Bayesian Analysis for a Redundant Repairable System with Imperfect Coverage

System characteristics of a redundant repairable system with two primary units and one standby are studied from a Bayesian viewpoint with different types of priors assumed for unknown parameters, in which the coverage factor is the same for an operating unit failure as that for a standby unit failure. Times to failure and times to repair of the operating and standby units are assumed to follow exponential distributions. When times to failure and times to repair with uncertain parameters, a Bayesian approach is adopted to evaluate system characteristics. Monte Carlo simulation is used to derive the posterior distribution for the mean time to system failure and the steady-state availability. Some numerical experiments are performed to illustrate the results derived in this paper.     

Statistical inference for the availability of repairable system with general repair,reboot delay,and switching railure

We studied the inferences of an availability system with reboot delay and standby switching failures in which the system consisted of two operating units and one warm standby. The system was studied under the assumption that the time-to-failure and the time-to-repair were assumed to follow an exponential and a general distribution. The reboot times are assumed to be exponentially distributed with parameter β. We constructed a consistent and asymptotically normal estimator of availability for such a repairable system. Based on this estimator, interval estimation and testing hypothesis were developed by using logit transformation. To implement the simulation inference for the system availability, we adopted two repair-time distributions—namely, lognormal and Weibull; and three types of Weibull distributions—characterized by their shape parameters—were considered. Finally, appropriate tables and figures of all simulation results have been included.     

Stochastic comparisons of two-unit repairable systems

Abstract

We consider two models of two-unit repairable systems: cold standby system and warm standby system. We suppose that the lifetimes and repair times of the units are all independent exponentially distributed random variables. Using stochastic orders we compare the lifetimes of systems under different assumptions on the parameters of exponential distributions. We also consider a cold standby system where the lifetimes and repair times of its units are not necessarily exponentially distributed.     

A multiple warm standby δ-shock system with a repairman having multiple vacations

In this article, a warm standby n-unit system is studied. The system is operational as long as there is one unit normal. The unit online, which has a lifetime distribution governed by a phase-type distribution, is also attacked by a shock from some external causes. Assume that shocks arrive according to a Poisson process. Whenever an interarrival time of shock is less than a threshold, the unit online fails. The lifetimes of the units in warm standby is exponentially distributed. A repairman who can take multiple vacations repairs the failed units based on the “first-in-first-out” rule. The repair times and the vacation times of repairman are governed by different phase-type distributions. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime; the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated. A numerical application is performed to illustrate the calculations.     

Reliability Evaluation for A Class of Multi-Unit Cold Standby Systems under Poisson Shocks

The reliability evaluation for a multi-units cold standby system with a switch-over under Poisson shocks is investigated. The random value of each shock is assumed to be i.i.d. with some known distribution. Each arrival of a shock has a random effect on the operating unit and on the switch-over. When the operating unit fails while the switch-over is normal, the next cold standby unit will start to operate immediately. The system fails only when all the units have failed or both the operating unit and the switch-over have failed. The reliability function and the mean time to the failure (MTTF) of the system are obtained.     

Applications of quadrivariate exponential distribution to a three-unit warm standby system with dependent structure

Two-unit warm standby systems have been elaborately dealt within the literature. However, the study of standby systems with more than two units, though very relevant in state-of-the-art practical situations, has received little attention because of mathematical intricacies involved in analyzing them. Also, such systems have been studied assuming: (i) the lifetime or repair time of the units to be exponential, or (ii) the lifetime and repair time to be independent. The present contribution is an improvement in the state-of-the-art in the sense that three-unit warm standby system with dependent structure is shown to be capable of comprehensive analysis.     

Copula-based reliability analysis for a parallel system with a cold standby

The traditional reliability models cannot well reflect the effect of performance dependence of subsystems on the reliability of system, and neglect the problems of initial reliability and standby redundancy. In this paper, the reliability of a parallel system with active multicomponents and a single cold-standby unit has been investigated. The simultaneously working components are dependent and the dependence is expressed by a copula function. Based on the theories of conditional probability, the explicit expressions for the reliability and the MTTF of the system, in terms of the copula function and marginal lifetime distributions, are obtained. Let the copula function be the FGM copula and the marginal lifetime distribution be exponential distribution, a system with two parallel dependent units and a single cold-standby unit is taken as an example. The effect of different degrees of dependence among components on system reliability is analyzed, and the system reliability can be expressed as the linear combination of exponential reliability functions with different failure rates. For investigating how the degree of dependence affects the mean lifetime, furthermore, the parallel system with a single cold standby, comprising different number of active components, is also presented. The effectiveness of the modeling method is verified, and the method presented provides a theoretical basis for reliability design of engineering systems and physics of failure.     

