Reliability Control of Complex Systems Through Penultimate Approximations

Proceedings of IFAC Conference On Manufacturing Modelling, Management, and Control, 7(1) (2013), 916-921
http://dx.doi.org/10.3182/20130619-3-RU-3018.00191

In statistical quality management, the control of the reliability of a system is a topic of primordial relevance in manufacture industry. After a brief reference to the importance of order statistics in reliability, the fact that any coherent system can be represented as either a series-parallel (SP) or a parallel-series (PS) system is referred. The lifetime of such a system can thus be written as the minimum of maxima or the maximum of minima, which reveals the relevant role of extreme value theory (EVT) in the field of reliability. Indeed, for large-scale coherent systems can be sensible to assume that the number of system components goes to infinity. And then, the possible non-degenerate extreme value distributions either for maxima or for minima are eligible candidates for the system reliability or at least for the finding of adequate lower and upper bounds for such a reliability. The main limiting results in EVT will be briefly mentioned, and the identification of the possible ultimate limit laws for maxima of a min-stable distribution and for minima of a max-stable distribution are identified. Considering a fixed large number of components, pre-asymptotic or penultimate models can lead to an improvement of the convergence rate and are provided. On the basis of a small-scale Monte-Carlo simulation study, we further make clear the attained gain in accuracy when a penultimate approximation is used instead of the ultimate limiting approximation.