{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T16:39:06Z","timestamp":1740155946142,"version":"3.37.3"},"reference-count":25,"publisher":"Wiley","license":[{"start":{"date-parts":[[2011,1,1]],"date-time":"2011-01-01T00:00:00Z","timestamp":1293840000000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 05-03745"],"award-info":[{"award-number":["DMS 05-03745"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Operations Research"],"published-print":{"date-parts":[[2011]]},"abstract":"<jats:p>We consider two coupled queues with a generalized processor sharing service discipline. The second queue has a much smaller Poisson arrival rate than the first queue, while the customer service times are of comparable magnitude. The processor sharing server devotes most of its resources to the first queue, except when it is empty. The fraction of resources devoted to the second queue is small, of the same order as the ratio of the arrival rates. We assume that the primary queue is heavily loaded and that the secondary queue is critically loaded. If we let the small arrival rate to the secondary queue be<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:mi>\u03b5<\/mml:mi><mml:mo>)<\/mml:mo><\/mml:math>, where<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mn>0<\/mml:mn><mml:mo>\u2264<\/mml:mo><mml:mi>\u03b5<\/mml:mi><mml:mo>\u226a<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:math>, then in this asymptotic limit the number of customers in the first queue will be large, of order<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:math>, while that in the second queue will be somewhat smaller, of order<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>\/<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:math>. We obtain a two-dimensional diffusion approximation for this model and explicitly solve for the joint steady state probability distribution of the numbers of customers in the two queues. This work complements that in (Morrison, 2010), which the second queue was assumed to be heavily or lightly loaded, leading to mean queue lengths that were<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:msup><mml:mrow><mml:mi>\u03b5<\/mml:mi><\/mml:mrow><mml:mrow><mml:mo>-<\/mml:mo><mml:mn>1<\/mml:mn><\/mml:mrow><\/mml:msup><mml:mo>)<\/mml:mo><\/mml:math>or<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>O<\/mml:mi><mml:mo>(<\/mml:mo><mml:mn>1<\/mml:mn><mml:mo>)<\/mml:mo><\/mml:math>, respectively.<\/jats:p>","DOI":"10.1155\/2011\/216790","type":"journal-article","created":{"date-parts":[[2011,3,15]],"date-time":"2011-03-15T15:31:18Z","timestamp":1300203078000},"page":"1-26","source":"Crossref","is-referenced-by-count":2,"title":["Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case"],"prefix":"10.1155","volume":"2011","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8106-8880","authenticated-orcid":true,"given":"Charles","family":"Knessl","sequence":"first","affiliation":[{"name":"Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 South Morgan Street Chicago, IL 60607-7045, USA"}]},{"given":"John A.","family":"Morrison","sequence":"additional","affiliation":[{"name":"Consultant Bell Laboratories, Alcatel-Lucent, 600 Mountain Avenue, Murray Hill, NJ 07974, USA"}]}],"member":"311","reference":[{"key":"2","doi-asserted-by":"publisher","DOI":"10.1080\/15326349908807169"},{"key":"3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02412240"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160300206"},{"key":"8","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1017\/S0269964800001376","volume":"3","year":"1989","journal-title":"Probability in the Engineering and Informational Sciences"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1137\/0145052"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1137\/0144074"},{"key":"27","doi-asserted-by":"publisher","DOI":"10.1007\/BF01149176"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1007\/BF00535168"},{"key":"13","doi-asserted-by":"publisher","DOI":"10.1239\/jap\/1091543429"},{"key":"16","doi-asserted-by":"publisher","DOI":"10.2307\/3213071"},{"volume":"79","year":"1983","series-title":"North-Holland Mathematics Studies","key":"4"},{"key":"23","doi-asserted-by":"publisher","DOI":"10.1109\/90.234856"},{"key":"24","doi-asserted-by":"publisher","DOI":"10.1109\/90.298432"},{"key":"20","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-009-9126-9"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-010-9174-1"},{"key":"25","doi-asserted-by":"publisher","DOI":"10.1214\/07-AAP438"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.1023\/A:1022816410813"},{"key":"11","doi-asserted-by":"publisher","DOI":"10.1109\/TIT.1982.1056473"},{"key":"14","doi-asserted-by":"publisher","DOI":"10.1137\/0141030"},{"volume":"171","year":"1979","series-title":"Lecture Notes in Economics and Mathematical Systems","key":"22"},{"year":"1967","key":"26"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1007\/s11134-007-9043-8"},{"volume":"32","year":"1966","series-title":"Die Grundlehren der mathematischen Wissenschaften","key":"18"},{"year":"1964","key":"1"},{"year":"1994","key":"12"}],"container-title":["Advances in Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2011\/216790.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2011\/216790.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2011\/216790.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,6,19]],"date-time":"2017-06-19T17:05:18Z","timestamp":1497891918000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/aor\/2011\/216790\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011]]},"references-count":25,"alternative-id":["216790","216790"],"URL":"https:\/\/doi.org\/10.1155\/2011\/216790","relation":{},"ISSN":["1687-9147","1687-9155"],"issn-type":[{"type":"print","value":"1687-9147"},{"type":"electronic","value":"1687-9155"}],"subject":[],"published":{"date-parts":[[2011]]}}}