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Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case

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Title: Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case
Author(s): Knessl, Charles; Morrison, John A.
Subject(s): couples queues arrival rate processor sharing
Abstract: 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 O(ε), where 0 ≤ ε « 1, then in this asymptotic limit the number of customers in the first queue will be large, of order O(ε−1), while that in the second queue will be somewhat smaller, of order O(ε−1/2). 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 O(ε−1) or O(1), respectively.
Issue Date: 2011
Publisher: Hindawi Publishing Corporation
Citation Info: Knessl, C. & Morrison, J. A. 2011. Two Coupled Queues with Vastly Different Arrival Rates: Critical Loading Case. Advances in Operations Research, 2011. DOI: 10.1155/2011/216790
Type: Article
Description: Copyright © 2011 C. Knessl and J. A. Morrison. 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. The original version is available through Hindawi Publishing Corporation at DOI: 10.1155/2011/216790.
URI: http://hdl.handle.net/10027/8197
ISSN: 1687-9147
Sponsor: The work of C. Knessl was partly supported by NSF Grant no. DMS 05-03745 and by NSA Grant no. H 98230-08-1-0102.
Date Available in INDIGO: 2012-03-15
 

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