{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T15:26:12Z","timestamp":1746804372540,"version":"3.40.5"},"reference-count":12,"publisher":"Wiley","license":[{"start":{"date-parts":[[2014,1,1]],"date-time":"2014-01-01T00:00:00Z","timestamp":1388534400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"name":"Basic and Frontier Research Foundation of Chongqing of China","award":["cstc2013jcyjA00008","2012ZD48"],"award-info":[{"award-number":["cstc2013jcyjA00008","2012ZD48"]}]},{"name":"Scientific Research Starting Foundation for Doctors of Chongqing University of Technology","award":["cstc2013jcyjA00008","2012ZD48"],"award-info":[{"award-number":["cstc2013jcyjA00008","2012ZD48"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Applied Mathematics"],"published-print":{"date-parts":[[2014]]},"abstract":"<jats:p>This paper develops a probabilistic decomposition method for an<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:msup><mml:mrow><mml:mtext>M<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math>\/G\/1 repairable queueing system with multiple vacations, in which the customers who arrive during server vacations enter the system with probability<jats:italic>p<\/jats:italic>. Such a novel method is used to analyze the main performance indices of the server, such as the unavailability and the mean failure number during<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mn>0<\/mml:mn><mml:mo>,<\/mml:mo><mml:mi>t<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>. It is derived that the structures of server indices are two convolution equations. Further, comparisons with existing methods indicate that our method is effective and applicable for studying server performances in single-server<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msup><mml:mrow><mml:mtext>M<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math>\/G\/1 vacation queues and their complex variants. Finally, a stochastic order and production system with a multipurpose production facility is numerically presented for illustrative purpose.<\/jats:p>","DOI":"10.1155\/2014\/241636","type":"journal-article","created":{"date-parts":[[2014,1,23]],"date-time":"2014-01-23T17:37:55Z","timestamp":1390498675000},"page":"1-9","source":"Crossref","is-referenced-by-count":1,"title":["Probabilistic Decomposition Method on the ServerIndices of an<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:msup><mml:mrow><mml:mtext>M<\/mml:mtext><\/mml:mrow><mml:mrow><mml:mi>\u03be<\/mml:mi><\/mml:mrow><\/mml:msup><\/mml:mrow><\/mml:math>\/G\/1 Vacation Queue"],"prefix":"10.1155","volume":"2014","author":[{"given":"Renbin","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Statistics, Chongqing University of Technology, Chongqing 400054, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","reference":[{"issue":"2","key":"1","first-page":"113","volume":"5","year":"1982","journal-title":"Acta Mathematicae Applicatae Sinica"},{"key":"2","doi-asserted-by":"publisher","DOI":"10.1016\/j.ejor.2003.11.033"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1016\/S1874-8651(10)60027-0"},{"issue":"1","key":"4","first-page":"44","volume":"18","year":"1995","journal-title":"Acta Mathematicae Applicatae Sinica"},{"key":"5","doi-asserted-by":"publisher","DOI":"10.1016\/S0360-8352(96)00313-0"},{"key":"6","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1017\/S0305004100030437","volume":"51","year":"1955","journal-title":"Proceedings of the Cambridge Philosophical Society"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2004.06.024"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2011.03.008"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2009.07.077"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1016\/0026-2714(94)90035-3"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1287\/opre.33.5.1117"},{"year":"1986","key":"12"}],"container-title":["Journal of Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/241636.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/241636.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/jam\/2014\/241636.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,22]],"date-time":"2017-06-22T02:33:51Z","timestamp":1498098831000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/jam\/2014\/241636\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014]]},"references-count":12,"alternative-id":["241636","241636"],"URL":"https:\/\/doi.org\/10.1155\/2014\/241636","relation":{},"ISSN":["1110-757X","1687-0042"],"issn-type":[{"type":"print","value":"1110-757X"},{"type":"electronic","value":"1687-0042"}],"subject":[],"published":{"date-parts":[[2014]]}}}