A stochastic model for a general load‐sharing system under overload condition

A load‐sharing parallel system functions if at least one unit in the system is functioning and the surviving units share the load. In most of research on load‐sharing system, the performance of the system has been studied only for the case when the lifetimes of components in the system follow exponential distributions. In this paper a load‐sharing parallel system is considered when the lifetimes of the units in the system are any continuous random variables. The reliability function of the system is derived and the problem of load allocation is also considered. Copyright © 2009 John Wiley & Sons, Ltd.     

Statistical inference for coherent systems with Weibull distributed component lifetimes under complete and incomplete information

Point estimators for the parameters of the component lifetime distribution in coherent systems are evolved assuming to be independently and identically Weibull distributed component lifetimes. We study both complete and incomplete information under continuous monitoring of the essential component lifetimes. First, we prove that the maximum likelihood estimator (MLE) under complete information based on progressively Type‐II censored system lifetimes uniquely exists and we present two approaches to compute the estimates. Furthermore, we consider an ad hoc estimator, a max‐probability plan estimator and the MLE for the parameters under incomplete information. In order to compute the MLEs, we consider a direct maximization of the likelihood and an EM‐algorithm–type approach, respectively. In all cases, we illustrate the results by simulations of the five‐component bridge system and the 10‐component parallel system, respectively.     

Mean residual life and increasing convex comparison of shock models

Two devices are subjected to shocks arriving according to a general counting process. Let M₁ and M₂ be the random number of shocks that cause the failure of the first and the second device, respectively. We find conditions on the counting process such that the mean residual life ordering, the increasing convex ordering and the expectation ordering between M₁ and M₂ are preserved in the random lifetimes of the two devices.     

A semiparametric Bayesian generalized linear mixed model for the reliability of Kevlar fibers

We analyze the reliability of NASA composite pressure vessels by using a new Bayesian semiparametric model. The data set consists of lifetimes of pressure vessels, wrapped with a Kevlar fiber, grouped by spool, subject to different stress levels; 10% of the data are right censored. The model that we consider is a regression on the log‐scale for the lifetimes, with fixed (stress) and random (spool) effects. The prior of the spool parameters is nonparametric, namely they are a sample from a normalized generalized gamma process, which encompasses the well‐known Dirichlet process. The nonparametric prior is assumed to robustify inferences to misspecification of the parametric prior. Here, this choice of likelihood and prior yields a new Bayesian model in reliability analysis. Via a Bayesian hierarchical approach, it is easy to analyze the reliability of the Kevlar fiber by predicting quantiles of the failure time when a new spool is selected at random from the population of spools. Moreover, for comparative purposes, we review the most interesting frequentist and Bayesian models analyzing this data set. Our credibility intervals of the quantiles of interest for a new random spool are narrower than those derived by previous Bayesian parametric literature, although the predictive goodness‐of‐fit performances are similar. Finally, as an original feature of our model, by means of the discreteness of the random‐effects distribution, we are able to cluster the spools into three different groups. Copyright © 2012 John Wiley & Sons, Ltd.     

Modeling and analysis of system reliability using phase‐type distribution closure properties

Phase‐type distribution closure properties are utilized to devise algorithms for generating reliability functions of systems with basic structures. These structures include series, parallel, K‐out‐of‐N, and standby structures with perfect/imperfect switch. The algorithms form a method for system reliability modeling and analysis based on the relationship between the system lifetime and component lifetimes for general structures. The proposed method is suitable for functional system reliability analysis, which can produce reliability functions of systems with independent components instead of only system reliability values. Once the system reliability function is obtained, other reliability measures such as the system's hazard function and mean time to failure can be obtained efficiently using only matrix algebra. Dimensional and numerical comparisons with computerized symbolic processing are also presented to show the superiority of the proposed method.     

Order statistics of uncertain random variables with application to <Emphasis Type="Italic">k</Emphasis>-out-of-<Emphasis Type="Italic">n</Emphasis> system

Uncertain random variables are tools to deal with a mixture of uncertainty and randomness. A new concept of order statistics associated with uncertain random variables is proposed, and is applied to analyze k-out-of-n systems with uncertain random lifetimes. The chance distributions of order statistics of uncertain random variables are derived from the operational law of uncertain random variables. Finally, the reliability of k-out-of-n systems with uncertain random lifetimes is discussed.     

