Queueing Systems

, Volume 5, Issue 4, pp 401–411 | Cite as

An extension to Norton's equivalent

  • Man-Tung T. Hsiao
  • Aurel A. Lazar
Short Communications

Abstract

The aggregation method for queueing networks known as the Norton's equivalent is interpreted as a conditional estimate of the intensities of associated point processes. For multi-class Markovian queueing networks, it is shown that a first-order equivalent system of an isolated station can be obtained via the conditional estimates of intensities of the arrival and departure processes to and from that station. Based on these conditional estimates, separation results for optimal flow control problems in queueing networks can be obtained. Several examples which illustrate these concepts are given. The results obtained here generalize those which require the “product form” networks.

Keywords

Norton's equivalent conditional estimates Markovian queueing networks 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S. Balsamo, and G. Iazeolla, An extension of Norton's theorem for queueing networks, IEEE Trans. on Software Engineering, SE-8 (1982) 298–305.Google Scholar
  2. [2]
    F. Basket, K.M. Chandy, R.R. Muntz and F.G. Palacios, Open, closed, and mixed networks of queues with different classes of customers, Journal of the ACM 22, no. 2 (April 1975) 248–260.Google Scholar
  3. [3]
    A. Brandwajn, Equivalence and decomposition in queueing systems-a unified approach, Performance Evaluation 5 (1985) 175–186.Google Scholar
  4. [4]
    P. Bremaud,Point Processes and Queues: Martingale Dynamics (Springer-Verlag, New York, 1981).Google Scholar
  5. [5]
    P. Bremaud, Théorie des files d'attente et de leurs réseaux, preprint, 1986.Google Scholar
  6. [6]
    K.M. Chandy, U. Herzog and L. Woo, Parametric analysis of queueing networks, IBM J. Res. Develop. 19 (1975) 36–42.Google Scholar
  7. [7]
    P.J. Courtois,Decomposability: Queueing and Computer Applications (Academic Press, New York, 1977).Google Scholar
  8. [8]
    M.T. Hsiao and A.A. Lazar, Bottleneck modeling and decentralized optimal flow control — I. Global objectives,Proc. Eighteenth Conf. on Information Sciences and Systems (Princeton University, Princeton, NJ, March 1984).Google Scholar
  9. [9]
    M.T. Hsiao and A.A. Lazar, Bottleneck modeling and decentralized optimal flow control — II. Individual objectives,Proc. Nineteenth Conf. on Information Sciences and Systems (Johns Hopkins University, Baltimore, MD, March 1985).Google Scholar
  10. [10]
    M.T. Hsiao and A.A. Lazar, Optimal flow control of multi-class queueing networks with decentralized information, CTR Technical Report, CU-CTR-TR-11, Center for Telecommunications Research, Columbia University, 1986.Google Scholar
  11. [11]
    M.T. Hsiao and A.A. Lazar, Optimal decentralized flow control of Markovian queueing networks with multiple controllers, CTR Technical Report, CU-CTR-TR-19, Center for Telecommunications Research, Columbia University, 1986.Google Scholar
  12. [12]
    P.S. Kritzinger, S. van Wyk and A.E. Krzesinski, A generalisation of Norton's theorem for multiclass queueing networks, Performance Evaluation 2 (1982) 98–107.Google Scholar
  13. [13]
    A. Kumar, Equivalent queueing networks and their use in approximate equilibrium analysis, The Bell System Technical Journal 62, No. 10 (1983) 2893–2910.Google Scholar
  14. [14]
    J. Walrand, A note on Norton's theorem, Journal of Applied Probability 20 (1983) 442–444.Google Scholar

Copyright information

© J.C. Baltzer A.G. Scientific Publishing Company 1989

Authors and Affiliations

  • Man-Tung T. Hsiao
    • 1
  • Aurel A. Lazar
    • 2
  1. 1.School of Electrical EngineeringPurdue UniversityWest LafayetteUSA
  2. 2.Department of Electrical EngineeringColumbia UniversityNew YorkUSA

Personalised recommendations