A Monte Carlo Technique for Obtaining System Reliability Confidence Limits from Component Test Data

A digital computer technique is developed, using a Monte Carlo simulation based on common probability models, with which component test data may be translated into approximate system reliability limits at any confidence level. The probability distributions from which the component failures are assumed to come are the exponential, Weibull (shape parameter K known), gamma (shape parameter ? known), normal, and lognormal. The components can be arranged in any system configuration, series, parallel, or both. Since reliability prediction is meaningful only when expressed with an associated confidence leve, this method provides a valuable and economical tool for the reliability analyst.     

A Monte-Carlo Technique for Estimating Lower Confidence Limits on System Reliability Using Pass-Fail Data

Several methods for estimating lower s-confidence limits (LCLs) for system reliability were examined using pass-fail data on the components. A new technique was used to obtain limits for selected systems. These limits were compared to those obtained by other methods. The new method was tested to verify its accuracy. Results indicate that the proposed technique is simple to understand, easy to implement, and accurate. One need only supply the component reliabilities, the number of component tests, and the desired level of s-confidence, to obtain, not only an estimated LCL of the system reliability, but also an idea of the accuracy of the estimate. Most of the other techniques are not valid in the case of zero-failures, whereas this method easily accommodates such a situation. This method is not restricted to series systems; it can easily handle parallel configurations.     

Approximate Tolerance Limits and Confidence Limits on Reliability for the Gamma Distribution

No exact method is known for determining tolerance limits or s-confidence limits for reliability for the gamma distribution when both parameters are unknown. Perhaps the simplest approximate method is to determine a tolerance limit assuming the shape parameter known and then replace the shape parameter with its ML estimate to obtain approximate limits. Simulated values of the true probability levels, achieved by this method, indicate that this method is not suitable, contrary to what has been anticipated. A second approach is to consider the corresponding tolerance limits assuming the distribution mean known and the shape parameter unknown, and then replace the distribution mean by the sample mean. This approach gives useful results for many practical cases. Simulated values of the true probability levels achieved are presented for some typical cases and limiting values are provided. This method appears satisfactory for all values of the shape parameter, for the common s-confidence levels, and moderate sample sizes.     

Bayesian Confidence Limits for the Reliability of Cascade Exponential Subsystems

The problem treated here is that of deriving exact Bayesian confidence intervals for the reliability of a cascade system consisting of N independent subsystems each having an exponential distribution of life with a failure rate which is estimated from life test data. The posterior probability density function of the system reliability is derived in closed form, using the method of the Mellin integral transform. The posterior distribution function is obtained, yielding Bayesian confidence limits on the total system reliability. These results, which are believed to be new for N > 3, have an immediate application to problems of reliability evaluation and test planning.     

A Comparison of Three Methods for Calculating Lower Confidence Limits on System Reliability Using Binomial Component Data

The Maximus, bootstrap, and Bayes methods can be useful in calculating lower s-confidence limits on system reliability using binomial component test data. The bootstrap and Bayes methods use Monte Carlo simulation, while the Maximus method is closed-form. The Bayes method is based on noninformative component prior distributions. The three methods are compared by means of Monte Carlo simulation using 20 simple through moderately complex examples. The simulation was generally restricted to the region of high reliability components. Sample coverages and average interval lengths are both used as performance measures. In addition to insights regarding the adequacy and desirability of each method, the comparison reveals the following regions of superior performance: 1. The Maximus method is generally superior for: a) moderate to large series systems of reliable components with small quantities of test data per component, and b) small series systems of repeated components. 2. The bootstrap method is generally superior for highly reliable and redundant systems. 3. The Bayes method is generally superior for: a) moderate to large series systems of reliable components with moderate to large numbers of component tests, and b) small series systems of reliable non-repeated components.     

Sequential Destruction Method for Monte Carlo Evaluation of System Reliability

Circumstances favoring the use of Monte Carlo methods for evaluating the reliability of large systems are discussed. A new method, that of Sequential Destruction (SD) is introduced. The SD method requires no preparatory topological analysis of the system, and remains viable when element failure probabilities are small. It applies to a variety of reliability measures and does not require element failures to be s-independent. The method can be used to improve the performance of selective sampling techniques. Substantial variance reductions, as well as computational savings, are demonstrated using a sample system with more than 100 elements.     

Confidence Limits for Redundant-System Availability

s-Confidence limits are established for the Availability of a standby redundant system (1-out-of-N:G system) for both hot and cold spares consisting of several identical units and repair facilities. The failure and repair rates of the units are s-independent, constant, and estimated from test data.     

Bayesian Confidence Limits for the Reliability of Mixed Exponential and Distribution-Free Cascade Subsystems

The problem treated here is the theoretical one of deriving exact Bayesian confidence intervals for the reliability of a system consisting of some independent cascade subsystems with exponential failure probability density functions (pdf) mixed with other independent cascade subsystems whose failure pdf's are unknown. The Mellin integral transform is used to derive the posterior pdf of the system reliability. The posterior cumulative distribution function (cdf) is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the derived pdf. A computer program written in Fortran IV is operational. It utilizes multiprecision to obtain the posterior pdf to any desired degree of accuracy in both functional and tabular form. The posterior cdf is tabulated at any desired increments to any required degree of accuracy.     

Sequential Testing of Components for System Reliability

It is often desirable to construct s-confidence limits for system reliability on the basis of data obtained from `pass-fail' tests on the components of the system. This paper presents a general method for sequentially testing the components that provides data from which these s-confidence limits can be easily derived. The method is applicable to any s-coherent system for which the reliability function is known. It is a generalization of a scheme given by Winterbottom and Verrall for systems composed of units arranged either in series or parallel.     

