Skip to main content
SpringerLink
Log in

Piecewise linear test functions for stability and instability of queueing networks

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

We develop the use of piecewise linear test functions for the analysis of stability of multiclass queueing networks and their associated fluid limit models. It is found that if an associated LP admits a positive solution, then a Lyapunov function exists. This implies that the fluid limit model is stable and hence that the network model is positive Harris recurrent with a finite polynomial moment. Also, it is found that if a particular LP admits a solution, then the network model is transient.

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.

Similar content being viewed by others

References

  1. F. Baccelli and S. Foss, Ergodicity of Jackson-type queueing networks, Queueing Systems 17 (1994) 5–72.

    Article  Google Scholar 

  2. D. Bertsimas, D. Gamarnik and J.N. Tsitsiklis, Stability conditions for multiclass fluid queueing networks, Technical Report, Massachusets Institute of Technology (1995).

  3. D. Bertsimas, I.Ch. Paschalidis and J.N. Tsitsiklis, Scheduling of multiclass queueing networks: Bounds on achievable performance, in: Workshop on Hierarchical Control for Real-Time Scheduling of Manufacturing Systems (Lincoln, NH, 1992).

  4. A.A. Borovkov, Limit theorems for queueing networks, Theory Probab. Appl. 31 (1986) 413–427.

    Article  Google Scholar 

  5. D.D. Botvich and A.A. Zamyatin, Ergodicity of conservative communication networks, Rapports de Recherche No. 1772, INRIA, Rocquencourt, France (1992).

    Google Scholar 

  6. M. Bramson, Instability of FIFO queueing networks, Ann. Appl. Probab. 4 (1994) 414–431.

    Google Scholar 

  7. H. Chen and A. Mandelbaum, Discrete flow networks: Bottlenecks analysis and fluid approximations, Math. Oper. Res. 16 (1991) 408–446.

    Google Scholar 

  8. H. Chen and H. Zhang, Stability of multiclass queueing networks under priority service disciplines, Preprint (1996).

  9. J.G. Dai and S.P. Meyn, Stability and convergence of moments for multiclass queueing networks via fluid models, IEEE Trans. Automat. Control 40 (1995) 1889–1904.

    Article  Google Scholar 

  10. J.G. Dai and Y. Wang, Nonexistence of Brownian models for certain multiclass queueing networks, Queueing Systems 13 (1993) 41–46.

    Article  Google Scholar 

  11. J.G. Dai and G. Weiss, Stability and instability of fluid models for reentrant lines, Math. Oper. Res. 21 (1996) 115–134.

    Article  Google Scholar 

  12. J.G. Dai, On the positive Harris recurrence for multiclass queueing networks: A unified approach via fluid limit models, Ann. Appl. Probab. 5 (1995) 49–77.

    Google Scholar 

  13. J.G. Dai, A fluid-limit model criterion for instability of multiclass queueing networks, Ann. Appl. Probab. (1996) 6.

  14. J.G. Dai and J.H. Vande Vate, Global stability of two-station queueing networks, in: Proceedings of Workshop on Stochastic Networks: Stability and Rare Events (Springer, New York, 1996).

  15. D. Down and S.P. Meyn, The structure of fluid models and instability of queueing networks, in: Proceedings of the European Control Conf., Rome, Italy (1995).

  16. D.G. Down and S.P. Meyn, A survey of Markovian methods for stability of networks, in: 11th Internat. Conf. on Analysis and Optimization of Systems, Sophia-Antipolis, France (1994).

  17. D.G. Down and S.P. Meyn, Piecewise linear test functions for stability of queueing networks, in: Proceedings of the 33rd Conf. on Decision and Control, Buena Vista, FL (1994).

  18. V. Dumas, A multiclass network with non-linear, non-convex, non-monotonic stability conditions, Technical Report, INRIA, Rocquencourt (1995).

  19. V. Dumas, Approches fluides pour la stabilité et l'instabilité de réseaux de files d'attente stochastiques à plusieurs classes de clients, Ph.D. thesis, L'Ecole Polytechnique (1996).

  20. P. Dupuis and R.J. Williams, Lyapunov functions for semimartingale reflecting Brownian motions, Ann. Probab. 22 (1994) 680–702.

    Google Scholar 

  21. G. Fayolle, On random walks arising in queueing systems: Ergodicity and transience via quadratic forms as Lyapunov functions – Part I, Queueing Systems 5 (1989) 167–184.

    Article  Google Scholar 

  22. G. Fayolle, V.A. Malyšev, M.V. Men'šikov and A.F. Sidorenko, Lyapunov functions for Jackson networks, Technical Report, INRIA, Rocquencourt (1991).

  23. S.G. Foss, Ergodicity of queueing networks, Siberian Math. J. (1992) 690–705. Translated from Sibirskii Matematicheskii Zhurnal 32 (1991) 184–203.

  24. F.G. Foster, On the stochastic matrices associated with certain queueing processes, Ann. Math. Statist. 24 (1953) 355–360.

