The minimal repair model is a fundamental framework for analyzing repairable systems. Optimizing the allocation of limited repair resources under this model is both important and challenging. This study introduces an efficient approach based on system signatures to address this problem. We first propose the generalized survival signature for repairable systems (rGSS) and derive a key mixture representation of system reliability. This extends the traditional signature concept beyond repairable series systems to general coherent systems. Leveraging the rGSS, we develop optimal repair allocation strategies between two specific components with comparative criticality via minimal cut (path) sets in arbitrary coherent systems, as well as among all components in four common system structures: Series, parallel, parallel-series, and series-parallel. Numerical examples are provided to illustrate the optimality of the proposed strategies and explore their potential for generalization.

We introduce a novel approach for modeling two-way functional trajectories, relying on Kendall’s tau representation of marginal covariance functions and utilizing the concept of product functional principal component analysis. The developed estimation procedure is intuitive and straightforward to implement. Theoretical results supporting its validity are also established. Numerical simulation studies validate its superior performance compared to recently developed methods. Furthermore, the application of this approach to analyze two-way air pollution trajectories demonstrates its practical superiority.

Computing the signature of a network is both significant and challenging. Addressing the limitations of existing methods in batch processing of large-scale network signatures, in this paper we propose a novel DNN (Deep Neural Network)-based framework for estimating the all-terminal signature and reliability for networks with varying topologies. Our framework involves constructing a DNN model with four efficient and compact network topological features (the numbers of nodes, and links, the node degrees and the link connectivity) as input features and the signature as the response. Additionally, we propose to estimate the all-terminal network reliability based on the signature estimated by the DNN, termed the two-stage DNN approach, which does not require the link reliability as one of the inputs, resulting in better estimation accuracy and generation performance compared to traditional DNN approaches. A case study is conducted and the results show that the estimation accuracy of our DNN model for the signature is satisfactory, and the two-stage DNN approach for network reliability outperforms existing DNN approaches in the literature.

The active redundancy technique is widely used in reliability engineering to enhance the performance of reliability systems. In this paper, we utilize the criticality ordering to study optimal allocations of active redundancies in coherent systems comprising independent components. Firstly, we establish optimal allocation strategies for two heterogeneous active redundancies for components ranked by the criticality ordering, which is characterized by the structure relationships induced from the associated minimal cut sets. Our findings demonstrate that, under certain conditions where the criticality ordering (and reliability performances) between any two original components is required, the stronger redundancy should be allocated to the more critical component. Secondly, we investigate the best allocation policies for a batch of homogeneous active spares for different components while preserving the criticality ordering. It is shown that allocating more active redundancies to the more critical component results in a more reliable coherent system. Additionally, we prove that under certain conditions, an imbalance in the numbers of allocated spares leads to even greater reliability in the system. Thirdly, we study optimal allocations for any two components in coherent systems when their locations are quantified by minimal path sets. The optimal allocation policies are further derived, most of which are similar to those obtained by applying minimal cut sets. Numerical examples are provided to illustrate the main findings, and we apply our findings to a real-world scenario in an airplane cockpit system to enhance its reliability.

Recently, two-way or longitudinal functional data analysis has attracted much attention in many fields. However, little is known on how to appropriately characterize the association between two-way functional predictor and scalar response. Motivated by a mortality study, in this paper, we propose a novel two-way functional linear model, where the response is a scalar and functional predictor is two-way trajectory. The model is intuitive, interpretable and naturally captures relationship between each way of two-way functional predictor and scalar-type response. Further, we develop a new estimation method to estimate the regression functions in the framework of weak separability. The main technical tools for the construction of the regression functions are product functional principal component analysis and iterative least square procedure. The solid performance of our method is demonstrated in extensive simulation studies. We also analyze the mortality dataset to illustrate the usefulness of the proposed procedure.