Random walks systems on complete graphs

We study two versions of random walks systems on complete graphs. In the first one, the random walks have geometrically distributed lifetimes so we define and identify a non-trivial critical parameter related to the proportion of visited vertices before the process dies out. In the second version, the lifetimes depend on the past of the process in a non-Markovian setup. For that version, we present results obtained from computational analysis, simulations and a mean field approximation. These three approaches match.     

Formal verification of tail distribution bounds in the HOL theorem prover

Tail distribution bounds play a major role in the estimation of failure probabilities in performance and reliability analysis of systems. They are usually estimated using Markov's and Chebyshev's inequalities, which represent tail distribution bounds for a random variable in terms of its mean or variance. This paper presents the formal verification of Markov's and Chebyshev's inequalities for discrete random variables using a higher‐order‐logic theorem prover. The paper also provides the formal verification of mean and variance relations for some of the widely used discrete random variables, such as Uniform(m), Bernoulli(p), Geometric(p) and Binomial(m, p) random variables. This infrastructure allows us to precisely reason about the tail distribution properties and thus turns out to be quite useful for the analysis of systems used in safety‐critical domains, such as space, medicine or transportation. For illustration purposes, we present the performance analysis of the coupon collector's problem, a well‐known commercially used algorithm. Copyright © 2008 John Wiley & Sons, Ltd.     

On allocating redundancies to k‐out‐of‐ n reliability systems

For two components in series and one redundancy with their lifetimes following the proportional hazard models, we build the likelihood ratio order and the hazard rate order for lifetimes of the redundant systems. Also, for k ‐out‐of‐ n system with components’ lifetimes having the arrangement increasing joint density and the redundancies having identically distributed lifetimes, allocating more redundancies to weaker components is shown to help improve the system's reliability. Copyright © 2014 John Wiley & Sons, Ltd.     

Two shock and wear systems under repair standing a finite number of shocks

A shock and wear system standing a finite number of shocks and subject to two types of repairs is considered. The failure of the system can be due to wear or to a fatal shock. Associated to these failures there are two repair types: normal and severe. Repairs are as good as new. The shocks arrive following a Markovian arrival process, and the lifetime of the system follows a continuous phase-type distribution. The repair times follow different continuous phase-type distributions, depending on the type of failure. Under these assumptions, two systems are studied, depending on the finite number of shocks that the system can stand before a fatal failure that can be random or fixed. In the first case, the number of shocks is governed by a discrete phase-type distribution. After a finite (random or fixed) number of non-fatal shocks the system is repaired (severe repair). The repair due to wear is a normal repair. For these systems, general Markov models are constructed and the following elements are studied: the stationary probability vector; the transient rate of occurrence of failures; the renewal process associated to the repairs, including the distribution of the period between replacements and the number of non-fatal shocks in this period. Special cases of the model with random number of shocks are presented. An application illustrating the numerical calculations is given. The systems are studied in such a way that several particular cases can be deduced from the general ones straightaway. We apply the matrix-analytic methods for studying these models showing their versatility.     

Some results for repairable systems with minimal repairs

In this paper, we consider a repairable system with minimal repairs whose number of repairs is a positive random variable with a given probability vector. Some preservation theorems and aging properties of repairable systems are established. Under the condition that at time t the system is working, a new random variable for the residual lifetime of the system is proposed. Some stochastic ordering results among the lifetimes and residual lifetimes of two systems are obtained. Similar results for coherent systems with independent components and exchangeable components were obtained in the previous literature. Copyright © 2013 John Wiley & Sons, Ltd.     

A weak‐derivative form for linear hyperbolic systems

Difference schemes for linear hyperbolic systems are considered. As a main result, a weak derivative form (WDF) of the governing equations is derived, which is also valid near flow discontinuities. The occurrence of one‐sided derivatives in the WDF structure indicated how to difference near discontinuities. When first‐order differencing is applied to the WDF result, the (linearly identical) schemes by Godunov, Roe, and Steger‐Warming are reproduced. The extension to nonlinear systems is via a local linearization. Choosing Roe's averaging reduces the WDF algorithm to Roe's scheme, whereas other nonlinear WDF schemes are possible. The suitability of various kinds of averaging is numerically investigated. For weak shocks a surprising lack of sensitivity of the method to a particular averaging is exhibited. However, for strong shocks and where the ordinary arithmetic average is used, a slightly more pronounced difference in performance exists between Roe's scheme and WDF. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004     

