{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T05:08:56Z","timestamp":1768712936169,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T00:00:00Z","timestamp":1615766400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["Nos. 12088101, 11534002, U1930402, 1930403, 51975546, 11872088"],"award-info":[{"award-number":["Nos. 12088101, 11534002, U1930402, 1930403, 51975546, 11872088"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Basic Program of China","award":["No. 2016YFA0301201"],"award-info":[{"award-number":["No. 2016YFA0301201"]}]},{"DOI":"10.13039\/501100002851","name":"CAEP","doi-asserted-by":"publisher","award":["CX20200035"],"award-info":[{"award-number":["CX20200035"]}],"id":[{"id":"10.13039\/501100002851","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The degradation and recovery processes are multi-scale phenomena in many physical, engineering, biological, and social systems, and determine the aging of the entire system. Therefore, understanding the interplay between the two processes at the component level is the key to evaluate the reliability of the system. Based on the principle of maximum entropy, an approach is proposed to model and infer the processes at the component level, and is applied to repairable and non-repairable systems. By incorporating the reliability block diagram, this approach allows for integrating the information of network connectivity and statistical moments to infer the hazard or recovery rates of the degradation or recovery processes. The overall approach is demonstrated with numerical examples.<\/jats:p>","DOI":"10.3390\/e23030348","type":"journal-article","created":{"date-parts":[[2021,3,15]],"date-time":"2021-03-15T11:38:58Z","timestamp":1615808338000},"page":"348","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Maximum Entropy Approach to Reliability of Multi-Component Systems with Non-Repairable or Repairable Components"],"prefix":"10.3390","volume":"23","author":[{"given":"Yi-Mu","family":"Du","sequence":"first","affiliation":[{"name":"Graduate School of China Academy of Engineering Physics, Beijing 100193, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7207-969X","authenticated-orcid":false,"given":"Jin-Fu","family":"Chen","sequence":"additional","affiliation":[{"name":"Graduate School of China Academy of Engineering Physics, Beijing 100193, China"},{"name":"Beijing Computational Science Research Center, Beijing 100193, China"}]},{"given":"Xuefei","family":"Guan","sequence":"additional","affiliation":[{"name":"Graduate School of China Academy of Engineering Physics, Beijing 100193, China"}]},{"given":"C. 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Rev."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1103\/PhysRev.108.171","article-title":"Information theory and statistical mechanics. II","volume":"108","author":"Jaynes","year":"1957","journal-title":"Phys. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"939","DOI":"10.1109\/PROC.1982.12425","article-title":"On the rationale of maximum-entropy methods","volume":"70","author":"Jaynes","year":"1982","journal-title":"Proc. IEEE"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1080\/00401706.1961.10489927","article-title":"Multi-component systems and structures and their reliability","volume":"3","author":"Birnbaum","year":"1961","journal-title":"Technometrics"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Rausand, M., Barros, A., and Hoyland, A. (2020). System Reliability Theory: Models, Statistical Methods, and Applications, John Wiley & Sons.","DOI":"10.1002\/9781119373940"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1410","DOI":"10.1016\/j.ijepes.2011.06.021","article-title":"Performance and reliability of electrical power grids under cascading failures","volume":"33","author":"Chang","year":"2011","journal-title":"Int. J. Electr. Power Energy Syst."},{"key":"ref_9","unstructured":"Mieghem, P.V. (2006). Performance Analysis of Communications Networks and Systems, Cambridge University Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1275","DOI":"10.1103\/RevModPhys.80.1275","article-title":"Critical phenomena in complex networks","volume":"80","author":"Dorogovtsev","year":"2008","journal-title":"Rev. Mod. Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"591","DOI":"10.1103\/RevModPhys.81.591","article-title":"Statistical physics of social dynamics","volume":"81","author":"Castellano","year":"2009","journal-title":"Rev. Mod. Phys."},{"key":"ref_12","unstructured":"Barrat, A., Barthelemy, M., and Vespignani, A. (2012). Dynamical Processes on Complex Networks, Cambridge University Press."