Two-component Parallel System Research Articles

An improved analytical solution to outcrossing rate for scalar nonstationary and non-gaussian processes

For time-dependent reliability methods, there is a lack of analytical solutions to outcrossing rates of nonstationary and non-Gaussian processes. This paper aims to propose an analytical method to determine this type of outcrossing rate with improved accuracy and efficiency. The novelty of this proposed method is that it can consider the probabilistic properties of non-Gaussian processes by combining a two-component parallel system model with higher-order moments-based reliability indexes. The main contributions of this method are: (1) compared with PHI2+ method, it is more accurate for the calculation of the outcrossing rate of nonstationary and non-Gaussian processes with nonlinear limit state functions; (2) compared with MPHI2 method, it is analytical and not only applicable for the nonstationary stochastic processes but also insensitive to time increments; and (3) it is not only more accurate than the existing methods for nonstationary and non-Gaussian processes but also more computation efficient than the Monte Carlo simulation method. The proposed method has shown its advantages for practical structures with neither Gaussian processes nor linear limit state functions, which are beneficial for both researchers and asset managers to evaluate the time-dependent reliability of structures accurately and to develop risk-informed maintenance schemes with a view to prolonging their service life.

Availability modelling and analysis of a two-component parallel system under stochastic dependence

Availability modelling and analysis of a two-component parallel system under stochastic dependence

Availability modeling and analysis of a two-component parallel system under stochastic dependence

Availability modeling and analysis of a two-component parallel system under stochastic dependence

Optimal Periodic Inspection Policy for a Parallel System Subject to Continuous Bi-Levels Degradation

Each system goes through several intermediate states between its initial state and the failure state, which are called degradation states. In addition, knowing these intermediate states allows us the building of an effective maintenance plan for the system under consideration. In this paper, we have considered a two-component parallel system where each component is subjected to two levels of degradation. Our first objective is to identify the mathematical (probabilistic) model associated with this system and this when the hypothesis of periodic inspection of the system is retained. Then, the main idea is to build the economic model encompassing the different average costs (inspection costs, preventive maintenance costs and corrective maintenance costs) spent during a renewal cycle. The latter will allow us to determine the optimal “inspection interval” while minimizing the average global-cost spent. Finally, in order to illustrate and analyze the sensitivity of the optimal inspection policy to the various maintenance costs, numerical examples are presented.

Application of adaptive surrogate models in time-variant fatigue reliability assessment of welded joints with surface cracks

The time-variant fatigue reliability of welded joints subjected to stochastic loading is assessed based on the PHI2 method, which takes the time-variant reliability problem as a two-component parallel system reliability problem. The complex fatigue process that takes place at the weld toe is modeled as a semi-elliptical surface crack growth process. Surrogate models, representing surface crack sizes at a time instant, are applied to solve the reliability problem involving time-consuming fatigue crack growth analyses. Polynomial regression models and Kriging interpolation models are both employed. Their corresponding adaptive procedures are also adopted to improve the reliability results. Some adjustments of the active refinement algorithm based on the Kriging model are proposed to make it more suitable for the present study. The time-variant fatigue reliability assessment of a T-plate welded joint is carried out. Without using the adaptive procedures, the Kriging models can better represent the crack sizes compared with the polynomial models. The outcrossing rate formulated by the PHI2 method is sensitive to the reliability index of the first component of the two-component parallel system. The effectiveness of the adaptive procedures, which improve the reliability results, is demonstrated.

A condition-based opportunistic maintenance policy integrated with energy efficiency for two-component parallel systems

Purpose: In order to improve the energy utilization and achieve sustainable development, this paper integrates energy efficiency into condition-based maintenance(CBM) decision-making for two-component parallel systems. The objective is to obtain the optimal maintenance policy by minimizing total cost.Design/methodology/approach: Based on energy efficiency, the paper considers the economic dependence between the two components to take opportunistic maintenance. Specifically, the objective function consists of traditional maintenance cost and energy cost incurred by energy consumption of components. In order to assess the performance of the proposed new maintenance policy, the paper uses Monte-Carlo method to evaluate the total cost and find the optimal maintenance policy.Findings: Simulation results indicate that the new maintenance policy is superior to the classical condition-based opportunistic maintenance policy in terms of total economic costs.Originality/value: For two-component parallel systems, previous researches usually simply establish a condition-based opportunistic maintenance model based on real deterioration data, but ignore energy consumption, energy efficiency (EE) and their contributions of sustainable development. This paper creatively takes energy efficiency into condition-based maintenance(CBM) decision-making process, and proposes a new condition-based opportunistic maintenance policy by using energy efficiency indicator(EEI).

