{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,12]],"date-time":"2026-05-12T08:40:32Z","timestamp":1778575232407,"version":"3.51.4"},"reference-count":37,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2024,6,24]],"date-time":"2024-06-24T00:00:00Z","timestamp":1719187200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100009110","name":"Natural Science Foundation of Xinjiang Uygur Autonomous Region","doi-asserted-by":"publisher","award":["2022D01C46"],"award-info":[{"award-number":["2022D01C46"]}],"id":[{"id":"10.13039\/100009110","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper considers a system with one robot and n safety units (one of which works while the others remain on standby), which is described by an integro-deferential equation. The system can fail in the following three ways: fails with an incident, fails safely and fails due to the malfunction of the robot. Using the C0\u2013semigroups theory of linear operators, we first show that the system has a unique non-negative, time-dependent solution. Then, we obtain the exponential convergence of the time-dependent solution to its steady-state solution. In addition, we study the asymptotic behavior of some time-dependent reliability indices and present a numerical example demonstrating the effects of different parameters on the system.<\/jats:p>","DOI":"10.3390\/axioms13070423","type":"journal-article","created":{"date-parts":[[2024,6,24]],"date-time":"2024-06-24T06:59:58Z","timestamp":1719212398000},"page":"423","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["C0\u2013Semigroups Approach to the Reliability Model Based on Robot-Safety System"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6680-7999","authenticated-orcid":false,"given":"Ehmet","family":"Kasim","sequence":"first","affiliation":[{"name":"College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Aihemaitijiang","family":"Yumaier","sequence":"additional","affiliation":[{"name":"College of Mathematics and Systems Science, Xinjiang University, Urumqi 830046, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,6,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1017\/S0305004100030437","article-title":"The analysis of non-markovian stochastic processes by the inclusion of supplementary variables","volume":"51","author":"Cox","year":"1955","journal-title":"Math. Proc. Camb. Phil. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1109\/TR.1963.5218202","article-title":"Time to failure and availability of parallel redundant systems with repair","volume":"R-12","author":"Gaver","year":"1963","journal-title":"IEEE Trans. Reliab."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/0026-2714(76)90140-2","article-title":"Some advancements in the analysis of two-unit parallel redundant systems","volume":"15","author":"Linton","year":"1976","journal-title":"Microelectron. Reliab."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1016\/0026-2714(86)90285-4","article-title":"Cost analysis of an electronic repairable redundant system with critical human errors","volume":"26","author":"Gupta","year":"1986","journal-title":"Microelectron. Reliab."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/BF02009636","article-title":"Availability analysis of a two-unit series system with shut-off rule and \u201cfirst-fail, first-repaired\u201d","volume":"9","author":"Shi","year":"1993","journal-title":"Acta. Math. Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1479","DOI":"10.1016\/0026-2714(94)00174-M","article-title":"Reliability of imperfect switching of cold stanby systems with multiple non-critical and critical errors","volume":"35","author":"Chung","year":"1995","journal-title":"Microelectron. Reliab."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1016\/j.anucene.2004.06.003","article-title":"Unavailability analysis of safety systems under aging by supplementary variables with imperfect repair","volume":"32","author":"Oliveira","year":"2005","journal-title":"Ann. Nucl. Energy"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"3052","DOI":"10.1016\/j.apm.2008.10.022","article-title":"Reliability analysis for a k\/n(f) system with repairable repair-equipment","volume":"33","author":"Zhang","year":"2009","journal-title":"Appl. Math. Model."},{"key":"ref_9","first-page":"3981","article-title":"Reliability analysis of polytube industry using supplementary variable technique","volume":"218","author":"Shakuntla","year":"2011","journal-title":"Appl. Math. Comput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1124","DOI":"10.1109\/TASE.2013.2266134","article-title":"Cost analysis of an engineering system involving subsystems in series configuration","volume":"10","author":"Singh","year":"2013","journal-title":"IEEE. Autom. Sci. Eng."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"12","DOI":"10.1016\/j.ress.2018.01.013","article-title":"Modeling of machine interference problem with unreliable repairman and standbys imperfect switchover","volume":"174","author":"Ke","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1108\/EC-01-2019-0003","article-title":"Fault-tolerant redundant repairable system with different failures and delays","volume":"37","author":"Shekhar","year":"2020","journal-title":"Eng. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"107240","DOI":"10.1016\/j.ress.2020.107240","article-title":"Reliability and availability analysis of a retrial system with mixed standbys and an unreliable repair facility","volume":"205","author":"Gao","year":"2021","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1108\/13552510210439784","article-title":"Robot systems reliability and safety: A review","volume":"8","author":"Dhillon","year":"2002","journal-title":"J. Qual. Maint. Eng."