Derivation of the Discrete-Time Lifetime Distribution for Circular k-out-of-n: G Balanced Systems

Purpose: This paper explores the discrete-time lifetime of circular κ-out-of-n: G balanced (CknGB) systems, inspired by multi-rotor UAV/UAM systems operating in shock environments. Specifically, the paper investigates the number of external shocks a system can endure before failure.<br/>Methods: The reliability and discrete-time lifetime of CknGB systems are defined using balance conditions, minimum tie-sets, and discrete-time Markov chains. The matrix-analytic method is then employed to derive the analytical form of the discrete-time system lifetime distribution.<br/>Results: The discrete-time system lifetime distribution is characterized as a discrete-time phase-type distribution (PH<SUB>d</SUB>) with a specific parameter set. Extensive numerical analysis is performed to examine the shape of the probability mass function in relation to various system parameters, such as κ, n, and r(unit reliability).<br/>Conclusion: The findings of this study can be extended to continuous-time system lifetimes. Future research directions include reliability prediction, maintenance optimization, and economic design for various systems, including multi-rotor UAV/UAM systems.