{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T20:04:12Z","timestamp":1769285052848,"version":"3.49.0"},"reference-count":29,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2026,1,22]],"date-time":"2026-01-22T00:00:00Z","timestamp":1769040000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["62573020"],"award-info":[{"award-number":["62573020"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Systems"],"abstract":"<jats:p>This study proposes a multi-level performance margin modeling and belief reliability framework for redundant systems. Starting from system performance, a \u201cperformance\u2013margin\u2013reliability\u201d linkage is established by defining the performance and margin of multi-level redundant systems and deriving performance, margin, and metric equations that account for failures. For complex redundant systems, a hierarchical Behavior Interaction Priority (BIP) modeling approach is developed to explicitly represent the normal and failure states of atomic component models. The effects of redundant components on the overall system are transformed into variations of performance parameters, enabling quantitative analysis of redundancy mechanisms. This paper proposes a boundary search algorithm for pruning optimization, which breaks through the computational bottleneck of non-analytic threshold sets in high-dimensional topological spaces. A case study on a power supply system with multi-level structural redundancy is conducted. Based on the proposed method, a performance-margin model of the redundant power supply system is constructed, critical states are analyzed, and system reliability is calculated. The results verify the effectiveness of the proposed margin-equation formulation and solution algorithm, offering practical guidance for reliability design of redundant systems.<\/jats:p>","DOI":"10.3390\/systems14010117","type":"journal-article","created":{"date-parts":[[2026,1,23]],"date-time":"2026-01-23T09:16:59Z","timestamp":1769159819000},"page":"117","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Performance Margin and Reliability Modeling Method for Multi-Level Redundant System"],"prefix":"10.3390","volume":"14","author":[{"given":"Tianyu","family":"Yang","sequence":"first","affiliation":[{"name":"School of Reliability and System Engineering, Beihang University, Beijing 100191, China"},{"name":"Science and Technology on Reliability and Environmental Engineering Laboratory, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6527-4251","authenticated-orcid":false,"given":"Ying","family":"Chen","sequence":"additional","affiliation":[{"name":"School of Reliability and System Engineering, Beihang University, Beijing 100191, China"},{"name":"Science and Technology on Reliability and Environmental Engineering Laboratory, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yujia","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Reliability and System Engineering, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Yaohui","family":"Guo","sequence":"additional","affiliation":[{"name":"School of Reliability and System Engineering, Beihang University, Beijing 100191, China"},{"name":"Science and Technology on Reliability and Environmental Engineering Laboratory, Beihang University, Beijing 100191, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,1,22]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Benabid, R., Merrouche, D., Bourenane, A., and Alzbutas, R. (2018). Reliability Assessment of Redundant Electrical Power Supply Systems Using Fault Tree Analysis, Reliability Block Diagram, and Monte Carlo Simulation Methods. Proceedings of the 2018 International Conference on Electrical Sciences and Technologies in Maghreb (CISTEM), IEEE.","DOI":"10.1109\/CISTEM.2018.8613431"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1521","DOI":"10.1016\/j.ress.2006.10.010","article-title":"State\/Event Fault Trees\u2014A Safety Analysis Model for Software-Controlled Systems","volume":"92","author":"Kaiser","year":"2007","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1016\/S0951-8320(96)00039-7","article-title":"Petri Nets for the Evaluation of Redundant Systems","volume":"55","author":"Ereau","year":"1997","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chen, Q., and Zhang, J. (2025). Evolution Model of Emergency Material Supply Chain Stress Based on Stochastic Petri Nets\u2014A Case Study of Emergency Medical Material Supply Chains in China. Systems, 13.","DOI":"10.3390\/systems13060423"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"128","DOI":"10.1109\/TDSC.2009.45","article-title":"A Rigorous, Compositional, and Extensible Framework for Dynamic Fault Tree Analysis","volume":"7","author":"Boudali","year":"2010","journal-title":"IEEE Trans. Dependable Secur. Comput."},{"key":"ref_6","first-page":"540","article-title":"Binary Decision Diagram-Based Reliability Evaluation of k-out-of-(n plus k) Warm Standby Systems Subject to Fault-Level Coverage","volume":"227","author":"Zhai","year":"2013","journal-title":"Proc. Inst. Mech. Eng. Part O-J. Risk Reliab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.cie.2016.04.017","article-title":"A Study of Interval Analysis for Cold-Standby System Reliability Optimization under Parameter Uncertainty","volume":"97","author":"Wang","year":"2016","journal-title":"Comput. Ind. Eng."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Ahmadi, M., Shekhar, A., and Bauer, P. (2022). Impact of the Various Components Consideration on Choosing Optimal Redundancy Strategy in MMC. Proceedings of the 2022 IEEE 20th International Power Electronics and Motion Control Conference (PEMC), IEEE.","DOI":"10.1109\/PEMC51159.2022.9962860"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1109\/24.159800","article-title":"Dynamic Fault-Tree Models for Fault-Tolerant Computer Systems","volume":"41","author":"Dugan","year":"1992","journal-title":"IEEE Trans. Reliab."