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.robot.2005.10.005","article-title":"Real-time safety for human-robot interaction","volume":"54","author":"Croft","year":"2006","journal-title":"Rob. Autom. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"169","DOI":"10.1016\/0026-2714(95)00103-9","article-title":"Availability analysis of a robot with safety system","volume":"36","author":"Dhillon","year":"1996","journal-title":"Microelectron. Reliab."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1016\/S0026-2714(96)00079-0","article-title":"Formulas for analyzing a redundant robot configuration with a built-in safety system","volume":"37","author":"Dhillon","year":"1997","journal-title":"Microelectron. Reliab."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0026-2714(96)00074-1","article-title":"Robotic systems probabilistic analysis","volume":"37","author":"Dhillon","year":"1997","journal-title":"Microelectron. Reliab."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/s10846-005-9024-2","article-title":"Stochastic analysis of a system with redundant robots, one built-in safety unit, and common-cause failures","volume":"45","author":"Dhillon","year":"2006","journal-title":"J. Intell. Rob. Syst."},{"key":"ref_20","first-page":"1077","article-title":"Performance analysis of a repairable robot safety system with standby, imperfect coverage and reboot delay","volume":"26","author":"Jain","year":"2013","journal-title":"Int. J. Eng. Trans. C Aspects"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1889","DOI":"10.1080\/00207721.2013.837592","article-title":"Overall availability and risk analysis of a general robot csafety device system","volume":"46","author":"Vanderperre","year":"2015","journal-title":"Int. J. Syst. Sci."},{"key":"ref_22","first-page":"188","article-title":"Well-posedness of the system consisting of two repairable units","volume":"3","author":"Gupur","year":"2001","journal-title":"Acta Anal. Funct. Appl."},{"key":"ref_23","first-page":"137","article-title":"Semigroup method for a mathematical model in reliability analysis","volume":"10","author":"Gupur","year":"2001","journal-title":"J. Syst. Sci. Syst. Eng."},{"key":"ref_24","first-page":"116","article-title":"Stability analysis of the system consising of a robot and its associated safety mechanism","volume":"33","author":"Guo","year":"2003","journal-title":"Math. Pract. Theory"},{"key":"ref_25","first-page":"170","article-title":"Analysis of exponential stability to the robot system with periodic repair function","volume":"12","author":"Chen","year":"2010","journal-title":"Acta. Anal. Funct. Appl."},{"key":"ref_26","first-page":"483","article-title":"Well-posedness of the model describing a repairable, standby, human & machine system","volume":"16","author":"Gupur","year":"2003","journal-title":"J. Syst. Sci. Complex."},{"key":"ref_27","first-page":"35","article-title":"Asymptotic property of the solution of a repairable, standby, human and machine system","volume":"28","author":"Gupur","year":"2006","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_28","first-page":"571","article-title":"Further result on a repairable, standby human and machine system","volume":"101","author":"Aili","year":"2016","journal-title":"Int. J. Pure Appl. Math."},{"key":"ref_29","first-page":"289","article-title":"Exponential stability analysis of a redundant robot configuration with a built-in safety systemfurther result on a repairable, standby human and machine system","volume":"131","author":"Zhang","year":"2019","journal-title":"J. Syst. Sci. Inf."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"1072","DOI":"10.1007\/s11424-019-7353-7","article-title":"Reliability and numerical analysis of a robot safety system","volume":"32","author":"Qiao","year":"2019","journal-title":"J Syst. Sci. Complex."},{"key":"ref_31","first-page":"9","article-title":"Reliability and controllability of the system consisting of a robot and its associated safety mechanism","volume":"40","author":"Li","year":"2020","journal-title":"J. Math. Res. Appl."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1108\/13552511111134619","article-title":"Reliability and availability analysis of a robot-safety system","volume":"17","author":"Dhillon","year":"2011","journal-title":"J. Qual. Maint. Eng."},{"key":"ref_33","unstructured":"Adams, R.A. (1975). Sobolev Spaces, Academic Press."},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Gupur, G. (2011). Functional Analysis Methods for Reliability Models, Springer.","DOI":"10.1007\/978-3-0348-0101-0"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Fattorini, H.O. (1984). The Cauchy Problem, Cambridge University Press.","DOI":"10.1017\/CBO9780511662799"},{"key":"ref_36","unstructured":"Nagel, R. (1986). One-Parameter Semigroups of Positive Operators (LNM 1184), Springer."},{"key":"ref_37","unstructured":"Engel, K.J., and Nagel, R. (2000). One-Parameter Semigroups for Linear Evolution Equations, Springer."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/423\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:03:26Z","timestamp":1760108606000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/7\/423"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,6,24]]},"references-count":37,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2024,7]]}},"alternative-id":["axioms13070423"],"URL":"https:\/\/doi.org\/10.3390\/axioms13070423","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,6,24]]}}}