Least‐squares finite element method on adaptive grid for PDEs with shocks

We apply the least‐squares finite element method with adaptive grid to nonlinear time‐dependent PDEs with shocks. The least‐squares finite element method is also used in applying the deformation method to generate the adaptive moving grids. The effectiveness of this method is demonstrated by solving a Burgers' equation with shocks. Computational results on uniform grids and adaptive grids are compared for the purpose of evaluation. The results show that the adaptive grids can capture the shock more sharply with significantly less computational time. For moving shock, the adaptive grid moves correctly with the shock. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Some results on information properties of coherent systems

This paper considers information properties of coherent systems when component lifetimes are independent and identically distributed. Some results on the entropy of coherent systems in terms of ordering properties of component distributions are proposed. Moreover, various sufficient conditions are given under which the entropy order among systems as well as the corresponding dual systems hold. Specifically, it is proved that under some conditions, the entropy order among component lifetimes is preserved under coherent system formations. The findings are based on system signatures as a useful measure from comparison purposes. Furthermore, some results on the system's entropy are derived when lifetimes of components are dependent and identically distributed. Several illustrative examples are also given.     

Modelling redundancy allocation for a fuzzy random parallel-series system

Due to subjective judgment, imprecise human knowledge and perception in capturing statistical data, the real data of lifetimes in many systems are both random and fuzzy in nature. Based on the fuzzy random variables that are used to characterize the lifetimes, this paper studies the redundancy allocation problems to a fuzzy random parallel-series system.Two fuzzy random redundancy allocation models (FR-RAM) are developed through reliability maximization and cost minimization, respectively. Some properties of the FR-RAM are obtained, in which an analytical formula of reliability with convex lifetimes is derived and the sensitivity of the reliability is discussed. To solve the FR-RAMs, we first address the computation of reliability. A random simulation method based on the derived analytical formula is proposed to compute the reliability with convex lifetimes. As for the reliability with nonconvex lifetimes, the technique of fuzzy random simulation together with the discretization method of fuzzy random variable is employed to compute the reliability, and a convergence theorem of the fuzzy random simulation is proved. Subsequently, we integrate the computation approaches of the reliability and genetic algorithm (GA) to search for the approximately optimal redundancy allocation of the models. Finally, some numerical examples are provided to illustrate the feasibility of the solution algorithm and quantify its effectiveness.     

System availability in a shock model under preventive repair and phase‐type distributions

A device that can fail by shocks or ageing under policy N of maintenance is presented. The interarrival times between shocks follow phase‐type distributions depending on the number of cumulated shocks. The successive shocks deteriorate the system, and some of them can be fatal. After a prefixed number k of nonfatal shocks, the device is preventively repaired. After a fatal shock the device is correctively repaired. Repairs are as good as new, and follow phase‐type distributions. The system is governed by a Markov process whose infinitesimal generator, stationary probability vector, and availability are calculated, obtaining well‐structured expressions due to the use of phase‐type distributions. The availability is optimized in terms of the number k of preventive repairs. Copyright © 2009 John Wiley & Sons, Ltd.     

Multi‐switching combination–combination synchronization of non‐identical fractional‐order chaotic systems

In this paper, multi‐switching combination–combination synchronization scheme has been investigated between a class of four non‐identical fractional‐order chaotic systems. The fractional‐order Lorenz and Chen's systems are taken as drive systems. The combination–combination of multi drive systems is then synchronized with the combination of fractional‐order Lü and Rössler chaotic systems. In multi‐switching combination–combination synchronization, the state variables of two drive systems synchronize with different state variables of two response systems simultaneously. Based on the stability of fractional‐order chaotic systems, the multi‐switching combination–combination synchronization of four fractional‐order non‐identical systems has been investigated. For the synchronization of four non‐identical fractional‐order chaotic systems, suitable controllers have been designed. Theoretical analysis and numerical results are presented to demonstrate the validity and feasibility of the applied method. Copyright © 2017 John Wiley & Sons, Ltd.     

Local limit theorems for occupancy models

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with d neighbors in a germ‐grain model, and the number of degree‐d vertices in an Erd?s‐Rényi random graph. In both cases, the error rate is optimal, up to logarithmic factors.