A General Model for the Reliability Analysis of Systems Under Various Preventive Maintenance Policies

Abstract

The problem treated is that of predicting the reliability characteristics of a complex system from data on individual components. A general model for systems maintained over a period of time is proposed, based on the idea that every system failure is induced by a component failure and corrected by the replacement of a single component. Moreover, it is assumed that components are sometimes replaced even when the system is operating correctly, in order to prevent unscheduled interruptions in operation. The assumptions which define the general model cover a number of different preventive maintenance policies, among them the following: (a) Block Changes: All components of a given type are replaced simultaneously, at times determined by a renewal process. (b) Individual Component Replacement on the Basis of Age: If a component reaches some given age without failing, it is preventively replaced. (c) System Check-Outs: If a component is used only intermittently and it fails while it is not being used, it does not induce a system failure until it is called into use. At regular intervals, those components which have failed without inducing system failure are located and replaced. (d) Marginal Testing: At regular intervals, a test is conducted to locate those components which are still operating satisfactorily but which are expected to fail in the near future. All components located by this test are replaced. It is assumed that preventive removals are regeneration points and that the performance of a component may be described by a distribution function $F(x: y)$, the probability that a component is removed by time $x$, given that it enters the system at $y$, where $x$ and $y$ are both measured from the time of the last preventive removal. $F(x: y)$ is the sum of $A(x: y)$ and $B(x: y)$, where $A(x: y)$ is the probability that the component is preventively removed by $x$ and $B(x: y)$ is the probability that the component induces a system failure by $x$. The integral equations which determine the following measures of system performance from $F(x: y), A(x: y)$, and $B(x: y)$ are developed: (1) the expected number of failures in a given time interval (2) the expected number of preventive removals in a given time interval (3) the reliability function; i.e., the probability of no failure in a given interval following a given system age. Results from Renewal Theory and the Theory of Regenerative Stochastic Processes, developed by W. L. Smith, are applied to the problem of exploring the asymptotic behavior of these quantities. Conditions sufficient for maintenance policies a, b, c, and d to meet the assumptions of the general model are precisely formulated, and the analysis necessary to derive $F(x: y), A(x: y)$, and $B(x: y)$ is carried out for each policy.