{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,30]],"date-time":"2026-05-30T00:26:33Z","timestamp":1780100793483,"version":"3.54.0"},"reference-count":32,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T00:00:00Z","timestamp":1716768000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Shanghai Sailing Program","award":["20YF1414800"],"award-info":[{"award-number":["20YF1414800"]}]},{"name":"Shanghai Sailing Program","award":["21QA1403400"],"award-info":[{"award-number":["21QA1403400"]}]},{"name":"Shanghai Rising-Star Program","award":["20YF1414800"],"award-info":[{"award-number":["20YF1414800"]}]},{"name":"Shanghai Rising-Star Program","award":["21QA1403400"],"award-info":[{"award-number":["21QA1403400"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>The Markov method is a common reliability assessment method. It is often used to describe the dynamic characteristics of a system, such as its repairability, fault sequence and multiple degradation states. However, the \u201ccurse of dimensionality\u201d, which refers to the exponential growth of the system state space with the increase in system complexity, presents a challenge to reliability assessments for complex systems based on the Markov method. In response to this challenge, a novel reliability assessment method for complex systems based on non-homogeneous Markov processes is proposed. This method entails the decomposition of a complex system into multilevel subsystems, each with a relatively small state space, in accordance with the system function. The homogeneous Markov model or the non-homogeneous Markov model is established for each subsystem\/system from bottom to top. In order to utilize the outcomes of the lower-level subsystem models as inputs to the upper-level subsystem model, an algorithm is proposed for converting the unavailability curve of a subsystem into its corresponding 2\u00d72 dynamic state transition probability matrix (STPM). The STPM is then employed as an input to the upper-level system\u2019s non-homogeneous Markov model. A case study is presented using the reliability assessment of the Reactor Protection System (RPS) based on the proposed method, which is then compared with the models based on the other two contrast methods. This comparison verifies the effectiveness and accuracy of the proposed method.<\/jats:p>","DOI":"10.3390\/s24113446","type":"journal-article","created":{"date-parts":[[2024,5,27]],"date-time":"2024-05-27T09:33:31Z","timestamp":1716802411000},"page":"3446","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["A Reliability Assessment Method for Complex Systems Based on Non-Homogeneous Markov Processes"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5860-1007","authenticated-orcid":false,"given":"Xiaolei","family":"Pan","sequence":"first","affiliation":[{"name":"College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China"},{"name":"Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0000-7277-7618","authenticated-orcid":false,"given":"Hongxiao","family":"Chen","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China"},{"name":"Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ao","family":"Shen","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China"},{"name":"Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Dongdong","family":"Zhao","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China"},{"name":"Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoyan","family":"Su","sequence":"additional","affiliation":[{"name":"College of Automation Engineering, Shanghai University of Electric Power, Shanghai 200090, China"},{"name":"Shanghai Key Laboratory of Power Station Automation Technology, Shanghai 200090, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2024,5,27]]},"reference":[{"key":"ref_1","unstructured":"International Atomic Energy Agency (2010). Development and Application of Level 1 Probabilistic Safety Assessment for Nuclear Power Plants, IAEA. IAEA Safety Standards Series No. SSG-3."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Pan, X., Di Maio, F., and Zio, E. (2017, January 20\u201322). A Benchmark of Dynamic Reliability Methods for Probabilistic Safety Assessment. Proceedings of the 2017 2nd International Conference on System Reliability and Safety (ICSRS), Milan, Italy.","DOI":"10.1109\/ICSRS.2017.8272801"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.3389\/fenrg.2021.750453","article-title":"Dynamic PRA Prospects for the Nuclear Industry","volume":"9","author":"Wiltbank","year":"2021","journal-title":"Front. Energy Res."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"55","DOI":"10.13182\/NT09-A4062","article-title":"A Benchmark System for Comparing Reliability Modeling Approaches for Digital Instrumentation and Control Systems","volume":"165","author":"Kirschenbaum","year":"2009","journal-title":"Nucl. Technol."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"104734","DOI":"10.1016\/j.jlp.2022.104734","article-title":"A Comprehensive Review on Dynamic Risk Analysis Methodologies","volume":"76","author":"Raveendran","year":"2022","journal-title":"J. Loss Prev. Proc."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"316","DOI":"10.1016\/j.anucene.2015.03.020","article-title":"Development of Reliability and Probabilistic Safety Assessment Program RiskA","volume":"83","author":"Wu","year":"2015","journal-title":"Ann. Nucl. Energy"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"108316","DOI":"10.1016\/j.ress.2021.108316","article-title":"Simulation Optimization Framework for Dynamic Probabilistic Safety Assessment","volume":"220","author":"Park","year":"2022","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/j.anucene.2012.08.001","article-title":"A Survey of Dynamic Methodologies for Probabilistic Safety Assessment of Nuclear Power plants","volume":"52","author":"Aldemir","year":"2013","journal-title":"Ann. Nucl. Energy"},{"key":"ref_9","unstructured":"Aldernir, T., Miller, D., Stovsky, M., Kirschenbaurr, J., Bucci, P., Fentiman, A., and Mangan, L. (2006). Current State of Reliability Modeling Methodologies for Digital Systems and Their Acceptance Criteria for Nuclear Power Plant Assessments (NUREG\/CR-6901)."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"91","DOI":"10.1007\/s41365-017-0255-2","article-title":"Biasing Transition Rate Method Based on Direct MC Simulation for Probabilistic Safety Assessment","volume":"28","author":"Pan","year":"2017","journal-title":"Nucl. Sci. Tech."