Showing papers in "IEEE Transactions on Reliability in 1972"
TL;DR: In this article, graphical methods are presented for analyzing accelerated life test data with the inverse power law model, when all test units are run to failure, and graphical methods for estimating its parameters from such complete data are given.
Abstract: In this paper, graphical methods are presented for analyzing accelerated life test data with the inverse power law model, when all test units are run to failure. The inverse power law model is described, and graphical methods for estimating its parameters from such complete data are given. These methods are illustrated with accelerated test data on time to breakdown of an insulating fluid. While the methods are presented with the inverse power law model, they can be used for analyzing many other accelerated life test situations. These methods are presented so that they can be used by individuals with a limited statistical background.
110 citations
TL;DR: A new way to transform the model of constrained optimization to a saddle point problem by using Lagrange multipliers and uses the Maximum principle to arrive at the optimal decision.
Abstract: A mathematical model is formulated for optimizing the reliability of a system subject to given linear constraints; the system has several stages in series; each stage has parallel redundancy to improve the reliability. Part I shows a new way to transform the model of constrained optimization to a saddle point problem by using Lagrange multipliers. Conditions are derived for maximizing the reliability function; Newton's method is used to solve the resulting multidimensional nonlinear algebraic equations. Further modifications are provided to avoid inverting the large Jacobian matrices; therefore this method is practical for large systems. Part II shows how to transform the model of constrained optimization to a multistage decision process and uses the Maximum principle to arrive at the optimal decision. This approach is easy to apply, formulate, and program. The solution can be obtained without fear of nonconvergence (very often experienced with earlier methods) besides providing considerable saving in computer time. Design alternatives can be easily considered.
94 citations
TL;DR: 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 and computes the system reliability based on the path information obtained in Phase 2.
Abstract: 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.
91 citations
TL;DR: In this paper, a simple algorithm for the solution of the redundancy optimization problem in the presence of multiple linear constraints is presented, where the solution to an r-constraint problem is obtained successively from the solutions of r-unconstrained problems.
Abstract: This paper provides a simple algorithm for the solution of the redundancy optimization problem in the presence of multiple linear constraints. Basicaily, the solution to an r-constraint problem is obtained successively from the solution of r-unconstrained problems. At each step, an active constraint is picked out, then the maximum gradient concept is used to find a closer point. The approach requires minimum effort and time for solution.
46 citations
TL;DR: A computer-aided procedure is described for analyzing the reliability of complicated networks, which breaks down a network into small subnetworks whose reliability can be more readily calculated.
Abstract: A computer-aided procedure is described for analyzing the reliability of complicated networks. This procedure breaks down a network into small subnetworks whose reliability can be more readily calculated. The subnetworks which are searched for are those with only two nodes; this allows the original network to be considerably simplified.
36 citations
TL;DR: In this article, an iterative procedure is given for joint maximum likelihood estimation of the three parameters of the Weibull population, where the likelihood function is forned and the maximum likelihood equations are obtained.
Abstract: An iterative procedure is given for joint maximum likelihood estimation of the three parameters of the Weibull population. For this population, the likelihood function is forned and the maximum likelihood equations are obtained. Simultaneous solution of these equations yields joint maximum likelihood estimators for the three parameters. The proposed iterative procedure is a modified quasilinearization algorithm for solving nonlinear equations with bounded variables. Numerical results are given in which the parameters are estimated from real and from simulated life test data drawn from the Weibull population.
35 citations
TL;DR: A multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss resulting from a particular choice of burn-in time and item replacement age is described.
