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We consider coherent systems subjected to Marshall-Olkin type shocks coming at random times and destroying components of the system. The paper combines two important models, coherent systems and Marshall-Olkin type shocks and studies the mean residual life (MRL) and the mean inactivity time (MIT) functions of coherent systems that is subjected to random shocks. The considered models and theoretical results are supported with examples and graphical representations. (C) 2016 Published by Elsevier B.V.

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... The MO type shock model in coherent systems was considered in [16]. The reliability, mean residual life and inactivity time of a coherent system subjected to the MO type shocks were discussed in [12]. Moreover, run shock and MO models were discussed in [42]. ...

Over the past two decades, a significant part of the statistical literature has been devoted to offer distinct univariate distributions belonging to the Marshall-Olkin family of distributions. It is because this family enjoys attractive statistical properties, providing consistently better fit than other generalized distributions with the same parental models, as well as wider applications. In this article, we provide a brief review of recent developments in Marshall-Olkin type distributions.

... Another examples are the utilization of the MRL functions of parallel system by Sadegh [12], the MRL for records by Raqab and Asadi [13], the MRL of a k-out-of-n:G system by Eryilmaz [14], the MRL of a (n − k + 1)-out-of-n system by Poursaeed [15], the MRL in reliability shock models by Eryilmaz [16], the MRL subjected to Marshall-Olkin type shocks by Bayramoglu and Ozkut [17], the MRL of coherent systems by Eryilmaz et al. [18] and Kavlak [19], the MRL for degrading systems by Zhao et al. [20], and the MRL of rail wagon bearings by Ghasemi and Hodkiewicz [21]. ...

We propose two new kernel-type estimators of the mean residual life function $m_X(t)$ of bounded or half-bounded interval supported distributions. Though not as severe as the boundary problems in the kernel density estimation, eliminating the boundary bias problems that occur in the naive kernel estimator of the mean residual life function is needed. In this article, we utilize the property of bijective transformation. Furthermore, our proposed methods preserve the mean value property, which cannot be done by the naive kernel estimator. Some simulation results showing the estimators' performances and a real data analysis will be presented in the last part of this article.

... Since the lifetime random variable S i is already defined as S i = min(T i , T n+1 ), i = 1, 2, ..., n, using Proposition 2 together with Lemma 3 of [23], one can easily derive the following Lemma. P (S r:n > t, S m:n > t + x) Proof. ...

... 0 and P m i = 1 p i = 1. Thus, equation (9) transforms to inequality Var½Z ø ( P m i = 1 p i v i ) 2 . Note that the right-hand side of this inequality is the variance of an exponential random variable with expectation P m i = 1 p i v i . ...

Reliability assessment of system suffering from random shocks is attracting a great deal of attention in recent years. Excluding internal factors such as aging and wear-out, external shocks which lead to sudden changes in the system operation environment are also important causes of system failure. Therefore, efficiently modeling the reliability of such system is an important applied problem. A variety of shock models are developed to model the inter-arrival time between shocks and magnitude of shocks. In a cumulative shock model, the system fails when the cumulative magnitude of damage caused by shocks exceed a threshold. Nevertheless, in the existing literatures, only the magnitude is taken into consideration, while the source of shocks is usually neglected. Using the same distribution to model the magnitude of shocks from different sources is too critical in real practice. To this end, considering a system subject to random shocks from various sources with different probabilities, we develop a generalized cumulative shock model in this article. We use phase-type distribution to model the variables, which is highly versatile to be used for modeling quantitative features of random phenomenon. We will discuss the reliability characteristics of such system in some detail and give some clear expressions under the one-dimensional case. Numerical example for illustration is also provided along with a summary.

... Durante et al. [4] studied Marshall-Olkin type copulas generated by a global shock. Bayramoglu and Ozkut [5] considered coherent systems subjected to Marshall-Olkin type shocks coming at random times and destroying components of the system. Shock models have been extensively studied in the literature. ...

In this paper, a new shock model called Marshall–Olkin run shock model is defined and studied. According to the model, two components are subject to shocks that may arrive from three different sources, and component i fails when it is subject to k consecutive critical shocks from source i or k consecutive critical shocks from source 3, i=1,2. Reliability and mean residual life functions of such components are studied when the times between shocks follow phase-type distribution.

