{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,21]],"date-time":"2026-05-21T01:04:30Z","timestamp":1779325470209,"version":"3.51.4"},"reference-count":18,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2020,4,7]],"date-time":"2020-04-07T00:00:00Z","timestamp":1586217600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>This paper deals with the N policy M\/G\/1 queue with working breakdowns. The supplementary variable and probability generating function techniques are implemented to develop the steady-state results. The stability condition of a stable queue, as well as several system performance measures, are also derived. A two-stage optimization method is employed to determine the optimal threshold N and the optimal joint values of two mean service rates until the stability constraint is satisfied. To demonstrate the effectiveness of two-stage optimization method, some numerical results are presented. Finally, we carry out sensitivity analysis for the expected cost function with numerical illustrations.<\/jats:p>","DOI":"10.3390\/sym12040583","type":"journal-article","created":{"date-parts":[[2020,4,9]],"date-time":"2020-04-09T14:42:03Z","timestamp":1586443323000},"page":"583","update-policy":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":16,"title":["Optimization Analysis of the N Policy M\/G\/1 Queue with Working Breakdowns"],"prefix":"10.3390","volume":"12","author":[{"given":"Tseng-Chang","family":"Yen","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Kuo-Hsiung","family":"Wang","sequence":"additional","affiliation":[{"name":"Department of Business Administration, Asia University, Wufeng, Taichung 41354, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jia-Yu","family":"Chen","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,7]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"393","DOI":"10.1057\/jors.1963.63","article-title":"Queueing systems with a removable service station","volume":"14","author":"Yadin","year":"1963","journal-title":"Oper. 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Manag."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1007\/s12597-011-0046-1","article-title":"Optimal strategy analysis of an N-policy two-phase MX\/M\/1 queueing system with server startup and breakdowns","volume":"48","author":"Vemuri","year":"2011","journal-title":"OPSEARCH"},{"key":"ref_14","first-page":"373","article-title":"Analysis of queue with two phases of service and m phases of repair for server breakdown under N-policy","volume":"16","author":"Singh","year":"2013","journal-title":"Int. J. Serv. Oper. Manag."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"5113","DOI":"10.1016\/j.apm.2014.03.012","article-title":"Cost optimization of a repairable M\/G\/1 queue with a randomized policy and single vacation","volume":"38","author":"Yang","year":"2014","journal-title":"Appl. Math. Modell."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"476","DOI":"10.1016\/j.ress.2018.08.011","article-title":"Reliability analysis of a retrial machine repair problem with warm standbys and a single server with N-policy","volume":"180","author":"Chen","year":"2018","journal-title":"Reliab. Eng. Syst. Saf."},{"key":"ref_17","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":"Proc. Camb. Philos. Soc."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Nocedal, J., and Wright, S.J. (1999). Numerical Optimization, Springer.","DOI":"10.1007\/b98874"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mdpi.com\/2073-8994\/12\/4\/583\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T09:16:05Z","timestamp":1760174165000},"score":1,"resource":{"primary":{"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/www.mdpi.com\/2073-8994\/12\/4\/583"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,7]]},"references-count":18,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["sym12040583"],"URL":"https:\/\/summer-heart-0930.chufeiyun1688.workers.dev:443\/https\/doi.org\/10.3390\/sym12040583","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,4,7]]}}}