    Google Scholar 

  25. J.M. Harrison and V. Nguyen, Brownian models of multiclass queueing networks: Current status and open problems, Queueing Systems 13 (1993) 5–40.

    Article  Google Scholar 

  26. C. Humes Jr., A regulator stabilization technique: Kumar and Seidman revisited, IEEE Trans. Automat. Control 39 (1994) 191–196.

    Article  Google Scholar 

  27. R.Z. Khas'minskii, Stochastic Stability of Differential Equations (Sijthoff & Noordhoff, Alphen aan den Rija, Netherlands, 1980).

    Google Scholar 

  28. J.F.C. Kingman, The ergodic behaviour of random walks, Biometrika 48 (1961) 391.

    Article  Google Scholar 

  29. P.R. Kumar and S.P. Meyn, Stability of queueing networks and scheduling policies, IEEE Trans. Automat. Control 40 (1995) 251–260.

    Article  Google Scholar 

  30. P.R. Kumar and S.P. Meyn, Duality and linear programs for stability and performance analysis of queueing networks and scheduling policies, IEEE Trans. Automat. Control 41 (1996) 4–17.

    Article  Google Scholar 

  31. P.R. Kumar, Re-entrant lines, Queueing Systems 13 (1993) 87–110.

    Article  Google Scholar 

  32. P.R. Kumar and T.I. Seidman, Dynamic instabilities and stabilization methods in distributed real-time scheduling of manufacturing systems, IEEE Trans. Automat. Control 35 (1990) 289–298.

    Article  Google Scholar 

  33. S. Kumar and P.R. Kumar, Performance bounds for queueing networks and scheduling policies, IEEE Trans. Automat. Control 39 (1994) 1600–1611.

    Article  Google Scholar 

  34. S. Kumar and P.R. Kumar, Fluctuation smoothing policies are stable for re-entrant lines, to appear in Discrete Event Dynamical Systems.

  35. J. LaSalle and S. Lefshetz, Stability by Liapunov's Direct Method (Academic Press, New York, NY, 1961).

  36. S.H. Lu and P.R. Kumar, Distributed scheduling based on due dates and buffer priorities, IEEE Trans. Automat. Control 36 (1991) 1406–1416.

    Article  Google Scholar 

  37. V.A. Malyšev, Classification of two-dimensional positive random walks and almost linear semimartingales, Soviet Math. Dokl. 13 (1972) 136–139.

    Google Scholar 

  38. V.A. Malyšev, Networks and dynamical systems, Adv. in Appl. Probab. 25 (1993) 140–175.

    Article  Google Scholar 

  39. V.A. Malyšev and M.V. Men'šikov, Ergodicity, continuity and analyticity of countable Markov chains, Trans. Moscow Math. Soc. 1 (1982) 1–48.

    Google Scholar 

  40. M.V. Menšikov, Ergodicity and transience conditions for random walks in the positive octant of space, Soviet Math. Dokl. 15 (1974) 1118–1121.

    Google Scholar 

  41. S.P. Meyn, Transience of multiclass queueing networks via fluid limit models, Ann. Appl. Probab. 5 (1994) 946–957.

    Google Scholar 

  42. S.P. Meyn and D. Down, Stability of generalized Jackson networks, Ann. Appl. Probab. 4 (1994) 124–148.

    Google Scholar 

  43. S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability (Springer, London, 1993).

  44. S.P. Meyn and R.L. Tweedie, Stability of Markovian processes III: Foster–Lyapunov criteria for continuous time processes, Adv. in Appl. Probab. 25 (1993) 518–548.

    Article  Google Scholar 

  45. S.P. Meyn and R.L. Tweedie, A survey of Foster–Lyapunov techniques for general state space Markov processes, in: Proceedings of the Workshop on Stochastic Stability and Stochastic Stabilization, Metz, France (1993).

  46. J.R. Perkins, C. Humes Jr. and P.R. Kumar, Distributed control of flexible manufacturing systems: stability and performance, IEEE Trans. Robotics Automat. 37 (1994) 132–141.

    Google Scholar 

  47. A.N. Rybko and A.L. Stolyar, On the ergodicity of stochastic processes describing open queueing networks, Technical Report, Moscow (1992).

  48. T.I. Seidman, First come first serve can be unstable, IEEE Trans. Automat. Control 39 (1994) 2166–2170.

    Article  Google Scholar 

  49. K. Sigman, The stability of open queueing networks, Stochastic Process. Appl. 35 (1990) 11–25.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Coordinated Science Laboratory, 1308 W. Main, Urbana, IL, 61801, USA

    D. Down & S.P. Meyn

Authors
  1. D. Down
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S.P. Meyn
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Down, D., Meyn, S. Piecewise linear test functions for stability and instability of queueing networks. Queueing Systems 27, 205–226 (1997). https://doi.org/10.1023/A:1019166115653

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019166115653

Search

Navigation