Abstract: An important problem facing a manufacturer is the determination of the amount of time to burn-in items (in order to eliminate early failures) and the age at which to replace items (to avoid failures due to wearout) This problem becomes difficult to solve if the time-to-failure distribution of an item is unknown and must be estimated from test and operational data This paper describes a method of statistical data analysis which is readily applied to the solution of this decision problem under a realistic but general loss (or gain) function The method is a multiparameter Bayesian analysis which requires multiple integration of the (multivariate) posterior of the parameters of the time-to-failure distribution to obtain the expected loss (or gain) resulting from a particular choice of burn-in time and item replacement age This integration is performed by a Monte Carlo Procedure using importance sampling An example demonstrates the flexibility of this method of analysis The data are a mixture of ``point'' and truncated data, which often create difficulties when using conventional methods of decision analysis In addition, since the method permits up to ten parameters for the family of time-to-failure distributions, a ``bathtub'' hazard rate function is used to generate the data for the example The results are presented in the form of Bayesian confidence intervals for the true hazard rate function and a presentation of the expected loss as a function of burn-in time and age at replacement
32 citations
TL;DR: In this article, a method for applying bilinear transformations to the identification of faulty components in linear electronic circuits is presented, where simple magnitude and phase measurements at a number of test frequencies are made and plotted on a set of predetermined loci in the complex transfer function plane.
Abstract: A method has been developed for applying bilinear transformations to the identification of faulty components in linear electronic circuits. Simple magnitude and phase measurements, at a number of test frequencies, are made and plotted on a set of predetermined loci in the complex transfer-function plane. The data for plotting the loci are determined either experimentally or by circuit analysis with a digital computer. The faulty component and the parameter value are then determined from the loci. The method has the advantage that it provides a graphical representation of the circuit behavior with a faulty component and also readily allows experimental error to be taken into account when plotting the measured data. The method is demonstrated with a practical transistor amplifier circuit.
29 citations
TL;DR: In this paper, the Laplace-Stieltjes transform of the time distribution to the first system-down and the mean time to first systemdown were derived by applying the relationship between Markov renewal processes and signal flow graph, and numerical examples were presented for the k-Erlang failure time distributions.
Abstract: We consider a standby-redundant model of two units, where we assume that one unit is operative and the other unit is in standby at time t = 0. If the operative unit fails, a unit in standby is put into preventive maintenance policy of the operative unit to maintain our the preventive maintenance policy of the operative unit to maintain our system with high reliability. Our concern for the system is the time to the first system-down. The Laplace-Stieltjes transform of the time distribution to the first system-down and the mean time to the first system-down are derived by applying the relationship between Markov renewal processes and signal flow graph. Further, the behavior after the system-down is investigated by using the results of Markov renewal processes. Finally, numerical examples are presented for the k-Erlang failure time distributions. The optimal preventive maintenance time is discussed.
26 citations
TL;DR: In this article, the Lapace-Stieltjes transform of the distribution of the time to first system faflure (TFSF) and the mean TFSF were derived.
Abstract: A standby-redundant repairable system having only two units and an imperfect switchover is considered. The Lapace-Stieltjes transform of the distribution of the time to first system faflure (TFSF) and the mean TFSF are derived. Numerical comparisons of the mean TFSF with both perfect and imperfect switching are made. All time distributions, except repair of the main units, are exponential.
25 citations
TL;DR: In this paper, Monte Carlo simulations are used to investigate the maximum likelihood estimators for the parameters of the Weibull distribution using multicensored samples for the case of unequal number of items censored, a single situation is considered.
Abstract: Monte Carlo simulations are used to investigate the maximum likelihood estimators for the parameters of the Weibull distribution using multicensored samples. For the case of unequal number of items censored, a single situation is considered. When the same number of items are removed at each test, several different estimators are compared for several values of the parameters.
TL;DR: In this article, the authors describe methods of fitting prior distributions to equipment MTBF =?, shows the priors fitted to different equipments, establishes data criteria for fitting prior distribution to?, and presents the results of a robustness analysis performed on the fitted priors.
Abstract: This paper describes methods of fitting prior distributions to equipment MTBF = ?, shows the priors fitted to different equipments, establishes data criteria for fitting prior distribution to ?, and presents the results of a robustness analysis performed on the fitted priors. Systematic procedures for fitting priors are shown for Type 1 data (number of failures in fixed time T) and Type 2 data (observed MTBF, number of failures not the same for all equipments), and specific data criteria, in the form of minimum values of n (number of equipments) and K (number of failures) are presented. The inverted-gamma prior-distributions were derived from operational failure data obtained from Tinker AFB. The equipments are primarily electronic, therefore, the time-to-failure distribution was assumed to be exponential; however, the methods are generally applicable whatever the form of the conditional distribution. The robustness analysis shows the effects of errors in estimating the parameters of the prior on the posterior distribution. In general, the effect of errors in estimating parameters of the prior was practically negligible for large values of K.
