{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:31:32Z","timestamp":1760149892888,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2023,9,21]],"date-time":"2023-09-21T00:00:00Z","timestamp":1695254400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"King Saud University","award":["RSP2023R464"],"award-info":[{"award-number":["RSP2023R464"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper explores the cumulative entropy of the lifetime of an n-component coherent system, given the precondition that all system components have experienced failure at time t. This investigation utilizes the system signature to compute the cumulative entropy of the system\u2019s lifetime, shedding light on a crucial facet of a system\u2019s predictability. In the course of this research, we unearth a series of noteworthy discoveries. These include formulating expressions, defining bounds, and identifying orderings related to this measure. Further, we propose a technique to identify a preferred system on the basis of cumulative Kullback\u2013Leibler discriminating information, which exhibits a strong relation with the parallel system. These findings contribute significantly to our understanding of the predictability of a coherent system\u2019s lifetime, underscoring the importance of this field of study. The outcomes offer potential benefits for a wide range of applications where system predictability is paramount, and where the comparative evaluation of different systems on the basis of discriminating information is needed.<\/jats:p>","DOI":"10.3390\/axioms12090899","type":"journal-article","created":{"date-parts":[[2023,9,22]],"date-time":"2023-09-22T05:32:45Z","timestamp":1695360765000},"page":"899","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Cumulative Entropy of Past Lifetime for Coherent Systems at the System Level"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3456-8393","authenticated-orcid":false,"given":"Mansour","family":"Shrahili","sequence":"first","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\/0000-0001-6363-1246","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"}]}],"member":"1968","published-online":{"date-parts":[[2023,9,21]]},"reference":[{"key":"ref_1","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_2","doi-asserted-by":"crossref","first-page":"1220","DOI":"10.1109\/TIT.2004.828057","article-title":"Cumulative residual entropy: A new measure of information","volume":"50","author":"Rao","year":"2004","journal-title":"IEEE Trans. Inf. 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