Compressing the Markov Chain State Space for Circular k-out-of-n: G Balanced Systems under Sequential External Shocks

Purpose: This study investigates the state-space explosion problem in discrete-time Markov chain (DTMC) models of circular k-out-of-n: G balanced systems under cumulative shock-induced system failures and develops a systematic state-space compression procedure based on transition-structure equivalence.<br/>Methods: States are layered according to the number of failed units, and sub-DTMCs within the same layer that have identical reachable structures and transition probabilities are merged. Transition probabilities to lower layers are summed to preserve probabilistic consistency.<br/>Results: The proposed compression algorithm substantially decreases the number of states in the DTMC compared with the original state space, while maintaining equivalent stochastic behavior. Numerical examples confirm consistent state-space reduction effects across various (n, k) configurations.<br/>Conclusion: The proposed compression method provides an efficient and exact approach for compressing DTMCs of cumulative-failure systems without relying on explicit symmetry groups. The results enable tractable reliability analysis for larger systems and provide a foundation for future analytical and computational extensions.