{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,12]],"date-time":"2025-11-12T03:21:44Z","timestamp":1762917704924,"version":"3.41.0"},"reference-count":21,"publisher":"Wiley","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Columbia College Chicago"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Operations Research"],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>We consider a storage allocation model with a finite number of storage spaces. There are<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>primary spaces and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>secondary spaces. All of them are numbered and ranked. Customers arrive according to a Poisson process and occupy a space for an exponentially distributed time period, and a new arrival takes the lowest ranked available space. We let<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M3\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M4\"><mml:mrow><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><\/mml:mrow><\/mml:math>denote the numbers of occupied primary and secondary spaces and study the joint distribution<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M5\"><mml:mi mathvariant=\"normal\">P<\/mml:mi><mml:mi mathvariant=\"normal\">r<\/mml:mi><mml:mi mathvariant=\"normal\">o<\/mml:mi><mml:mi mathvariant=\"normal\">b<\/mml:mi><mml:mo stretchy=\"false\">[<\/mml:mo><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mi>k<\/mml:mi><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>=<\/mml:mo><mml:mi>r<\/mml:mi><mml:mo stretchy=\"false\">]<\/mml:mo><\/mml:math>in the steady state. The joint process<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M6\"><mml:mo stretchy=\"false\">(<\/mml:mo><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">1<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo>,<\/mml:mo><mml:msub><mml:mrow><mml:mi>N<\/mml:mi><\/mml:mrow><mml:mrow><mml:mn mathvariant=\"normal\">2<\/mml:mn><\/mml:mrow><\/mml:msub><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math>behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rate<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M7\"><mml:mrow><mml:mi>\u03bb<\/mml:mi><\/mml:mrow><\/mml:math>becomes large, and the storage capacities<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M8\"><mml:mrow><mml:mi>m<\/mml:mi><\/mml:mrow><\/mml:math>and<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M9\"><mml:mrow><mml:mi>R<\/mml:mi><\/mml:mrow><\/mml:math>are scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.<\/jats:p>","DOI":"10.1155\/2016\/1925827","type":"journal-article","created":{"date-parts":[[2016,8,31]],"date-time":"2016-08-31T21:02:48Z","timestamp":1472677368000},"page":"1-56","source":"Crossref","is-referenced-by-count":1,"title":["Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution"],"prefix":"10.1155","volume":"2016","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0727-084X","authenticated-orcid":true,"given":"Eunju","family":"Sohn","sequence":"first","affiliation":[{"name":"Department of Science and Mathematics, Columbia College Chicago, 623 South Wabash Avenue, Chicago, IL 60605, USA"}]},{"given":"Charles","family":"Knessl","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA"}]}],"member":"311","reference":[{"volume":"1","year":"1997","key":"12"},{"key":"21","doi-asserted-by":"publisher","DOI":"10.1137\/0153030"},{"key":"13","first-page":"5","volume":"14","year":"1937","journal-title":"Electra Nachr.-Tech"},{"key":"1","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/0304-4149(86)90007-4","volume":"22","year":"1986","journal-title":"Stochastic Processes and Their Applications"},{"key":"4","doi-asserted-by":"publisher","DOI":"10.1137\/0214032"},{"year":"1984","key":"14"},{"key":"15","doi-asserted-by":"publisher","DOI":"10.2307\/3215389"},{"key":"17","doi-asserted-by":"publisher","DOI":"10.1016\/j.aml.2007.03.006"},{"key":"7","doi-asserted-by":"publisher","DOI":"10.1214\/aoap\/1019487357"},{"key":"9","doi-asserted-by":"publisher","DOI":"10.1111\/1467-9590.t01-1-00232"},{"key":"10","doi-asserted-by":"publisher","DOI":"10.1023\/b:ques.0000035314.35882.5b"},{"key":"16","doi-asserted-by":"crossref","first-page":"835","DOI":"10.1017\/S0001867800002810","volume":"40","year":"2008","journal-title":"Advances in Applied Probability"},{"key":"18","doi-asserted-by":"publisher","DOI":"10.1017\/S0956792516000048"},{"volume":"2","year":"1989","key":"5"},{"year":"1986","key":"3"},{"year":"1978","key":"2"},{"year":"2001","key":"20"},{"key":"6","doi-asserted-by":"publisher","DOI":"10.1017\/cbo9780511801655"},{"key":"19","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032770"},{"issue":"3-4","key":"11","doi-asserted-by":"crossref","first-page":"281","DOI":"10.3233\/ASY-2001-458","volume":"27","year":"2001","journal-title":"Asymptotic Analysis"},{"key":"8","doi-asserted-by":"publisher","DOI":"10.1137\/s003613990037515x"}],"container-title":["Advances in Operations Research"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2016\/1925827.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2016\/1925827.xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/downloads.hindawi.com\/journals\/aor\/2016\/1925827.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T15:52:14Z","timestamp":1749570734000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.hindawi.com\/journals\/aor\/2016\/1925827\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"references-count":21,"alternative-id":["1925827","1925827"],"URL":"https:\/\/doi.org\/10.1155\/2016\/1925827","relation":{},"ISSN":["1687-9147","1687-9155"],"issn-type":[{"type":"print","value":"1687-9147"},{"type":"electronic","value":"1687-9155"}],"subject":[],"published":{"date-parts":[[2016]]}}}