Two workload properties for Brownian networks

As one approach to dynamic scheduling problems for open stochastic processing networks, J.M. Harrison has proposed the use of formal heavy traffic approximations known as Brownian networks. A key step in this approach is a reduction in dimension of a Brownian network, due to Harrison and Van Mieghem [21], in which the "queue length" process is replaced by a "workload" process. In this paper, we establish two properties of these workload processes. Firstly, we derive a formula for the dimension of such processes. For a given Brownian network, this dimension is unique. However, there are infinitely many possible choices for the workload process. Harrison [16] has proposed a "canonical" choice, which reduces the possibilities to a finite number. Our second result provides sufficient conditions for this canonical choice to be valid and for it to yield a non-negative workload process. The assumptions and proofs for our results involve only first-order model parameters.