Two workload properties for Brownian networks

M. Bramson, R. J. Williams

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)191-221
Number of pages31
JournalQueueing Systems
Issue number3
StatePublished - Nov 2003

Bibliographical note

Funding Information:
∗Research supported in part by NSF Grant DMS-9971248. ∗∗Research supported in part by NSF Grant DMS-0071408, a John Simon Guggenheim Fellowship, and a gift from the David and Holly Mendel Fund. During part of the period in which this research was conducted, R.J. Williams was supported by the Operations, Information and Technology Program of the Graduate School of Business, Stanford University.


  • Brownian control problem
  • Brownian network
  • Dynamic scheduling
  • Equivalent workload formulation
  • Leontief matrix
  • Open multiclass queueing network
  • Open stochastic processing network
  • Unitary network


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