The Annals of Applied Probability

The heavy traffic limit of an unbalanced generalized processor sharing model

Kavita Ramanan and Martin I. Reiman

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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 $\mathcal{M}$ of the so-called fluid limit associated with this model is shown to have the form $\mathcal{M}=\{x\in\mathbb{R}_{+}^{J}:x_{j}=0\mbox{ for }j\in\mathcal{S}\}$, where $\mathcal{S}$ 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 $\mathcal{M}$. 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 $\mathcal{M}=\mathbb{R}_{+}^{J}$ 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.

Article information

Ann. Appl. Probab., Volume 18, Number 1 (2008), 22-58.

First available in Project Euclid: 9 January 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60F17: Functional limit theorems; invariance principles
Secondary: 60K25: Queueing theory [See also 68M20, 90B22] 90B22: Queues and service [See also 60K25, 68M20] 68M20: Performance evaluation; queueing; scheduling [See also 60K25, 90Bxx]

Heavy traffic diffusion approximations generalized processor sharing fluid limits invariant manifold state space collapse queueing networks Skorokhod problem Skorokhod map extended Skorokhod problem nonsemimartingale comparison principle


Ramanan, Kavita; Reiman, Martin I. The heavy traffic limit of an unbalanced generalized processor sharing model. Ann. Appl. Probab. 18 (2008), no. 1, 22--58. doi:10.1214/07-AAP438.

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