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1103\/RevModPhys.87.925","article-title":"Epidemic processes in complex networks","volume":"87","author":"Castellano","year":"2015","journal-title":"Rev. Mod. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"47","DOI":"10.1103\/RevModPhys.74.47","article-title":"Statistical Mechanics of Complex Networks","volume":"74","author":"Albert","year":"2002","journal-title":"Rev. Mod. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1016\/j.physrep.2005.10.009","article-title":"Complex networks: Structure and dynamics","volume":"424","author":"Boccaletti","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"046119","DOI":"10.1103\/PhysRevE.64.046119","article-title":"Topology of technology graphs: Small world patterns in electronic circuits","volume":"64","author":"Cancho","year":"2001","journal-title":"Phys. Rev. E Stat. Nonlinear Soft Matter Phys."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"604","DOI":"10.1002\/qre.2594","article-title":"Identification of the influential parts in a complex mechanical product from a reliability perspective using complex network theory","volume":"36","author":"Yin","year":"2020","journal-title":"Qual. Reliab. Eng. Int."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"052409","DOI":"10.1103\/PhysRevE.94.052409","article-title":"Network model of human aging: Frailty limits and information measures","volume":"94","author":"Farrell","year":"2016","journal-title":"Phys. Rev. E"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"231","DOI":"10.1016\/j.physa.2017.08.028","article-title":"Reliability estimation of a N-M-cold-standby redundancy system in a multicomponent stress-strength model with generalized half-logistic distribution","volume":"490","author":"Liu","year":"2017","journal-title":"Physics A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"50","DOI":"10.1016\/j.physa.2018.02.028","article-title":"Reliability analysis in interdependent smart grid systems","volume":"500","author":"Peng","year":"2018","journal-title":"Physics A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"120","DOI":"10.1016\/j.physa.2016.01.083","article-title":"Reliability analysis of interdependent lattices","volume":"452","author":"Limiao","year":"2016","journal-title":"Physics A"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.exger.2017.08.027","article-title":"Unifying aging and frailty through complex dynamical networks","volume":"107","author":"Rutenberg","year":"2018","journal-title":"Exp. Gerontol."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"121865","DOI":"10.1016\/j.physa.2019.121865","article-title":"Reliability analysis of CPS systems under different edge repairing strategies","volume":"532","author":"Peng","year":"2019","journal-title":"Physics A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"440","DOI":"10.1038\/30918","article-title":"Collective dynamics of small-world networks","volume":"393","author":"Watts","year":"1998","journal-title":"Nature"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"26","DOI":"10.1109\/TIT.1980.1056144","article-title":"Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy","volume":"26","author":"Shore","year":"1980","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"2999","DOI":"10.1002\/sim.3136","article-title":"A likelihood-based method for real-time estimation of the serial interval and reproductive number of an epidemic","volume":"27","author":"White","year":"2008","journal-title":"Stat. Med."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Kucharski, A.J., Russell, T.W., Diamond, C., Liu, Y., Edmunds, J., Funk, S., Eggo, R.M., Sun, F., Jit, M., and Munday, J.D. (2020). Early dynamics of transmission and control of COVID-19: A mathematical modelling study. Lancet Infect. Dis.","DOI":"10.1101\/2020.01.31.20019901"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1016\/j.physa.2010.11.027","article-title":"Link prediction in complex networks: A survey","volume":"390","author":"Zhou","year":"2011","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2018.05.002","article-title":"Structure-oriented prediction in complex networks","volume":"750","author":"Ren","year":"2018","journal-title":"Phys. Rep."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/348\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T05:35:56Z","timestamp":1760160956000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/3\/348"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,15]]},"references-count":30,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,3]]}},"alternative-id":["e23030348"],"URL":"https:\/\/doi.org\/10.3390\/e23030348","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,15]]}}}