Frank Proschan

Theoretical explanation of observed decreasing failure rate

Pooled data on the times of successive failures of the air conditioning system of a fleet of jet airplanes seemed to indicate that the life distribution had a decreasing failure rate. More refined analysis showed that the failure distribution for each airplane separately was exponential, but with a different failure rate. Using the theorem that a mixture of distributions each having a non-increasing failure rate (e.g., a mixture of exponential distributions) itself has a non-increasing failure rate, the apparent decreasing failure rate of the pooled air-conditioning life distribution was satisfactorily explained. The present study has implications in other areas where an observed decreasing failure rate may well be the result of mixing exponential distributions having different parameters.

Periodic Replacement with Increasing Minimal Repair Costs at Failure

When an expensive one unit system fails or breaks down, it is often more practical to perform “minimal repair” than to make a replacement or perform a complete overhaul. Instead replacements or complete overhauls are made periodically at fixed multiples of some predetermined time T . In this paper we treat a model for this minimal repair-periodic replacement policy, and consider the problems of determining: (1) the period T which minimizes the total expected cost of repair and replacement over a fixed time horizon [0, t ), and (2) the period T which minimizes the total expected cost per unit time over an infinite time horizon.

12 Mean residual life: Theory and applications

Publisher Summary This chapter discusses the theory and applications of the mean residual life (MRL). MRL has been used as far back as the third century A.D. The MRL function is like the density function, the moment generating function, or the characteristic function: for a distribution with a finite mean, the MRL completely determines the distribution via an inversion formula. Not only is the MRL used for parametric modeling but also for nonparametric modeling. Large nonparametric classes of life distributions such as decreasing mean residual life (DMRL) and new better than used in expectation (NBUE) have been defined using MRL. Actuaries apply MRL to setting rates and benefits for life insurance. In the biomedical setting researchers analyze survivorship studies by MRL. IMRL distributions have been found useful as models in the social sciences for the lifelengths of wars and strikes. MRL functions occur naturally in other areas such as optimal disposal of an asset, renewal theory, dynamic programming, and branching processes. The chapter defines more formally the MRL function and surveys some of the key theory.

APPLICATIONS OF THE HAZARD RATE ORDERING IN RELIABILITY AND ORDER STATISTICS

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector (T, - .., T,) to the vector (T1,. ? ., T,), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards (A1r(t), A2r(t)), the more diverse (A1, 2) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k-out-of-n system. It is shown that if TkIn is the lifetime of a k-out-of-n system, then Tkl n is greater than Tk +ln in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T1, - * *, Tn, we let Tk:n represent the kth order statistic (in increasing order). Then it is shown that Tk +1: is greater than Tk:n in the hazard rate ordering for all k = 1, - - -, n - 1. The result does not, however, extend to the stronger likelihood ratio order.

Testing to Determine the Underlying Distribution Using Randomly Censored Data.

Abstract : For right-censored data, a goodness-of-fit procedure is developed for testing whether the underlying distribution is a specified functions G. The test statistic C is the one-sample limit of Efrons (1967) two-sample statistic W. The test based on C is compared with recently proposed competitors due to Koziol and Green (1976) and Hyde (1977). The comparisons are on the basis of applicability, the extent to which the censoring distribution can affect the inference, and power. It is shown that in certain situations the C test compares favourably with the tests of Koziol-Green and Hyde.

LAPLACE ORDERING AND ITS APPLICATIONS

Two arbitrary life distributions F and G can be ordered with respect to their Laplace transforms. We say F is Laplace-smaller than C if foJ e-sF(t)dt 0. Interpretations of this ordering concept in reliability, operations research, and economics are described. General preservation properties are presented. Using these preservation results we derive useful inequalities and discuss their applications to M/G/1 queues, time series, coherent systems, shock models and cumulative damage models.

Stochastic order for redundancy allocations in series and parallel systems

The problem of where to allocate a redundant component in a system in order to optimize the lifetime of a system is an important problem in reliability theory which also poses many interesting questions in mathematical statistics. We consider both active redundancy and standby redundancy, and investigate the problem of where to allocate a spare in a system in order to stochastically optimize the lifetime of the resulting system. Extensive results are obtained in particular for series and parallel systems.

27 Nonparametric concepts and methods in reliability

Publisher Summary This chapter reveals the nonparametric analysis in reliability and several classes of life distributions. Classes of life distributions are based on the notions of aging afford nonparametric statisticians an opportunity to consider problems of a character somewhat different from the usual. The chapter summarizes the definitions, physical interpretations, useful geometric characterizations, probabilistic properties of and logical relationships, and implication .among these classes of life distributions and formulates a variety of classes of life distributions based on notions of aging. The reliability operations considered are: (1) formation of coherent systems, (2) addition of independent life lengths (convolution of life distributions), (3) selection of a life length observation from one of a set of distributions (mixture of distributions), and (4) subjecting a device to shocks. The chapter also surveys nonparametric inference for these classes along with the discussion on the total time- on-test plots and empirical mean residual life functions.

Theory of Maintained Systems: Distribution of Time to First System Failure

The distribution of time to first system failure is considered for systems whose repairable components are separately maintained. It is shown that if repairable components have exponential failure law, repair distributions have decreasing repair rate, and nonrepairable components have increasing failure rate, then the distribution of time to first system failure is new better than used. Improved bounds are given for the exponential repair case.

Relationship Between System Failure Rate and Component Failure Rates

A simple sufficient condition is given for a system to have an increasing failure rate when the identical components comprising it have an increasing failure rate. Systems which function if and only if at least k of the n components function (“k out of n” systems) satisfy this condition. For systems of non-identical components, upper and lower bounds on failure rate are obtained in terms of component failure rates. These bounds are increasing functions of time for “k out of n” structures having components with increasing failure rates.