Harold E. Ascher

Evaluation of Repairable System Reliability Using the ``Bad-As-Old'' Concept

It is usually assumed that the underlying distribution of times to failure of systems is the exponential distribution. This is justified on the basis of the bathtub curve or Drenicks theorem, but the bathtub curve is merely a statement of plausibility and conflicts with Drenicks theorem. Even if exponentiality is not assumed, it is usually assumed that a system under study is as-good-as-new after repair. This is not a plausible assumption to make for a complex system. If failure data are available they should be tested for trend among successive failure times. If a trend exists, a time dependent (nonhomogeneous) Poisson process (called bad-as-old model in this paper) should be fitted and tested for adequacy. This paper is not intended to provide a rigorous, definitive treatment of bad-as-old models. Rather, it has three main purposes: 1) to point out the glaring, but somehow usually overlooked, inconsistency between the commonly accepted concept of wearout of repairable systems and the a priori use of renewal processes for modeling these systems; 2) to outline basic procedures for evaluating data from repairable systems and for formulating bad-as-old probabilistic models; and 3) to present the results of Monte Carlo simulations, which illustrate the grossly misleading results which can occur if independence of successive failure times is invalidly assumed.

Spurious exponentiality observed when incorrectly fitting a distribution to nonstationary data

Failure data for a repairable system can be represented either by a set of chronologically ordered arrival times at which the system failed, or by a set of interarrival times defined as the times observed between successive failures (ignoring repair times in both cases). The two representations are mathematically equivalent if the chronological order of the interarrival times is maintained. Methods aimed at describing the distribution of the observed interarrival times are meaningful only if the interarrival times are identically distributed. In all other cases, such analyses are meaningless and often result in maximally misleading impressions about the system behavior, as demonstrated here by several examples. That is, when the information in the chronological order of interarrival times is ignored, they often appear spuriously exponential, leading to the impression that the system can be modeled using a homogeneous Poisson process. Misunderstandings of this nature can be avoided by applying an appropriate test for trend before attempting to fit a distribution to the interarrival times. If evidence of trend is determined, then a nonstationary model such as the nonhomogeneous Poisson process should be fitted using the chronologically ordered data.

Regression Analysis of Repairable Systems Reliability

The regression model for survival analysis introduced by Cox (1972) has had major impact on the biometry field. It is surprising, therefore, that Cox’s model, which is equally applicable to the reliability of nonrepairable items, has almost never been applied to such problems. This model has recently been extended to the analysis of multiple events, such as recurring infections experienced by a single subject, by Prentice, Williams and Peterson (1981), (PWP 1981). Some modifications to the PWP model are useful for optimum application to reliability problems but basically their model applies to repairable systems by simply replacing words like “infections” with “repairable failures.”

A set-of-numbers is NOT a data-set

Evans (1997) and Rees (1997) have emphasized that great care is needed to obtain good data because, otherwise, garbage in leads to garbage out. This tutorial demonstrates that, even when one has good data, the results still are incorrect if data analysis is performed incorrectly. A central issue in correct statistical analysis is determining the context within which the data arose; and resolving the inherent ambiguities in interpreting failure-data makes it essential to incorporate such a context into reliability data analyses. When this is ignored, as is usually the case (a set-of-numbers is treated as if it were an entire data-set, thus ignoring other essential information), even good data in results in garbage out.

The Use of Regression Techniques for Matching Reliability Models to the Real World

Similar models are often used in various disciplines. For example, models for time to an event or for times between successive events are needed in biometry and sociology applications, as well as in reliability. The specific circumstances of a particular discipline may suggest a particular family of distribution functions, e.g., the Weibull distribution, when modeling time to an event. Alternatively, a specific point process, e.g. the power law process (a nonhomogeneous Poisson process of specific functional form, see Ascher and Feingold (1984)) may be appropriate in a particular reliability application dealing with times between successive failures of a repairable system. In a biometry application, in which times between successive nonfatal illnesses of a patient are studied, another point process might be suggested. In practice, however, instead of considering that models are suggested by circumstances, there is far too much reliance on a priori specification of models. For example, in hardware reliability applications it is usually assumed that the exponential distribution is the appropriate model to use, regardless of the application. If this model is generalized at all, the “generalization” usually is restricted to using a Weibull distribution. In fact, one or the other of these distributions is usually invoked even when no distribution whatsoever is the appropriate model! That is, when dealing with a repairable system—and most systems are designed to be repaired rather than replaced after failure—the correct model is a sequence of distribution functions, i.e., a point process. Distribution functions and point processes are not equivalent models, even in the most special cases. A homogeneous Poisson process (HPP) can be defined as a nonterminating sequence of independent and identically exponentially distributed times between events. Ascher and Feingold (1979, 1984) show that there are important distinctions between the exponential distribution and HPP models.

MIL-STD-781C: A Vicious Circle

Reliability practitioners have confused theorists about the true area of application of MIL-STD-781C. This has occurred because theorists have neglected to explain important basic concepts to practitioners. This paper analyzes the situation in detail.

Comment on: Sequential Tests of Hypotheses for System-Reliability Modeled by a 2-Parameter Weibull Distribution

A different point of view is presented on the simple to show that content of the original paper. K hx(x)= K K 1 (2) oK

Comments on: Reliability Discipline and Technology in Historical Perspective - 1930-1984

Too little attention is paid to accurate models for repairable systems.