$k$-out-of-$n$:G System Reliability With Imperfect Fault Coverage

Systems requiring very high levels of reliability, such as aircraft controls or spacecraft, often use redundancy to achieve their requirements. Reliability models for such redundant systems have been widely treated in the literature. These models describe k-out-of-n:G systems, where n is the number of components in the system, and k is the minimum number of components that must work if the overall system is to work. Most of this literature treats the perfect fault coverage case, meaning that the system is perfectly capable of detecting, isolating, and accommodating failures of the redundant elements. However, the probability of accomplishing these tasks, termed fault coverage, is frequently less than unity. Correct modeling of imperfect coverage is critical to the design of highly reliable systems. Even very high values of coverage, only slightly less than unity, will have a major impact on the overall system reliability when compared to the ideal system with perfect coverage. The appropriate coverage modeling approach depends on the system design architecture, particularly the technique(s) used to select among the redundant elements. This paper demonstrates how coverage effects can be computed, using both combinatorial, and recursive techniques, for four different coverage models: perfect fault coverage (PFC), element level coverage (ELC), fault level coverage (FLC), and one-on-one level coverage (OLC). The designation of PFC, ELC, FLC, and OLC to distinguish types of coverage modeling is suggested in this paper.