Generalized Jackson Networks

  • Hong Chen
  • David D. Yao
Chapter
Part of the Stochastic Modelling and Applied Probability book series (SMAP, volume 46)

Abstract

In this chapter we consider a queueing network that generalizes the Jackson network studied in Chapter 2, by allowing renewal arrival processes that need not be Poisson and i.i.d. service times that need not follow exponential distributions. (However, we do not allow the service times of the network to be state-dependent; in this regard, the network is more restrictive than the Jackson network. Nevertheless, this network has been conventionally referred to as the generalized Jackson network.) Unlike the Jackson network, the stationary distribution of a generalized Jackson network usually does not have an explicit analytical form. Therefore, approximations and limit theorems that support such approximations are usually sought for the generalized Jackson networks.

Keywords

Reflection Mapping Closed Network Jackson Network Nonnegative Orthant Workload Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Berman, A. and R.J. Plemmons. (1979). Nonnegative Matrices in the Mathematical Sciences,Academic Press.Google Scholar
  2. [2]
    Bernard, A. and A. El Kharroubi. (1991). Régulation de processus dans le premier orthant de Rn. Stochastics and Stochastics Reports, 34, 149–167.MathSciNetMATHGoogle Scholar
  3. [3]
    Chen, H. and A. Mandelbaum. (1991a). Leontief systems, RBV’s and RBM’s, in The Proceedings of the Imperial College Workshop on Applied Stochastic Processes, ed. by M.H.A. Davis and R.J. Elliotte, Gordon and Breach Science Publishers.Google Scholar
  4. [4]
    Chen, H. and A. Mandelbaum. (1991b). Discrete flow networks: bottleneck analysis and fluid approximations, Mathematics of Operations Research, 16, 408–446.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Chen, H. and A. Mandelbaum. (1991c). Stochastic discrete flow networks: diffusion approximations and bottlenecks, Annals of Probability, 19, 1463–1519.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Chen, H. and A. Mandelbaum. (1994). Hierarchical modelling of stochastic networks, Part II: strong approximations, in D.D. Yao (ed.), Stochastic Modeling and Analysis of Manufacturing Systems, 107–131, Springer-Verlag.Google Scholar
  7. [7]
    Chen, H. and X. Shen. (2000). Computing the stationary distribution of SRBM in an orthant. In preparation.Google Scholar
  8. [8]
    Chen, H. and W. Whitt. (1993). Diffusion approximations for open queueing networks with service interruptions, Queueing Systems, Theory and Applications, 13, 335–359MathSciNetMATHGoogle Scholar
  9. [9]
    Cottle, R.W., J.S. Pang and R.E. Stone. (1992). The Linear Complementarity Problem,Academic Press.Google Scholar
  10. [10]
    Dai, J.G. and J.M. Harrison. (1991), Steady-state analysis of reflected Brownian motions: characterization, numerical methods and a queueing application, Annals of Applied Probability, 1, 16–35.MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    Dai, J.G. and J.M. Harrison. (1992), Reflected Brownian motion in an orthant: numerical methods and a queueing application, Annals of Applied Probability, 2, 65–86.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    Harrison, J. M. and M.I. Reiman. (1981). Reflected Brownian motion on an orthant, Annals of Probability, 9, 302–308.MathSciNetMATHCrossRefGoogle Scholar
  13. [13]
    Harrison, J.M. and R.J. Williams. (1987). Brownian models of open queueing networks with homogeneous customer populations, Stochastics, 22, 77–115.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Harrison, J.M., R.J. Williams, and H. Chen. (1990). Brownian models of closed queueing networks with homogeneous customer populations, Stochastics and Stochastics Reports, 29, 37–74.MATHGoogle Scholar
  15. [15]
    Horvath, L. (1992). Strong approximations of open queueing networks, Mathematics of Operations Research, 17, 487–508.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Johnson, D. P. (1983). Diffusion Approximations for Optimal Filtering of Jump Processes and for Queueing Networks, Ph. D Dissertation, University of Wisconsin.Google Scholar
  17. [17]
    Kaspi, H. and A. Mandelbaum. (1992). Regenerative closed queueing networks. Stochastics and Stochastic Reports, 39, 239–258.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Mandelbaum, A. (1989). The dynamic complementarity problem (preprint).Google Scholar
  19. [19]
    Reiman, M.I. (1984). Open queueing networks in heavy traffic, Mathematics of Operations Research, 9, 441–458.MathSciNetMATHCrossRefGoogle Scholar
  20. [20]
    Zhang, H. (1997). Strong approximations of irreducible closed queueing networks, Advances in Applied Probability, 29, 498–522.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Hong Chen
    • 1
  • David D. Yao
    • 2
  1. 1.Faculty of Commerce and Business AdministrationUniversity of British ColumbiaVancouverCanada
  2. 2.Department of Operations Research and Industrial EngineeringColumbia UniversityNew YorkUSA

Personalised recommendations