{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,3]],"date-time":"2023-04-03T04:02:21Z","timestamp":1680494541634},"reference-count":41,"publisher":"Institute for Operations Research and the Management Sciences (INFORMS)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics of OR"],"published-print":{"date-parts":[[2019,8]]},"abstract":" We consider a single-server queue that serves a finite population of [Formula: see text] customers that will enter the queue (require service) only once, also known as the [Formula: see text] queue. This paper presents a method for analyzing heavy-traffic behavior by using uniform acceleration, which simultaneously lets [Formula: see text] and the service rate grow large, while the initial resource utilization approaches one. A key feature of the model is that, as time progresses, more customers have joined the queue, and fewer customers can potentially join. This diminishing population gives rise to a class of reflected stochastic processes that vanish over time and hence do not have a stationary distribution. We establish that, when the arrival times are exponentially distributed, by suitably rescaling space and time, the queue-length process converges to a Brownian motion with a negative quadratic drift, a stochastic-process limit that captures the effect of the diminishing population. When the arrival times are generally distributed, our techniques provide information on the typical queue length and the first busy period. <\/jats:p>","DOI":"10.1287\/moor.2018.0947","type":"journal-article","created":{"date-parts":[[2019,5,3]],"date-time":"2019-05-03T20:41:50Z","timestamp":1556916110000},"page":"821-864","source":"Crossref","is-referenced-by-count":5,"title":["Heavy-Traffic Analysis Through Uniform Acceleration of Queues with Diminishing Populations"],"prefix":"10.1287","volume":"44","author":[{"ORCID":"http:\/\/orcid.org\/0000-0001-8431-0636","authenticated-orcid":false,"given":"Gianmarco","family":"Bet","sequence":"first","affiliation":[{"name":"Department of Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ Eindhoven, Netherlands;"}]},{"given":"Remco","family":"van der Hofstad","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ Eindhoven, Netherlands;"}]},{"given":"Johan S. H.","family":"van Leeuwaarden","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ Eindhoven, Netherlands"}]}],"member":"109","reference":[{"key":"B1","volume-title":"Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables","volume":"55","author":"Abramowitz M","year":"1964"},{"key":"B2","doi-asserted-by":"publisher","DOI":"10.1214\/aop\/1024404421"},{"key":"B3","doi-asserted-by":"publisher","DOI":"10.1214\/EJP.v15-817"},{"key":"B4","doi-asserted-by":"publisher","DOI":"10.1214\/11-AOP680"},{"key":"B5","volume-title":"Convergence of Probability Measures","author":"Billingsley P","year":"2009"},{"key":"B6","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511814068"},{"key":"B7","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20168"},{"key":"B8","doi-asserted-by":"publisher","DOI":"10.1007\/s10955-006-9168-x"},{"key":"B9","doi-asserted-by":"publisher","DOI":"10.1016\/j.jairtraman.2013.09.004"},{"key":"B10","doi-asserted-by":"publisher","DOI":"10.1090\/cbms\/107"},{"key":"B11","doi-asserted-by":"publisher","DOI":"10.1002\/0471722162"},{"key":"B12","volume-title":"Markov Processes: Characterization and Convergence","author":"Ethier SN","year":"1985"},{"key":"B13","doi-asserted-by":"publisher","DOI":"10.1080\/03461238.1956.10414944"},{"issue":"4","key":"B14","doi-asserted-by":"crossref","first-page":"1316","DOI":"10.1214\/aos\/1017938928","volume":"27","author":"Groeneboom P","year":"1999","journal-title":"Ann. Statist."},{"key":"B15","doi-asserted-by":"publisher","DOI":"10.1007\/s00186-010-0330-5"},{"key":"B16","doi-asserted-by":"publisher","DOI":"10.1287\/mnsc.1070.0802"},{"key":"B17","doi-asserted-by":"publisher","DOI":"10.1287\/opre.2014.1338"},{"key":"B19","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-014-9428-4"},{"key":"B20","doi-asserted-by":"publisher","DOI":"10.2307\/3518347"},{"key":"B21","doi-asserted-by":"publisher","DOI":"10.2307\/1426324"},{"key":"B22","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-05265-5"},{"key":"B23","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032718"},{"key":"B24","doi-asserted-by":"publisher","DOI":"10.1214\/13-AAP985"},{"key":"B25","doi-asserted-by":"publisher","DOI":"10.1137\/1024098"},{"key":"B26","volume-title":"Probability Theory: A Comprehensive Course","author":"Klenke A","year":"2007"},{"key":"B27","doi-asserted-by":"publisher","DOI":"10.1016\/0304-4149(94)90060-4"},{"key":"B28","doi-asserted-by":"publisher","DOI":"10.1287\/moor.20.1.33"},{"key":"B29","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1032265215"},{"key":"B31","doi-asserted-by":"publisher","DOI":"10.1287\/moor.10.2.305"},{"key":"B32","doi-asserted-by":"publisher","DOI":"10.2307\/3212264"},{"key":"B33","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-009-5970-5"},{"key":"B34","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1143936140"},{"key":"B35","doi-asserted-by":"publisher","DOI":"10.1016\/j.orl.2014.10.008"},{"key":"B36","doi-asserted-by":"publisher","DOI":"10.1137\/1101022"},{"key":"B37","doi-asserted-by":"publisher","DOI":"10.1002\/rsa.20450"},{"key":"B38","doi-asserted-by":"publisher","DOI":"10.1017\/9781316779422"},{"key":"B39","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/1293113157"},{"key":"B40","doi-asserted-by":"publisher","DOI":"10.1287\/moor.5.1.67"},{"key":"B41","doi-asserted-by":"publisher","DOI":"10.1007\/b97479"},{"key":"B42","doi-asserted-by":"publisher","DOI":"10.1214\/07-PS122"},{"key":"B44","doi-asserted-by":"publisher","DOI":"10.1023\/A:1019116804750"}],"container-title":["Mathematics of Operations Research"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubsonline.informs.org\/doi\/pdf\/10.1287\/moor.2018.0947","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,2]],"date-time":"2023-04-02T12:24:32Z","timestamp":1680438272000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubsonline.informs.org\/doi\/10.1287\/moor.2018.0947"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,8]]},"references-count":41,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2019,8]]}},"alternative-id":["10.1287\/moor.2018.0947"],"URL":"https:\/\/doi.org\/10.1287\/moor.2018.0947","relation":{},"ISSN":["0364-765X","1526-5471"],"issn-type":[{"value":"0364-765X","type":"print"},{"value":"1526-5471","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,8]]}}}