ON THE PREDICTION OF FUTURE FAILURES FOR A REPAIRABLE EQUIPMENT SUBJECT TO OVERHAULS

This paper deals with the prediction, from a Bayes viewpoint, of future failures for a repairable equipment subjected both to minimal repairs and periodic overhauls. The effect of major overhauls on the reliability of the equipment is modeled by a proportional age reduction model, while the failure process between two successive overhaul epochs is modeled by the power law process. Prediction both of the future failure times and of the number of failures in a future time interval are provided on the basis of the observed data and of a number of suitable prior densities, which reflect different degrees of belief on the failure mechanism and overhaul effectiveness. Finally, a numerical application illustrates the proposed prediction procedures and their use in assessing the adequacy of the model to describe the observed data set.     

A Study on Reliability for A Two-Item Cold Standby Markov Repairable System with Neglected Failures

This article studies reliability for a Markov repairable two-item cold standby system with neglected failures. In the system, if a failed time of the system is too short (less than a given critical value) to cause the system to fail, then the failed time may be omitted from the downtime record, i.e., the failure effect could be neglected. In ion-channel modeling, this situation is called the time interval omission problem. The availability indices and the mean downtime are presented as two measures of reliability for this repairable system. Some numerical examples are shown to illustrate the results obtained in this article.     

Optimal preventive policies for imperfect maintenance models with replacement first and last

ABSTRACT

This paper proposes preventive replacement policies for an operating system which may continuously works for N jobs with random working times and is imperfectly maintained upon failure. As a failure occurs, the system suffers one of the two types of failures based on some random mechanism: type-I (repairable or minor) failure is rectified by a minimal repair, or type-II (non repairable or catastrophic) failure is removed by a corrective replacement. A notation of preventive replacement last model is considered in which the system is replaced before any type-II failure at an operating time T or at number N of working times, whichever occurs last. Comparisons between such a preventive replacement last and the conventional replacement first are discussed in detail. For each model, the optimal schedule of preventive replacement that minimizes the mean cost rate is presented theoretically and determined numerically. Because the framework and analysis are general, the proposed models extend several existing results.     

Maximum likelihood and bayes prediction of current system lifetime

The Power Law Process is often used to analyse failure data of repairable systems undergoing development testing where the system failure intensity decreases as a result of repeated application of corrective actions. At the end of the development program, the system failure intensity is assumed to remain constant and the current system lifetime is assumed to be exponentially distributed. In this paper, prediction limits on the current system lifetime have been derived both in the maximum likelihood and Bayesian context. Exact values and a closed form approximation of percentage points of the pivotal quantity used in the classical approach are given in the case of failure truncated testing. For both failure and time truncated testing, the Bayesian approach is developed both when no prior knowledge is available and when information on the reliability growth rate can be given. A numerical example is also given.     

Age Replacement Policy with a Safety Constraint via the Bayesian Method

Consider a system that is subject to shocks that arrive according to a non homogeneous Poisson process. As the shocks occur, the system has m + 1 failure modes including the following: (i) a non repairable failure (catastrophic) mode that calls for a replacement and (ii) m repairable failure (non catastrophic) modes that are rectified by minimal repairs. In this article, we propose an age-replacement model with minimal repair based on using the natural conjugate prior of Bayesian method. In addition, a safety constraint is considered to control the risk of occurring catastrophic failures in a specified time interval. The minimum-cost replacement policy is studied in terms of its existence and safety constraint. A numerical example is also presented to illustrate the proposed model.     

Generalized hazard-based goodness-of-fit tests for a repairable system

This paper examines the problem of goodness-of-fit concerning the distribution of the initial failure times of a repairable system. In particular, the class of intensity-based smooth goodness-of-fit tests studied in Pen ¨ a (1998a,b) and Agustin and Pen ¨ a (2000, 2001) is considered. Specific members of the family are derived and are shown to be generalizations of existing tests for independent and identically distributed observations. The results of a Monte Carlo simulation study are presented to illustrate the potential of these tests as powerful directional and omnibus tests.     

