{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,3]],"date-time":"2026-01-03T06:44:42Z","timestamp":1767422682966,"version":"build-2065373602"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2023,3,23]],"date-time":"2023-03-23T00:00:00Z","timestamp":1679529600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Sciences and Engineering Research Council of Canada","award":["06536-2018","RSP2023R464"],"award-info":[{"award-number":["06536-2018","RSP2023R464"]}]},{"name":"King Saud University in Riyadh, Saudi Arabia","award":["06536-2018","RSP2023R464"],"award-info":[{"award-number":["06536-2018","RSP2023R464"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The measurement of uncertainty across the lifetimes of engineering systems has drawn more attention in recent years. It is a helpful metric for assessing how predictable a system\u2019s lifetime is. In these circumstances, Renyi entropy, a Shannon entropy extension, is particularly appealing. In this paper, we develop the system signature to give an explicit formula for the Renyi entropy of the residual lifetime of a coherent system when all system components have lived to a time t. In addition, several findings are studied for the aforementioned entropy, including the bounds and order characteristics. It is possible to compare the residual lifespan predictability of two coherent systems with known signatures using the findings of this study.<\/jats:p>","DOI":"10.3390\/axioms12040320","type":"journal-article","created":{"date-parts":[[2023,3,24]],"date-time":"2023-03-24T03:16:46Z","timestamp":1679627806000},"page":"320","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Renyi Entropy of the Residual Lifetime of a Reliability System at the System Level"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4692-3663","authenticated-orcid":false,"given":"Mhamed","family":"Mesfioui","sequence":"first","affiliation":[{"name":"D\u00e9partement de Math\u00e9matiques et d\u2019Informatique, Universit\u00e9 du Qu\u00e9bec \u00e0 Trois-Rivi\u00e8res, 3351 Boulevard des Forges, Trois-Rivi\u00e8res, QC G9A 5H7, Canada"},{"name":"Department of Statistics, United Arab Emirates University, Al Ain 15551, United Arab Emirates"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6363-1246","authenticated-orcid":false,"given":"Mohamed","family":"Kayid","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3456-8393","authenticated-orcid":false,"given":"Mansour","family":"Shrahili","sequence":"additional","affiliation":[{"name":"Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2023,3,23]]},"reference":[{"key":"ref_1","unstructured":"Clausius, R. (1879). The Mechanical Theory of Heat, MacMillan and Co."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Basaran, C. (2023). Introduction to Unified Mechanics Theory with Applications, Springer Nature.","DOI":"10.1007\/978-3-031-18621-9"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"111383","DOI":"10.1016\/j.matdes.2022.111383","article-title":"Modeling fatigue of pre-corroded body-centered cubic metals with unified mechanics theory","volume":"224","author":"Lee","year":"2022","journal-title":"Mater. 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