Support vector machine (SVM) methods are widely used for classification and regression analysis. In many engineering applications, only one class of data is available, and then one-class SVM methods are employed. In reliability applications, the one-class data may be failure data since the data are recorded during reliability experiments when only failures occur. Different from the problems handled by existing one-class SVM methods, there is a bias constraint in the SVM model in this work and the constraint comes from the probability of failure estimated from the failure data. In this study, a new one-class SVM regression method is proposed to accommodate the bias constraint. The one class of failure data is maximally separated from a hypersphere whose radius is determined by the known probability of failure. The proposed SVM method generates regression models that directly link the states of failure modes with design variables, and this makes it possible to obtain the joint probability density of all the component states of an engineering system, resulting in a more accurate prediction of system reliability during the design stage. Three examples are given to demonstrate the effectiveness of the new one-class SVM method.

Ebrahimi (2001) proposes an interesting model for assessing a system's reliability, expressing the failure time of the system in terms of a deterioration process and covariates. He also provides illustrative examples and gives some properties of the model. In this note, we give conditions for negative ageing of the system's lifetime, and we correct one of his statements. Moreover, for the lifetimes of two systems of the same kind, some stochastic comparisons are presented.

If τ is the lifetime of a coherent system, then the signature of the system is the vector of probabilities that the lifetime coincides with the ith order statistic of the component lifetimes. The signature can be useful in comparing different systems. In this treatment we give a characterization of the signature of a system with independent identically distributed components in terms of the number of path sets in the system as well as in terms of the number of what we call ordered cut sets. We consider, in particular, the signatures of indirect majority systems and compare them with the signatures of simple majority systems of the same size. We note that the signature of an indirect majority system of size r × s = n is symmetric around , and use this to show that the expected lifetime of an r × s = n indirect majority system exceeds that of a simple (direct) majority system of size n when the components are exponentially distributed with the same parameter.

We study a reliability system subject to shocks generated by a renewal point process. When a shock occurs, components fail independently of each other with equal probabilities that are random numbers drawn from a distribution that may differ from shock to shock. We first consider the case of a parallel system and derive closed expressions for the Laplace-Stieltjes transform and the expectation of the time to system failure and for its density in the case that the distribution function of the renewal process possesses a density. We then treat a more general system structure, which has some very important special cases, such as k-out-of-n:F systems, and derive analogous formulae.

We present general criteria for analyzing the crossing characteristics of R I, the reliability function of an m-of-n system of components operating within a laboratory (or test-bench) environment, and R O, the reliability function of the same system now operating subject to an external environment. Inside the laboratory the components' lifetimes may be dependently distributed, and the external environment is modeled using the general approach of Lindley and Singpurwalla (1986). Our techniques, which utilize results basic to the theory of order statistics, apply to broad classes of external environment models.

We consider some important systems in reliability theory situated in a random environment, where shocks occur and cause component failure in a specific way. We study some appropriate coefficients, which play an important role in the reduction of our systems to a linear combination of parallel subsystems.

We compare R 1(t), the reliability function of a redundant m-of-n system operating within the laboratory, with R D(t), the reliability function of the same system operating subject to environmental effects. Within the laboratory, all component lifetimes are independent and identically distributed according to G(α + 1, λ), a gamma distribution with index α + 1 and scale λ. Outside the laboratory, we adopt the model of Lindley and Singpurwalla (J. Appl. Prob. 23 (1986), 418-431) and assume that, conditional on a positive random variable η which models the effect of the common environment, all component lifetimes are independent and identically distributed according to G(α + 1, λη). When α is a non-negative integer we prove that for R D(t) to underestimate (resp. overestimate) R 1(t) for all t sufficiently close to zero, it is necessary and sufficient that E(η (n-m + 1)(α+1)) > 1 (resp. E(η (n-m + 1)(α+1)) < 1). In the case in which n = 2, m= 1 and α = 0 we obtain a special case of a result of Currit and Singpurwalla (J. Appl. Prob. 26 (1988), 763-771). As an application, we obtain a necessary and sufficient condition under which R D(t) initially understimates (or overestimates) R 1(t) when η follows a gamma or an inverse Gaussian distribution.

We consider a consecutive-k-out-of-n system situated in a random environment where shocks occur and cause components to fail with certain probabilities. Under certain reasonable assumptions we compute the moment generating function of the time of failure of the system and the distribution of the number of shocks required to cause failure of the system.

In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.