This paper examines the lifetime distributions of circular $k$-out-of-$n$: G balanced systems operating in a shock environment, providing a unified framework for both discrete- and continuous-time perspectives. The system remains functioning only if at least $k$ operating units satisfy a predefined balance condition (BC). Building on this concept, we demonstrate that the shock numbers to failure (SNTF) follow a discrete phase-type distribution by modeling the system's stochastic dynamics with a finite Markov chain and applying BC-based state space consolidation. Additionally, we develop a computationally efficient method for directly computing multi-step transition probabilities of the underlying Markov chain. Next, assuming the inter-arrival times between shocks follow a phase-type distribution, we establish that the continuous-time system lifetime, or the time to system failure (TTF), also follows a phase-type distribution with different parameters. Extensive numerical studies illustrate the impact of key parameters-such as the number of units, minimum requirement of the number of operating units, individual unit reliability, choice of balance condition, and inter-shock time distribution-on the SNTF, TTF, and their variability.

翻译：本文研究了在冲击环境下运行的循环$k$-out-of-$n$: G平衡系统的寿命分布，为离散时间和连续时间视角提供了一个统一框架。仅当至少$k$个运行单元满足预定义的平衡条件时，系统才能保持正常工作。基于此概念，我们通过使用有限马尔可夫链对系统的随机动态进行建模，并应用基于平衡条件的状态空间合并，证明了失效冲击次数服从离散位相型分布。此外，我们开发了一种计算高效的方法，用于直接计算底层马尔可夫链的多步转移概率。接着，假设冲击到达间隔时间服从位相型分布，我们建立了连续时间系统寿命（即系统失效时间）也服从具有不同参数的位相型分布。大量的数值研究阐明了关键参数——如单元数量、运行单元数量的最低要求、单个单元可靠性、平衡条件的选择以及冲击间隔时间分布——对失效冲击次数、系统失效时间及其变异性的影响。