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| Main Authors: | , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.10479 |
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| _version_ | 1866929715893239808 |
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| author | Baik, Seung Min Cho, Yongkyu |
| author_facet | Baik, Seung Min Cho, Yongkyu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_10479 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lifetime Analysis of Circular $k$-out-of-$n$: G Balanced Systems in a Shock Environment Baik, Seung Min Cho, Yongkyu Systems and Control Performance Probability 60J10, 90B25 (Primary) 62N05 (Secondary) G.3; C.4 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. |
| title | Lifetime Analysis of Circular $k$-out-of-$n$: G Balanced Systems in a Shock Environment |
| topic | Systems and Control Performance Probability 60J10, 90B25 (Primary) 62N05 (Secondary) G.3; C.4 |
| url | https://arxiv.org/abs/2502.10479 |