Insufficiency of Instantaneous Strength Determinations for Failure Rate Prediction

The failure of an element subjected to a steady applied stress is associated with the time-wise deterioration of the element's strength. At the instant of failure, the element can no longer perform to specifications. The strength of the element can be obtained at a specified point in time by abruptly modifying the applied stress to a level which causes the element to fail. The instantaneous strengths determined by this procedure are investigated for the possibility of reducing the test time necessary in obtaining the failure rate associated with exponentially distributed failure times. A simple, physically plausible class of strength deterioration functions is considered and the distributional character of the instantaneous strengths at any time instant is determined. It is found that when the underlying distribution of time to failure is exponential, the instantaneous strengths have distributions which are invariant with time. This result indicates that no clue to the form of strength deterioration can be obtained by testing for the instantaneous strengths. Thus the conclusion is reached, in the case of the exponential failure distribution, that instantaneous destruction tests for strength determination are ineffective for assessing the underlying failure rate.     

A reliability physics model for parallel system

Reliability analysis of a nonrepairable 2-unit parallel system is carried out using the stress-strength model of failure physics. The analysis is carried out for correlated strengths and altered stress distribution depending upon the number of components surviving. The analysis includes both the cases of deterministic and random cycle times. In the case of random cycle times, Poisson distributed stress cycle occurrences have been considered. Various system characteristics, such as failure time distribution, reliability and moments of time to failure of the system, have been evaluated for both deterministic and random cycle times. Two particular cases, namely (i) bivariate exponential distribution for strength variables and univariate exponential distributions for stress variables and (ii) bivariate normal distribution for strength variables and univariate normal distributions for stress variables, have also been considered.     

Bayesian estimation of reliability function and hazard rate

The Bayes estimates of reliability function and hazard rate function of the finite range failure model have been developed based on life tests that are terminated at a preassigned time point or after a certain number of failures have occurred, taking the order of observations into consideration. For the prior distribution of the parameter involved, the uniform, exponential and inverted gamma densities have been considered. As an example, failure data for a V600 indicator tube used in aircraft radar sets, which fit well the finite range failure model, have been considered as the current distribution for obtaining the Bayes estimates of the reliability function.     

Availability analysis of a repairable standby human-machine system

This paper presents a model representing a two units active and one unit on standby human-machine system with general failed system repair time distribution. In addition, the model takes into consideration the occurrence of common-cause failures. The method of linear ordinary differential equation is presented to obtain general expressions for system steady state availability for failed system repair time distributions such as Gamma, Weibull, lognormal, exponential, and Rayleigh. Generalized expressions for system reliability, time-dependent availability, mean time to failure, and system variance of time to failure are also presented. Selected plots are presented to demonstrate the impact of human error on system steady state availability, reliability, time-dependent availability, and mean time to failure.     

Initial Provisioning of a Standby System with Deteriorating and Repairable Spares

A technique is developed for finding the time dependent operating probabilities used by reliability systems designers for provisioning a system with N + k identical units, k of which are called spares and N called operating units, and s repair facilities. System failure occurs when less than N units are operational. Units fail with exponential interfailure times and are repaired with exponential service time. Idle spares fail due to deterioration at a rate possibly different from that of the operating units. Graphs are presented which show the minimum numbers of spares needed to achieve system reliabilities of 0.90 and 0.99 as a function of time. The technique is applicable for finding, numerically, the first passage time distribution for any system modeled by birth and death processes.     

Preventive Replacement of a 1-Unit System with a Wearout State

This paper is concerned with a 1-unit system with 3 types of states: usual, wearout, and failure; the hazard rate is constant in the usual state and is monotonically increasing and unbounded in the wearout state. In order to deal with this system in detail, we introduce two models: model 1 in which the period of normal operation is fixed and model 2 in which it is a random variable with a negative exponential distribution function. The models have different preventive replacement policies based on both age and state. Moreover we give the conditions under which these policies are effective.     

