On modular decompositions of system signatures

Considering a semicoherent system made up of $n$ components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the $n$-tuple whose $k$-th coordinate is the probability that the $k$-th component failure causes the system to fail. This $n$-tuple, which depends only on the structure of the system and not on the distribution of the component lifetimes, is a very useful tool in the theoretical analysis of coherent systems. It was shown in two independent recent papers how the structural signature of a system partitioned into two disjoint modules can be computed from the signatures of these modules. In this work we consider the general case of a system partitioned into an arbitrary number of disjoint modules organized in an arbitrary way and we provide a general formula for the signature of the system in terms of the signatures of the modules. The concept of signature was recently extended to the general case of semicoherent systems whose components may have dependent lifetimes. The same definition for the $n$-tuple gives rise to the probability signature, which may depend on both the structure of the system and the probability distribution of the component lifetimes. In this general setting, we show how under a natural condition on the distribution of the lifetimes, the probability signature of the system can be expressed in terms of the probability signatures of the modules. We finally discuss a few situations where this condition holds in the non-i.i.d. and nonexchangeable cases and provide some applications of the main results

Reliability of systems with dependent components based on lattice polynomial description

Reliability of a system is considered where the components' random lifetimes may be dependent. The structure of the system is described by an associated "lattice polynomial" function. Based on that descriptor, general framework formulas are developed and used to obtain direct results for the cases where a) the lifetimes are "Bayes-dependent", that is, their interdependence is due to external factors (in particular, where the factor is the "preliminary phase" duration) and b) where the lifetimes' dependence is implied by upper or lower bounds on lifetimes of components in some subsets of the system. (The bounds may be imposed externally based, say, on the connections environment.) Several special cases are investigated in detail

Minimal repair of failed components in coherent systems

The minimal repair replacement is a reasonable assumption in many practical systems. Under this assumption a failed component is replaced by another one whose reliability is the same as that of the component just before the failure, i.e., a used component with the same age. In this paper we study the minimal repair in coherent systems. We consider both the cases of independent and dependent components. Three replacement policies are studied. In the first one, the first failed component in the system is minimally repaired while, in the second one, we repair the component which causes the system failure. A new technique based on the relevation transform is used to compute the reliability of the systems obtained under these replacement policies. In the third case, we consider the replacement policy which assigns the minimal repair to a fixed component in the system. We compare these three options under different stochastic criteria and for different system structures. In particular, we provide the optimal strategy for all the coherent systems with 1-4 independent and identically distributed components

The joint survival signature of coherent systems with shared components

The concept of joint bivariate signature, introduced by Navarro et al. [13], is a useful tool for quantifying the reliability of two systems with shared components. As with the univariate system signature, introduced by Samaniego [17], its applications are limited to systems with only one type of components, which restricts its practical use. Coolen and Coolen-Maturi [2] introduced the survival signature, which generalizes Samaniego’s signature and can be used for systems with multiple types of components. This paper introduces a joint survival signature for multiple systems with multiple types of components and with some components shared between systems. A particularly important feature is that the functioning of these systems can be considered at different times, enabling computation of relevant conditional probabilities with regard to a system’s functioning conditional on the status of another system with which it shares components. Several opportunities for practical application and related challenges for further development of the presented concept are briefly discussed, setting out an important direction for future research

Minimal repair of failed components in coherent systems

© 2019 This document is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ This document is the accepted version of a published work that appeared in final form in European Journal of Operational ResearchThe minimal repair replacement is a reasonable assumption in many practical systems. Under this as- sumption a failed component is replaced by another one whose reliability is the same as that of the component just before the failure, i.e., a used component with the same age. In this paper we study the minimal repair in coherent systems. We consider both the cases of independent and dependent compo- nents. Three replacement policies are studied. In the first one, the first failed component in the system is minimally repaired while, in the second one, we repair the component which causes the system fail- ure. A new technique based on the relevation transform is used to compute the reliability of the systems obtained under these replacement policies. In the third case, we consider the replacement policy which assigns the minimal repair to a fixed component in the system. We compare these three options un- der different stochastic criteria and for different system structures. In particular, we provide the optimal strategies for all the coherent systems with 1–4 independent and identically distributed components

Nonparametric Predictive Inference for System Failure Time

This thesis presents the use of signatures within nonparametric predictive inference (NPI) for the failure time of a coherent system with a single type of components, given failure times of tested components that are exchangeable with those in the system. NPI is based on few modelling assumptions and here leads to lower and upper survival functions. We also illustrate comparison of reliability of two systems, by directly considering the random failure times of the systems. This includes explicit consideration of the difference between failure times of two systems. In this method we assume that the signature is precisely known. In addition, we show how bounds for these lower and upper survival functions can be derived based on limited information about the system structure, which can reduce computational effort substantially for specific inferential questions. It is illustrated how one can base reliability inferences on a partially known signature, assuming that bounds for the probabilities in the signature are available. As a further step in the development of NPI, we present the use of survival signatures within NPI for the failure time of a coherent system which consists of different types of components. It is assumed that, for each type of component, additional components which are exchangeable with those in the system have been tested and their failure times are available. Throughout this thesis we assume that the system is coherent, we start with a system consisting of a single type of components, then we extend for a system consisting of different types of components

The survival signature for quantifying system reliability: an introductory overview from practical perspective

The structure function describes the functioning of a system dependent on the states of its components, and is central to theory of system reliability. The survival signature is a summary of the structure function which is sufficient to derive the system’s reliability function. Since its introduction in 2012, the survival signature has received much attention in the literature, with developments on theory, computation and generalizations. This paper presents an introductory overview of the survival signature, including some recent developments. We discuss challenges for practical use of survival signatures for large systems

Nonparametric predictive inference for system failure time based on bounds for the signature

System signatures provide a powerful framework for reliability assessment for systems consisting of exchangeable components. The use of signatures in nonparametric predictive inference has been presented and leads to lower and upper survival functions for the system failure time, given failure times of tested components. However, deriving the system signature is computationally complex. This article presents how limited information about the signature can be used to derive bounds on such lower and upper survival functions and related inferences. If such bounds are sufficiently decisive they also indicate that more detailed computation of the system signature is not required