A simple recursive Markov chain model to determine the optimal replacement policies under general repairs

Abstract

A repairable system (machine) is subject to failure. At each failure epoch, a general repair is performed. Such repairs return the system to a working condition somewhere between ‘good-as-new’ (a perfect repair) and ‘bad-as-old’ (a minimal repair). An important question for such systems is at what time should a major overhaul be conducted to return the unit to a ‘good-as-new’ state. Two policies are investigated in this research: (1) overhaul at fixed intervals and (2) overall at variable intervals by conducting each overhaul on the first failure following a predetermined time. We show in this research that both of these policies can be analyzed using a simple recursive Markov model thereby permitting the best of these policies to be identified in any particular situation. The Markov model is quite general, permitting a wide variety of structures to be investigated. Scope and purpose Considerable research has been conducted on the issue of periodic replacement times for failing systems. Typically in such analysis, each random failure is assumed to be repaired minimally and the replacement (or major overhaul) is assumed to refresh the failure intensity of the system. More recently, there has been recognition that repairs often serve to improve the system by partially resetting its failure intensity. Such a repair effect is known as a general (or an imperfect) repair. Previous research has extended well-known renewal functions into generalized renewal functions in order to estimate the impact of these general repair processes. In this paper we propose an alternative approach based on Markov chains which yields a simpler, more flexible model.