Читать книгу Reliability Analysis, Safety Assessment and Optimization - Enrico Zio - Страница 37

For a system composed of n components, the system is operational if and only if at least k of the n components are operational. We call this type of system as k-out-of-n: G system, where G is short for Good. For a system composed of n components, the system fails if and only if at least k of the n components are failed. We call this type of system a k-out-of-n: F system. According to the definition, the series system is a 1-out-of-n: F system, where F is short for Failed. The parallel system is a 1-out-of-n: G system. We will mainly present the reliability of the k-out-of-n: G system here.

Assume that the n components are identical and independent. Denote R as the reliability of each component, F as the unreliability of each component, q=1−p. Let Pi be the probability so that exactly i components are functional. In a k-out-of-n: G system, the number of functional components follows the binomial distribution with parameter n and R. The probability that exactly i components are functional, Pi, is

(1.53)

The reliability of the system is the probability that the number of functional components is greater than or equal to k. Thus, the system reliability, Rs, is calculated by

(1.54)

If the components are not identical, the system reliability should be calculated by enumerating all combinations of working components.