On the application and extension of system signatures in engineering reliability

Following a review of the basic ideas in structural reliability, including signature‐based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes m < n with exchangeable component lifetimes, it is shown that the distribution of an m‐component system's lifetime can be written as a mixture of the distributions of k‐out‐of‐n systems. When the system has n components, the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R–34 (1985) 69–72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

On the conditional residual lifetime of coherent systems under double regularly checking

In this paper, we consider a coherent system with n independent and identically distributed components under the condition that the system is monitored at time instances t₁ and t₂ (t₁ < t₂). First, various mixture representations for reliability function of the conditional residual lifetime of the coherent system are derived under different scenarios at times t₁ and t₂ (t₁ < t₂). Several stochastic comparisons between two systems are also made based on the proposed conditional random variables. Then, we consider the conditional residual lifetime of the functioning components of the system given that j components have failed at time t₁ and the system has failed at time t₂. Some stochastic comparisons on the proposed conditional residual lifetimes are investigated. Several illustrative graphs and examples are also provided.     

Signature based analysis of m‐Consecutive‐k‐out‐of‐n: F systems with exchangeable components

In this article, we study reliability properties of m‐consecutive‐k‐out‐of‐n: F systems with exchangeable components. We deduce exact formulae and recurrence relations for the signature of the system. Closed form expressions for the survival function and the lifetime distribution as a mixture of the distribution of order statistics are established as well. These representations facilitate the computation of several reliability characteristics of the system for a given exchangeable joint distribution or survival function. Finally, we provide signature‐based stochastic ordering results for the system's lifetime and investigate the IFR preservation property under the formulation of m‐consecutive‐k‐out‐of‐n: F systems. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Dynamic signatures and their use in comparing the reliability of new and used systems

The signature of a system with independent and identically distributed (i.i.d.) component lifetimes is a vector whose ith element is the probability that the ith component failure is fatal to the system. System signatures have been found to be quite useful tools in the study and comparison of engineered systems. In this article, the theory of system signatures is extended to versions of signatures applicable in dynamic reliability settings. It is shown that, when a working used system is inspected at time t and it is noted that precisely k failures have occurred, the vector s [0,1]^n‐k whose jth element is the probability that the (k + j)th component failure is fatal to the system, for j = 1,2,2026;,n ‐ k, is a distribution‐free measure of the design of the residual system. Next, known representation and preservation theorems for system signatures are generalized to dynamic versions. Two additional applications of dynamic signatures are studied in detail. The well‐known “new better than used” (NBU) property of aging systems is extended to a uniform (UNBU) version, which compares systems when new and when used, conditional on the known number of failures. Sufficient conditions are given for a system to have the UNBU property. The application of dynamic signatures to the engineering practice of “burn‐in” is also treated. Specifically, we consider the comparison of new systems with working used systems burned‐in to a given ordered component failure time. In a reliability economics framework, we illustrate how one might compare a new system to one successfully burned‐in to the kth component failure, and we identify circumstances in which burn‐in is inferior (or is superior) to the fielding of a new system. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

Tail hazard rate ordering properties of order statistics and coherent systems

We study tail hazard rate ordering properties of coherent systems using the representation of the distribution of a coherent system as a mixture of the distributions of the series systems obtained from its path sets. Also some ordering properties are obtained for order statistics which, in this context, represent the lifetimes of k‐out‐of‐n systems. We pay special attention to systems with components satisfying the proportional hazard rate model or with exponential, Weibull and Pareto type II distributions. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Evaluating methods for the reliability of a large 2‐dimensional rectangular k‐within‐consecutive‐(r,s)‐out‐of‐(m,n):F system

A 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system consists of m × n components, and fails if and only if k or more components fail in an r × s submatrix. This system can be treated as a reliability model for TFT liquid crystal displays, wireless communication networks, etc. Although an effective method has been developed for evaluating the exact system reliability of small or medium‐sized systems, that method needs extremely high computing time and memory capacity when applied to larger systems. Therefore, developing upper and lower bounds and accurate approximations for system reliability is useful for large systems. In this paper, first, we propose new upper and lower bounds for the reliability of a 2‐dimensional rectangular k‐within‐consecutive‐(r, s)‐out‐of‐(m, n):F system. Secondly, we propose two limit theorems for that system. With these theorems we can obtain accurate approximations for system reliabilities when the system is large and component reliabilities are close to one. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

Component importance for linear consecutive‐ k ‐Out‐of‐ n and m ‐Consecutive‐ k ‐Out‐of‐ n systems with exchangeable components

