A recursive algorithm for reliability evaluation of multi-state hierarchical systems with stochastic dependent components

• Published in: Reliability engineering & system safety Vol. 266; p. 111653
• Main Authors: Jiang, Chen, Sun, Muxia, Wang, Luyao, Wang, Zisheng, Li, Yan-Fu
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.02.2026
• Subjects: Bayesian network; Multi-state hierarchical system; Recursive algorithm; Reliability; Stochastic dependence; Stochastic dependence; Reliability; Multi-state hierarchical system; Recursive algorithm; Bayesian network
• ISSN: 0951-8320
• DOI: 10.1016/j.ress.2025.111653

Multi-state hierarchical systems (MSHSs), composed of recursively nested subsystems, are prevalent in engineering applications. However, their reliability evaluation remains challenging, especially when components exhibit stochastic dependencies. Existing methods either assume mutual independence – which oversimplifies real-world systems – or suffer from high computational cost and limited structural generality. In this work, we propose a computationally efficient recursive algorithm based on Bayesian Networks (BNs) for evaluating the reliability of generalized MSHSs with dependent components. Unlike traditional methods that rely on global system representations, our approach leverages the system’s hierarchical architecture by assigning a local BN to each structural level, thereby capturing intra-level dependencies while maintaining scalable computation. The algorithm proceeds in a bottom-up manner to iteratively compute marginal and conditional state distributions, ultimately yielding the system-level reliability. The method, to our knowledge, offers the fastest known performance for MSHSs with stochastic dependence. Numerical experiments and two case studies demonstrate that the proposed algorithm reduces computation time by up to 95% compared to the Universal Generating Function (UGF) and Multivalued Decision Diagram (MDD) approaches, and by up to 99.5% compared to the Monte Carlo Simulation (MCS) method, particularly in systems with inter-subsystem dependence. These results highlight the proposed method’s strong generality, structural adaptability, and significant computational advantage in complex reliability modeling. •Propose a recursive BN-based algorithm for reliability evaluation of MSHSs with arbitrary structures and stochastic dependencies.•Address component dependencies without structural assumptions, ensuring broad applicability to complex systems.•Provide simplified implementation and complexity analysis for typical system structures (series, parallel, bridge).•Achieve 20×–200× computational speedup over UGF, MDD, and MCS in extensive experiments and case studies.•Demonstrate practical value via application to a computing cluster hosting large language models (LLMs) and Autonomous driving system (ADS).