Skip to main content
SpringerLink
Log in

On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process

  • Published:
Queueing Systems Aims and scope Submit manuscript

Abstract

In this paper, sufficient conditions are given for the existence of limiting distribution of a nonhomogeneous countable Markov chain with time-dependent transition intensity matrix. The method of proof exploits the fact that if the distribution of random process Q=(Q t ) t≥0 is absolutely continuous with respect to the distribution of ergodic random process Q°=(Q° t ) t≥0, then \(Q_t \xrightarrow[{t \to \infty }]{{law}}\pi \) where π is the invariant measure of Q°. We apply this result for asymptotic analysis, as t→∞, of a nonhomogeneous countable Markov chain which shares limiting distribution with an ergodic birth-and-death process.

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. V.M. Abramov, On the asymptotic distribution of the maximum number of infectives in epidemic models with immigrations, J. Appl. Probab. 31 (1994) 606–613.

    Google Scholar 

  2. V.M. Abramov, A large closed queueing network with autonomous service and bottleneck, Queueing Systems 35(1–3) (2000) 23–54.

    Google Scholar 

  3. V.M. Abramov, Some results for large closed queueing networks with and without bottleneck: Up-and down-crossings approach, Queueing Systems 38(2) (2001) 149–184.

    Google Scholar 

  4. S. Asmussen and H. Thorisson, A Markov chain approach to periodic queues, J. Appl. Probab. 24 (1987) 215–225.

    Google Scholar 

  5. N. Bambos and J. Walrand, On stability of state-dependent queues and acyclic queueing networks, Adv. in Appl. Probab. 21 (1989) 681–701.

    Google Scholar 

  6. T. Collings and C. Stoneman, The M/M/∞ queue with varying arrival and departure rates, Oper. Res. 24 (1976) 760–773.

    Google Scholar 

  7. E. Gelenbe and D. Finkel, Stationary deterministic flows: The single server queue, Theoret. Comput. Sci. 52 (1987) 269–280.

    Google Scholar 

  8. I.I. Geterontidis, On certain aspects of homogeneous Markov systems in continuous time, J. Appl. Probab. 27 (1990) 530–544.

    Google Scholar 

  9. I.I. Geterontidis, Periodic strong ergodicity in non-homogeneous Markov systems, J. Appl. Probab. 28 (1991) 58–73.

    Google Scholar 

  10. I.I. Geterontidis, Cyclic strong ergodicity in non-homogeneous Markov systems, SIAM J. Matrix Anal. Appl. 13 (1992) 550–566.

    Google Scholar 

  11. D.B. Gnedenko, On a generalization of Erlang formulae, Zastos. Matem. 12 (1971) 239–242.

    Google Scholar 

  12. B.V. Gnedenko and A.D. Soloviev, On the conditions of the existence of final probabilities for a Markov process, Math. Operationsforsh. Statist. 4 (1973) 379–390.

    Google Scholar 

  13. B.L. Granovsky and A.I. Zeifman, The decay function of nonhomogeneous birth-and-death process, with application to mean-field model, Stochastic Process. Appl. 72 (1997) 105–120.

    Google Scholar 

  14. B.L. Granovsky and A.I. Zeifman, Nonstationary Markovian queues, J. Math. Sci. 99(4) (2000) 1415–1438.

    Google Scholar 

  15. L. Green and P. Kolesar, Testing a validity of the queueing model of a police patrol, Managm. Sci. 35 (1989) 127–148.

    Google Scholar 

  16. J. Johnson and D. Isaacson, Conditions for strong ergodicity using intensity matrices, J. Appl. Probab. 25 (1988) 34–42.

    Google Scholar 

  17. Yu.M. Kabanov, R.Sh. Liptser and A.N. Shiryaev, Absolutely continuity and singularity of locally absolutely continuous probability distributions II, Math. USSR Sbornik 36(1) (1980) 31–58.