An accelerated life test plan for a two-component parallel system under ramp-stress loading using masked data

PurposeMost of the literature on the design of accelerated life test (ALT) plan focus on a single system (subsystem) totally disregarding its internal configuration. Many a times it is not possible to identify the components that cause the system failure or that the cause can only be identified by a subset of its components resulting in a masked observation. The purpose of this paper is to deal with the planning of ramp-stress accelerated life testing for a high-reliability parallel system comprising two dependent components using masked failure data. Such a testing may prove to be useful in a twin-engine aircraft. A ramp-stress results when stress applied on the system increases linearly with time.Design/methodology/approachA parallel system with two dependent components is taken with dependency modeled by Gumbel-Hougaard copula. The stress-life relationship is modeled using inverse power law, and cumulative exposure model is assumed to model the effect of changing stress. The method of maximum likelihood is used for estimating design parameters. The optimal plan consists in finding optimal stress rate using D-optimality criterion.FindingsThe optimal plan consists in finding optimal stress rate using D-optimality criterion by minimizing the reciprocal of the determinant of Fisher information matrix. The proposed plan has been explained using a numerical example and carrying out a sensitivity analysis.Originality/valueThe model formulated can help reliability engineers obtain reliability estimates quickly of high-reliability products that are likely to last for several years.

Some reliability characteristics and stochastic ordering of series and parallel systems of bivariate generalized exponential distribution

ABSTRACTIn this paper, the reliability properties of two-component parallel and series systems are considered for bivariate generalized exponential (BVGE) distribution introduced by Kundu and Gupta [Bivariate generalized exponential distribution. J Multivar Anal. 2009;100:581–593]. For this model, the moments and mean residual life functions of these systems and the regression mean residual life function are derived. Stochastic comparisons of series and parallel systems of BVGE distribution are investigated. Moreover, various ordering results for the comparisons of series and parallel systems arising from BVGE random vectors are obtained with respect to the usual stochastic, reversed hazard rate and likelihood ratio orderings.

Degradation model analysis of two-component series and parallel systems

ABSTRACTMany high-reliability and long-life products to form a system in series or parallel. Based on the assumption that the degradation process of the performance characteristics is Wiener procession, this article proposes a degradation model of the series and parallel systems. Here, a bivariate constant-stress degradation model is established, assuming Wiener procession and utilizing the copula function to describe the correlation among the performance characteristics. The adoption of an inference function for margins method for estimating the model's parameter ensures the accuracy of the estimation results as well as reduces the difficulties of parameter estimation effectively. In the numerical example research, we estimate the parameters of the degradation models of Gaussian Copula and Frank Copula using the rubidium spectral lamp as an example. Furthermore, we also adopt the Monte Carlo simulation method on the effects of copula function's estimation. We find that the Frank Copula more applicable than Gaussian Copula. For the two-component series and parallel systems composed by the rubidium spectral lamp, we get the reliability curve, the failure probability density curve and the MTBF value of the series system and parallel system. These curves are compared and analysed to achieve the conclusions.

