Brownian models of open processing networks: canonical representation of workload

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.