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"103479","DOI":"10.1016\/j.probengmech.2023.103479","article-title":"Monte Carlo and Variance Reduction Methods for Structural Reliability Analysis: A Comprehensive Review","volume":"73","author":"Song","year":"2023","journal-title":"Probabilistic Eng. Mech."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1041","DOI":"10.1108\/EC-09-2022-0583","article-title":"Reliability Analysis of Repairable Multistate Phased Mission Systems with Markov Approach Based on States","volume":"40","author":"Yilmaz","year":"2023","journal-title":"Eng. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109411","DOI":"10.1016\/j.ress.2023.109411","article-title":"Reliability Analysis of The Multi-state System with Nonlinear Degradation Model under Markov Environment","volume":"238","author":"Lyu","year":"2023","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"48","DOI":"10.1016\/j.ress.2014.08.007","article-title":"Markov Analysis of Redundant Standby Safety Systems under Periodic Surveillance Testing","volume":"133","author":"Hellmich","year":"2015","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/s41872-018-0054-8","article-title":"Markov Modeling and Reliability Allocation in Wind Turbine for Availability Enhancement","volume":"7","author":"Rajeevan","year":"2018","journal-title":"Life Cycle Reliab. Saf. Eng."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Bessman, J.C. (2020, January 27\u201330). Non-Homogeneous Markov Models and Their Application in Reliability. Proceedings of the 2020 Annual Reliability and Maintainability Symposium (RAMS), Palm Springs, CA, USA.","DOI":"10.1109\/RAMS48030.2020.9153655"},{"key":"ref_17","unstructured":"Chen, G., Gaebler, J.D., Peng, M., Sun, C., and Ye, Y. (2021). An Adaptive State Aggregation Algorithm for Markov Decision Processes. arXiv."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1016\/j.jtbi.2016.04.003","article-title":"Simplification of Irreversible Markov Chains by removal of States with Fast Leaving Rates","volume":"400","author":"Jia","year":"2016","journal-title":"J. Theor. Biol."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"108373","DOI":"10.1016\/j.ress.2022.108373","article-title":"Analysis of Simplification in Markov State-based Models for Reliability Assessment of Complex Safety Systems","volume":"221","author":"Liang","year":"2022","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zuo, W., and Li, K. (2021, January 24\u201326). Three-State Markov Chain Based Reliability Analysis of Complex Traction Power Supply Systems. Proceedings of the 2021 5th International Conference on System Reliability and Safety (ICSRS), Palermo, Italy.","DOI":"10.1109\/ICSRS53853.2021.9660623"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"872","DOI":"10.1016\/j.ress.2008.09.007","article-title":"Dynamic Fault Tree analysis Using Monte Carlo Simulation in Probabilistic Safety Assessment","volume":"94","author":"Gopika","year":"2009","journal-title":"Reliab. Eng. Syst. Safe"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"3445","DOI":"10.1002\/qre.3196","article-title":"A Hybrid Computing Framework for Risk-oriented Reliability Analysis in Dynamic PSA Context: A Case Study","volume":"39","author":"Jiang","year":"2023","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_23","unstructured":"Abdulmunem, A.H., and Al-Khafaji, Z. (2023, January 4\u20135). Using Markov Models and Fault Tree for Finding the Reliability of Some Engineering Problems. Proceedings of the 2023 6th International Conference on Engineering Technology and its Applications (IICETA), Al-Najaf, Iraq."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"107409","DOI":"10.1016\/j.anucene.2020.107409","article-title":"Periodic Surveillance Test Strategies to Effectively Enhance the Availability of Safety-critical Systems in NPPs Using the Multi-state Based Availability Model","volume":"142","author":"Son","year":"2020","journal-title":"Ann. Nucl. Energy"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1016\/j.jtbi.2012.07.007","article-title":"Simplification of Reversible Markov Chains by Removal of States with Low Equilibrium Occupancy","volume":"311","author":"Ullah","year":"2012","journal-title":"J. Theor. Biol."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"032402","DOI":"10.1103\/PhysRevE.96.032402","article-title":"Simplification of Markov Chains with Infinite State Space and the Mathematical Theory of Random Gene Expression Bursts","volume":"96","author":"Jia","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_27","unstructured":"(2000). Functional Safety of Electrical\/Electronic\/Programmable Electronic Safety Related Systems (Standard No. IEC 61508)."},{"key":"ref_28","first-page":"100605","article-title":"Application of Dynamic Fault Tree in Reliability Assessment of Reactor Protection System","volume":"044","author":"Li","year":"2021","journal-title":"Nucl. Tech."},{"key":"ref_29","unstructured":"Eide, S.A., Calley, M.B., Gentillon, C.A., Wierman, T.E., Rasmuson, D., and Marksberry, D. (1999, January 22\u201325). Westinghouse Reactor Protection System Unavailability, 1984\u20131995. Proceedings of the PSA \u201899, Washington, DC, USA."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"e12201","DOI":"10.1002\/prs.12201","article-title":"Development of Dynamic Fault Tree Model for Reactor Protection System","volume":"40","author":"Fahmy","year":"2021","journal-title":"Process Saf. Prog."},{"key":"ref_31","unstructured":"Yu, Q. (2012). Rechearch on Reliability Evaluation and Life Test Methods for Sppace Relays. [Ph.D. Thesis, Harbin Institute of Technology]."},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Pham, H. (2011). Safety and Risk Modeling and Its Applications, Springer.","DOI":"10.1007\/978-0-85729-470-8"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/11\/3446\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T14:49:25Z","timestamp":1760107765000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/24\/11\/3446"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,5,27]]},"references-count":32,"journal-issue":{"issue":"11","published-online":{"date-parts":[[2024,6]]}},"alternative-id":["s24113446"],"URL":"https:\/\/doi.org\/10.3390\/s24113446","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,5,27]]}}}