Brownian networks are a class of stochastic system models that can arise as heavy traffic approximations for stochastic processing networks. In earlier work we developed the ‘equivalent workload formulation’ of a generalized Brownian network: denoting by Z(t) the state vector of the generalized Brownian network at time t, one has a lower dimensional state descriptor W(t) = MZ(t) in the equivalent workload formulation, where M is an arbitrary basis matrix for a linear space ℳ that is orthogonal to the space of so-called ‘reversible displacements.’ Here we use the special structure of a stochastic processing network to develop a more extensive interpretation of the equivalent workload formulation associated with its Brownian network approximation. In particular, we (i) characterize and interpret the notion of a reversible displacement, and (ii) show how the basis matrix M can be constructed from the basic optimal solutions of a certain dual linear program. The latter provides a mechanism for reducing the choices for M from an infinite set to a finite one (when the workload dimension exceeds one). We illustrate our results for an example of a closed stochastic processing network.