Conditional failure occurrence rates for semi-Markov chains*

ABSTRACT

In the present paper, we aim at providing plug-in-type empirical estimators that enable us to quantify the contribution of each operational or/and non-functioning state to the failures of a system described by a semi-Markov model. In the discrete-time and finite state space semi-Markov framework, we study different conditional versions of an important reliability measure for random repairable systems, the failure occurrence rate, which is based on counting processes. The identification of potential failure contributors through the conditional counterparts of the failure occurrence rate is of paramount importance since it could lead to corrective actions that minimize the occurrence of the more important failure modes and therefore improve the reliability of the system. The aforementioned estimators are characterized by appealing asymptotic properties such as strong consistency and asymptotic normality. We further obtain detailed analytical expressions for the covariance matrices of the random vectors describing the conditional failure occurrence rates. As particular cases we present the failure occurrence rates for hidden (semi-) Markov models. We illustrate our results by means of a simulated study. Different applications are presented based on wind, earthquake and vibration data.     

A combinatorial estimation approach for storage reliability with initial failures based on periodic testing data

Storage reliability that measures the ability of products in a dormant state to keep their required functions is studied in this paper. Unlike the operational reliability, storage reliability for certain types of products may not be always 100% at the beginning of storage since there are existing possible initial failures that are normally neglected in the models of storage reliability. In this paper, a new combinatorial approach, the nonparametric measure for the estimates of the number of failed products and the current reliability at each testing time in storage, and the parametric measure for the estimates of the initial reliability and the failure rate based on the exponential reliability function, is proposed for estimating and predicting the storage reliability with possible initial failures. The proposed method has taken into consideration that the initial failure and the reliability testing data, before and during the storage process, are available for providing more accurate estimates of both initial failure probability and the probability of storage failures. When storage reliability prediction that is the main concern in this field should be made, the nonparametric estimates of failure numbers can be used into the parametric models for the failure process in storage. In the case of exponential models, the assessment and prediction method for storage reliability is provided in this paper. Finally, numerical examples are given to illustrate the method. Furthermore, a detailed comparison between the proposed method and the traditional method, for examining the rationality of assessment and prediction on the storage reliability, is presented. The results should be useful for planning a storage environment, decision-making concerning the maximum length of storage, and identifying the production quality.     

Discrete time cold standby repairable system: Combinatorial analysis

ABSTRACT

In this article, we obtain exact expression for the distribution of the time to failure of discrete time cold standby repairable system under the classical assumptions that both working time and repair time of components are geometric. Our method is based on alternative representation of lifetime as a waiting time random variable on a binary sequence, and combinatorial arguments. Such an exact expression for the time to failure distribution is new in the literature. Furthermore, we obtain the probability generating function and the first two moments of the lifetime random variable.     

Statistical analysis of a two-unit dependent repairable system

Maximum likelihood estimators are investigated a for lr-out-of 2:G repairable system when failures and repairs are statistically dependent. Under two censoring schemes Accelerated Life Testing Procedure (ALTP) is used to obtain the estimates. Information matrices are supplied and a special case when the failure and repair phenomenon are independent of each other is considered.     

On the least favorable configuration in mult1hgmial selection problems

The Modulated Power Law process (MPLP) has been recently used to analyse repairable mechanical units whose failure pattern incorporates both renewal-type behaviour and time trend. In this paper, a Bayes approach is used to develop significance tests for testing for a reduction of the MPLP to simpler models, namely the Gamma Renewal and the Power Law processes, which are special cases of the MPLP model. Furthermore, significance tests to compare the effect of the repair actions on the future reliability or the time trend in two independent MPLP samples are proposed. Large simulation studies have been carried out, both to measure the exact size of the proposed tests and to investigate their power in a largely repeated use of each test. Finally, numerical examples are given to illustrate the proposed procedure.     

Prediction of the residual failure processes based on the process history

In the reliability area, the concept of the residual lifetime of a non repairable system is very important and its property has been intensively studied. In this article, we define the “residual failure process” for a repairable system and study its stochastic properties thoroughly. The detailed discussions are given when the corresponding failure process is a renewal process. An illustrative example is also discussed.     

Exact and approximate runs distributions

Let R = R_n denote the total (and unconditional) number of runs of successes or failures in a sequence of n Bernoulll (p) trials, where p is assumed to be known throughout. The exact distribution of R is related to a convolution of two negative binomial random variables with parameters p and q (=1-p). Using the representation of R as the sum of 1 - dependent indicators, a Berry - Esséen theorem is derived; the obtained rate of sup norm convergence is O(n^-½). This yields an unconditional version of the classical result of Wald and Wolfowitz (1940). The Stein - Chen method for m - dependent random variables is used, together with a suitable coupling, to prove a Poisson limit theorem for R. but with the limiting support set being the set of odd integers, Total variation error bounds (of order O(p) are found for the last result. Applications are indicated.     

