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February 2000 Brownian models of open processing networks: canonical representation of workload
J. Michael Harrison
Ann. Appl. Probab. 10(1): 75-103 (February 2000). DOI: 10.1214/aoap/1019737665


A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an “equivalent workload formulation” of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t) = MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of “heavy traffic ” for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which insures that $M$ admits a factorization related to the notion of resource pooling.


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J. Michael Harrison. "Brownian models of open processing networks: canonical representation of workload." Ann. Appl. Probab. 10 (1) 75 - 103, February 2000.


Published: February 2000
First available in Project Euclid: 25 April 2002

zbMATH: 1131.60306
MathSciNet: MR1765204
Digital Object Identifier: 10.1214/aoap/1019737665

Primary: 60J70 , 60K25 , 60K25 , 90B15

Keywords: Brownian approximation , dynamic control , equivalent workload formulation , heavy traffic , Queueing theory

Rights: Copyright © 2000 Institute of Mathematical Statistics


Vol.10 • No. 1 • February 2000
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