... The problem of reliability evaluation in a system failing according to a MO model has been addressed by different techniques. Analytical expressions can be obtained for specific topologies, for example, for the k-out-of-n system (see [11]- [13]), but for general networks, even in the exchangeable case, we only rely on simulation techniques [14]. Up to our knowledge, the only work dealing with the nonexchangeable case is [15], that proposed different rare-event simulation techniques to compute the static network reliability. ...

The Marshall–Olkin (MO) copula model has emerged as the standard tool for capturing dependence between components in failure analysis in reliability. In this model, shocks arise at exponential random times, that affect one or several components inducing a natural correlation in the failure process. However, because the number of parameter of the model grows exponentially with the number of components, MO suffers of the “curse of dimensionality.” MO models are usually intended to be applied to design a network before its construction; therefore, it is natural to assume that only partial information about failure behavior can be gathered, mostly from similar existing networks. To construct such an MO model, we propose an optimization approach to define the shock’s parameters in the MO copula, in order to match marginal failures probabilities and correlations between these failures. To deal with the exponential number of parameters of this problem, we use a column-generation technique. We also discuss additional criteria that can be incorporated to obtain a suitable model. Our computational experiments show that the resulting MO model produces a close estimation of the network reliability, especially when the correlation between component failures is significant. IEEE

... Zhang and Meeker (2013) obtained mixture representations of the reliability functions of the residual life and inactivity time of a coherent system with n independent and identically distributed components, given that before time t 1 , exactly r (r < n) components have failed and at time t 2 , the system is either still working or has failed. Some recent discussions on the mean residual life of systems can be found in Navarro and Gomis (2016), Bayramoglu and Ozkut (2016), and Bayramoglu Kavlak (2017). ...

Mean residual life is a useful dynamic characteristic to study reliability of a system. It has been widely considered in the literature not only for single unit systems but also for coherent systems. This article is concerned with the study of mean residual life for a coherent system that consists of multiple types of dependent components. In particular, the survival signature based generalized mixture representation is obtained for the survival function of a coherent system and it is used to evaluate the mean residual life function. Furthermore, two mean residual life functions under different conditional events on components’ lifetimes are also defined and studied.

... Although the concept of residual life has been well studied in the literature under active redundancy ( [1], [2], [5], [6], [8], [10], [3], [13], [14], [16]), it has not been considered for a system with a standby unit. In the case of cold standby redundancy, the standby redundant component neither degrade nor fail while in standby. ...

In this paper, we define and study two different residual life random variables corresponding to
a single unit system equipped with a cold standby unit. We obtain the conditional survival functions when the lifetimes of active and standby units are dependent. Some properties of the associated mean residual life functions are also investigated. Graphical illustrations are presented to observe time dependent behaviors of associated mean residual life function.

... An interesting extension of the concept of VRL at the system level for more complex systems, can be considered for the case where the system has a (n − k + 1)-out-of-n structure, k = 1, 2, . . . , n (see Asadi and Bayramoglu 2006) as well as the MOSE type (n − k + 1)-out-of-n: G system (see Bayramoglu and Ozkut 2016). ...

As a measure of maximum dispersion from the mean, upper bounds on variance have applications in all areas of theoretical and applied mathematical sciences. In this paper, we obtain an upper bound for the variance of a function of the residual life random variable . Since one of the most important types of system structures is the parallel structure, we give an upper bound for the variance of a function of this system consisting of identical and independent components, under the condition that, at time , , of its components are still working. Here we characterize the Pareto distribution through Cauchy’s functional equation for mean residual life. It is shown that the underlying distribution function can be recovered from the proposed mean and variance residual life function of the system for . Moreover, we see that the variance residual lifetime of the components of the system is not necessarily a decreasing function of and increasing of for , unlike their mean residual lifetime. As an application, the variance of for all is investigated and also a real data analysis is presented.

In this paper, we investigate optimal age-based preventive maintenance (PM) policies for an ( n- k + 1)-out-of- n system whose components are exposed to fatal shocks that arrive from various sources. We consider two different scenarios for the system failure. In the first one, it is assumed that the shock process is of the type of Marshall-Olkin where each shock affects one component of the system and puts it down, and one shock affects all components and destroys all of them. In the second scenario, it is assumed that the system is subject to an extended type of Marshall-Olkin shock process where the shocks arriving at random times may cause the breakdown of 1, 2, …, or n components. Under each scenario for the components failure, we investigate an optimal age-based PM model for the system by imposing the related cost function. Then, in each case, we explore the optimal PM time that minimizes the mean cost per unit of time. Some numerical results are presented to illustrate the applications of the proposed models.

