The Annals of Applied Probability

Brownian models of open processing networks: canonical representation of workload

J. Michael Harrison

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Abstract

A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an “equivalent workload formulation” of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t) = MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of “heavy traffic ” for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which insures that $M$ admits a factorization related to the notion of resource pooling.

Article information

Source
Ann. Appl. Probab. Volume 10, Number 1 (2000), 75-103.

Dates
First available in Project Euclid: 25 April 2002

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1019737665

Digital Object Identifier
doi:10.1214/aoap/1019737665

Mathematical Reviews number (MathSciNet)
MR1765204

Zentralblatt MATH identifier
1131.60306

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

Keywords
Queueing theory heavy traffic dynamic control Brownian approximation equivalent workload formulation

Citation

Harrison, J. Michael. Brownian models of open processing networks: canonical representation of workload. Ann. Appl. Probab. 10 (2000), no. 1, 75--103. doi:10.1214/aoap/1019737665. http://projecteuclid.org/euclid.aoap/1019737665.


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References

  • [1] Atkins, D. and Chen, H. (1995). Performance evalution of scheduling control of queueing networks: fluid model heuristics. Queueing Systems 21 391-413.
  • [2] Avram, F., Bertsimas, D. and Ricard, M. (1995). Fluid models of sequencing problems in open queueing networks: an optimal control approach. In Stochastic Networks (F. P. Kelly and R. J. Williams, eds.) 199-234. Springer, New York.
  • [3] Chen, H. and Yao, D. D. (1993). Dynamic scheduling of a multiclass fluid network. Oper. Res. 41 1104-1115.
  • [4] Chevalier, P. B. and Wein, L. M. (1993). Scheduling networks of queues: heavy traffic analysis of a multistation closed network. Oper. Res. 41 743-758.
  • [5] Eng, D., Humphrey, J. and Meyn, S. P. (1996). Fluid network models: linear programs for control and performance bounds. In Thirteenth World Congress of International Federation of Automatic Control, San Francisco, 1996.
  • [6] Harrison, J. M. (1988). Brownian models of queueing networks with heterogeneous customer populations. In Stochastic Differential Systems, Stochastic Control Theory and Applications (W. Fleming and P. L. Lions, eds.) 147-186. Springer, New York.
  • [7] Harrison, J. M. (1995). Balanced fluid models of multiclass queueing networks: a heavy traffic conjecture. In Stochastic Networks (F. P. Kelly and R. J. Williams, eds.) 1-20. Springer, New York.
  • [8] Harrison, J. M. (1996). The BIGSTEP approach to flow management in stochastic processing networks. In Stochastic Networks: Theory and Applications (F. Kelly, S. Zachary and I. Ziendins, eds.) Oxford Univ. Press.
  • [9] Harrison, J. M. (2000). BIGSTEP control of a processing network with two servers working in parallel. Ann. Appl. Probab. To appear.
  • [10] Harrison, J. M. and Van Mieghem, J. A. (1997). Dynamic control of Brownian networks: State space collapse and equivalent workload formulations. Ann. Appl. Probab. 7 747- 771.
  • [11] Harrison, J. M. and Wein, L. M. (1989). Scheduling networks of queues: heavy traffic analysis of a simple open network. Queueing Systems 5 265-280.
  • [12] Harrison, J. M. and Wein, L. M. (1990). Scheduling networks of queues: heavy traffic analysis of a two-station closed network. Oper. Res. 38 1052-1064.
  • [13] Kelly, F. P. and Laws, C. N. (1993). Dynamic routing in open queueing networks: Brownian models, cut constraints and resource pooling. Queueing Systems 13 47-86.
  • [14] Laws, C. N. (1990). Dynamic routing in queueing networks. Ph.D. dissertation, Cambridge Univ.
  • [15] Laws, C. N. (1992). Resource pooling in queueing networks with dynamic routing. Adv. in Appl. Probab. 24 699-726.
  • [16] Laws, C. N. and Louth, G. M. (1990). Dynamic scheduling of a four-station queueing network. Probab. Engng. Inform. Sci. 4 131-156.
  • [17] Wein, L. M. (1990). Scheduling networks of queues: heavy traffic analysis of a two-station network with controllable inputs. Oper. Res. 38 1065-1078.
  • [18] Wein, L. M. (1991). Brownian networks with discretionary routing. Oper. Res. 39 322-340.
  • [19] Wein, L. M. (1992). Scheduling networks of queues: heavy traffic analysis of a multistation network with controllable inputs. Oper. Res. 40 S312-S334.
  • [20] Weiss, G. (1995). On optimal draining of re-entrant fluid lines. In Stochastic Networks (F. P. Kelly and R. J. Williams, eds.) 91-103. Springer, New York.