    Google Scholar 

  18. S. Karlin, A First Course in Stochastic Processes (Academic Press, New York/London, 1968).

    Google Scholar 

  19. C. Knessl, B. Matkovsky, Z. Schuss and C. Tier, Asymptotic analysis of a state-dependent M/G/1 queueing system, SIAM J. Appl. Math. 46(3) (1986) 483–505.

    Google Scholar 

  20. Y. Kogan and R.Sh. Liptser, Limit non-stationary behavior of large closed queueing networks with bottlenecks, Queueing Systems 14 (1993) 33–55.

    Google Scholar 

  21. A.J. Lemoine, On queues with periodic Poisson input, J. Appl. Probab. 18 (1981) 889–900.

    Google Scholar 

  22. R.Sh. Liptser and A.N. Shiryayev, Theory of Martingales (Kluwer Academic, Dordrecht, 1989).

    Google Scholar 

  23. A. Mandelbaum and W. Massey, Strong approximations for time dependent queues, Math. Oper. Res. 20 (1995) 33–64.

    Google Scholar 

  24. A. Mandelbaum and G. Pats, State-dependent queues: Approximations and applications, in: IMA Volumes in Mathematics and Its Applications, eds. F. Kelly and R.J. Williams, Vol. 71 (Springer, Berlin, 1995) pp. 239–282.

    Google Scholar 

  25. A. Mandelbaum and G. Pats, State-dependent stochastic networks. Part I: Approximations and applications with continuous diffusion limits, Ann. Appl. Probab. 8(2) (1998) 569–646.

    Google Scholar 

  26. B.H. Margoluis, A sample path analysis of the M t /M t /c queue, Queueing Systems 31(1) (1999) 59–93.

    Google Scholar 

  27. W. Massey, Asymptotic analysis of the time-dependent M/M/1 queues, Math. Oper. Res. 10(2) (1985) 305–327.

    Google Scholar 

  28. A.N. Shiryayev, Probability (Springer, Berlin, 1984).

    Google Scholar 

  29. H. Thorisson, Periodic regeneration, Stochastic Process. Appl. 20 (1985) 85–104.

    Google Scholar 

  30. A. Zeifman, Some estimates of the rate of convergence for birth-and-death processes, J. Appl. Probab. 28 (1991) 268–277.

    Google Scholar 

  31. A. Zeifman, On stability of continuous-time nonhomogeneous Markov chains, Soviet Math. 35(7) (1991) 29–34.

    Google Scholar 

  32. A. Zeifman, On the ergodicity of nonhomogeneous birth and death processes, J. Math. Sci. 72(1) (1994) 2893–2899.

    Google Scholar 

  33. A. Zeifman, On the estimation of probabilities for birth-and-death processes, J. Appl. Probab. 32 (1995) 623–634.

    Google Scholar 

  34. A. Zeifman, Upper and lower bounds on the rate of convergence for nonhomogeneous birth-and-death processes, Stochastic Process. Appl. 59 (1995) 159–173.

    Google Scholar 

  35. A. Zeifman, Stability of birth and death processes, J. Math. Sci. 91(3) (1998) 3023–3031.

    Google Scholar 

  36. A. Zeifman and D. Isaacson, On strong ergodicity for nonhomogeneous continuous-time Markov chains, Stochastic Process. Appl. 50 (1994) 263–273.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, and

    V. Abramov

  2. College of Judea and Samaria, 44837, Ariel, Israel

    V. Abramov

  3. Department of Electrical Engineering-Systems, Tel Aviv University, 69978, Tel Aviv, Israel

    R. Liptser

Authors
  1. V. Abramov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Liptser
    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

Abramov, V., Liptser, R. On Existence of Limiting Distribution for Time-Nonhomogeneous Countable Markov Process. Queueing Systems 46, 353–361 (2004). https://doi.org/10.1023/B:QUES.0000027990.74497.b2

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:QUES.0000027990.74497.b2

Search

Navigation