The computation of the reliability function of a (complex) coherent system is a difficult task. Hence, sometimes, we should simply work with some bounds (approximations). The computation of these ...bounds has been widely studied in the case of coherent systems with independent and identically distributed (IID) components. However, few results have been obtained in the case of heterogeneous (non ID) components. In this paper, we derive explicit bounds for systems with heterogeneous (independent or dependent) components. Also some stochastic comparisons are obtained. Some illustrative examples are included where we compare the different bounds proposed in the paper.
We consider coherent and mixed systems with exchangeable components whose lifetimes have positive and finite variances. We present sharp lower and upper bounds on the variance of the system lifetime, ...expressed in terms of the system signature and the variance of a single component. The bounds are attained for the power and Pareto distributions of the component lifetimes.
We consider semicoherent and mixed systems with exchangeable components. We present sharp lower and upper bounds on various dispersion measures (in particular, variance and median absolute deviation) ...of the system lifetime, expressed in terms of the system signature and the dispersion of a single component lifetime. We construct joint exchangeable distributions of component lifetimes with two-point marginals which attain the bounds in the limit.
Component importance measures are relevant to improve the system design and to develop optimal replacement policies. Birnbaum's importance measure is one of the most relevant measures. If the ...components are (stochastically) independent, this measure can be defined using several equivalent expressions. However, in many practical situations, the independence assumption is unrealistic. It also turns out that in the case of dependent components, different Birnbaum's measure definitions lead to different concepts. In this paper, we extend Birnbaum's importance measure to the case of dependent components in a way allowing us to obtain relevant properties including connections and comparisons with other measures proposed and studied recently. The dependence is modeled through copulas and the new measure is based on the contribution of the component to the system reliability.
In the paper we propose a method how to assess sharp bounds on change in the expected value and variance of the individual loss distributions of the risks in the flood catastrophe model. The method ...does not involve the use of the whole model, only the river discharge distributions and above mentioned loss distribution moments are required. We present the method on a case study inspired by floods in Poland.
We consider semicoherent and mixed systems with exchangeable components. We present sharp lower and upper bounds on various dispersion measures (in particular, variance and median absolute deviation) ...of the system lifetime, expressed in terms of the system signature and the dispersion of a single component lifetime. We construct joint exchangeable distributions of component lifetimes with two-point marginals which attain the bounds in the limit.
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