On existence of limiting distribution for time-nonhomogeneous countable Markov process

Queueing Systems

Volumn 46, Issue 3-4, 2004, Pages 353-361

  Abramov, V ^a,b   Liptser, R ^a  

^a TEL AVIV UNIVERSITY   (Israel)

^b ARIEL UNIVERSITY   (Israel)

Author keywords

Birth and death process; Countable Markov process; Existence of the limiting distribution

Indexed keywords

ASYMPTOTIC ANALYSIS;

BIRTH AND DEATH PROCESS; ERGODICS; INVARIANT MEASURE; LIMITING DISTRIBUTIONS; NON-HOMOGENEOUS; TIME DEPENDENT; TRANSITION INTENSITY MATRIX;

MARKOV CHAINS;

EID: 3543132404     PISSN: 02570130     EISSN: 15729443     Source Type: Journal    
DOI: 10.1023/b:ques.0000027990.74497.b2     Document Type: Conference Paper

References (36)

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.
2. V.M. Abramov, A large closed queueing network with autonomous service and bottleneck, Queueing Systems 35(1-3) (2000) 23-54.
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.
4. S. Asmussen and H. Thorisson, A Markov chain approach to periodic queues, J. Appl. Probab. 24 (1987) 215-225.
5. N. Bambos and J. Walrand, On stability of state-dependent queues and acyclic queueing networks, Adv. in Appl. Probab. 21 (1989) 681-701.
6. T. Collings and C. Stoneman, The M/M/∞ queue with varying arrival and departure rates, Oper. Res. 24 (1976) 760-773.
7. E. Gelenbe and D. Finkel, Stationary deterministic flows: The single server queue, Theoret. Comput. Sci. 52 (1987) 269-280.
8. I.I. Geterontidis, On certain aspects of homogeneous Markov systems in continuous time, J. Appl. Probab. 27 (1990) 530-544.
9. I.I. Geterontidis, Periodic strong ergodicity in non-homogeneous Markov systems, J. Appl. Probab. 28 (1991) 58-73.
10. I.I. Geterontidis, Cyclic strong ergodicity in non-homogeneous Markov systems, SIAM J. Matrix Anal. Appl. 13 (1992) 550-566.
11. D.B. Gnedenko, On a generalization of Erlang formulae, Zastos. Matem. 12 (1971) 239-242.
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.
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.
14. B.L. Granovsky and A.I. Zeifman, Nonstationary Markovian queues, J. Math. Sci. 99(4) (2000) 1415-1438.
15. L. Green and P. Kolesar, Testing a validity of the queueing model of a police patrol, Managm. Sci. 35 (1989) 127-148.
16. J. Johnson and D. Isaacson, Conditions for strong ergodicity using intensity matrices, J. Appl. Probab. 25 (1988) 34-42.
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.
18. S. Karlin, A First Course in Stochastic Processes (Academic Press, New York/London, 1968).
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.
20. Y. Kogan and R.Sh. Liptser, Limit non-stationary behavior of large closed queueing networks with bottlenecks, Queueing Systems 14 (1993) 33-55.
21. A.J. Lemoine, On queues with periodic Poisson input, J. Appl. Probab. 18 (1981) 889-900.
22. R.Sh. Liptser and A.N. Shiryayev, Theory of Martingales (Kluwer Academic, Dordrecht, 1989).
23. A. Mandelbaum and W. Massey, Strong approximations for time dependent queues, Math. Oper. Res. 20 (1995) 33-64.
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.
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.
26. t/c queue, Queueing Systems 31(1) (1999) 59-93.
27. W. Massey, Asymptotic analysis of the time-dependent M/M/1 queues, Math. Oper. Res. 10(2) (1985) 305-327.
28. A.N. Shiryayev, Probability (Springer, Berlin, 1984).
29. H. Thorisson, Periodic regeneration, Stochastic Process. Appl. 20 (1985) 85-104.
30. A. Zeifman, Some estimates of the rate of convergence for birth-and-death processes, J. Appl. Probab. 28 (1991) 268-277.
31. A. Zeifman, On stability of continuous-time nonhomogeneous Markov chains, Soviet Math. 35(7) (1991) 29-34.
32. A. Zeifman, On the ergodicity of nonhomogeneous birth and death processes, J. Math. Sci. 72(1) (1994) 2893-2899.
33. A. Zeifman, On the estimation of probabilities for birth-and-death processes, J. Appl. Probab. 32 (1995) 623-634.
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.
35. A. Zeifman, Stability of birth and death processes, J. Math. Sci. 91(3) (1998) 3023-3031.
36. A. Zeifman and D. Isaacson, On strong ergodicity for nonhomogeneous continuous-time Markov chains, Stochastic Process. Appl. 50 (1994) 263-273.