AbstractPareto distribution is an important distribution in extreme value theory. In this paper, we consider parallel systems with Pareto components and study the effect of heterogeneity on skewness of such systems. It is shown that, when the lifetimes of components have different shape parameters, the parallel system with heterogeneous Pareto component lifetimes is more skewed than the system with independent and identically distributed Pareto components. However, for the case when the lifetimes of components have different scale parameters, the result gets reversed in the sense of star ordering. We also establish the relation between star ordering and dispersive ordering by extending the result of Deshpande and Kochar [(1983). Dispersive ordering is the same as tail ordering. Advances in Applied Probability 15(3): 686–687] from support $(0, \infty )$ to general supports $(a, \infty )$, $a > 0$. As a consequence, we obtain some new results on dispersion of order statistics from heterogeneous Pareto samples with respect to dispersive ordering.

Taking repair action has proven to be an effective and flexible way to maintain the proper functioning of reliability systems. As a generalization of the minimal repair policy, the relevation is one of models for describing the repair effect. However, there are few studies in the literature dealing with the optimal allocation problem of repair resources to coherent systems because of their complex distribution theory. In this study, we tackle the allocation problem of a single relevation resource for coherent systems. Sufficient conditions based on the orderings among components lifetimes and repair effects are established for improving the system reliability by distinguishing the structural relationships of the minimal path/cut sets, which answer the problems proposed by Belzunce et al. Several numerical examples are also presented to illustrate the main results.

The generalized aggregation n-ary sumation i=1nWi phi(Xi,ai)arises in many research fields including applied probability, actuarial science, and reliability theory, where phi is a bivariate kernel function andais a parameter vector. One of its remarkable features is that bothXandWare dependent in many practical situations. Therefore, studying the stochastic properties of generalized aggregations under various dependence structures is an interesting and meaningful problem. In this paper, by using left tail weakly stochastic arrangement increasing, right tail weakly stochastic arrangement increasing, and comonotonicity to characterize the dependent structures amongXorW, we establish the increasing convex ordering and the expectation ordering of generalized aggregations to investigate the effects of the arrangement and heterogeneity amonga(i)'s. Numerical examples and three practical applications are presented to illustrate our results as well.

Phasor measurement units (PMUs) have been integrated into the smart grid for monitoring the operational state of system and improving the reliability. Due to the high cost of PMU installment, the optimal placement strategies have attracted considerable attention in the literature. However, the impacts of cyber threats on the placement have been largely ignored owing to the cyber complexities. This paper initializes the study on the optimal PMU placement in a smart grid under the cyber threats. A probabilistic model is developed for assessing the unobservable risk of the power grid. We characterize the impacts of several cyber factors on the PMU placements including the number of directly attacked PMUs, the dependence among attack outcomes, and risk propagation. We further study the impacts of cyber attacks on the allocation strategies under a bi-level placement model. In particular, a novel 'greedy' algorithm for PMU placement is introduced with the presence of cyber risks. Our studies show that the cyber risk can significantly increase the unobservability risk of a power system which in turn requires additional PMU allocations, and the dependence among cyber attacks can lead to more unobservable risk.

This paper studies stochastic comparisons between a population and subpopulations in both multiplicative and additive frailty models. The comparisons between a population and its baseline in stochastic ordering are conducted as a special case. We build equivalent characterizations of some common stochastic orders between a population and a subpopulation, in terms of the frailty of the subpopulation and the first two moments of frailty variable. Some examples and applications are discussed as well.

In this article, we tackle the problem of using survival signatures to compare heterogeneous systems with ordered components. The scenarios where systems have the same size and those where they have different sizes are studied. Several sufficient conditions comparing two system lifetimes in the sense of the usual stochastic order are provided. Several numerical examples are presented to illustrate the main results.

The PMU network has been utilized to monitor the system operation of a smart grid in recent decades. This brings a significant cyber risk for the power grid as the attacker can manipulate the PMU network to introduce the false positive and false negative observation errors. A novel risk model is proposed for the PMU networks by considering the observation errors. It is discovered that ignoring the due observation errors can severely underestimate the PMU risks as shown by the theoretical and simulation studies. The risk mitigation strategies for improving the safety levels of PMUs with the observation errors are studied. It is found that the optimal mitigation strategy can be different when the observation errors are present. Numerical examples are presented for illustrations as well.