Statistical estimation for a failure model with damage accumulation in a case of small samples

A failure model with damage accumulation is considered. Damages occur according to a Poisson process and they degenerate into failures in a random time. The rate of the Poisson process and the degeneration time distribution are unknown. Two sample populations are available: a sample of intervals between damages and a sample of degeneration times. The case of small samples is considered. The purpose is to estimate the expectation and the distribution of the number of damages and failures at time t. We consider the plug-in and resampling estimators of the above mentioned characteristics. The expectations and variances of the suggested estimators are investigated. The numerical examples show that the resampling estimator has some advantages.     

A two-phase method for controlling Erlang-failure processes with high reliability

Monitoring a failure process and measuring its performance are important issues for complex nonrepairable and repairable systems. For a highly reliable process, traditional methods for reliability monitoring and performance measuring become inapplicable. This paper proposes a new two-phase controlling method for monitoring and measuring an Erlang-failure process (EFP). In the first-phase controlling method, a control chart is used to monitor the EFP condition. When special causes of variation have been removed from the EFP and all of the failure times plotted on the control chart lie within the control limits, the EFP is considered to be in control. However, the in-control EFP still likely carries out bad or out-of-lifetime-specification conditions. Thus, its lifetime-specification limit is taken into consideration as the second-phase controlling method for measuring the in-control EFP performance. We propose a lifetime-capability index. Its value has a one-to-one corresponding relationship with the lifetime-conforming rate, which indicates the lifetime performance of this EFP. Without collecting additional data efforts, in-control data gathered from the control chart in the first phase is employed to estimate the lifetime-capability index. To realize main lifetime-capability of the EFP impacting on downstream customers, the lower confidence bound of the estimate of the lifetime-capability index, capturing its minimum lifetime capability, is considered. The advantage of this two-phase method for controlling the failure processes can motivate the manufacturer to develop a reliability-monitoring technique, establish an adequate reliability improvement program and implement an appropriate analysis to ensure its lifetime performance meeting the customers requirement.     

Rotation test with pairwise distance measures of sample vectors in a GLM

One way to cope with high-dimensional data even in small samples is the use of pairwise distance measures—such as the Euclidean distance—between the sample vectors. This is usually done with permutation tests. Here we propose the application of exact parametric rotation tests which are no longer restricted by the finite number of possible permutations of a sample. The method is presented in the framework of the general linear model.     

Comparison of two independent life tests subject to type II censoring

The problem of comparing two life distributions is considered. It Is assumed that each life test Is stopped after a predetermined number of failures without regard to the number of failures which have occurred in the other life test. A generalized Wilcoxon-rlann-Whitney test is proposed. Small and large sample distributional results are obtained for the test statistic. Under the assumption that the life distributions differ by at most a scale parameter an exact confidence interval for that parameter is obtained. An algorithm for computing the interval is given.     

The optimal choice of negative binomial charts for monitoring high-quality processes

Good control charts for high quality processes are often based on the number of successes between failures. Geometric charts are simplest in this respect, but slow in recognizing moderately increased failure rates p. Improvement can be achieved by waiting until r>1 failures have occurred, i.e. by using negative binomial charts. In this paper we analyze such charts in some detail. On the basis of a fair comparison, we demonstrate how the optimal r is related to the degree of increase of p. As in practice p will usually be unknown, we also analyze the estimated version of the charts. In particular, simple corrections are derived to control the nonnegligible effects of this estimation step.     

A Bivariate Optimal Replacement Policy for a System With Age-dependent Minimal Repair and Cumulative Repair-cost Limit

In this article, a repairable system with age-dependent failure type and minimal repair based on a cumulative repair-cost limit policy is studied, where the information of entire repair-cost history is adopted to decide whether the system is repaired or replaced. As the failures occur, the system has two failure types: (i) a Type-I failure (minor) type that is rectified by a minimal repair, and (ii) a Type-II failure (catastrophic) type that calls for a replacement. We consider a bivariate replacement policy, denoted by (n,T), in which the system is replaced at life age T, or at the n-th Type-I failure, or at the kth Type-I failure (k < n and due to a minor failure at which the accumulated repair cost exceeds the pre-determined limit), or at the first Type-II failure, whichever occurs first. The optimal minimum-cost replacement policy (n,T)* is derived analytically in terms of its existence and uniqueness. Several classical models in maintenance literature could be regard as special cases of the presented model. Finally, a numerical example is given to illustrate the theoretical results.