Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

A characterisation of (max,+)-linear queueing systems

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

The (max,+)-algebra has been successfully applied to many areas of queueing theory, like stability analysis and ergodic theory. These results are mainly based on two ingredients: (1) a (max,+)-linear model of the time dynamic of the system under consideration, and (2) the time-invariance of the structure of the (max,+)-model. Unfortunately, (max,+)-linearity is a purely algebraic concept and it is by no means immediate if a queueing network admits a (max,+)-linear representation satisfying (1) and (2). In this paper we derive the condition a queueing network must meet if it is to have a (max,+)-linear representation. In particular, we study (max,+)-linear systems with time-invariant transition structures. For this class of systems, we find a surprisingly simple necessary and sufficient condition for (max,+)-linearity, based on the flow of customers through the network.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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. E. Altman, B. Gaujal and A. Hordijk, Admission control in stochastic event graphs, IEEE Trans. Automat. Control, to appear.

  2. F. Baccelli, Ergodic theory of stochastic Petri networks, Ann. Probab. 20 (1992) 375–396.

    Google Scholar 

  3. F. Baccelli, G. Cohen, G. Olsder and J.-P. Quadrat, Synchronization and Linearity (Wiley, New York, 1992).

    Google Scholar 

  4. F. Baccelli, S. Foss and J. Mairesse, Stationary ergodic Jackson networks: Results and counterexamples, in: Stochastic Networks, chapter 1 (Oxford Univ. Press, Oxford, 1996) pp. 281–307.

    Google Scholar 

  5. F. Baccelli, E. Gelenbe and B. Plateau, An end-to-end approach to the resequencing problem, J. Assoc. Comput. Mach. 31 (1984) 474–485.

    Google Scholar 

  6. F. Baccelli, S. Hasenfuß and V. Schmidt, Transient and stationary waiting times in (max,+)-linear systems with Poisson input, Queueing Systems 26 (1997) 301–342.

    Article  Google Scholar 

  7. F. Baccelli and D. Hong, Analytic expansions of (max,+) Lyapunov exponents, Technical Report no. 3427, INRIA, Sophia Antipolis (1998).

    Google Scholar 

  8. F. Baccelli and P. Konstantopoulos, Estimates of cycle times in stochastic Petri nets, in: Lecture Notes in Control and Information Science, Vol. 177 (Springer, Berlin, 1992) pp. 1–20.

    Google Scholar 

  9. F. Baccelli and V. Schmidt, Taylor series expansions for Poisson-driven (max,+)-linear systems, Ann. Appl. Probab. 6 (1996) 138–185.

    Article  Google Scholar 

  10. D. Cheng, Tandem queues with general blocking: a unified model and comparison results, J. Discrete Event Dynamic Systems 2 (1993) 207–234.

    Article  Google Scholar 

  11. R. Cuninghame-Green, Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, Vol. 166 (Springer, Berlin, 1979).

    Google Scholar 

  12. H. Daduna, Exchangeable items in repair systems: Delay times, Oper. Res. 38 (1990) 349–354.

    Article  Google Scholar 

  13. S. Ermakov and N. Krivulin, Efficient algorithms for tandem queueing system simulation, Appl. Math. Lett. 7 (1994) 45–49.

    Article  Google Scholar 

  14. P. Glasserman and D. Yao, Structured buffer-allocation problems, J. Discrete Event Dynamic Systems 6 (1996) 9–41.

    Article  Google Scholar 

  15. B. Hajek, Extremal splittings of point processes, Math. Oper. Res. 10(4) (1985) 543–556.

    Google Scholar 

  16. B. Heidergott, A differential calculus for random matrices with applications to (max,+)-linear stochastic systems, Technical Report 98-10, Delft University of Technology (1998) (submitted).

  17. A. Jean-Marie, Waiting time ditributions in Poisson-driven deterministic systems, Technical Report no. 3083, INRIA, Sophia Antipolis (1997).

    Google Scholar 

  18. N. Krivulin, Unbiased estimates for gradients of stochastic network performance measures, Acta Appl. Math. 33 (1993) 21–43.

    Article  Google Scholar 

  19. N. Krivulin, A max-algebra approach to modeling and simulation of tandem queueing systems, Math. Comput. Modelling 22 (1995) 25–37.

    Article  Google Scholar 

  20. N. Krivulin, The max-plus algebra approach in modelling of queueing networks, in: Proc. of 1996 Summer Computer Simulation Conf., Portland (July 21–25, 1996) pp. 485–490.

  21. J. Mairesse, Products of irreducible random matrices in the (max,+) algebra, Adv. in Appl. Probab. 29 (1997) 444–477.

    Article  Google Scholar 

  22. J. Resing, R. de Vries, G. Hooghiemstra, M. Keane and G. Olsder, Asymptotic behavior of random discrete event systems, Stochastic Process. Appl. 36 (1990) 195–216.

    Article  Google Scholar 

  23. J. Vincent, Some ergodic results on stochastic iterative discrete event systems, J. Discrete Event Dynamic Systems 7 (1997) 209–232.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. EURANDOM, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Bernd Heidergott

Authors
  1. Bernd Heidergott
    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

Heidergott, B. A characterisation of (max,+)-linear queueing systems. Queueing Systems 35, 237–262 (2000). https://doi.org/10.1023/A:1019102429650

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1019102429650

Search

Navigation