This paper deals with residual lifetimes of sequential k-out-of-n systems when component lifetimes are dependent. Given the first j component failure times, an unbiased predictor for the (j + i)-th component failure time is derived. Maximum likelihood prediction of future component failure times is also studied. Estimation of the nuisance parameters is investigated under the conditional proportional hazard rates model with the Exponential baseline distribution and the Gumbel-Hougaard copula. Illustrative examples are also given.

There are several failure modes may cause system failed in reliability and survival analysis. It is usually assumed that the causes of failure modes are independent each other, though this assumption does not always hold. Dependent competing risks modes from Marshall-Olkin bivariate Weibull distribution under Type-I progressive interval censoring scheme are considered in this paper. We derive the maximum likelihood function, the maximum likelihood estimates, the 95% Bootstrap confidence intervals and the 95% coverage percentages of the parameters when shape parameter is known, and EM algorithm is applied when shape parameter is unknown. The Monte-Carlo simulation is given to illustrate the theoretical analysis and the effects of parameters estimates under different sample sizes. Finally, a data set has been analyzed for illustrative purposes.

For practical applications in reliability analysis, the assumption of dependence among lifetimes of components of the system is more realistic than the assumption of independence. This paper investigates the residual lifetimes of coherent systems in situation where there exists a stochastic dependence among the components of the system. A stochastic comparison among residual lives of k-out-of-n systems with exchangeable components is conducted. The mean residual life function of a k-out-of-n system with exchangeable components is investigated. Special examples in the case of a system consisting of two dependent components with a joint Gumbel's bivariate exponential distribution and for a system having n components with Marshall and Olkin's multivariate exponential distribution, illustrating the behavior of the mean residual life function are provided. The extension to general coherent systems with exchangeable components using properties of Samaniego's signature are given.

In this paper we introduce a new probability model known as type 2 Marshall–Olkin bivariate Weibull distribution as an extension
of type 1 Marshall–Olkin bivariate Weibull distribution of Marshall–Olkin (J Am Stat Assoc 62:30–44, 1967). Various properties
of the new distribution are considered. Bivariate minification processes with the two types of Weibull distributions as marginals
are constructed and their properties are considered. It is shown that the processes are strictly stationary. The unknown parameters
of the type 1 process are estimated and their properties are discussed. Some numerical results of the estimates are also given.
KeywordsType 1 and type 2 Marshall–Olkin bivariate Weibull distribution–Marshall–Olkin bivariate exponential distribution–Bivariate minification process–Stationary process–Estimation

In this article we study profust reliability of non-repairable coherent systems through the concept of system signature. We obtain explicit expressions for the profust reliability and mean time to fuzzy failure of coherent systems. We compute and present mean time to failure and mean time to fuzzy failure of all coherent systems with three and four components. Finally, we illustrate the results for a well known class of coherent systems called m-consecutive-k-out-of-n:F.

This paper investigates properties of a new parametric distribution generated by Marshall and Olkin (199712.
Marshall , A. W. ,
Olkin , I. ( 1997 ). A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families . Biometrika 84 : 641 – 652 . [CrossRef], [Web of Science ®]View all references) extended family of distributions based on the Lomax model. We show that the proposed distribution can be expressed as a compound distribution with mixing exponential model. Simple sufficient conditions for the shape behavior of the density and hazard rate functions are given. The limiting distributions of the sample extremes are shown to be of the exponential and Fréchet type. Finally, utilizing maximum likelihood estimation, the proposed distribution is fitted to randomly censored data.

We define the survival and mean residual life function of system consisting of n identical and independent components having se- ries or parallel structure. Let Xi, i = 1,2,...,n be the survival time of i th component, such that X1,X2,...,Xn are independent, identically dis- tributed random variables with continuous distribution functionF. LetXi:n, i = 1,2,...,n be the i th smallest among X1,X2,...,Xn. The mean resid- ual life function of system having parallel structure function is defined as n(t) = E(Xn:n t | X1:n > t), which can be interpreted as the condi- tional expectation of residual life length of the system given X1:n > t — none of the components of the system fails at time t. The inverse formula is obtained; i.e.. it is shown that knowledge of n and n 1 for some n, amounts to knowing F. Similar residual life function is defined for a sys- tem with functioning series structure, and the inverse formula is given. For parallel structure it is also considered regressing Xn:n on X1:n wich can be interpreted as the best predictor of the life length of the system knowing the time when weakest component fails. Some extensions of obtained results to a systems having more complex structure are discussed.

Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a system's lifetime distribution as a function of the system's “signature,” that is, as a function of the vector p= (p1, … , pn), where pi is the probability that the system fails upon the occurrence of the ith component failure. Sufficient conditions are provided for the lifetime of one system to be larger than that of another system in three different senses: stochastic ordering, hazard rate ordering, and likelihood ratio ordering. Further, a new preservation theorem for hazard rate ordering is established. In the final section, the notion of system signature is used to examine a recently published conjecture regarding componentwise and systemwise redundancy. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 507–523, 1999

In the present paper, we consider a (n − k+1)-out-of-n system with identical components where it is assumed that the lifetimes of the components are independent and have a common
distribution function F. We assume that the system fails at time t or sometime before t, t>0. Under these conditions, we are interested in the study of the mean time elapsed since the failure of the components.
We call this as the mean past lifetime (MPL) of the components at the system level. Several properties of the MPL are studied.
It is proved that the relation between the proposed MPL and the underlying distribution is one-to-one. We have shown that
when the components of the system have decreasing reversed hazard then the MPL of the system is increasing with respect to
time. Some examples are also provided.
KeywordsTruncated expectation-Order statistics-Reversed hazard rate-Parallel system-Reliability function

A representation is derived for the failure rate of an arbitrary s-coherent system when the lifetimes of its components are s-independently distributed according to a common absolutely continuous distribution F. The system failure rate is written explicitly as a function of F and its failure rate. The representation is used in several examples, including an example showing that the closure theorem for k-out-of-n systems in i.i.d. IFR components proven by Barlow & Proschan cannot be extended to all s-coherent systems. The class of s-coherent systems for which such closure obtains is characterized.

This paper investigates some ordering properties of the residual lives and the inactivity times of coherent systems with dependent exchangeable absolutely continuous components, based on the stochastically ordered signatures between systems, extending the results of Li and Zhang [2008. Some stochastic comparisons of conditional coherent systems. Applied Stochastic Models in Business and Industry 24, 541–549] for the case of independent and identically distributed components.

Marshall-Olkin bivariate semi-Pareto distribution (MO-BSP) and Marshall-Olkin bivariate Pareto distribution (MO-BP) are introduced and studied. AR(1) and AR(k) time series models are developed with minification structure having MO-BSP stationary marginal distribution. Various characterizations are investigated. Copyright Springer-Verlag 2004

Recently it has been observed that the generalized exponential distribution can be used quite effectively to analyze lifetime data in one dimension. The main aim of this paper is to define a bivariate generalized exponential distribution so that the marginals have generalized exponential distributions. It is observed that the joint probability density function, the joint cumulative distribution function and the joint survival distribution function can be expressed in compact forms. Several properties of this distribution have been discussed. We suggest to use the EM algorithm to compute the maximum likelihood estimators of the unknown parameters and also obtain the observed and expected Fisher information matrices. One data set has been re-analyzed and it is observed that the bivariate generalized exponential distribution provides a better fit than the bivariate exponential distribution.

A new class of bivariate distributions is presented in this paper. The procedure used in this paper is based on a latent random variable with exponential distribution. The model introduced here is of Marshall-Olkin type. A mixture of the proposed bivariate distributions is also discussed. The results obtained here generalize those of the bivariate exponential distribution present in the literature.

SUMMARY A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.

In the classical Marshall–Olkin model, the system is subjected to two types of shocks coming at random times, and destroying components of the system. In statistics and reliability engineering literature, there are numerous papers dealing with various extensions of this model. However, none of these works takes into account the system structure, i.e., in existing shock models usually the system structure is not considered. In this work, we consider a new shock model involving the system structure. More precisely, we consider a coherent system which is subjected to Marshall–Olkin type shocks. We investigate the reliability, and mean time to failure (MTTF) of such systems subjected to shocks coming at random times. Numerical examples and graphs are provided, and an extension to a general model is discussed.