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Popa, C., Stefanov, O., Goia, I., and Nistor, F. (2025). A Hybrid Fault Tree\u2013Fuzzy Logic Model for Risk Analysis in Multimodal Freight Transport. Systems, 13.","DOI":"10.3390\/systems13060429"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1235","DOI":"10.1109\/TSG.2011.2180937","article-title":"Reliability Evaluation of Phasor Measurement Unit Using Monte Carlo Dynamic Fault Tree Method","volume":"3","author":"Zhang","year":"2012","journal-title":"IEEE Trans. Smart Grid"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/j.ejor.2013.07.020","article-title":"A Novel Optimal Preventive Maintenance Policy for a Cold Standby System Based on Semi-Markov Theory","volume":"232","author":"Zhong","year":"2014","journal-title":"Eur. J. Oper. Res."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"6471","DOI":"10.1080\/03610926.2014.944660","article-title":"An Alternative Approach to Reliability Analysis of Cold Standby Systems","volume":"45","author":"Fathizadeh","year":"2016","journal-title":"Commun. Stat.\u2014Theory Methods"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1513","DOI":"10.1108\/EC-05-2019-0241","article-title":"Reliability Analysis for Complex Systems Based on Generalized Stochastic Petri Nets and EDA Approach Considering Common Cause Failure","volume":"37","author":"Zeng","year":"2019","journal-title":"Eng. Comput."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Tannous, O., Xing, L., and Dugan, J.B. (2011). Reliability Analysis of Warm Standby Systems Using Sequential BDD. Proceedings of the Annual Reliability and Maintainability Symposium (Rams), 2011 proceedings, IEEE.","DOI":"10.1109\/RAMS.2011.5754426"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"795","DOI":"10.1109\/TR.2017.2717928","article-title":"Reliability Evaluation for Demand-Based Warm Standby Systems Considering Degradation Process","volume":"66","author":"Jia","year":"2017","journal-title":"IEEE Trans. Reliab."},{"key":"ref_17","first-page":"557","article-title":"An Enhanced Decision Diagram-Based Method for Common-Cause Failure Analysis","volume":"227","author":"Mo","year":"2013","journal-title":"Proc. Inst. Mech. Eng. Part O-J. Risk Reliab."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"895","DOI":"10.3390\/signals3040053","article-title":"An Improved d-MP Algorithm for Reliability of Logistics Delivery Considering Speed Limit of Different Roads","volume":"3","author":"Yeh","year":"2022","journal-title":"Signals"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"920","DOI":"10.1109\/TSMC.2016.2633808","article-title":"Optimizing Dynamic Performance of Multistate Systems With Heterogeneous 1-Out-of-N Warm Standby Components","volume":"48","author":"Levitin","year":"2018","journal-title":"IEEE Trans. Syst. Man Cybern.-Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"397","DOI":"10.1016\/j.ress.2019.05.002","article-title":"Dynamic Demand Satisfaction Probability of Consecutive Sliding Window Systems with Warm Standby Components","volume":"189","author":"Levitin","year":"2019","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"571","DOI":"10.1016\/j.cja.2016.04.004","article-title":"Measuring Reliability under Epistemic Uncertainty: Review on Non-Probabilistic Reliability Metrics","volume":"29","author":"Kang","year":"2016","journal-title":"Chin. J. Aeronaut."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1007\/s10700-012-9138-5","article-title":"Belief Reliability: A New Metrics for Products\u2019 Reliability","volume":"12","author":"Zeng","year":"2013","journal-title":"Fuzzy Optim. Decis. Mak."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"108697","DOI":"10.1016\/j.cie.2022.108697","article-title":"Belief Reliability Modeling for the Two-Phase Degradation System with a Change Point under Small Sample Conditions","volume":"173","author":"Wang","year":"2022","journal-title":"Comput. Ind. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1016\/j.ins.2022.05.022","article-title":"Belief Reliability Analysis of Multi-State Deteriorating Systems under Epistemic Uncertainty","volume":"604","author":"Li","year":"2022","journal-title":"Inf. Sci."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"521","DOI":"10.1109\/TR.2023.3298018","article-title":"Epistemic Uncertainty Propagation and Reliability Evaluation of Feedback Control System","volume":"73","author":"Chen","year":"2024","journal-title":"IEEE Trans. Reliab."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"108958","DOI":"10.1016\/j.ress.2022.108958","article-title":"Hybrid Uncertainty Quantification of Dependent Competing Failure Process with Chance Theory","volume":"230","author":"Chen","year":"2023","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Basu, A., Bozga, M., and Sifakis, J. (2006). Modeling heterogeneous real-time components in BIP. Proceedings of the Fourth IEEE International Conference on Software Engineering and Formal Methods (SEFM 2006), IEEE.","DOI":"10.1109\/SEFM.2006.27"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Basu, A., Bensalem, S., Bozga, M., Bourgos, P., and Sifakis, J. (2012). Rigorous System Design: The BIP Approach, Springer. Lecture Notes in Computer Science (MEMICS 2011).","DOI":"10.1007\/978-3-642-25929-6_1"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1315","DOI":"10.1109\/TC.2008.26","article-title":"The Algebra of Connectors\u2014Structuring Interaction in BIP","volume":"57","author":"Bliudze","year":"2008","journal-title":"IEEE Trans. Comput."}],"container-title":["Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2079-8954\/14\/1\/117\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T05:14:02Z","timestamp":1769231642000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2079-8954\/14\/1\/117"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,1,22]]},"references-count":29,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2026,1]]}},"alternative-id":["systems14010117"],"URL":"https:\/\/doi.org\/10.3390\/systems14010117","relation":{},"ISSN":["2079-8954"],"issn-type":[{"value":"2079-8954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,1,22]]}}}