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The mean function of a repairable system that is subjected to an imperfect repair policy.

Article Excerpt
1. Introduction

Consider a repairable system which is maintained and repaired upon each failure. Let the counting process {N(t); t [greater than or equal to] 0} be the number of failures in the interval [0, t]. We refer to the process {N(t); t [greater than or equal to] 0} as the "system functioning process." Given that the repair time is negligible, an important figure for such system is the mean function M(t) which is the expected number of failures up to time t. If the probability of simultaneous failures is zero, then we refer to M(t) as the cumulative intensity function. For such cases, the intensity function for the process {N(t); t [greater than or equal to] 0} is simply M'(t) = d(M(t))/dt.

Assume that the system is monitored. As soon as it fails, a repair starts on it and when the repair is completed the system is reinstated to operation. Traditionally, M{t) is determined under various types of repair policies which differ in cost and the quality of repair. At one extreme is a renewal process corresponding to the perfect repair policy in which the repaired system is brought to the "good-as-new" state. At the other extreme is a non-homogeneous Poisson process corresponding to the minimal repair policy which leaves the revived system "as old as it was" immediately prior to failure. In the former case the effective age of the system is returned to zero, whereas in the latter case the effective age of the system is unchanged from the effective age at failure. Between these two extremes, there are a host of other intermediate repair policies that we can adopt as a compromise between saving resources and achieving high quality repair. See, for example, Barlow and Davis (1977), Barlow and Proschan (1981), Ascher and Feingold (1984), Kijima (1989), Ebrahimi (1993), Presnell et al. (1994), Singpurwalla (1995), Basu and Rigdon (2000), Guo et al (2000, 2001), Lindquist (2006) and many references cited there.

As is typical in many applications, failures of a repairable system over a specific period of time may depend on the behavior of several observable explanatory variables (covariates), and it is most prudent, whenever possible, to develop repair policies by using these covariates. Specifically, suppose that a system functioning process {N(t); t [greater than or equal to] 0} is determined by a finite number of covariates. Usually, there are two groups of covariates. The first group consists of covariates whose behaviors are independent of the repair policies. We call them uncontrollable covariates. The second group consists of covariates whose behaviors are dependent on the repair policies. We refer to these covariates as controllable covariates. For example, if [X.sub.1] (t) and [X.sub.2] (t) are the stress and strength of a system at time t respectively, then the system will fail as soon as the system stress exceeds the strength. Therefore, the failure of the system depends on both [X.sub.1](t) and [X.sub.2](t). For this situation, it is clear that [X.sub.1](t) and [X.sub.2](t) are uncontrollable and controllable covariates respectively. In this paper we concentrate on controllable covariates. It should be noted that the models and methodologies proposed in this article do not incorporate any maintenance schedule or cost. The main aim of this paper is to develop a repair policy and to assess M(t) based on several controllable covariates. However, all the results and statistical tools presented in this paper can be extended to more than one covariate. To avoid complexity, our focus is on having only one controllable covariate.

In general, one must deal with three separate problems.

1. Choosing an appropriate stochastic model for the covariate.

2. Developing repair policies based on the covariate.

3. Assessing M(t), the mean function for a repair policy for a functioning repairable system N(t).

In analyses of repairable systems, assessing the mean function M(t) for a certain repair policy has been a major goal of many researchers and many methods and techniques have been published to enable this goal to be achieved. The most commonly used methodologies to assess the reliability of a repairable system, M(t), are based on experimental data on the failure times of either the case where a single repairable system is observed or the case where several systems of the same kind are observed by its manufacturer, see Basu and Rigdon (2000) and Lindquist (2006) for more details. However, because of the tremendous emphasis on quality improvement in industry driven by global competition and increasing customer expectations and because of pressures to limit development time and budgets, tests must be conducted under severe time constraints. Consequently, in many scientific investigations, experimental data is limited to a few failures. Thus, one of the great challenges in reliability engineering is to develop methods to assess M(t) within a reasonable time frame.

Our goal in this article is to explore the relationship between the covariate process (observable) and functioning system process {N(t); t [greater than or equal to] 0} (unobservable) and consequently assess M(t) based on the stochastic behavior of this covariate. In fact, in recent years advances in sensing and measurement technologies have allowed researchers to collect data on covariates over finer units of time that are related to the time to failure. Frequently, the data are a very rich source of information and offer many advantages over the...