Age replacement policies for two-component systems with stochastic dependence

Purpose – The purpose of this paper is to investigate age replacement policies for two-component parallel system with stochastic dependence. The stochastic dependence considered, is modeled by a one-sided domino effect. The failure of component 1 at instant t may induce the failure of component 2 at instant t+τ with probability p 1→2. The time delay τ is a random variable with known probability density function h p 1→2 (.). The system is considered in a failed state when both components are failed. The proposed replacement policies suggest to replace the system upon failure or at age T whichever occurs first. Design/methodology/approach – In the first policy, costs and durations associated with maintenance activities are supposed to be constant. In the second replacement policy, the preventive replacement cost depends on the system’s state and age. The expected cost per unit of time over an infinite span is derived and numerical examples are presented. Findings – In this paper and especially in the second policy, the authors find that the authors can get a more economical policy if the authors consider that the preventive replacement cost is not constant but depends on T. Originality/value – In this paper, the authors take into account of the stochastic dependence between system components. This dependence affects the global reliability of the system and replacement’s periodicity. It can be used to measure the performance of the system et introduced into design phase of the system.

On the mean remaining strength at the system level for some bivariate survival models based on exponential distribution

In this paper, we consider the mean remaining strength at the system level of two-component parallel and series systems in the case of possibly dependent components which are subject to a common stress. We assume that the strengths of the components have FGM, Freund’s, Marshall–Olkin and Block–Basu’s bivariate exponential models. A numerical study based on the generated data set is performed to obtain the maximum likelihood estimates of the mean remaining strength for two-component parallel system in the stress–strength setup.

Study of some measures of dependence between order statistics and systems

Let X = ( X 1 , X 2 , … , X n ) be a random vector, and denote by X 1 : n , X 2 : n , … , X n : n the corresponding order statistics. When X 1 , X 2 , … , X n represent the lifetimes of n components in a system, the order statistic X n − k + 1 : n represents the lifetime of a k -out-of- n system (i.e., a system which works when at least k components work). In this paper, we obtain some expressions for the Pearson’s correlation coefficient between X i : n and X j : n . We pay special attention to the case n = 2 , that is, to measure the dependence between the first and second failure in a two-component parallel system. We also obtain the Spearman’s rho and Kendall’s tau coefficients when the variables X 1 , X 2 , … , X n are independent and identically distributed or when they jointly have an exchangeable distribution.

Analytical derivation of the outcrossing rate in time-variant reliability problems

The usual approach to time-variant reliability problems is based on the computation of the outcrossing rate through the limit state surface and its time integration. The so-called PHI2 method allows one to compute the outcrossing rate by solving a two-component parallel system reliability problem using the First Order Reliability Method (FORM). Following this approach, the present paper provides new analytical expressions of the outcrossing rate and their implementation. The corresponding improvement of the PHI2 method in terms of accuracy is shown. The method is first validated using a simple time-variant reliability problem, for which an analytical expression of the associated outcrossing rate exists. Then, it is applied to evaluate the reliability of a corroded steel beam submitted to a midspan random load.

A Bivariate Distribution for the Reliability Analysis of Failure Data in Presence of a Forewarning or Primer Event

This article presents a bivariate distribution for analyzing the failure data of mechanical and electrical components in presence of a forewarning or primer event whose occurrence denotes the inception of the failure mechanism that will cause the component failure after an additional random time. The characteristics of the proposed distribution are discussed and several point estimators of parameters are illustrated and compared, in case of complete sampling, via a large Monte Carlo simulation study. Confidence intervals based on asymptotic results are derived, as well as procedures are given for testing the independence between the occurrence time of the forewarning event and the additional time to failure. Numerical applications based on failure data of cable insulation specimens and of two-component parallel systems are illustrated.

On the inspection policy of a two-component parallel system with failure interaction

In this paper we study a two-component standby system which can successfully operate upon a demand if at least one component is not failed. We assume that failures can be detected only by periodic inspections. We consider that the failure of one component can modify the (conditional) failure probability of the component still alive with probability p and do not interact with probability 1−p. For that failure interaction scheme we obtain the system reliability function for the case of staggered inspections. We compare staggered and non-staggered inspections through numerical examples considering constant hazard rates.

Nonparametric reliability modeling for parallel systems

In the present study, we consider a two-component parallel system. We do not know the probability distributions of the lifetimes of each component that characterize their behavior. Using the conventional nonparametric kernel estimate, we introduce three different approaches to estimate nonparametrically the cumulative distribution function or reliability function of the parallel system. In addition to the analytical formulation of these estimates, we utilize the criteria of mean integrated squared error to evaluate and compare each of the proposed estimates. A numerical simulation illustrates and compares three proposed methods of estimating nonparametrically the reliability behavior of a two-component parallel system.

