Skip to main content
Log in

Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

In this paper, we study the stability conditions of the MMAP[K]/G[K]/1/LCFS preemptive repeat queue. We introduce an embedded Markov chain of matrix M/G/1 type with a tree structure and identify conditions for the Markov chain to be ergodic. First, we present three conventional methods for the stability problem of the queueing system of interest. These methods are either computationally demanding or do not provide accurate information for system stability. Then we introduce a novel approach that develops two linear programs whose solutions provide sufficient conditions for stability or instability of the queueing system. The new approach is numerically efficient. The advantages and disadvantages of the methods introduced in this paper are analyzed both theoretically and numerically.

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

Access this article

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. S. Asmussen and G. Koole, Marked point processes as limits of Markovian arrival streams, J. Appl. Probab. 30 (1993) 365–372.

    Google Scholar 

  2. H. Chen and H. Zhang, Stabiliy of multiclass queueing networks under priority service disciplines, Oper. Res. 48 (2000) 26–37.

    Google Scholar 

  3. J.W. Cohen, The Single Server Queue (North-Holland, Amsterdam, 1982).

    Google Scholar 

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

    Google Scholar 

  5. G. Fayolle, V.A Malyshev and M.V. Menshikov, Topics in the Constructive Theory of Countable Markov Chains (Cambridge Univ. Press, Cambridge, 1995).

    Google Scholar 

  6. F.R. Gantmacher, The Theory of Matrices (Chelsea, New York, 1959).

    Google Scholar 

  7. K. Goebel and W.A. Kirk, Topics in Metric Fixed Point Theory (Cambridge Univ. Press, Cambridge, 1990).

    Google Scholar 

  8. Q.-M. He, Classification ofMarkov processes of M / G / 1 type with a tree structure and its applications to queueing systems, Oper. Res. Lett. 26 (2000) 67–80.

    Google Scholar 

  9. Q.-M. He, Classification of Markov processes of matrix M / G / 1 type with a tree structure and its applications to the MMAP[K]/PH[K]/1 queue, Stochastic Models 16 (2000) 407–433.

    Google Scholar 

  10. Q.-M. He, A fixed point approach to the classification ofMarkov chains with a tree stucture, Stochastic Models 19(1) (2003) 76–114.

    Google Scholar 

  11. Q.-M. He and A.S. Alfa, The discrete timeMMAP[K]/PH[K]/1/LCFS-GPR queues and its variants, in: Advances in Algorithmic Methods for Stochastic Models – Proc. of the 3rd Internat. Conf. on Matrix Analytic Methods, eds. G. Latouche and P.G. Taylor (Notable Publications, New Jersey, 2000) pp. 167–190.

    Google Scholar 

  12. Q.-M. He and M.F. Neuts, Markov arrival processes with marked transitions, Stochastic Process. Appl. 74(1) (1998) 37–52.

    Google Scholar 

  13. 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(1) (1996) 4–17.

    Google Scholar 

  14. R. Loynes, The stability of a queue with non-independent inter-arrival and service times, Proc. Cambridge Philos. Soc. 58 (1962) 497–520.

    Google Scholar 

  15. S.P. Meyn and R. Tweedie, Markov Chains and Stochastic Stability (Springer, Berlin, 1993).

    Google Scholar 

  16. M.F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach (Johns Hopkins Univ. Press, Baltimore, MD, 1981).

    Google Scholar 

  17. M.F. Neuts, Structured Stochastic Matrices of M / G / 1 Type and Their Applications (Marcel Dekker, New York, 1989).

    Google Scholar 

  18. T. Takine, B. Sengupta and R.W. Yeung, A generalization of the matrix M/ G / 1 paradigm for Markov chains with a tree structure, Stochastic Models 11 (1995) 411–421.

    Google Scholar 

  19. B. Van Houdt and C. Blondia, Stability and performance of stack algorithms for random access communication modeled as a tree structured QBD Markov chain, Stochastic Models 17(3) (2001) 1–28.

    Google Scholar 

  20. R.W. Yeung and B. Sengupta, Matrix product-form solutions for Markov chains with a tree structure, Adv. in Appl. Probab. 26(4) (1994) 965–987.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

He, QM., Li, H. Stability Conditions of the MMAP [ K ]/ G [ K ]/1/ LCFS Preemptive Repeat Queue. Queueing Systems 44, 137–160 (2003). https://doi.org/10.1023/A:1024420505098

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1024420505098

Navigation