Queueing Systems

, Volume 46, Issue 3–4, pp 353–361 | Cite as

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

  • V. Abramov
  • R. Liptser


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=(Qt)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.

countable Markov process existence of the limiting distribution birth-and-death process 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [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. [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. [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. [4]
    S. Asmussen and H. Thorisson, A Markov chain approach to periodic queues, J. Appl. Probab. 24 (1987) 215–225.Google Scholar
  5. [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. [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. [7]
    E. Gelenbe and D. Finkel, Stationary deterministic flows: The single server queue, Theoret. Comput. Sci. 52 (1987) 269–280.Google Scholar
  8. [8]
    I.I. Geterontidis, On certain aspects of homogeneous Markov systems in continuous time, J. Appl. Probab. 27 (1990) 530–544.Google Scholar
  9. [9]
    I.I. Geterontidis, Periodic strong ergodicity in non-homogeneous Markov systems, J. Appl. Probab. 28 (1991) 58–73.Google Scholar
  10. [10]
    I.I. Geterontidis, Cyclic strong ergodicity in non-homogeneous Markov systems, SIAM J. Matrix Anal. Appl. 13 (1992) 550–566.Google Scholar
  11. [11]
    D.B. Gnedenko, On a generalization of Erlang formulae, Zastos. Matem. 12 (1971) 239–242.Google Scholar
  12. [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. [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. [14]
    B.L. Granovsky and A.I. Zeifman, Nonstationary Markovian queues, J. Math. Sci. 99(4) (2000) 1415–1438.Google Scholar
  15. [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. [16]
    J. Johnson and D. Isaacson, Conditions for strong ergodicity using intensity matrices, J. Appl. Probab. 25 (1988) 34–42.Google Scholar
  17. [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. [18]
    S. Karlin, A First Course in Stochastic Processes (Academic Press, New York/London, 1968).Google Scholar
  19. [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. [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. [21]
    A.J. Lemoine, On queues with periodic Poisson input, J. Appl. Probab. 18 (1981) 889–900.Google Scholar
  22. [22]
    R.Sh. Liptser and A.N. Shiryayev, Theory of Martingales (Kluwer Academic, Dordrecht, 1989).Google Scholar
  23. [23]
    A. Mandelbaum and W. Massey, Strong approximations for time dependent queues, Math. Oper. Res. 20 (1995) 33–64.Google Scholar
  24. [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. [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. [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. [27]
    W. Massey, Asymptotic analysis of the time-dependent M/M/1 queues, Math. Oper. Res. 10(2) (1985) 305–327.Google Scholar
  28. [28]
    A.N. Shiryayev, Probability (Springer, Berlin, 1984).Google Scholar
  29. [29]
    H. Thorisson, Periodic regeneration, Stochastic Process. Appl. 20 (1985) 85–104.Google Scholar
  30. [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. [31]
    A. Zeifman, On stability of continuous-time nonhomogeneous Markov chains, Soviet Math. 35(7) (1991) 29–34.Google Scholar
  32. [32]
    A. Zeifman, On the ergodicity of nonhomogeneous birth and death processes, J. Math. Sci. 72(1) (1994) 2893–2899.Google Scholar
  33. [33]
    A. Zeifman, On the estimation of probabilities for birth-and-death processes, J. Appl. Probab. 32 (1995) 623–634.Google Scholar
  34. [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. [35]
    A. Zeifman, Stability of birth and death processes, J. Math. Sci. 91(3) (1998) 3023–3031.Google Scholar
  36. [36]
    A. Zeifman and D. Isaacson, On strong ergodicity for nonhomogeneous continuous-time Markov chains, Stochastic Process. Appl. 50 (1994) 263–273.Google Scholar

Copyright information

© Kluwer Academic Publishers 2004

Authors and Affiliations

  • V. Abramov
    • 1
    • 2
  • R. Liptser
    • 3
  1. 1.Department of Mathematics, The Faculty of Exact SciencesTel Aviv UniversityTel Aviv, and
  2. 2.College of Judea and SamariaArielIsrael
  3. 3.Department of Electrical Engineering-SystemsTel Aviv UniversityTel AvivIsrael

Personalised recommendations