8 - Large Systems Related Problems

Abstract

Domains of attraction for limit reliability functions of two-state systems are introduced. They are understood as the conditions that the reliability functions of the particular components of the system have to satisfy in order that the system limit reliability function is one of the limit reliability functions from the previously fixed class for this system. Exemplary theorems concerned with domains of attraction for limit reliability functions of homogeneous series systems are presented and the application of one of them is illustrated. A practically important problem of accuracy of the asymptotic approach to large systems reliability evaluation, concerned with the speed of convergence of system reliability sequence, is discussed. This problem is illustrated by analysing the speed of convergence of the homogeneous series–parallel system reliability sequences to its limit reliability function. Series–‘m out of n’ systems and ‘m out of n’–series systems are defined and exemplary theorems on their limit reliability functions are presented and applied to the reliability evaluation of an illumination system and a rope elevator. Hierarchical series–parallel and parallel–series systems of any order are defined, their reliability functions are determined and limit theorems on their reliability functions are applied to reliability evaluation of exemplary hierarchical systems of order 2. Applications of the asymptotic approach in large series systems reliability improvement are also presented.