On a conjecture of Barlow and Proschan concerning reliability bounds (Q1096995)

The author applies to the well-known textbook of \textit{R. E. Barlow} and \textit{F. Proschan}, Statistical theory of reliability and life testing. (1975; Zbl 0379.62080). In this book various bounds for the reliability of systems with independent components are given. Let \(l_ 1\) \((u_ 1)\) be the so-called ``path-cut'' lower (upper) bound (which are in the book denoted by \(l_{\phi}\), \(u_{\phi})\) and \(l_ 2\) \((u_ 2)\) the ``minimax'' lower (upper) bound. The author gives a comprehensive answer to the question whether \(l_ 1\) or \(l_ 2\) respectively \(u_ 1\) or \(u_ 2\) are better if the reliability of all components is ``large''. It is shown that for ``large'' reliability of all components \(l_ 1\) is better than \(l_ 2\) (except series systems, for which \(l_ 1=l_ 2)\) and necessary and sufficient conditions are given that for ``large'' reliability of all components \(u_ 2\) is a better upper bound than \(u_ 1\). Further, the case of ``small'' reliabilities of all components is dealt with in a similar manner.