{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,29]],"date-time":"2026-06-29T20:47:46Z","timestamp":1782766066055,"version":"3.54.5"},"reference-count":32,"publisher":"EDP Sciences","issue":"1","license":[{"start":{"date-parts":[[2018,4,18]],"date-time":"2018-04-18T00:00:00Z","timestamp":1524009600000},"content-version":"vor","delay-in-days":107,"URL":"https:\/\/www.edpsciences.org\/en\/authors\/copyright-and-licensing"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["RAIRO-Oper. Res."],"accepted":{"date-parts":[[2018,1,14]]},"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>In this paper, we analyze an <jats:italic>M<\/jats:italic>\u2215<jats:italic>G<\/jats:italic>\u22151 queue operating in multi-phase random environment with <jats:italic>Min<\/jats:italic>(<jats:italic>N<\/jats:italic>, <jats:italic>V<\/jats:italic>) vacation policy. In operative phase <jats:italic>i<\/jats:italic>, <jats:italic>i<\/jats:italic> = 1, 2, \u2026, <jats:italic>n<\/jats:italic>, customers are served according to the discipline of First Come First Served (FCFS). When the system becomes empty, the server takes a vacation under the <jats:italic>Min<\/jats:italic>(<jats:italic>N<\/jats:italic>, <jats:italic>V<\/jats:italic>) policy, causing the system to move to vacation phase 0. At the end of a vacation, if the server finds no customer waiting, another vacation begins. Otherwise, the system jumps from the phase 0 to some operative phase <jats:italic>i<\/jats:italic> with probability <jats:italic>q<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>i<\/jats:italic> = 1, 2, \u2026, <jats:italic>n<\/jats:italic>. And whenever the number of the waiting customers in the system reaches <jats:italic>N<\/jats:italic>, the server interrupts its vacation immediately and the system jumps from the phase 0 to some operative phase <jats:italic>i<\/jats:italic> with probability <jats:italic>q<\/jats:italic><jats:sub><jats:italic>i<\/jats:italic><\/jats:sub>, <jats:italic>i<\/jats:italic> = 1, 2, \u2026, <jats:italic>n<\/jats:italic>, too. Using the method of supplementary variable, we derive the distribution for the stationary system size at arbitrary epoch. We also obtain mean system size, the results of the cycle analysis and the sojourn time distribution. In addition, some special cases and numerical examples are presented.<\/jats:p>","DOI":"10.1051\/ro\/2018006","type":"journal-article","created":{"date-parts":[[2018,1,16]],"date-time":"2018-01-16T08:51:59Z","timestamp":1516092719000},"page":"61-77","source":"Crossref","is-referenced-by-count":7,"title":["On an M\/G\/1 queue in random environment with <i>Min<\/i>(<i>N<\/i>, <i>V<\/i>) policy"],"prefix":"10.1051","volume":"52","author":[{"given":"Jianjun","family":"Li","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Liwei","family":"Liu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"250","published-online":{"date-parts":[[2018,4,18]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/j.ejor.2008.01.024","volume":"195","author":"Baykal-Gursoy","year":"2009","journal-title":"Eur. J. Oper. 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