Bayes sequential estimation in a life test and asymptotic properties

For a life test without replacement on M machines, assuming an exponential distribution for failure times, the Bayes sequential procedure for estimating the failure rate is studied. Estimation error is assumed to be measured by one of a family of loss functions, and the cost of sampling consists of a cost per machine failure c, >, 0 and a cost per unit time c > 0. Assuming a conjugate prior on 9, the Bayes sequential procedure and its asymptotic Bayesian and sampling theory properties are obtained as c1, z9 - 0 and M + jointly.     

Improving the bridge structure by using linear failure rate distribution

In this paper, a system of five components is studied; one of these components is a bridge network component. Each of these components has a non-constant failure rate. The system components have linear failure rate lifetime distribution. The given system is improved by using three methods: reduction, warm standby with perfect switch and warm standby with imperfect switch. The reliability equivalence factors of the bridge structure system are obtained. The γ-fractiles are obtained to compare the original system with these improved systems. Finally, we present numerical results to show the difference between these methods.     

On the survival time of a repairable duplex system sustained by a cold standby unit subjected to a priority rule

We analyze the survival time of a general duplex system sustained by a cold standby unit subjected to a priority rule. The analysis is based on advanced complex function theory (sectionally holomorphic functions). As an example, we consider Weibull–Gnedenko and Erlang distributions for failure and repair. Several graphs are displaying the survival function.     

Statistical analysis of middle censored competing risks data with exponential distribution

In this paper, we consider some problems of estimation and reconstruction based on middle censored competing risks data. It is assumed that the lifetime distributions of the latent failure times are independent and exponential distributed with different parameters and also that the censoring mechanism is independent. The maximum likelihood estimators (MLEs) of the unknown parameters are obtained. We then use the asymptotic distribution of the MLEs to construct approximate confidence intervals. Based on gamma priors, Lindley's approximation method is applied to obtain the Bayesian estimates of the unknown parameters under squared error loss function. Since it is not possible to construct the credible intervals, we propose and implement the Gibbs sampling technique to construct the credible intervals. Several point reconstructors for failure time of censored units are provided. Finally, a simulation study is given by Monte-Carlo simulations to evaluate the performances of the different methods and a data set is analysed to illustrate the proposed procedures.     

Estimation of parameters of 1-out-of-2:g repairable system

Moment estimators of l-out-of-2:G repairable system are supplied under four sampling schemes assuming that the failure and repair time distribution of the units are exponential with unknown parameters λ, μ respectively. Information metrices of the estimators are supplied. Also it is shown that the estimators are asymptotically normally distributed in every sampling scheme.     

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.     

An Availability System with General Repair Distribution: Statistical Inference

This article we study the statistical inferences of an availability system with imperfect coverage. The time-to-failure and time-to-repair of the active and standby components are assumed to be exponential and general distribution, respectively. Assume that the coverage factor is the same for an active-component failure as that for a standby-component failure. Firstly, we propose a consistent and asymptotically normal (CAN) estimator of availability for such repairable system. Based on the CAN estimator of the system availability, interval estimation and testing (hypothesis) are performed. To implement the simulation inference for the system availability, we adopt two repair-time distributions, such as lognormal and Weibull distribution, in which three types of Weibull distribution are considered according to the shape parameter β. The component holds the decreasing repair rate (DRR), constant repair rate (CRR), and increasing repair rate (IRR) if β < 1, β = 1, and β > 1, respectively. Finally, all simulation results are displayed by appropriate tables and curves for understanding performance of the statistical inference procedures presented in this article.     

