Assessing component reliability using lifetime data from systems

ABSTRACT

In this paper, we introduce a competing risks model for the lifetimes of components that differs from the classical competing risks models by the fact that it is not directly observable which component has failed. We propose two statistical methods for estimating the reliability of components from failure data on a system. Our methods are applied to simulated failure data, in order to illustrate the performance of the methods.     

A model for k-out-of-m load-sharing systems

ABSTRACT

In a load-sharing system, the failure of a component affects the residual lifetime of the surviving components. We propose a model for the load-sharing phenomenon in k-out-of-m systems. The model is based on exponentiated conditional distributions of the order statistics formed by the failure times of the components. For an illustration, we consider two component parallel systems with the initial lifetimes of the components having Weibull and linear failure rate distributions. We analyze one data set to show that the proposed model may be a better fit than the model based on sequential order statistics.     

Estimation of impulse response functions in two-output systems

Abstract

A single input–double output (SIDO) linear time-invariant (LTI) system is considered, whose impulse response function (IRF) is assumed to have one unknown component. The problem is to estimate this unknown component after observations of the second component. Both IRF’s components are supposed to be L₂-integrable, and the estimation is made by cross-correlating the outputs, given that the input is a standard Wiener process on R. Weak asymptotic normality of appropriately centred estimators is proved.     

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.     

Dependent masking and system life data analysis: Bayesian inference for two-component systems

Data from field operations of a system is often used to estimate the reliability of components. Under ideal circumstances, this system field data contains the time to failure along with information on the exact component responsible for the system failure. However, in many cases, the exact component causing the failure of the system cannot be identified, and is considered to be masked. Previously developed models for estimation of component reliability from masked system life data have been based upon the assumption that masking occurs independently of the true cause of system failure. In this paper we develop a Bayesian methodology for estimating component reliabilities from masked system life data when the probability of masking is dependent upon the true cause of system failure. The Bayesian approach is illustrated for the case of a two-component system of exponentially distributed components.     

Performance analysis of machine repair system with single working vacation

Abstract

This paper studies a machine repair problem with repairman’s single working vacation in which repairman works with a lower repair rate rather than completely terminating repair during vacation period. Employing Markov process theory and matrix analytical method, various system performance measures are obtained in transient and stationary regimes. Moreover, we deduce the system reliability, the mean time to failure, the repairman’s busy period and the waiting time of failed machine by using the probabilistic properties of phase type distribution. Further, some numerical examples are provided. Finally, a cost model is developed to determine the optimum value of operating machines.     

Direct and Indirect Effects Based on Difference-in-Differences With an Application to Political Preferences Following the Vietnam Draft Lottery

Abstract

We propose a difference-in-differences approach for disentangling a total treatment effect within specific subpopulations into a direct effect and an indirect effect operating through a binary mediating variable. Random treatment assignment along with specific common trend and effect homogeneity assumptions identify the direct effects on the always and never takers, whose mediator is not affected by the treatment, as well as the direct and indirect effects on the compliers, whose mediator reacts to the treatment. In our empirical application, we analyze the impact of the Vietnam draft lottery on political preferences. The results suggest that a high draft risk due to the draft lottery outcome leads to an increase in mild preferences for the Republican Party, but has no effect on strong preferences for either party or on specific political attitudes. The increase in Republican support is mostly driven by the direct effect not operating through the mediator that is military service.     

Nonparametric model checking for k-out-of-n systems

It is an important problem in reliability analysis to decide whether for a given k-out-of-n system the static or the sequential k-out-of-n model is appropriate. Often components are redundantly added to a system to protect against failure of the system. If the failure of any component of the system induces a higher rate of failure of the remaining components due to increased load, the sequential k-out-of-n model is appropriate. The increase of the failure rate of the remaining components after a failure of some component implies that the effects of the component redundancy are diminished. On the other hand, if all the components have the same failure distribution and whenever a failure occurs, the remaining components are not affected, the static k-out-of-n model is adequate. In this paper, we consider nonparametric hypothesis tests to make a decision between these two models. We analyze test statistics based on the profile score process as well as test statistics based on a multivariate intensity ratio and derive their asymptotic distribution. Finally, we compare the different test statistics.     

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.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

A note on the mean residual life of a coherent system with a cold standby component

In this paper, we investigate the effect of a cold standby component on the mean residual life (MRL) of a system. When the system fails, a cold standby component is immediately put in operation. We particularly focus on the coherent systems in which, after putting the standby component into operation, the failure of the system is due to the next component failure. For these systems, we define MRL functions and obtain their explicit expressions. Also some stochastic ordering results are provided. Such systems include k-out-of-n systems. Hence, our results extend some results in literature.     