TL;DR: This note presents a straightforward method for system reliability evaluation where probability is treated as a point in a Cartesian frame and formulas are derived to evaluate the reliability of series, parallel, series-parallel, and bridge networks.
Abstract: The evaluation of system reliability becomes very diffilcult when the number of components in a system is large or the system configuration is complex. This note presents a straightforward method for system reliability evaluation. Probability is treated as a point in a Cartesian frame and formulas are derived to evaluate the reliability of series, parallel, series-parallel, and bridge networks. The values obtained by the classical method agree closely with those obtained by this parametric method.
TL;DR: This tutorial paper presents the problem of optimal allocation of parallel redundancy under multiple linear constraints by the method of dynamic programming.
Abstract: This tutorial paper presents the problem of optimal allocation of parallel redundancy under multiple linear constraints by the method of dynamic programming. The problem is approached with a standard dynamic programming formulation; the concept of dominating system configurations is introduced to reduce the number of computations involved; and a simple procedure is presented. Solutions to three numerical examples are presented.
TL;DR: The application of the Bayesian approach depends on how the reliability decision is conceptualized as mentioned in this paper, and the difficulties of interpreting the probability function and assigning prior distributions restrict the presentation of a unified philosophy.
Abstract: The rudiments of Bayesian philosophy are introduced, and the mathematics of its application are surveyed; there is no uniformity of thought concerning either. The more extreme Bayesian philosophy, which allows subjective probabilities, is a means of plausible reasoning, or of making inferences, through inductive logic. Because inferences concerning reliability concepts are important in the decision process, this philosophy has a place in the reliability field. The difficulties of interpreting the probability function and of assigning prior distributions restrict the presentation of a unified philosophy. Thus, only techniques for describing prior probabilities under various circumstances can be given. Application of the Bayesian approach depends on how the reliability decision is conceptualized.
TL;DR: In this article, the reliability of MOS integrated circuits based on p-channel enhancement-mode transistors is described, and their failure modes and mechanisms are discussed. But the results of experimental studies of the effects of variations in construction, processing, and levels of stress are presented, and are compared with other available information on mOS integrated circuit reliability.
Abstract: This paper presents information on the reliability of MOS integrated circuits based on p-channel enhancement-mode transistors, and describes their failure modes and mechanisms. The principal failure mechanisms were ion migration at the surface and oxide shorting. The results of experimental studies of the effects of variations in construction, processing, and levels of stress are presented, and are compared with other available information on MOS integrated circuit reliability. The failure rate for commercially available complex MOS arrays is on the order of 0.001 to 0.01 per 1000 h of operating life at 125°C for arrays containing approximately 600 p-channel transistors. This corresponds to a failure rate on the order of 5 × 10?6 to 5 × 10?5 per equivalent gate per 1000 h. The effects of device complexity, operating temperature, and other factors are discussed. A reliability prediction equation for MOS integrated circuits is derived from available information. An overall activation energy for functional failure mechanisms of approximately 5 kcal/mole (?0.2 eV/molecule) is considered applicable to typical MOS integrated circuits. Thus, the failure rate of MOS devices operated at 50°C ambient temperature can be predicted to be on the order of 10?6 to 10?5 per equivalent gate per 1000 h.
TL;DR: The objective is to show that there is some substance to the classical statistician's opposition to Bayesian inference and that the issues are pertinent and meaningful to the reliability engineer.
Abstract: There appear to be two important developing trends in reliability. One is away from the use of statistics by reliability engineers; the other is toward increased use of Bayesian techniques. One source of the latter may be disillusionment with what is regarded as classical statistics; however, one source of the former may be dissatisfaction with what has been called Bayesian statistics. The purpose of this paper is to discuss these trends and to present a personal view of the issues between classical and Bayesian statistics. The objective is to show that there is some substance to the classical statistician's opposition to Bayesian inference and that the issues are pertinent and meaningful to the reliability engineer.
