Reliability Estimation of <formula formulatype="inline"> <tex Notation="TeX">$k$</tex> </formula>-out-of-<formula formulatype="inline"> <tex Notation="TeX">$n$</tex> </formula> Pairs:G Balanced Systems With Spatially Distributed Units

Abstract

The applications of $k$ -out-of- $n$ pairs:G Balanced systems with spatially distributed units are emerging in aerospace and military industries. A $k$ -out-of- $n$ pairs:G Balanced system has $n$ pairs of units distributed evenly in a circular configuration. The system operates when at least $k$ pairs of units operate in a balanced arrangement. The reliability estimation of such systems is important since their failures are most likely to result in losses in property and humans such as the case of unmanned aerial vehicles (UAVs) and balanced engine systems in planetary descent vehicles. In this paper we present methods for reliability estimation of different types of $k$ -out-of- $n$ pairs:G Balanced systems in two scenarios: 1) unbalanced systems are considered as failed and 2) unbalanced systems are rebalanced. We develop a systematic approach for enumerating the complete set of successful events, which are ordered sequences of failures described by system state transition paths, and obtain closed form expressions for calculating the probabilities of successful events. The developed methods can be easily generalized to other systems with spatially distributed units. It is found that there exists an optimal redundancy configuration for $k$ -out-of- $n$ pairs:G Balanced systems when unbalanced systems are considered as failed; and that system reliability can be increased by rebalancing unbalanced system.