A hierarchical state ordering method for heterogeneous k-out-of-n:F systems with multi-state load-sharing components

k-out-of-n systems are crucial models for ensuring reliability in critical applications by providing redundancy and fault tolerance that mitigate system failure risks. While homogeneous k-out-of-n systems composed of independent components generally demonstrate strong structural properties and can be computed efficiently, the computational challenges of heterogeneous k-out-of-n systems with load-sharing remain underexplored. A key issue is the exponential growth of the state space, which lacks effective reduction methods. This paper investigates a k-out-of-n:F system with heterogeneous multi-state components and load-sharing, proposing a novel state ordering problem aimed at designing and rearranging system states to reduce computational complexity in reliability assessments. We introduce an ordered set with a rank function that establishes a new order relation for the state space and calculates state indexes through a hierarchical mapping. This state ordering approach, enhanced by insights from order statistics, enables a block-wise representation of the transition rate matrix, significantly decreasing computational time for reliability and RUL calculations. A case study of a four-component bridge foundation system demonstrates that our approach reduces the transition rate matrix size to about one-fifth, indicating a cubic reduction in computational complexity when using phase-type methods.