TL;DR: In this paper, a Bayes analysis of system availability is carried out on the basis of snapshot data obtained on each of the N component subsystems using a natural conjugate prior probability density function (pdf), yielding the posterior pdf of subsystem availability.
Abstract: A Bayes analysis of system availability A is carried out on the basis of snapshot data obtained on each of the N component subsystems Application of Bayes formula, using a natural conjugate prior probability density function (pdf), yields the posterior pdf of subsystem availability Determination of the posterior pdf of system availability requires the derivation of the pdf of the product of N independent random variables, which is accomplished through the use of the Mellin integral transform From the knowledge of the posterior pdf of A, confidence limits on system availability can be obtained
TL;DR: In this article, a Bayesian decision model is constructed around a conjugate probability density function for the Weibull hazard rate, and prior, posterior, and preposterior analysis of this decision model are discussed.
Abstract: A two-parameter Weibull distribution is assumed to be the appropriate statistical life-model of an engineering device. The hazard rate of this device is the relevant quantity in terms of which statistical decisions are to be made. A Bayesian decision model is constructed around a conjugate probability density function for the Weibull hazard rate. Prior, posterior, and preposterior analysis of this decision model are discussed. The results indicate the rational decision before and after sampling, and permit the optimization of sequential single-item sampling schemes. Such schemes are of particular importance in the reliability testing of high-cost equipment.
TL;DR: In this article, a Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators.
Abstract: For life testing procedures, a Bayesian analysis is developed with respect to a random intensity parameter in the Poisson distribution. Bayes estimators are derived for the Poisson parameter and the reliability function based on uniform and gamma prior distributions of that parameter. A Monte Carlo procedure is implemented to make possible an empirical mean-squared error comparison between Bayes and existing minimum variance unbiased, as well as maximum likelihood, estimators. As expected, the Bayes estimators have mean-squared errors that are appreciably smaller than those of the other two.
TL;DR: In this paper, a Bayesian reliability estimation technique known as the ''empirical Bayes approach'' is developed which uses previous experience nce to get a bayesian point estimator.
Abstract: A Bayesian reliability estimation technique known as the ``empirical Bayes approach'' is developed which uses previous experience nce to get a Bayesian point estimator. The techniques require no knowledge of the form of the unknown prior distribution and are robust to assumptions about its form. Empirical Bayes techniques are applicable to situations in which prior, independent observations of the random variable X from the random couple (?, X) are available where ? is the observed parameter of interest distributed in accordance with the unknown prior distribution. Performance comparisons of the empirical Bayes and other well established techniques are developed by examples for the binomial, exponential, Normal, and Poisson situations which often occur in reliability problems. In all cases the empirical Bayes estimator performed better than the classical estimator in minimizing the average squared error.
TL;DR: In this paper, a mathematical model for the reliability of modularly redundant systems with unequal failure rates for the operating and standby units is presented. But the model does not consider the failure modes of the active and standby unit.
Abstract: A mathematical model is established for the reliability of modularly redundant systems with unequal failure rates for the operating and standby units. The failure modes include failures of the active and standby units, three types of switch failures, and failures on system recovery. System reliability is considered for cases of both similar and dissimilar units, and for various restrictions on the failure parameters. It is shown that the most important failure parameters are those related to catastrophic failures, and that putting more reliable units as basic units, which operate initially, is important when switches and recovery are imperfect.
TL;DR: In this article, empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter.
Abstract: In part I empirical Bayes estimation procedures are introduced and employed to obtain an estimator for the unknown random scale parameter of a two-parameter Weibull distribution with known shape parameter. In part II, procedures are developed for estimating both the random scale and shape parameters. These estimators use a sequence of maximum likelihood estimates from related reliability experiments to form an empirical estimate of the appropriate unknown prior probability density function. Monte Carlo simulation is used to compare the performance of these estimators with the appropriate maximum likelihood estimator. Algorithms are presented for sequentially obtaining the reduced sample sizes required by the estimators while still providing mean squared error accuracy compatible with the use of the maximum likelihood estimators. In some cases whenever the prior pdf is a member of the Pearson family of distributions, as much as a 60% reduction in total test units is obtained. A numerical example is presented to illustrate the procedures.
