Lifetime analysis of circular k-out-of-n: G balanced systems in a shock environment

The increasing deployment of UAV systems in harsh environments underscores the need for reliability models that account not only for the number of functioning units but also for their spatial or geometric balance. Motivated by such systems, this study investigates the lifetime distribution of circular k-out-of-n: G balanced systems under stochastic external shocks. While preserving the geometric concept of the balance conditions, we develop an analytical framework that models both discrete-time (shock numbers to system failure, SNTF) and continuous-time (time to system failure, TTF) lifetimes consolidating the concept of geometric balance condition, minimum tie-set, finite Markov chain imbedding approach and phase-type distributions. To ensure scalability, we propose a computationally efficient method for directly calculating multi-step transition probabilities of the embedded Markov chain, thereby enabling fast and accurate evaluation of system lifetime distributional indices. Extensive numerical experiments validate the analytical results and illustrate how key factors—such as unit reliability, system size, and balance condition—affect the system’s lifetime characteristics.