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Workload Interpretation for Brownian Models of Stochastic Processing Networks

Graduate School of Business, Stanford University, Stanford, California 94305, USA
Department of Mathematics, University of California, San Diego, La Jolla, California 92093, USA
harrison_michael{at}gsb.stanford.edu, http://faculty-gsb.stanford.edu/harrison
williams{at}math.ucsd.edu, http://www.math.ucsd.edu/williams

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 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.

Key Words: Brownian network model; workload; state space collapse; reversible displacements; dual problem; stochastic control; singular control; stochastic processing network
History: Received: December 6, 2005; revision received: August 20, 2006;