TL;DR: In this paper, the authors apply Bayes' Equation to hypotheses concerning reliability and obtain posterior probabilities for the reliability hypotheses, which are consistent with the prior beliefs and the available test results.
Abstract: The rudiments of applying Bayes' Equation to hypotheses concerning reliability are introduced in a simple manner. The application is a means of obtaining posterior probabilities, for the reliability hypotheses, which are consistent with the prior beliefs and the available test results. The posterior distributions, from which decision theory could formally arrive at optimal estimates, are greatly dependent on the prior distributions. Thus, the discussion centers about the desired properties of a prior and its effects on the posterior for various data situations. Formulations for both continuous-conjugate and discrete representations of the prior beliefs are discussed and contrasted. The use of discrete priors offers many advantages over the use of continuous-conjugate priors.
TL;DR: A comterm map method permits machine computation of the probability of occurrence of the event associated with the output for each logic gate of a multistate logic tree.
Abstract: A comterm map method permits machine computation of the probability of occurrence of the event associated with the output for each logic gate of a multistate logic tree. The method is general and applies to any tree configuration, except where feedback occurs from higher to lower gates. Any state of two or more mutually exclusive component states may be used one or more times as primary inputs to the logic gates. The method usually requires much less memory than a minterm or Karnaugh map method by providing map space for only those selected components called ``common.''
TL;DR: In this paper, a branch and bound algorithm is extended to parallel standby redundancy where components are subject to both open and short circuit failures, and the algorithm is then modified to consider parallel-series arrangemememts of such standbys.
Abstract: A branch and bound algorithm is extended to parallel standby redundancy wherein components are subject to both open and short circuit failures. The algorithm is then modified to consider parallel-series arrangememts of such standbys, and is also applied to solving problems where there is a choice of components at each stage. In the last section, the two types of failures are separated in the constraints of the problems.
TL;DR: In this paper, the authors present a procedure for the generation of cost-minimized designs of circuits and systems, and suggest a policy by which the designer, with a reasonable computational effort, can find a set of good implementations.
Abstract: We present a procedure for the generation of cost-minimized designs of circuits and systems. Suppose a designer has decided upon the topology of his product. Also suppose he knows the cost and quality of the different grades of the N components required to implement the product. The designer then faces the following problem: How should he proceed to find the combination of grades that will give him the desired manufacturing yield at minimum product cost? We discuss the problem and suggest a policy by which the designer, with a reasonable computational effort, can find a set of ``good'' implementations. The suggested policy is applied to an electronic amplifier. The results are quite encouraging.
TL;DR: This tutorial paper shows, via an example, how optimum sampling plans can be derived using Bayesian decision theory, both nonsequential and sequential sampling plans.
Abstract: This tutorial paper shows, via an example, how optimum sampling plans can be derived using Bayesian decision theory. Both nonsequential and sequential sampling plans are described. The Bayesian method is contrasted with classical methods. The salient difference between Bayesian and classical methods is that the Bayesian approach allows one formally to pose and answer the question: ``How many should I test?''.
TL;DR: In this article, the authors discuss empirically derived procedures for selecting two types of semiconductor diodes for undersea cable operation based on test data on which the selection is based.
Abstract: The nonredundant transistorized repeaters used in undersea cable systems are intended to operate on the ocean floor without maintenance for twenty years. To provide the best possible assurance that this high level of reliability is in fact attained, semiconductor transistors and diodes manufactured for cable use are screened to identify and eliminate the early failures. Devices surviving the screening are then subjected, for at least six months, to real time aging which simulates the worst-case use-conditions. The devices are removed from the aging environment and certain of the devices are selected for use in undersea cable operation. It is the purpose of this paper to discuss the empirically derived procedures for selecting two types of semiconductor diodes. The test data on which the selection is based consist, in each case, of period measurements of four electrical parameters while the diodes are aging. There was apparently no accepted and well formulated statistical model for the data on which a conventional technique could be based. Thus more unconventional procedures for device selection had to be constructed. The selection procedures have been implemented and perform well.