Dynamic Reliability Model of Components Under Random Load

The dynamic reliability model of components is developed using order statistics, and probability differential equations. The relationship between reliability and time, and that between the hazard rate and time, are discussed in this paper. First, according to the statistical meaning of random load application, the cumulative distribution function, and probability density function of equivalent load, when random load is applied at multiple times, are derived. Further, the reliability model of components under repeated random load, is developed. Then, the loading process described under a Poisson process, the dynamic reliability model of components without strength degeneration, and that with strength degeneration are developed respectively. Finally, the reliability, and the hazard rate of components are discussed. The result shows that, when strength doesn't degenerate, the reliability of components decreases over time, and the hazard rate of components decreases over time, too. When strength degenerates, the reliability of components decreases over time more obviously, and the hazard rate curve is bathtub-shaped.      

Bayes estimation of the linear hazard-rate model

In life testing, the failure-time distributions are often specified by choosing an appropriate hazard-rate function. The class of life-time distribution characterized by a linear hazard-rate includes the one-parameter exponential and Rayleigh distributions. Usually the parameters of the linear hazard-rate model are estimated by the method of least squares. This work is concerned with Bayes estimation of the two-parameters from a type-2 censored sample. Monte Carlo simulation is used to compare the Bayes risk of the regression estimator with the minimum Bayes risk. Discrete mixtures of decreasing failure rate distributions are known to have decreasing failure rates. The authors prove that the result holds for continuous mixtures as well     

Bayes estimation of the piece-wise exponential distribution

A Bayes method to infer an unknown failure time distribution is presented. The method is based on the piecewise exponential distribution and a relationship between values of the failure rate in successive intervals; it provides smooth estimates of the survival and hazard functions of the distribution. This is accomplished in a model-based framework without resorting to smoothing procedures that require ad-hoc specification of parameters that have no bearing on the data. It is a useful procedure whenever the failure rate is anticipated to be reasonably continuous. Incorporating these beliefs into the model allows a more rational solution to the nonparametric estimation problem. The advantages are illustrated using a real data-set where a smooth estimate of the failure rate is obtained. The method can be used with any possibly-censored data-set and is easily implemented on a microcomputer. The Bayes solution is compared with the classical solution for the problem     

Hazard Rates and Generalized Beta Distributions

This paper considers the behavior of the hazard rates of the Generalized gamma, and beta of the first and second kind. The hazard functions include strictly decreasing, constant, strictly increasing, ? and ? shaped hazard rates. By considering the generalized distributions a unified development for such distributions as beta type 1, beta type 2, Burr types 3 and 12, power, Weibull, gamma, Lomax, Fisk, uniform, Rayleigh, and exponential are included as special cases. The results are conveniently summarized in three figures. The generalized distributions considered in this paper are seen to provide models for all of the different shaped hazard rates mentioned above. This flexibility permits the data to determine the nature of the hazard function without its being inadvertently imposed through the selection of an improper model. For example, the selection of a Weibull distribution permits a decreasing, constant, or increasing hazard rate, but not a ? or ? shaped one. The use of the generalized gamma or either of the generalized beta functions considered in section II does permit realization of these additional shapes for the hazard rate.     

Evaluation of Conditional Failure Density from Hazard Rate

Further to the recent discussion regarding the concepts of hazard rate and conditional failure density, it is shown that the latter may be expressed in terms of the former. Moreover, six functions, namely, failure time distribution function, failure time density, reliability function, hazard rate, conditional failure distribution, and conditional failure density, are shown to be equivalent to the extent that if one of them is known, the other five are completely determined. The results are summarized in a table.     

The failure rate function estimated from parameter drift measurements

A theoretical model for prediction of the component drift failure rate as a function of time from component parameter drift rates is described. The model assumes statistical independence of the initial value and the drift of the parameter. To use the model it is necessary to know the distribution of the initial value of the component parameter, the component parameter drift function and the distribution of the functional parameters. Further the concept of “component working lifetimes” is discussed. Two different definitions are suggested, both based on the assumptions of a Weibull distribution for the wear-out lifetimes and an exponential distribution to give the earlier “constant” failure rate.     

