2008/01/16 by Kavita Ramanan, Martin I. Reiman
Business, Management and Accounting · Mathematics · Social Sciences · #Advanced Queuing Theory Analysis #Supply Chain and Inventory Management #Transportation Planning and Optimization #math.PR #msc:60F05 #msc:60F17 #msc:60K25 #msc:68M20 #msc:90B22
paper · pdf · doi:10.1214/07-aap438
published as Annals of Applied Probability 2008, Vol. 18, No. 1, 22-58 · Published in at http://dx.doi.org/10.1214/07-AAP438 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/01/16 · arxiv created 2008/01/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work considers a server that processes J classes using the generalized processor sharing discipline with base weight vector α=(α1, …, αJ) and redistribution weight vector β=(β1, …, βJ). The invariant manifold M of the so-called fluid limit associated with this model is shown to have the form M=\x∈ℝ+J:xj=0 for j\inS\, where S is the set of strictly subcritical classes, which is identified explicitly in terms of the vectors α and β and the long-run average work arrival rates γj of each class j. In addition, under general assumptions, it is shown that when the heavy traffic condition ∑j=1Jγj=∑j=1Jαj holds, the functional central limit of the scaled unfinished work process is a reflected diffusion process that lies in M. The reflected diffusion limit is characterized by the so-called extended Skorokhod map and may fail to be a semimartingale. This generalizes earlier results obtained for the simpler, balanced case where γj=αj for j=1, …, J, in which case M=ℝ+J and there is no state-space collapse. Standard techniques for obtaining diffusion approximations cannot be applied in the unbalanced case due to the particular structure of the GPS model. Along the way, this work also establishes a comparison principle for solutions to the extended Skorokhod map associated with this model, which may be of independent interest.