There has been significant progress recently in our understanding of the stationary measures of the exclusion process on Z. The corresponding situation in higher dimensions remains largely a mystery. In this paper we give necessary and sufficient conditions for a product measure to be stationary for the exclusion process on an arbitrary set, and apply this result to find examples on Z(d) and on homogeneous trees in which product measures are stationary even when they are neither homogeneous nor reversible. We then begin the task of narrowing down the possibilities for existence of other stationary measures for the process on Zd. In particular, we study stationary measures that are invariant under translations in all directions orthogonal to a fixed nonzero vector. We then prove a number of convergence results as t -> infinity for the measure of the exclusion process. Under appropriate initial conditions, we show convergence of such measures to the above stationary measures. We also employ hydrodynamics to provide further examples of convergence.

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.