The Editors-in-Chief have retracted this article [1] because its results are invalid. It also shows considerable overlap with an article by Lu and Sun [2] that was simultaneously under consideration. Additionally, the article shows evidence of authorship manipulation. The authors have not responded to any correspondence regarding this retraction.

This paper studies the variability of both series and parallel systems comprised of heterogeneous (and dependent) components. Sufficient conditions are established for the star and dispersive orderings between the lifetimes of parallel [series] systems consisting of dependent components having multiple-outlier proportional hazard rates and Archimedean [Archimedean survival] copulas. We also prove that, without any restriction on the scale parameters, the lifetime of a parallel or series system with independent heterogeneous scaled components is larger than that with independent homogeneous scaled components in the sense of the convex transform order. These results generalize some corresponding ones in the literature to the case of dependent scenarios or general settings of components lifetime distributions.

In this paper we treat a two-stage grouping procedure of building a k-out-of-n system from several clusters of components. We use a static framework in which the component reliabilities are fixed. Under such a framework, we address the impact of the selecting strategies, the sampling probabilities, and the component reliabilities on the constructed system's reliability. An interesting finding is that the level of component reliabilities could be identified as a decisive factor in determining how the selecting strategies and the component reliabilities affect the system reliability. The new results generalize and extend those established earlier in the literature such as Di Crescenzo and Pellerey (2011), Hazra and Nanda (2014), Navarro, Pellerey, and Di Crescenzo (2015), and Hazra, Finkelstein, and Cha (2017). Several Monte Carlo simulation experiments are provided to illustrate the theoretical results.

The effect of system structure on relative ageing properties of coherent systems has been extensively studied in terms of the increasing [reversed] hazard ratio. In this paper, we investigate the effects of dependence and heterogeneity among components on the relative ageing of series and parallel systems. Numerical examples are provided as illustrations.

This paper studies a maintenance model for an one-unit degenerative system with multiple failure states based on the proportional hazards and proportional reversed hazards models. The authors investigate how the variation of system configuration parameters have an impact on both operating and repair times and hence the system performance. Furthermore, the authors also derive the explicit expression for the long-run average cost per unit time. An algorithm to locate the optimal number of repairs in a renewal cycle is discussed as well.

In this paper, we investigate the equivalence among stochastic orderings between random variables under the scenario where the ratio of density functions or survival functions is log-concave. Some sufficient and necessary conditions that may be easily checked for the usual stochastic order, the hazard rate order, the mean residual lifetime order, the harmonic mean residual lifetime order and the mean inactivity time order are presented. As for illustration, we also provide several numerical examples and three applications.

In the reliability context, the geometric distribution is a natural choice to model the lifetimes of some equipment and components when they are measured by the number of completed cycles of operation or strokes, or in case of periodic monitoring of continuous data. This paper aims at investigating how the heterogeneity among the parameters affects some characteristics such as the distribution and hazard rate functions of spacings arising from independent heterogeneous geometric random variables. First, refined representations of the distribution functions are provided for both the second spacing and sample range from heterogeneous geometric sample. Second, stochastic comparisons are carried out on the second spacings and sample ranges for two sets of independent and heterogeneous geometric random variables in the sense of the usual stochastic and hazard rate orderings. The results established here not only fill the gap on the study of stochastic properties of spacings from heterogeneous geometric samples, but also are expected to be applied in the fields of statistics and reliability.

Using boundary behaviors of solutions for certain Laplace equation proved by Yan and Ychussie (Adv. Difference Equ. 2015:226, 2015) and applying a new method to dispose of the impulsive term with finite mass subject presented by Shi and Liao (J. Inequal. Appl. 2015:363, 2015) from another point of view, we prove that there exists a supra-open in \((X,\tau)\) for each \(V \in\sigma\) in which the modified equilibrium equation has normal families of solutions. Moreover, we establish a new expression of a harmonic multifunction for the above equation. As applications, we not only prove the existence of normal families of solutions for modified equilibrium equations but also obtain several characterizations and fundamental properties of these new classes of superharmonic multifunctions.