This article extends A. W. Marshall and I. Olkin’s bivariate exponential distribution [ibid. 62, 30-44 (1967; Zbl 0147.381)] such that it is absolutely continuous and need not be memoryless. The new marginal distribution has an increasing failure rate, and the joint distribution exhibits an aging pattern. It offers an advantage in separately identifying the shock arrival rates and their impacts. Regarding estimation of the model, both maximum likelihood and method-of- moments-type estimation are considered. The former is more efficient but computationally more demanding, whereas the latter is simpler in computation but less efficient. The trade-off between computational burden and efficiency is gauged through Monte Carlo simulations, and it turns out to be favorable for the method-of-moments-type estimation.

Given a coherent reliability system, let Z be the age of the machine at breakdown, and I the set of parts dead by time Z. It is proved that if all lifetime distributions are non-atomic and share the same essential extrema, and if the incidence matrix of the minimal cut sets has rank equal to the number of parts, then the joint distribution of Z and I determines uniquely the lifetime distribution of each part. A Newton-Kantorovic iterative method is presented for the computation of those distributions. The relaxation of the assumptions and the statistical problem, where instead of the joint distribution of Z and I one has an empirical estimate of this joint distribution are dealt with informally.

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution. © Australian Statistical Publishing Association Inc. 1999. Published by Blackwell Publishers Ltd.

In classical Marshall–Olkin type shock models and their modifications a system of two or more components is subjected to shocks that arrive from different sources at random times and destroy the components of the system. With a distinctive approach to the Marshall–Olkin type shock model, we assume that if the magnitude of the shock exceeds some predefined threshold, then the component, which is subjected to this shock, is destroyed; otherwise it survives. More precisely, we assume that the shock time and the magnitude of the shock are dependent random variables with given bivariate distribution. This approach allows to meet requirements of many real life applications of shock models, where the magnitude of shocks is an important factor that should be taken into account. A new class of bivariate distributions, obtained in this work, involve the joint distributions of shock times and their magnitudes. Dependence properties of new bivariate distributions have been studied. For different examples of underlying bivariate distributions of lifetimes and shock magnitudes, the joint distributions of lifetimes of the components are investigated. The multivariate extension of the proposed model is also discussed.

Consecutive-k-out-of-n system and its generalisations have attracted substantial interest due to their applications to model telecommunications systems, oil pipeline systems, heating systems, etc. One of the most important generalisation of this system is an m-consecutive-k-out-of-n:F system. An m-consecutive-k-out-of-n:F system with overlapping runs is a system consisting of n linearly ordered components and fails if and only if there are at least m overlapping runs of k consecutive failures. For m = 1, the system is same with the usual consecutive-k-out-of-n:F system. In this paper, we study the reliability properties of such systems via system signature. In particular, we obtain a closed form expression for the signature of this system and use it to compute and evaluate several reliability characteristics of the system.

The concept of “signature” is a useful tool to study the reliability properties of a coherent system. In this paper, we consider a coherent system consisting of n components and assume that the system is not working at time t. Mixture representations of the inactivity times (IT) of the system and IT of the components of the system are obtained under different scenarios on the signatures of the system. Some stochastic comparisons are made on IT of the coherent systems with same type and different type of components and some aging properties of the IT of the system and its components are investigated. It is proved, under some conditions on the vector of signatures of the system, that when the components of the system have decreasing reversed hazard rate, the mean of the IT (MIT) of the system and the MIT of the components of the system are increasing in time. Several examples and illustrative graphs are also provided.

This article extends Marshall and Olkin's bivariate exponential distribution such that it is absolutely continuous and need not be memoryless. The new marginal distribution has an increasing failure rate, and the joint distribution exhibits an aging pattern. It offers an advantage in separately identifying the shock arrival rates and their impacts. Regarding estimation of the model, both maximum likelihood and method-of-moments-type estimation are considered. The former is more efficient but computationally more demanding, whereas the latter is simpler in computation but less efficient. The trade-off between computational burden and efficiency is gauged through Monte Carlo simulations, and it turns out to be favorable for the method-of-moments-type estimation.

In this paper we obtain several mixture representations of the reliability
function of the inactivity time of a coherent system under the condition that
the system has failed at time t (> 0) in terms of the reliability
functions of inactivity times of order statistics. Some ordering properties of
the inactivity times of coherent systems with independent and
identically distributed components are obtained, based on the stochastically
ordered coefficient vectors between systems.

A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. This paper presents some meaningful derivations of a multivariate exponential distribution that serves to indicate conditions under which the distribution is appropriate. Two of these derivations are based on “shock models,” and one is based on the requirement that residual life is independent of age. It is significant that the derivations all lead to the same distribution.For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. A multivariate Weibull distribution is obtained through a change of variables.

