Lifetime Analysis of Circular $k$-out-of-$n$: G Balanced Systems in a Shock Environment

TL;DR

The paper addresses lifetime analysis for circular $k$-out-of-$n$: G balanced systems in shock environments by embedding the system in a two-stage finite Markov chain framework and employing balance-condition-based state consolidation. It proves that the discrete-time lifetime (SNTF) is a discrete phase-type distribution and that the continuous-time lifetime (TTF) is a phase-type distribution when inter-shock times are PH, providing efficient methods to compute multi-step transitions and moments. Through a descriptive case and extensive numerics, it shows how parameters such as $n$, $k$, unit reliability $r$, balance condition, and inter-shock timing influence $M$, $Z$, and their variability, with BC3 generally yielding higher vulnerability. The framework advances scalable reliability analysis for geometry-based balanced systems and informs design and maintenance of UAV/UAM and aerospace propulsion systems under shock environments.

Abstract

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.

Figures (12)

• Figure 1: A real-world motivation of the target system and its abstracted graphical model EFT
• Figure 2: Three different types of balance conditions cho2023
• Figure 3: Concept of the rebalancing by turning on/off the operational units
• Figure 4: Comparison between the sizes of the full state space $\mathcal{X}$ and the consolidated state spaces $\bar{\mathcal{X}}_\mathrm{BC}$'s for different $k$ and $n$ values where BC3 is applied as a balance condition
• Figure 5: The transition probability diagram for a CknGB system with $k=2$ and $n=4$ where BC3 is applied as a balance condition and $\tilde{r}\equiv1-r$

Theorems & Definitions (6)

• Remark 1
• Remark 2
• Proposition 1
• proof
• Proposition 2
• proof