On reliability analysis of a load-sharing k-out-of-n: G system with interacting Markov subsystems

Dependency plays a key role in the design stage of multi-component manufacturing systems. To capture stochastic dependencies between subsystems, this paper considers a load-sharing $k $k-out-of- $n $n: G system which contains non-identical multi-state subsystems. The state evolution of each subsystem follows a continuous-time homogeneous Markov chain. Most existing research on $k $k-out-of- $n $n: G systems focuses on independent subsystems. However, a subsystem failure usually leads to a higher failure rate for each surviving subsystem within the system. This paper aims to study the interaction among subsystems and analyse the system reliability performance. The transition rate matrix for a given subsystem is assumed to depend on the total number of the remaining surviving subsystems, which can be represented as the sum of a baseline transition rate matrix and a stochastic dependency matrix. This paper proves that the transition rate matrix of the system is the generalised Kronecker sum of transition rate matrices of subsystems, and develops an explicit method to calculate the generalised Kronecker sum. The theory of aggregated stochastic processes is employed to obtain closed-form formulas for reliability indexes. A case study of multi-engine aircraft systems is provided where numerical examples are given to illustrate the developed model and obtained results.