This paper conducts stochastic comparison on general residual life and general inactivity time of (n − k + 1)-out-of-n systems and investigates the stochastic behavior of the general inactivity time of a system with units having decreasing reversed hazard rate. These results strengthen some conclusions in both Khaledi and Shaked (2006) and Hu et al. (2007).

A system with n independent components which has a k-out-of-n: G structure operates if at least k components operate. Parallel systems are 1-out-of-n: G systems, that is, the system goes out of service when all of its components fail. This paper investigates the mean residual life function of systems with independent and nonidentically distributed components. Some examples related to some lifetime distribution functions are given. We present a numerical example for evaluating the relationship between the mean residual life of the k-out-of-n: G system and that of its components.

Recently Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527] introduced a new bivariate distribution using generalized exponential and exponential distributions. They discussed several interesting properties of this new distribution. Unfortunately, they did not discuss any estimation procedure of the unknown parameters. In this paper using the similar idea as of Sarhan and Balakrishnan [2007. A new class of bivariate distribution and its mixture. Journal of Multivariate Analysis 98, 1508–1527], we have proposed a singular bivariate distribution, which has an extra shape parameter. It is observed that the marginal distributions of the proposed bivariate distribution are more flexible than the corresponding marginal distributions of the Marshall–Olkin bivariate exponential distribution, Sarhan–Balakrishnan's bivariate distribution or the bivariate generalized exponential distribution. Different properties of this new distribution have been discussed. We provide the maximum likelihood estimators of the unknown parameters using EM algorithm. We reported some simulation results and performed two data analysis for illustrative purposes. Finally we propose some generalizations of this bivariate model.

Let T be a lifetime random variable. In order to study the properties of T in reliability theory and survival analysis, several measures are proposed in the literature. Among these measures, hazard rate, mean residual lifetime, reversed hazard rate and the mean past lifetime (MPL) play important roles. In the present paper, we focus mainly on the MPL. We investigate its properties in connection with other reliability measures. Some results on partial ordering and characterization are also given. Finally, we deal with its statistical estimation.

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

This paper investigates coherent systems with independent and identical components. Stochastic comparison on the residual life and the inactivity time of two systems with stochastically ordered signatures is conducted. Copyright © 2008 John Wiley & Sons, Ltd.

Consider a binary, monotone system of n components. The assessment of the parameter vector, θ, of the joint distribution of the lifetimes of the components and hence of the reliability of the system is often difficult due to scarcity of data. It is therefore important to make use of all information in an efficient way. For instance, prior knowledge is often of importance and can indeed conveniently be incorporated by the Bayesian approach. It may also be important to continuously extract information from a system currently in operation. This may be useful both for decisions concerning the system in operation as well as for decisions improving the components or changing the design of similar new systems. As in Meilijson [12], life-monitoring of some components and conditional life-monitoring of some others is considered. In addition to data arising from this monitoring scheme, so-called autopsy data are observed, if not censored. The probabilistic structure underlying this kind of data is described, and basic likelihood formulae are arrived at. A thorough discussion of an important aspect of this probabilistic structure, the inspection strategy, is given. Based on a version of this strategy a procedure for preventive system maintenance is developed and a detailed application to a network system presented. All the way a Bayesian approach to estimation of θ is applied. For the special case where components are conditionally independent given θ with exponentially distributed lifetimes it is shown that the weighted sum of products of generalized gamma distributions, as introduced in Gåsemyr and Natvig [7], is the conjugate prior for θ. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 551–577, 2001.

The three-parameter gamma and three-parameter Weibull distributions are commonly used for analysing any lifetime data or skewed data. Both distributions have several desirable properties, and nice physical interpretations. Because of the scale and shape parameters, both have quite a bit of flexibility for analysing different types of lifetime data. They have increasing as well as decreasing hazard rate depending on the shape parameter. Unfortunately both distributions also have certain drawbacks. This paper considers a three-parameter distribution which is a particular case of the exponentiated Weibull distribution originally proposed by Mudholkar, Srivastava & Freimer (1995) when the location parameter is not present. The study examines different properties of this model and observes that this family has some interesting features which are quite similar to those of the gamma family and the Weibull family, and certain distinct properties also. It appears this model can be used as an alternative to the gamma model or the Weibull model in many situations. One dataset is provided where the three-parameter generalized exponential distribution fits better than the three-parameter Weibull distribution or the three-parameter gamma distribution.