Measuring the relative importance of components in a mechanical system is useful for various purposes. In this article, we study Birnbaum and Barlow‐Proschan importance measures for two frequently studied system designs: linear consecutive k ‐out‐of‐ n and m ‐consecutive‐ k ‐out‐of‐ n systems. We obtain explicit expressions for the component importance measures for systems consisting of exchangeable components. We illustrate the results for a system whose components have a Lomax type lifetime distribution. © 2013 Wiley Periodicals, Inc. Naval Research Logistics, 2013     

Generalizing the survival signature to unrepairable homogeneous multi‐state systems

The notion of signature has been widely applied for the reliability evaluation of technical systems that consist of binary components. Multi‐state system modeling is also widely used for representing real life engineering systems whose components can have different performance levels. In this article, the concept of survival signature is generalized to a certain class of unrepairable homogeneous multi‐state systems with multi‐state components. With such a generalization, a representation for the survival function of the time spent by a system in a specific state or above is obtained. The findings of the article are illustrated for multi‐state consecutive‐k‐out‐of‐n system which perform its task at three different performance levels. The generalization of the concept of survival signature to a multi‐state system with multiple types of components is also presented. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 593–599, 2017     

Parametric inference for component distributions from lifetimes of systems with dependent components

In system reliability analysis, for an n ‐component system, the estimation of the performance of the components in the system is not straightforward in practice, especially when the components are dependent. Here, by assuming the n components in the system to be identically distributed with a common distribution belonging to a scale‐family and the dependence structure between the components being known, we discuss the estimation of the lifetime distributions of the components in the system based on the lifetimes of systems with the same structure. We develop a general framework for inference on the scale parameter of the component lifetime distribution. Specifically, the method of moments estimator (MME) and the maximum likelihood estimator (MLE) are derived for the scale parameter, and the conditions for the existence of the MLE are also discussed. The asymptotic confidence intervals for the scale parameter are also developed based on the MME and the MLE. General simulation procedures for the system lifetime under this model are described. Finally, some examples of two‐ and three‐component systems are presented to illustrate all the inferential procedures developed here. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Failure profiles of coherent systems

In this paper we first introduce and study the notion of failure profiles which is based on the concepts of paths and cuts in system reliability. The relationship of failure profiles to two notions of component importance is highlighted, and an expression for the density function of the lifetime of a coherent system, with independent and not necessarily identical component lifetimes, is derived. We then demonstrate the way that failure profiles can be used to establish likelihood ratio orderings of lifetimes of two systems. Finally we use failure profiles to obtain bounds, in the likelihood ratio sense, on the lifetimes of coherent systems with independent and not necessarily identical component lifetimes. The bounds are relatively easy to compute and use. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2004     

A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components

We give necessary and sufficient conditions based on signatures to obtain distribution‐free stochastic ordering properties for coherent systems with exchangeable components. Specifically, we consider the stochastic, the hazard (failure) rate, the reversed hazard rate, and the likelihood ratio orders. We apply these results to obtain stochastic ordering properties for all the coherent systems with five or less exchangeable components. Our results extend some preceding results. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

The “signature” of a coherent system and its application to comparisons among systems

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, … , p_n), where p_i 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     

A note on signature‐based expressions for the entropy of mixed r‐out‐of‐ n systems

We provide an expression for the Shannon entropy of mixed r‐out‐of‐ n systems when the lifetimes of the components are independent and identically distributed. The expression gives the system's entropy in terms of the system signature, the distribution and density functions of the lifetime model, and the information measures of the beta distribution. Bounds for the system's entropy are obtained by direct applications of the concavity of the entropy and the information inequality.Copyright © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 202–206, 2014     

On the availability of R out of N repairable systems

An R out of N repairable system consisting of N components and operates if at least R components are functioning. Repairable means that failed components are repaired, and upon repair completion they are as good as new. We derive formulas for the expected up‐time, expected down‐time, and the availability of the system, using Markov renewal processes. We assume that either the repair times of the components are generally distributed and the components' lifetimes are exponential or vice versa. The analysis is done for systems with either cold or warm stand‐by. Numerical examples are given for several life time and repair time distributions. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 483–498, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10025     

