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Advances in Operations Research
Volume 2016, Article ID 1925827, 56 pages
http://dx.doi.org/10.1155/2016/1925827
Research Article

Asymptotic Analysis of a Storage Allocation Model with Finite Capacity: Joint Distribution

1Department of Science and Mathematics, Columbia College Chicago, 623 South Wabash Avenue, Chicago, IL 60605, USA
2Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, IL 60607-7045, USA

Received 7 August 2015; Accepted 26 January 2016

Academic Editor: Hsien-Chung Wu

Copyright © 2016 Eunju Sohn and Charles Knessl. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider a storage allocation model with a finite number of storage spaces. There are primary spaces and 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 and denote the numbers of occupied primary and secondary spaces and study the joint distribution in the steady state. The joint process behaves as a random walk in a lattice rectangle. We study the problem asymptotically as the Poisson arrival rate becomes large, and the storage capacities and are scaled to be commensurably large. We use a singular perturbation analysis to approximate the forward Kolmogorov equation(s) satisfied by the joint distribution.