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.