{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,29]],"date-time":"2026-05-29T15:07:04Z","timestamp":1780067224516,"version":"3.54.0"},"reference-count":39,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics of OR"],"published-print":{"date-parts":[[2026,5]]},"abstract":"<jats:p>A single-server queue with renewal arrivals and generally distributed independent and identically distributed service times is considered. Customers are served using the longest job first (LJF) scheduling algorithm with first in, first out being used as a tie-breaking rule. We introduce a fluid model for the evolution of a measure-valued state descriptor of this queue, and we investigate its properties. We also prove a fluid limit theorem justifying our fluid model as the first order approximation of the queueing system under consideration. Finally, we compare LJF fluid models and fluid limits with their counterparts for the longest remaining service time first (LRTF) service discipline, a preemptive variant of LJF. It turns out that the queue limiting behavior under these two protocols differs significantly, suggesting implementing LJF rather than LRTF in single-server systems.<\/jats:p>","DOI":"10.1287\/moor.2024.0415","type":"journal-article","created":{"date-parts":[[2025,5,15]],"date-time":"2025-05-15T12:49:31Z","timestamp":1747313371000},"page":"1227-1252","source":"Crossref","is-referenced-by-count":1,"title":["Fluid Limits for Longest Job First Queues"],"prefix":"10.1287","volume":"51","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-3073-959X","authenticated-orcid":false,"given":"\u0141ukasz","family":"Kruk","sequence":"first","affiliation":[{"name":"Institute of Mathematics, Maria Curie-Sk\u0142odowska University, Lublin 20-031, Poland"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"109","reference":[{"key":"B1","unstructured":"Arisha A, Young P, El Baradie M (2002) Flow shop scheduling problem: A computational study.\n                      Proc. Sixth Internat. 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Accessed August 17, 2024, https:\/\/digitalthinkerhelp.com\/longest-remaining-time-first-scheduling\/."},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1287\/moor.1090.0409"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1145\/1639562.1639593"},{"key":"B12","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1015345295"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.23.6.567"},{"key":"B14","doi-asserted-by":"publisher","DOI":"10.1002\/9780470316658"},{"key":"B15","unstructured":"Gate Vidyalay. Longest job first algorithm | LRTF scheduling. Accessed February 3, 2024, https:\/\/www.gatevidyalay.com\/longest-job-first-algorithm-lrtf-scheduling\/."},{"key":"B16","unstructured":"GeeksforGeeks. Longest job first (LJF) CPU scheduling algorithm. Accessed February 3, 2024, https:\/\/www.geeksforgeeks.org\/longest-job-first-ljf-cpu-scheduling-algorithm\/."},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1137\/0117039"},{"key":"B18","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-011-9267-5"},{"key":"B19","unstructured":"Gromoll HC, Kochalski KD (2023) Critical fluid limit of a gated processor sharing queue. 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Ao SI, Castillo O, Douglas C, Feng DD, Lee Y-A, eds.\n                      Proc. Internat. MultiConference Engineers Comput. Sci.\n                      , vol. I (Newswood Limited, Hong Kong), 18\u201322."},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2023.0090"},{"key":"B30","doi-asserted-by":"publisher","DOI":"10.1287\/moor.2015.0768"},{"key":"B31","doi-asserted-by":"publisher","DOI":"10.1214\/105051604000000314"},{"key":"B32","unstructured":"Kumar M, Sharma SC (2016) Priority aware longest job first (PA-LJF) algorithm for utilization of the resource in cloud environment. Hoda MN, ed.\n                      Thi\n                      rd Internat. Conf. Comput. 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