Estimation of components reliability in a parallel system using masked system life data

In this paper we estimate the reliability of the individual components that belong to a parallel system using masked-system life test data. In particular, we compute the maximum likelihood and Bayes estimates of component reliabilities. The considered system consists of J independent components having non-identical complementary exponential lifetime distributions. It is assumed for Bayes estimates that the unknown parameters are independent random variables having symmetrical triangle probability density functions. The approaches are illustrated with a two-component parallel system. A numerical simulation study is introduced to show how one can use the present approaches to compute these estimations for a practical problem. Also in the numerical simulation study a comparison between Bayes and maximum likelihood estimates is introduced.

Asymptotic properties of maximum likelihood estimates for a bivariate exponential distribution and mixed censored data

This article investigates asymptotic properties of the maximum likelihood estimators (MLE) of parameters in the bivariate exponential distribution (BVE) of Marshall and Olkin (1967) based on the following mixed censored data. In life-testing two-component parallel systems (A, B), a cost-saving procedure is to stop the testing experiment after observing the first r failure times of component A. The resulting data are (X (1), Y * [1]), (X (2), Y * [2]), …,(X (r) , Y * [ r ]), (X * (r+1), Y * [r+1]), …, (X * ( n ), Y * [ n ]), where X (1)≤X (2)≤⋯≤X ( r ) are ordered lifetimes from component A, and Y [ i ] is the concomitant order statistic corresponding to X ( i ) from component B. The data X * (r+1), …, X * ( n ), and Y * [ i ], i=1, …, n are all censored at time X ( r )=x ( r ). Because of the complexity of the data type and the irregularity of the BVE distribution of (X, Y), there are no immediately applicable asymptotic results for the MLE. This article provides a rigorous treatment of the asymptotic behavior of the MLE, with a numerical example illustrating the mathematical derivations.

The asymptotics of maximum-likelihood estimates of parameters based on a data type where the failure and the censoring time are dependent

This article derives the asymptotic results of the maximum-likelihood estimates of the parameters in the general bivariate continuous distribution for the data type, in which the failure time and the censoring variables are dependent. This data type is motivated from life-testing two-component parallel systems. Because the duration of collecting all system failure times can be too long to justify its experimental cost, the testing experiment is terminated at the rth smallest failure time, X ( r) , of one component. The resulting bivariate censored data are (X ∗ (1), Y ∗ [i]) , where X ∗ (i) = X (i) if i ⩽ r; X ∗ (i) = X (r) if i > r, Y ∗ [i] = Y [i] if Y [i] ⩽ X (r); Y ∗ [i] = X (r), if Y [i] ⩽ X (r) , and Y [ i] 's are the concomitants of the ordered statistics X ( i) 's. The failure time Y [ i] is dependent on X ( r) due to the induced dependence in concomitant ordered statistics. Our procedures can be applied to derive asymptotics in other complicated data types, where the life-testing experiment is stopped at the rth smallest failure time of max( X, Y) or min( X, Y).

A new plan for life-testing two-component parallel systems

Although life-testing of parallel systems provides more information than testing series systems, it can be very time consuming, which limits its usage in practice. In this article, a new plan of life-testing two-component ( A and B) parallel systems is proposed. This plan terminates life-testing experiments at the rth smallest order statistics X ( r) of component A data. Our data type consists of type-II censored data ( i) , i = 1, 2, …, n from component A and their concomitants Y [ i] randomly censored at X ( r) from component B. Compared to the plan with complete samples, where the experiment is terminated until observing W ( n) , the maximum of W i = max( X i , Y i ), i = 1, 2, …, n, our new plan will shorten the test duration and save some unfailed components. General procedures of constructing the likelihood function and of deriving the expected number of failed components and testing duration are presented and illustrated under Marshall and Olkin's bivariate exponential distribution. A follow-up study shows that the loss of accuracy in maximum likelihood estimation is not severe compared with the gain of shortening the testing duration and saving some unfailed components.