Multi-Component Systems and Structures and Their Reliability

A number of recent publications have dealt with problems of analyzing the performance of multi-component systems and evaluating their reliability. For example, a comprehensive theory of two-terminal networks was presented in I by Moore and Shannon who, among other results, have developed methods for obtaining highly reliable systems using components of low reliability; some of their procedures are credited to earlier work by von Neumann 2. Several of the concepts and results of the present paper are generalizations of the corresponding concepts and results of the Moore-Shannon paper. A discussion of complex systems interpreted as Boolean functions may be found in the paper 3 by Mine. The present study deals with general classes of systems which contain two-terminal networks and most other kinds of systems considered previously as special cases, and investigates their combinatorial properties and their reliability. These classes consist, with several variants, of systems such that the more components that perform the greater the probability that the system performs. For such systems it is shown that, if each component has reliability p and the reliability of the system is denoted by h(p), then under mild additional assumptions h is an S-shaped function, i.e., its graph has the shape indicated in Fig. 3.2.4.1. Some of the consequences are these: there exists a critical value of p such that above that value the reliability of the system is greater than the reliability of a single component and below that value it is smaller; for p small the system has a reliability comparable to that of a series system, and for p large to that of a parallel system; by repeatedly iterating the system, i.e., by using replicas of the system instead of single components, one obtains systems with reliability arbitrarily close to 1 if one starts with component reliability above the critical value, but with reliability arbitrarily close to 0 if one starts below that critical value.