On optimal system designs in reliability‐economics frameworks

Reliability Economics is a field that can be defined as the collection of all problems in which there is tension between the performance of systems of interest and their cost. Given such a problem, the aim is to resolve the tension through an optimization process that identifies the system which maximizes some appropriate criterion function (e.g. expected lifetime per unit cost). In this paper, we focus on coherent systems of n independent and identically distributed (iid) components and mixtures thereof, and characterize both a system's performance and cost as functions of the system's signature vector (Samaniego, IEEE Trans Reliabil (1985) 69–72). For a given family of criterion functions, a variety of optimality results are obtained for systems of arbitrary order n. Approximations are developed and justified when the underlying component distribution is unknown. Assuming the availability of an auxiliary sample of N component failure times, the asymptotic theory of L‐estimators is adapted for the purpose of establishing the consistency and asymptotic normality of the proposed estimators of the expected ordered failure times of the n components of the systems under study. These results lead to the identification of ε‐optimal systems relative to the chosen criterion function. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Some comparisons of residual life of coherent systems with exchangeable components

Most of the research, on the study of the reliability properties of technical systems, assume that the components of the system operate independently. However, in real life situation, it is more reasonable to assume that there is dependency among the components of the system. In this article, we give sufficient conditions based on the signature and the joint distribution of component lifetimes to obtain stochastic ordering results for coherent and mixed systems with exchangeable components. Some stochastic orders on dynamic (or conditional) signature of coherent systems are also provided. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 549–556, 2014     

The NBUL class of life distribution and replacement policy comparisons

Let X and X_τ denote the lifetime and the residual life at age τ of a system, respectively. X is said to be a NBUL random variable if X_τ is smaller than X in Laplace order, i.e., X_τ ≤_L X. We obtain some characterizations for this class of life distribution by means of the lifetime of a series system and the residual life at random time. We also discuss preservation properties for this class of life distribution under shock models. Finally, under the assumption that the lifetimes have the NBUL property, we make stochastic comparisons between some basic replacement policies. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 578–591, 2001.     

The joint signature of coherent systems

We investigate the joint signature of $m$ coherent systems, under the assumption that the components have independent and identically distributed lifetimes. The joint signature, for a particular ordering of failure times, is an $m$‐dimensional matrix depending solely on the composition of the systems and independent of the underlying distribution function of the component lifetimes. The elements of the $m$‐dimensional matrix are formulated based on the joint signatures of numerous series of parallel systems. The number of the joint signatures involved is an exponential function of the number of the minimal cut sets of each original system and may, therefore, be significantly large. We prove that although this number is typically large, a great number of the joint signatures are repeated, or removed by negative signs. We determine the maximum number of different joint signatures based on the number of systems and components. It is independent of the number of the minimal cut sets of each system and is polynomial in the number of components. Moreover, we consider all permutations of failure times and demonstrate that the results for one permutation can be of use for the others. Our theorems are applied to various examples. The main conclusion is that the joint signature can be computed much faster than expected.     

Stochastic framework for partially degradation systems with continuous component degradation‐rate‐interactions

Many conventional models that characterize the reliability of multicomponent systems are developed on the premise that for a given system, the failures of its components are independent. Although this facilitates mathematical tractability, it may constitute a significant departure from what really takes place. In many real‐world applications, system components exhibit various degrees of interdependencies, which present significant challenges in predicting degradation performance and the remaining lifetimes of the individual components as well as the system at large. We focus on modeling the performance of interdependent components of networked systems that exhibit interactive degradation processes. Specifically, we focus on how the performance level of one component affects the degradation rates of other dependent components. This is achieved by using stochastic models to characterize how degradation‐based sensor signals associated with the components evolve over time. We consider “Continuous‐Type” component interactions that occur continuously over time. This type of degradation interaction exists in many applications, in which interdependencies occur on a continuum. We use a system of stochastic differential equations to capture such “Continuous‐Type” interaction. In addition, we utilize a Bayesian approach to update the proposed model using real‐time sensor signals observed in the field and provide more accurate estimation of component residual lifetimes. © 2014 Wiley Periodicals, Inc. Naval Research Logistics 61: 286–303, 2014     

Consecutive k‐within‐m‐out‐of‐n:F system with exchangeable components

As a generalization of k‐out‐of‐n:F and consecutive k‐out‐of‐n:F systems, the consecutive k‐within‐m‐out‐of‐n:F system consists of n linearly ordered components such that the system fails iff there are m consecutive components which include among them at least k failed components. In this article, the reliability properties of consecutive k‐within‐m‐out‐of‐n:F systems with exchangeable components are studied. The bounds and approximations for the survival function are provided. A Monte Carlo estimator of system signature is obtained and used to approximate survival function. The results are illustrated and numerics are provided for an exchangeable multivariate Pareto distribution. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009