Diffusion approximations for open multiclass queueing networks: sufficient conditions involving state space collapse |
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Authors: | Williams RJ |
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Institution: | (1) Department of Mathematics, University of California, San Diego, La Jolla, CA 92093-0112, USA |
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Abstract: | Certain diffusion processes known as semimartingale reflecting Brownian motions (SRBMs) have been shown to approximate many
single class and some multiclass open queueing networks under conditions of heavy traffic. While it is known that not all
multiclass networks with feedback can be approximated in heavy traffic by SRBMs, one of the outstanding challenges in contemporary
research on queueing networks is to identify broad categories of networks that can be so approximated and to prove a heavy
traffic limit theorem justifying the approximation. In this paper, general sufficient conditions are given under which a heavy
traffic limit theorem holds for open multiclass queueing networks with head-of-the-line (HL) service disciplines, which, in
particular, require that service within each class is on a first-in-first-out (FIFO) basis. The two main conditions that need
to be verified are that (a) the reflection matrix for the SRBM is well defined and completely- S, and (b) a form of state space collapse holds. A result of Dai and Harrison shows that condition (a) holds for FIFO networks
of Kelly type and their proof is extended here to cover networks with the HLPPS (head-of-the-line proportional processor sharing)
service discipline. In a companion work, Bramson shows that a multiplicative form of state space collapse holds for these
two families of networks. These results, when combined with the main theorem of this paper, yield new heavy traffic limit
theorems for FIFO networks of Kelly type and networks with the HLPPS service discipline.
This revised version was published online in June 2006 with corrections to the Cover Date. |
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Keywords: | multiclass queueing networks heavy traffic FIFO Kelly type head-of-the-line-proportional processor sharing semimartingale reflecting Brownian motions diffusions completely-S |
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