Bayesian Confidence Limits for the Reliability of Mixed Cascade and Parallel Independent Exponential Subsystems

This paper deals with the theoretical problem of derving Bayesian confidence intervals for the reliability of a system consisting of both cascade and parallel subsystems where each subsystem is independent and has an exponential failure probability density function (pdf). This approach is applicable when test data are available for each individual subsystem and not for the enfire system. The Mellin integral transform is used to analyze the system in a step-by-step procedure until the posterior pdf of the system reliability is obtained. The posterior cumulative distribution function is then obtained in the usual manner by integrating the pdf, which serves the dual purpose of yielding system reliability confidence limits while at the same time providing a check on the accuracy of the derived pdf. A computer program has been written in FORTRAN IV to evaluate the confidence limits. An example is presented which uses the computer program.     

Lower Confidence Limits and a Test of Hypotheses for System Availability

Any estimate of system availability calculated from time-to-failure and time-to-repair test data will be subject to some degree of uncertainty due to the uncertainty associated with the sample estimates of MTTF and MTTR. Although decisions about the true availability of the system should take this uncertainty into account, a point estimate of availability is usually the only statistic calculated. This paper derives techniques for placing a lower confidence limit on system availability and for deciding if the true system availability differs significantly from a specified value when MTTF and MTTR are estimated from test data. These techniques could be used to analyze existing test data or to design a test program for demonstrating system availability and/or detecting significant deviations from specified values of system availability. To facilitate utilization of these techniques, curves of the lower 0.90 and 0.95 confidence limits and power curves for the test of hypotheses at the 0.05 and 0.10 levels of significance are presented. Examples illustrating the use of the curves are given.     

Approximate Confidence Intervals for Reliability of a Series System

Two solutions are proposed for estimating s-confidence intervals for reliability of a repairable series system comprised of non-constant failure rate components: 1) the system is treated as a sum of renewal processes with the mean and variance of total number of system failures being computed from the moments of failure times of the components; and 2) a pseudo-Bayesian solution is derived for the mean and variance of the log-reliability of a system of Weibull components. In both solution approaches, the central limit theorem is invoked for a sum of component random variables determined from test data such as number of failures or log-reliabilities. s-Confidence limits are then approximated using Gaussian probability tables. The intervals derived yield close-to-exact frequency limits, depending on such variables as number of test failures, number of components, and component parameters.     

A Method for Computing Complex System Reliability

The computation of reliability becomes quite tedious when one has to deal with a non series-parallel system. In this paper a proposed method is developed by taking the system as a probabilistic graph in which a component of the system is represented by a branch. The proposed method is composed of three phases: Phase 1 involves the reduction of all series, parallel, and series-parallel components to an irreducible non series-parallel system. In Phase 2 the algorithm enumerates all possible paths from the source to the sink of the graph. Phase 3 then computes the system reliability based on the path information obtained in Phase 2. An example of the use of the method to compute system reliability is given.     

A Direct Method for Maximizing the System Reliability

A simple computational procedure has been developed for allocating redundancy among subsystems so as to achieve maximum reliability of a multistage system subject to multiple constraints which need not be linear. The computational time is quite short. Two examples are shown.     

A Confidence Interval for the Barlow-Scheuer Reliability Growth Model

Barlow & Scheuer proposed a useful scheme for estimating reliability growth of a system undergoing developmental testing and offered a conservative lower s-confidence bound. This paper shows how a less conservative lower s-confidence bound can be found by using an equivalent model for system reliability.     

Tables for Exact Lower Confidence Limits for Reliability and Quantiles, Based on Least-Squares Estimators of Weibull Parameters

For the 2-parameter Weibull distribution, this paper gives tables to obtain exact lower s-confidence limits for reliability on the basis of the least-squares method and median plotting positions. The tables use a method based on the ancillary property of these estimators. They apply to samples of size N = 3(1)13, censored after the first m observations, m = 3(1)N. The same tables enable one to obtain lower s-confidence limits for population quantiles. The use of the tables is illustrated with a numerical example.     

Confidence Limits for Weibull Regression With Censored Data

The response variable in an experiment follows a 2-parameter Weibull distribution having a scale parameter that varies inversely with a power of a deterministic, externally controlled, variable generically termed a stress. The shape parameter is invariant with stress. A numerical scheme is given for solving a pair of nonlinear simultaneous equations for the maximum likelihood (ML) estimates of the common shape parameter and the stress-life exponent. Interval and median unbiased point estimates for the shape parameter, stress-life exponent and a specified percentile at any stress, are expressed in terms of percentage points of the sampling distributions of pivotal functions of the ML estimates. A numerical example is given.     

Confidence Intervals for Reliability From Stress-Strength Relationships

A new distribution-free procedure obtains s-confidence intervals for the reliability in the stress-strength model. Based on the coverage probability and average-length criteria, a simulation study compared the procedure with other methods. Generally the proposed intervals perform best in maintaining nominal coverage probabilities.     

A Simple Method for Reliability Evaluation of a Communication System

Very few techniques exist for reliability evaluation of communication systems where links as well as nodes have certain probability of failure. This correspondence describes a technique by which the reliability expression for such a system can be conveniently derived. It is also shown that using the concept of this correspondence, it is possible to extend all the existing reliability-evaluation algorithms to communication systems with little effort.