{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,10]],"date-time":"2025-11-10T13:55:01Z","timestamp":1762782901108,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2023,4,27]],"date-time":"2023-04-27T00:00:00Z","timestamp":1682553600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>This work considers a preemptive priority queueing system with vacation, where the single server may break down with imperfect coverage. Various combinations of server vacation priority queueing models have been studied by many scholars. A common assumption in these models is that the server will only resume its normal service rate after the vacation is over. However, such speculation is more limited in real-world situations. Hence, in this study, the vacation will be interrupted if a customer waits for service in the system at the moment of completion of service during vacation. The stationary probability distribution is derived by using the probability generating function approach. We also develop varieties of performance measures and provide a simple numerical example to illustrate these measures. Optimization analysis is finally carried out, including cost optimization and tri-object optimization.<\/jats:p>","DOI":"10.3390\/computation11050089","type":"journal-article","created":{"date-parts":[[2023,4,28]],"date-time":"2023-04-28T02:27:56Z","timestamp":1682648876000},"page":"89","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Preemptive Priority Markovian Queue Subject to Server Breakdown with Imperfect Coverage and Working Vacation Interruption"],"prefix":"10.3390","volume":"11","author":[{"given":"Tzu-Hsin","family":"Liu","sequence":"first","affiliation":[{"name":"Department of Finance, Chaoyang University of Technology, 168, Jifeng E. Rd., Wufeng District, Taichung City 41349, Taiwan"}]},{"given":"He-Yao","family":"Hsu","sequence":"additional","affiliation":[{"name":"Ph.D. Program of Business Administration in Industrial Development, Department of Business Administration, Chaoyang University of Technology, 168, Jifeng E. Rd., Wufeng District, Taichung City 41349, Taiwan"}]},{"given":"Jau-Chuan","family":"Ke","sequence":"additional","affiliation":[{"name":"Department of Applied Statistics, National Taichung University of Science and Technology, No. 129, Sec. 3, Sanmin Rd., North District, Taichung City 404336, Taiwan"}]},{"given":"Fu-Min","family":"Chang","sequence":"additional","affiliation":[{"name":"Department of Finance, Chaoyang University of Technology, 168, Jifeng E. Rd., Wufeng District, Taichung City 41349, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2023,4,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Tian, N., and Zhang, Z.G. (2006). 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