The Posterior Distribution of the Parameters of Component Lifetimes Based on Autopsy Data in a Shock Model

In this paper we consider a binary, monotone system whose component states are dependent through the possible occurrence of independent common shocks, i.e. shocks that destroy several components at once. The individual failure of a component is also thought of as a shock. Such systems can be used to model common cause failures in reliability analysis. The system may be a technological one, or a human being. It is observed until it fails or dies. At this instant, the set of failed components and the failure time of the system are noted. The failure times of the components are not known. These are the so-called autopsy data of the system. For the case of independent components, i.e. no common shocks, Meilijson (1981), Nowik (1990), Antoine et al . (1993) and GTsemyr (1998) discuss the corresponding identifiability problem, i.e. whether the component life distributions can be determined from the distribution of the observed data. Assuming a model where autopsy data is known to be enough for identifia bility, Meilijson (1994) goes beyond the identifiability question and into maximum likelihood estimation of the parameters of the component lifetime distributions based on empirical autopsy data from a sample of several systems. He also considers life-monitoring of some components and conditional life-monitoring of some other. Here a corresponding Bayesian approach is presented for the shock model. Due to prior information one advantage of this approach is that the identifiability problem represents no obstacle. The motivation for introducing the shock model is that the autopsy model is of special importance when components can not be tested separately because it is difficult to reproduce the conditions prevailing in the functioning system. In Gåsemyr & Natvig (1997) we treat the Bayesian approach to life-monitoring and conditional life- monitoring of components     

A superiority problem in misspecified restricted linear models

Abstract

In this article we study the relationship between principal component analysis and a multivariate dependency measure. It is shown, via simulated examples and real data, that the information provided by principal components is compatible with that obtained via the dependency measure δ. Furthermore, we show that in some instances in which principal component analysis fails to give reasonable results due to nonlinearity among the random variables, the dependency statistic δ still provides good results. Finally, we give some ideas about using the statistic δ in order to reduce the dimensionality of a given data set.     

Optimization of continuous and discrete scheduled times for a cumulative damage system with age-dependent maintenance

Abstract

This paper presents a preventive replacement problem when a system is operating successive works with random times and suffering stochastic shocks. The works cause random amount additive damage to the system, and the system fails whenever the cumulative damage reaches a failure level threshold. As an external shock occurs, the system experiences one of the two types of shocks with age-dependent maintenance mechanism: type-I (minor) shock is rectified by a minimal repair, or type-II (catastrophic) shock causes the system to fail. To control the deterioration process, preventive replacement is scheduled to replace the system at a continuous age T or at a discrete number N of working cycles, whichever occurs first, and corrective replacement is performed immediately whenever the system fails due to either shock or damage. The optimal preventive replacement schedule that minimizes the expected cost rate is discussed analytically and computed numerically. The proposed model provides a general framework for analyzing maintenance policies and extends several existing results.     

Reliability evaluation of a Semi-Markov repairable system under alternative environments

ABSTRACT

To accurately describe the performance of repairable systems operating under alternative environments, for example, mild/harsh, working/idling, maximum/minimum level demand etc., a Semi-Markov process with a finite state space and two different Semi-Markov kernels is introduced. The state set of the system which is regarded as acceptable might depend on the environments. Two important reliability indices, the availability and time to the first system failure, are obtained via Markov renewal theory, transform and matrix methods. The results and numerical examples are also provided for two special cases: (1) when sojourn times under alternative environments are constants and (2) when sojourn times under environments have exponential distributions.     

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.     

A family of distributions to model load sharing systems

The main characteristic of a load sharing system is that after the failure of one component the surviving components have to shoulder extra load and hence are prone to failure at an earlier time than what is expected under the original model. In others, the failure of one component may release extra resources to the survivors, thus delaying the system failure. In this paper we consider such m component systems and some observation schemes and identifiability issues under them. Then we construct a general semiparametric multivariate family of distributions which explicitly models this phenomenon through proportional conditional hazards. We suggest estimates for the constant of proportionality. We propose a nonparametric test for the hypothesis that the failures take place independently according to the common distribution against the alternative hypothesis that the second failure takes place earlier than warranted, study its properties and illustrate its use.     

On the Kullback–Leibler information of hybrid censored data

ABSTRACT

A hybrid censoring is a mixture of Type I and Type II censoring where the experiment terminates when either rth failure or predetermined censoring time comes first or later. In this article, we consider order statistics of the Type I censored data and provide a simple expression for their Kullback–Leibler (KL) information. Then, we provide the expressions for the KL information of the Type I and Type II hybrid censored data.     

A sampling plan for one-shot systems considering destructive inspection

Abstract

The paper is concerned with an acceptance sampling problem under destructive inspections for one-shot systems. The systems may fail at random times while they are operating (as the systems are considered to be operating when storage begins), and these failures can only be identified by inspection. Thus, n samples are randomly selected from N one-shot systems for periodic destructive inspection. After storage time T, the N systems are replaced if the number of working systems is less than a pre-specified threshold k. The primary purpose of this study is to determine the optimal number of samples n*, extracted from the N for destructive detection and the optimal acceptance number k*, in the sample under the constraint of the system interval availability, to minimize the expected cost rate. Numerical experiments are studied to investigate the effect of the parameters in sampling inspection on the optimal solutions.