Estimation for a probabilistic stress-strength model

Estimation for an unknown strength distribution is considered in two situations: (1) independent identically distributed stresses are applied to the component until it fails (no cumulative damage); and (2) each applied stress causes damage to the component and damage cumulates until the component fails. Both situations lead to mixtures of probability distributions, with the strength distribution playing the role of the mixing distribution. Based on the observation of cycles to failure of several independent components and on the theory of mixtures of distributions, estimators of the mixing distributions are obtained using linear programming. In particular, the solution to the linear-programming problem yields a probability mass function which approximates the unknown strength distribution. From this estimate of the strength distribution, an estimate of the mean strength of the item can be obtained by the usual computation of the mean of a probability distribution. Hence, the results provide a method of estimating the mean strength, or other parameters of the strength distribution, without requiring observations directly on the strengths of the test components     

Optimum life cycle and labour costs estimation of repairable equipment

This paper presents a mathematical model to estimate optimum life cycle cost of an equipment with known mean time to failure and mean time to repair. In addition, several equipment labour cost estimation models are presented when mean time to failure and mean time to repair follow distributions such as exponential, Rayleigh, Weibull and gamma.     

Stochastic behavior of a two-unit cold standby redundant system subject to random failure

The present paper deals with a stochastic model of a two-unit cold standby redundant system subject to random failure. The random failure occurs at random times which follow an exponential distribution. Using a regenerative point technique in the Markov-renewal process, several reliability characteristics are obtained. The mean time to system failure function is studied graphically.     

The power function distribution: A useful and simple distribution to assess electrical component reliability

Statistical distributions have long been employed in the assessment of semiconductor device and product reliability. The use of the exponential distribution which is frequently preferred over mathematically more complex distributions, such as the Weibull and the lognormal among others, suggests that most engineers favour the application of simpler models to obtain failure rates and reliability figures quickly. It is therefore proposed that the power function distribution be considered as a simple alternative which, in some circumstances, may exhibit a better fit for failure data and provide more appropriate information about reliability and hazard rates. The statistical analyses of data obtained from two types of destructive tests demonstrate the applicability of the power function distribution.     

Optimum simple step-stress accelerated life-tests with competingcauses of failure

Optimum simple step-stress accelerated life tests (ALTs) for products with competing causes of failure are presented. The life distribution of each failure cause, which is independent of the others, is assumed to be exponential with a mean that is a log-linear function of the stress, and a cumulative exposure model is assumed. Optimum plans for time-step and failure-step ALTs are obtained which minimize the sum over all failure causes of asymptotic variances of the maximum likelihood estimators of the log mean lives at design stress. The competing causes of failure affect the optimum test plan only through the product of two ratios-the ratio of the sums of the mean lives and the ratio of the sums of the failure rates over all failure causes at low and high stress levels. The effect of this product (of two ratios) is studied     

Failure Time for a Redundant Repairable System of Two Dissimilar Elements

The distribution of time to failure for a system consisting of two dissimilar elements or subsystems operating redundantly and susceptible to repair is discussed. It is assumed that the times to failure for the two system elements are independent random variables from possibly different exponential distributions, and that the repair times peculiar to each element are independently distributed in an arbitrary fashion. For this basic model a derivation is given of the Laplace-Stieltjes transform of the distribution function of time to system failure, i.e, the time until both elements are simultaneously down for repair, measured from an instant at which both are operating. An explicit formula is given for the mean or expected time to system failure, a natural approximation to the latter is exhibited, and numerical comparisons indicate the quality of this approximation for various repair time distributions. In a second model the possibility of system failures due to overloading the remaining element after a single element failure is explicitly recognized. The assumptions made for the basic model are augmented by a stochastic process describing the random occurrence of overloads. Numerical examples are given. Finally, it is shown how the above models may be easily modified to account for delays in initiating repairs resulting from only occasional system surveillance, and to account for random catastrophic failures.     

A 2-level environmental-stress-screening (ESS) model: amixed-distribution approach

Environmental stress screening (ESS) is used to reduce, if not eliminate, the occurrence of some types of failures from the field by fixing them before the product is deployed. This paper models a 2-level ESS program where screening is performed at the part and unit levels. The parts are screened for a specified duration before being assembled into a unit. Defects induced during the assembly process are screened at the unit level. These parts and connections are assumed to come from either a good or a substandard population, and their times-to-failure distributions are modeled by mixed distributions. The optimal screening durations are obtained by minimizing the life-cycle cost. The model is simple to use and its viability is illustrated using mixed exponential distributions. The implementation of screens at various levels depends on costs and failure distribution characteristics