18,555 research outputs found

    Aging functions and multivariate notions of NBU and IFR

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    For d≥2, let X=(X1, …, Xd) be a vector of exchangeable continuous lifetimes with joint survival function F\overline{F}. For such models, we study some properties of multivariate aging of F\overline{F} that are described by means of the multivariate aging function BFB_{\overline{F}}, which is a useful tool for describing the level curves of F\overline{F}. Specifically, the attention is devoted to notions that generalize the univariate concepts of New Better than Used and Increasing Failure Rate. These multivariate notions are satisfied by random vectors whose components are conditionally independent and identically distributed having univariate conditional survival function that is New Better than Used (respectively, Increasing Failure Rate). Furthermore, they also have an interpretation in terms of comparisons among conditional survival functions of residual lifetimes, given a same history of observed survivals

    On modular decompositions of system signatures

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    Considering a semicoherent system made up of nn components having i.i.d. continuous lifetimes, Samaniego defined its structural signature as the nn-tuple whose kk-th coordinate is the probability that the kk-th component failure causes the system to fail. This nn-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 nn-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

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    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
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