for purposes of the weak convergence, each with its own advantages. (We can work with the “natural time scales” of the deterioration process, or with that of the sampling process, or with something in between.) The diffusion process limit is obtained when the random sequences (time, quality) are appropriately correlated. The limit process is of the form of a filtering problem for white-noise corrupted observations of a function of a Markov chain, but the limit problem is somewhat nonstandard since the effective noise covariance and the signal part of the effective observation might depend on the current conditional probability, due to the nature of the “scaling.”