Achievable Limits on the Reliability of $k$-out-of- $n$:G Systems Subject to Imperfect Fault Coverage

Systems which must be designed to achieve very low probabilities of failure often use redundancy to meet these requirements. However, redundant
k
-out-of-
n
:G systems which are subject to imperfect fault coverage have an optimum level of redundancy,
n

_opt
. That is to say, additional redundancy in excess of
nopt
will result in an increase, not a decrease, in the probability of system failure. This characteristic severely limits the level of redundancy considered in the design of highly reliable systems to modest values of
n
. Correct modeling of imperfect coverage is critical to the design of highly reliable systems. Two distinctly different
k
-out-of-
n
:G imperfect fault coverage reliability models are discussed in this paper: Element Level Coverage (ELC), and Fault Level Coverage (FLC). ELC is the appropriate model when each component can be assigned a given coverage level, while FLC is the appropriate model for systems using voting as the primary means of selection among redundant components. It is shown, over a wide range of realistic coverage values and relatively high component reliabilities, that the optimal redundancy level for ELC systems is 2 and 4 for FLC systems. It is also shown that the optimal probability of failure for FLC systems exceeds that of ELC systems by several orders of magnitude. New functions for computing the mean time to failure for i.i.d.
k
-out-of-
n
:G ELC, and FLC systems are also presented.