## The Annals of Applied Probability

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

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

#### 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

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1214/aoap/1019737665

**Mathematical Reviews number (MathSciNet)**

MR1765204

**Zentralblatt MATH identifier**

1131.60306

**Subjects**

Primary: 60K25: Queueing theory [See also 68M20, 90B22] 60K25: Queueing theory [See also 68M20, 90B22] 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 90B15: Network models, stochastic

**Keywords**

Queueing theory heavy traffic dynamic control Brownian approximation equivalent workload formulation

#### 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. https://projecteuclid.org/euclid.aoap/1019737665