Two Workload Properties for Brownian Networks

Authors:
M. Bramson;R. J. Williams
Affiliations:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA bramson@math.umn.edu;Department of Mathematics, University of California at San Diego, La Jolla, CA 92093-0112, USA williams@math.ucsd.edu
Venue:
Queueing Systems: Theory and Applications
Year:
2003

Citing 11
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Abstract

As one approach to dynamic scheduling problems for open stochastic processing networks, J.M. Harrison has proposed the use of formal heavy traffic approximations known as Brownian networks. A key step in this approach is a reduction in dimension of a Brownian network, due to Harrison and Van Mieghem [21], in which the “queue length” process is replaced by a “workload” process. In this paper, we establish two properties of these workload processes. Firstly, we derive a formula for the dimension of such processes. For a given Brownian network, this dimension is unique. However, there are infinitely many possible choices for the workload process. Harrison [16] has proposed a “canonical” choice, which reduces the possibilities to a finite number. Our second result provides sufficient conditions for this canonical choice to be valid and for it to yield a non-negative workload process. The assumptions and proofs for our results involve only first-order model parameters.