Comment on: a hazard function approximation used in reliability theory

In a recent paper, Keren & Mehrez have shown that in the case of a geometric probability mass function, the approximation to the cumulative hazard function converges to a finite value (1.606695), when the actual value of the cumulative hazard function should be diverging. In this commentary, we show that gives incorrect results, and that the corresponding results for the approximation are actually diverging.     

A simple approximation to the renewal function [reliability theory]

The authors present a simple, easy-to-understand approximation to the renewal function that is easy to implement on a personal computer. The key idea is that, for small values of time, the renewal function is almost equal to the cumulative distribution function of the interrenewal time, whereas for larger values of time an asymptotic expansion depending only on the first and second moment of the interrenewal time can be used. The relative error is typically smaller than a few percent for Weibull interrenewal times. The simple approximation methods works very well with one term if not too much accuracy is required (e.g. in the block replacement problem) or if the interrenewal (failure) distribution is not exactly known (e.g. only the first two moments are known). Although the accuracy of the simple approximation can be improved by increasing the number of terms, this strategy is not advocated since speed and simplicity are lost. If high accuracy is required, it is better to use another approximating method (e.g. power series expansion or cubic splines method)     

Compound Poisson approximation in reliability theory

The use of the compound Poisson local formulation of Stein's method to obtain reliability bounds is discussed. In the context of the consecutive-k-out-of-n model, the approximations obtained are comparable with the best known for the system, which rely heavily on its special structure. In contrast, Stein's method for the compound Poisson approach can be applied equally effectively in many more-complicated models     

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.     

Bayesian analysis of reliability and hazard rate function of a mixture model

The present article deals with the Bayesian analysis for a modified (general) Mukherji-Islam model [A finite range distribution of failure times, Naval Research Logistics Quarterly, 1983, 30, 487–491] on the basis of failure time data x₁, x₂, …, x_r for a prefixed number of failures. Under the use of different prior densities for the parameter and the squared error loss function (SELF), the Bayesian estimates of hazard rate and the reliability functions are obtained, considering uniform, Gamma and inverted gamma densities as prior failure densities.     

布尔函数与图论结合在系统可靠性分析中的应用

提出了一种运用布尔函数与图论相结合，对系统进行可靠性分析的新方法。应用图论，可以表赢系统可靠性与各组成部件的相互关系；运用布尔函数，可以表现系统中特殊部件之间失效的相互影响作用：将这些关系和作用进一步地综合，就可以对整个系统进行分析。为说明该方法，结合具体事例进行了分析。     

Development of statistical theory in reliability engineering: A survey

Statistical theory and its applications have played a significant role in the advancement of methodology and techniques for solving various problems in reliability engineering. Different criteria of aging have been introduced for modelling and characterizations of wear properties. Some real life engineering reliability operations arising in maintenance and replacement policies formulations are investigated as convolution, mixture and formation of coherent systems. Shock models are studied for shocks arriving according to Poisson process and causing damages with specific form of aging. Cumulative damage models have been also studied by many researchers. Statistical hypotheses testing are used to test whether a certain, censored or uncensored data has exponential or a specific aging propety.In this paper we present a unified state of the art type of survey for the research and developments in these problems. In particular we expose the main recent contributions, trace the trend of research and highlight the possible direction of further research.     

Mixture models in profust reliability theory

Conventional (i.e. probist) reliability theory is based on the probability assumption and the binary-state assumption, whereas profust reliability theory is based on the probability assumption and the fuzzy-state assumption. The mixture model is an important form of typical systems in profust reliability theory. In this paper we show that a mixture model can be converted into a series system, and study the IFR and DFR preservabilities for mixture models. Various profust reliability bounds are given for the mixture model. In addition, we also show how to construct typical profust lifetime distributions for mixture models.     

Computational algebra applications in reliability theory

Reliability analysts are typically forced to choose between using an `algorithmic programming language' or a `reliability package' for analyzing their models and lifetime data. This paper shows that computational languages can be used to bridge the gap to combine the flexibility of a programming language with the ease of use of a package. Computational languages facilitate the development of new statistical techniques and are excellent teaching tools. This paper considers three diverse reliability problems that are handled easily with a computational algebra language: system reliability bounds; lifetime data analysis; and model selection     

A practical reliability metric for block codes used on binary-inputchannels

A reliability metric (RM) for a block code is defined to be a function that operates on both the decoder input (a block of channel output) and the decoder output (the codeword estimate) and produces a real number as a measure of the reliability of the decoder decision. The best RM has the disadvantage of depending on the codeword probabilities. Thus, the ideal RM is defined as the value that would be computed for equally likely codewords. The implementation of the ideal RM is too costly for most applications. The author proposes an easily implemented RM for binary-input memoryless channels (for state-observable channels with a freely evolving state such as some fading channels) when the codewords consist of n 2^m-ary symbols, each of which is transmitted serially by m uses of the channel. Simulation results for some BCH codes and some Reed-Solomon codes used in a simple ARQ system show that the proposed RM performs nearly as well as the ideal RM and much better than a previously proposed practical RM     

一种基于证据理论的可靠性信息融合方法

针对产品同一状态信息的融合问题,提出了基于证据理论的多个验后信息加权融合法。该方法在充分考虑多个验后分布一致度的基础上,结合D-S证据理论计算权重因子,再通过加权融合法的一般思路得到一个综合的验后分布形式,进一步提高了统计推断的精度和置信度,最后以实例计算验证了方法的有效性和可行性。     

The Bayes predictive approach in reliability theory

A strong motivation for reliability analyses is to support decision-making relative to the construction and operation of systems involving an economic/environmental risk. The Bayes approach to making decisions in face of uncertainty about mission survival is presented step by step. The authors show how the decision maker defines his own predictive probability distribution on the system time to failure and ranks the couples [decision taken, observed value of system failure time] by means of a loss function. They also introduce the minimum-expected-loss principle as a leading criterion for decision making. Finally, they address the more general case in which the final decision can be delayed in favor of collecting more information and derive the optimal termination procedure for life testing. For selecting the best course of action in a Bayes reliability frame there is neither need nor room for estimation of probability distribution parameters     

Polynomial approximation of Maliuzhinets' function

Approximate formulas for the Maliuzhinets functions which give the solutions for the wedge diffraction with face impedance, are derived using the Chebyshev polynomials for various wedge angles. The coefficients of the polynomials are listed in a table. Although the formulas are derived for real variables, it is found that they can be used for the restricted range of complex variables. Since the validity of the formulas depends on both real and imaginary part when applied to complex variables, the equi-error contour of the formulas is given for the special case where the exact solution is available. An alternative but empirical formula for Maliuzhinets' functions is also obtained, which can be applied to arbitrary wedge angles.     

A new technique for active minimal cut set selection used in substation reliability evaluation

This paper presents a new technique for the selection of active minimal cut sets in substation reliability evaluation. This technique is presented in this paper using a simple substation configuration. The developed technique can be used in the development of computer programs for substation reliability evaluation. This developed technique is further illustrated by application to other practical system examples.     

System reliability with common-cause hazard to obey an exponential power model

A model is proposed for the analysis of systems subject to common-cause failures that are not considered constant and assume that they obey an exponential power model. To represent common-cause failures, we use the Marshall, Olkin formulation of the multivariate exponential distribution. That is, the components are subject to failure by Poisson failure processes that govern simultaneous failure of a specific subset of the components. The method for calculating system reliability requires that a procedure exists for determining system reliability from component reliabilities under the statistically-independent component assumption. The paper includes examples to illustrate the method.