## The Annals of Applied Probability

### Brownian models of open processing networks: canonical representation of workload

J. Michael Harrison

#### Abstract

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.

#### Article information

Source
Ann. Appl. Probab. Volume 10, Number 1 (2000), 75-103.

Dates
First available in Project Euclid: 25 April 2002

http://projecteuclid.org/euclid.aoap/1019737665

Digital Object Identifier
doi:10.1214/aoap/1019737665

Mathematical Reviews number (MathSciNet)
MR1765204

Zentralblatt MATH identifier
1131.60306

#### Citation

Harrison, J. Michael. Brownian models of open processing networks: canonical representation of workload. Ann. Appl. Probab. 10 (2000), no. 1, 75--103. doi:10.1214/aoap/1019737665. http://projecteuclid.org/euclid.aoap/1019737665.

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