The Annals of Applied Probability

A broader view of Brownian networks

J. Michael Harrison

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This paper describes a general type of stochastic system model that involves three basic elements: activities, resources, and stocks of material. A system manager chooses activity levels dynamically based on state observations, consuming some materials as inputs and producing other materials as outputs, subject to resource capacity constraints. A generalized notion of heavy traffic is described, in which exogenous input and output rates are approximately balanced with nominal activity rates derived from a static planning problem. A Brownian network model is then proposed as a formal approximation in the heavy traffic parameter regime. The current formulation is novel, relative to models analyzed in previous work, in that its definition of heavy traffic takes explicit account of the system manager's economic objective.

Article information

Ann. Appl. Probab., Volume 13, Number 3 (2003), 1119-1150.

First available in Project Euclid: 6 August 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 90B15: Network models, stochastic 60K25: Queueing theory [See also 68M20, 90B22]
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

Stochastic processing networks Brownian networks heavy traffic stochastic control


Harrison, J. Michael. A broader view of Brownian networks. Ann. Appl. Probab. 13 (2003), no. 3, 1119--1150. doi:10.1214/aoap/1060202837.

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  • [1] BILLINGSLEY, P. (1999). Convergence of Probability Measures, 2nd ed. Wiley, New York.
  • [2] BRAMSON, M. and WILLIAMS, R. J. (2000). On dy namic scheduling of stochastic networks in heavy traffic and some new results for the workload process. In Proceedings of the 39th IEEE Conference on Decision and Control. IEEE, New York.
  • [3] BRAMSON, M. and WILLIAMS, R. J. (2002). Two workload properties for Brownian networks. Unpublished manuscript.
  • [4] HARRISON, J. M. (1988). Brownian models of queueing networks with heterogeneous customer populations. In Stochastic Differential Sy stems, Stochastic Control Theory and Applications (W. Fleming and P. L. Lions, eds.) 147-186. Springer, New York.
  • [5] HARRISON, J. M. (1998). Heavy traffic analysis of a sy stem with parallel servers: Asy mptotic analysis of discrete-review policies. Ann. Appl. Probab. 8 822-848.
  • [6] HARRISON, J. M. (2000). Brownian models of open processing networks: Canonical representation of workload. Ann. Appl. Probab. 10 75-103.
  • [7] HARRISON, J. M. (2003). Stochastic networks and activity analysis. In Analy tic Methods in Applied Probability. In Memory of Fridrich Karpelevich (Y. Suhov, ed.). Amer. Math. Soc., Providence, RI.
  • [8] HARRISON, J. M. and VAN MIEGHEM, J. A. (1997). Dy namic control of Brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Probab. 7 747-771.
  • [9] KARATZAS, I. and SHREVE, S. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Springer, New York.
  • [10] WILLIAMS, R. J. (2000). On dy namic scheduling of stochastic networks in heavy traffic. Presentation to Workshop on Stochastic Networks, Univ. Wisconsin, Madison.