Point predictors of the latent failure times of censored units in progressively Type-II censored competing risks data from the exponential distributions

In this paper, we discuss the problem of predicting times to the latent failures of units censored in multiple stages in a progressively Type-II censored competing risks model. It is assumed that the lifetime distribution of the latent failure times are independent and exponential-distributed with the different scale parameters. Several classical point predictors such as the maximum likelihood predictor, the best unbiased predictor, the best linear unbiased predictor, the median unbiased predictor and the conditional median predictor are obtained. The Bayesian point predictors are derived under squared error loss criterion. Moreover, the point estimators of the unknown parameters are obtained using the observed data and different point predictors of the latent failure times. Finally, Monte-Carlo simulations are carried out to compare the performances of the different methods of prediction and estimation and one real data is used to illustrate the proposed procedures.     

Progressively Type-II censored competing risks data for exponential distributions based on sequential order statistics

We consider the progressively Type-II censored competing risks model based on sequential order statistics. It is assumed that the latent failure times are independent and the failure of each unit influences the lifetime distributions of the latent failure times of surviving units. We provide explicit expressions for the likelihood function of the available data under the conditional proportional hazard rate (CPHR) and the power trend conditional proportional hazard rate (PTCPHR) models. Under CPHR and PTCPHR models and assumption that the baseline distributions of the latent failure times are exponential, classical and Bayesian estimates of the unknown parameters are provided. Monte Carlo simulations are then performed for illustrative purposes. Finally, two datasets are analyzed.     

Bayesian estimation of reliability using time-to-failure distribution of parametric degradation models

This paper focusses on computing the Bayesian reliability of components whose performance characteristics (degradation – fatigue and cracks) are observed during a specified period of time. Depending upon the nature of degradation data collected, we fit a monotone increasing or decreasing function for the data. Since the components are supposed to have different lifetimes, the rate of degradation is assumed to be a random variable. At a critical level of degradation, the time to failure distribution is obtained. The exponential and power degradation models are studied and exponential density function is assumed for the random variable representing the rate of degradation. The maximum likelihood estimator and Bayesian estimator of the parameter of exponential density function, predictive distribution, hierarchical Bayes approach and robustness of the posterior mean are presented. The Gibbs sampling algorithm is used to obtain the Bayesian estimates of the parameter. Illustrations are provided for the train wheel degradation data.     

Analysis of hybrid censored competing risks data

In this paper, we consider the analysis of hybrid censored competing risks data, based on Cox's latent failure time model assumptions. It is assumed that lifetime distributions of latent causes of failure follow Weibull distribution with the same shape parameter, but different scale parameters. Maximum likelihood estimators (MLEs) of the unknown parameters can be obtained by solving a one-dimensional optimization problem, and we propose a fixed-point type algorithm to solve this optimization problem. Approximate MLEs have been proposed based on Taylor series expansion, and they have explicit expressions. Bayesian inference of the unknown parameters are obtained based on the assumption that the shape parameter has a log-concave prior density function, and for the given shape parameter, the scale parameters have Beta–Gamma priors. We propose to use Markov Chain Monte Carlo samples to compute Bayes estimates and also to construct highest posterior density credible intervals. Monte Carlo simulations are performed to investigate the performances of the different estimators, and two data sets have been analysed for illustrative purposes.     

Order restricted Bayesian inference for exponential simple step-stress model

A step-stress model has received a considerable amount of attention in recent years. In the usual step-stress experiment, a stress level is allowed to increase at each step to get rapid failure of the experimental units. The expected lifetime of the experimental unit is shortened as the stress level increases. Although extensive amount of work has been done on step-stress models, not enough attention has been paid to analyze step-stress models incorporating this information. We consider a simple step-stress model and provide Bayesian inference of the unknown parameters under cumulative exposure model assumption. It is assumed that the lifetime of the experimental units are exponentially distributed with different scale parameters at different stress levels. It is further assumed that the stress level increases at each step, hence the expected lifetime decreases. We try to incorporate this restriction using the prior assumptions. It is observed that different censoring schemes can be incorporated very easily under a general setup. Monte Carlo simulations have been performed to see the effectiveness of the proposed method, and two datasets have been analyzed for illustrative purposes.     

Stochastic analysis of a complex system with an auxiliary unit

This paper considers the stochastic analysis of a complex system having three units. The first two units are designated as unit-1 and unit-2 while the third unit is called an auxiliary unit (unit-3) which enhances the range of transmission and efficiency of unit-1. Joint distributions of failure and repair times are bivariate exponential (B.V.E.) with different parameters. Regenerative point technique is used to obtain various reliability characteristics of the system under study. Behaviour of some characteristics has also been studied through graphs.