Skip to main content
SpringerLink
Log in

Stability of generalized Jackson networks with infinite supply of work

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

A general Jackson network (GJN) with infinite supply of work is considered. By fluid limit model, the author finds that the Markov process describing the dynamics of the GJN with infinite supply of work is positive Harris recurrent if the corresponding fluid model is stable. Furthermore, the author proves that the fluid model is stable if the usual traffic condition holds.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. R. Jackson, Networks of waiting lines, Operations Research, 1957, 5(4): 518–521.

    Google Scholar 

  2. J. R. Jackson, Jobshop-like queueing systems, Management Science, 1963, 10(1): 131–142.

    Google Scholar 

  3. F. P. Kelly, Reversibility and Stochastic Networks, Wiley, New York, 1979.

    Google Scholar 

  4. J. M. Harrison, Brownian models of queueing networks with heterogeneous customer populations, in Stochastic Differential Systems, Stochastic Control Theory and Applications, Proceedings of the IMA (Ed. by W. Fleming and P. L. Lions), Vol.10, Springer-Verlag, New York, 1988, 147–186.

  5. K. Sigman, The stability of open queueing networks, Stochastic Processes and Their Applications, 1990, 35(1): 11–25.

    Article  Google Scholar 

  6. S. P. Meyn and D. Down, Stability of generalized Jackson networks, Annals of Applied Probability, 1994, 4(1): 124–148.

    Article  Google Scholar 

  7. J. G. Dai, On the positive Harris recurrence for multiclass queueing networks, Annals of Applied Probability, 1995, 5(1): 49–77.

    Article  Google Scholar 

  8. H. Chen and D. Yao, Fundamentals of Queueing Networks, Springer-Verlag, New York, 2001.

    Google Scholar 

  9. M. I. Reiman, Open queueing networks in heavy traffic, Mathematics of Operations Research, 1984, 9(2): 441–458.

    Google Scholar 

  10. D. Gamarnik and A. Zeevi, Validity of heavy traffic steady-state approximations in generalized Jackson networks, Annals of Applied Probability, 2006, 16(1): 56–90.

    Article  Google Scholar 

  11. N. Gans and Y. Zhou, A call-routing problem with service-level constraints, Operations Research, 2003, 51(2): 255–271.

    Article  Google Scholar 

  12. Y. Levy and U. Yechiali, Utilization of idle time in an M = G = 1 queueing system, Management Science, 1975, 22(2): 202–211.

    Google Scholar 

  13. G. Weiss, Jackson networks with unlimited supply of work and full utilization, Journal of Applied Probability, 2005, 42(3): 879–882.

    Article  Google Scholar 

  14. I. J. B. F. Adan and G. Weiss, A two node Jackson network with infinite supply of work, Probability in Engineering and Informational Sciences, 2005, 19(2): 191–212.

    Google Scholar 

  15. I. J. B. F. Adan and G. Weiss, Analysis of a simple Markovian re-entrant line with infinite supply of work under the LBFS Policy, Queueing Systems, 2006, 54(3): 169–183.

    Article  Google Scholar 

  16. Y. Guo and H. Zhang, On the stability of a simple re-entrant line with infinite supply, OR Trans- actions, 2006, 10(2): 75–85.

    Google Scholar 

  17. Y. Guo and H. Zhang, Stability of Re-entrant Lines with Infinite Supply, Working Paper, Academy of mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 2006.

  18. Y. Guo and J. Yang, Stability of a 2-station-5-class re-entrant line with infinite supply of work, Accepted by Asia-Pacific Journal of Operational Research, 2007.

  19. G. Weiss, Stability of a simple re-entrant line with infinite supply of work–the case of exponential processing times, Journal of the Operations Research Society of Japan, 2004, 47(4): 304–313.

    Google Scholar 

  20. G. Weiss and A. Kopzon, A push pull queueing system, Operations Research Letters, 2002, 30(6): 351–359.

    Article  Google Scholar 

  21. S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986.

    Google Scholar 

  22. H. Chen and H. Zhang, Stability of multiclass queueing networks under priority service disciplines, Operation Research, 2000, 48(1): 26–37.

    Article  Google Scholar 

  23. M. H. A. Davis, Piecewise deterministic Markov processes: a general class of nondiffusion models, Journal of Royal Statist. Soc. Series B, 1984, 46(3): 353–388.

    Google Scholar 

  24. H. Chen, Fluid approximations and stability of multiclass queueing networks I: Work-conserving disciplines, Annals of Applied Probability, 1995, 5(3): 637–665.

    Article  Google Scholar 

  25. J. M. Harrison and M. I. Reiman, Reflected Brownian motion on an orthant, Annals of Probability, 1981, 9(2): 302–308.

    Article  Google Scholar 

  26. H. Chen and A. Mandelbaum, Leontief systems, RBV's, and RBM's, in Proceedings of the Imperial College Workshop on Applied Stochastic Processes (ed. by M. H. A. Davis and R. J. Elliotte), Fordon and Breach Science Publishers, New York, 1991.

  27. H. Chen and A. Mandelbaum, Discrete flow networks: Bottleneck analysis and fluid approximation, Mathematics of Operations Research, 1991, 16(2): 408–446.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Yongjiang GUO

Authors
  1. Yongjiang GUO
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongjiang GUO.

Rights and permissions

Reprints and permissions

About this article

Cite this article

GUO, Y. Stability of generalized Jackson networks with infinite supply of work. J Syst Sci Complex 21, 283–295 (2008). https://doi.org/10.1007/s11424-008-9112-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-008-9112-z

Keywords

Search

Navigation