Marshall–Olkin semi-Burr and Marshall–Olkin Burr distributions are introduced and studied. Their various characteristics in
reliability analysis are derived. Applications in time series analysis are discussed.

The signature of a system is a useful tool in a variety of applications including the evaluation of the reliability characteristics of systems, and the comparison of the performance of competing systems. We study the evaluation and application of signatures of systems involving two common failure criteria which are common in real life applications. The failure or survival of these systems generally depends on the number of consecutively failed or working components, or total number of failed or working components in the whole system. We provide a method for obtaining the signatures of such systems. Applications of the results are also presented.

Sharp upper and lower bounds are obtained for the reliability functions and the expectations of lifetimes of coherent systems based on dependent exchangeable absolutely continuous components with a given marginal distribution function, by use of the concept of Samaniego's signature. We first show that the distribution of any coherent system based on exchangeable components with absolutely continuous joint distribution is a convex combination of distributions of order statistics (equivalent to the k-out-of-n systems) with the weights identical with the values of the Samaniego signature of the system. This extends the Samaniego representation valid for the case of independent and identically distributed components. Combining the representation with optimal bounds on linear combinations of distribution functions of order statistics from dependent identically distributed samples, we derive the corresponding reliability and expectation bounds, dependent on the signature of the system and marginal distribution of dependent components. We also present the sequences of exchangeable absolutely continuous joint distributions of components which attain the bounds in limit. As an application, we obtain the reliability bounds for all the coherent systems with three and four exchangeable components, expressed in terms of the parent marginal reliability function and specify the respective expectation bounds for exchangeable exponential components, comparing them with the lifetime expectations of systems with independent and identically distributed exponential components.

In the study of reliability of the technical systems and subsystems, parallel systems play a very important role. In the present paper, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. It is assumed that at time t the system has failed. Under these conditions, we obtain the mean past lifetime (MPL) of the components of the system. Some properties of MPL are studied. It is shown that the underlying distribution function F can be recovered from the proposed MPL. Also, a comparison between two parallel systems are made based on their MPLs in the case where the components of the system are ordered in terms of reversed hazard rate. Finally a characterization of the uniform distribution is given based on MPL.

Consider a system of n components that has the property that there exists a number r(r<n), such that if it is known that at most r components have failed, the system is still functioning with probability 1. Suppose that such a system is equipped with a warning light that comes up at the time of the failure of the rth component. The system is still working then, and we are interested in its residual life. In this paper we obtain some results which stochastically compare the residual lives of such systems with the same type, or with different types, of components. Some applications are given. In particular, we derive upper and lower bounds on the expected residual lives of such systems given that the warning light has not come up yet, and given that the component hazard rate functions are bounded from below or from above by a known constant.

Two different exchangeable samples are considered and these two samples are assumed to be independent of each other. From these two samples a new sample is combined and treated as a single set of observations. The distribution of a single order statistic and the joint distribution of two order statistics for a new mixed sample are derived and expressed in terms of joint distribution functions. As a special case the distribution of a single order statistic and the joint distribution of two nonadjacent order statistics from exchangeable random variables are obtained. The results presented in this paper allows widespread applications in modelling of various lifetime data, biomedical sciences, reliability and survival analysis, actuarial sciences etc., where the assumption of independence of data cannot be accepted and the exchangeability is a more realistic assumption.

A class of generalized bivariate Marshall–Olkin distributions, which includes as special cases the Marshall–Olkin bivariate exponential distribution and the Marshall–Olkin type distribution due to Muliere and Scarsini (1987) [19] are examined in this paper. Stochastic comparison results are derived, and bivariate aging properties, together with properties related to evolution of dependence along time, are investigated for this class of distributions. Extensions of results previously presented in the literature are provided as well.

In this paper we show that the Marshall-Olkin extended Weibull distribution can be obtained as a compound distribution with mixing exponential distribution. In addition, we provide simple sufficient conditions for the shape of the hazard rate function of the distribution. Moreover, we extend the considered distribution to accommodate randomly right censored data. Finally, application of the extended distribution to a data set representing the remission times of bladder cancer patients is given and its goodness-of-fit is demonstrated.

In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose a new definition for the mean residual life (MRL) function of the system, and obtain several properties of that system.