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Some estimates of the rate of convergence for birth and death processes

Published online by Cambridge University Press:  14 July 2016

A. I. Zeifman*
Affiliation:
Vologda State Pedagogical Institute
*
*Postal address: Vologda State Pedagogical Institute, Vologda, S. Orlova, 6, 160600, USSR.

Abstract

The ergodic properties of birth and death processes are studied. We obtain some explicit estimates for the rate of convergence by the methods of theory of differential equations.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1991 

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References

[1]Daleckij, Ju. L, and Krein, M. G. (1974) Stability of solutions of differential equations in Banach space. Amer. Math. Soc. Trans. 43.Google Scholar
[2]Lindvall, T. (1979) A note on coupling of birth and death processes. J. Appl. Prob. 16, 505512.Google Scholar
[3]Losinsky, S. M. (1958) Error estimate for numerical integration of ordinary differential equations. Izv. VUZ. Math. N 5, 5290 (in Russian).Google Scholar
[4]Natvig, B. (1974) On the transient probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Prob. 11, 345354.Google Scholar
[5]Van Doorn, E. (1981) The transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Prob. 18, 499506.10.2307/3213296Google Scholar
[6]Van Doorn, E. (1981) Stochastic Monotonicity and Queueing, Applications of Birth-Death Processes. Lecture Notes in Statistics 4, Springer-Verlag, Berlin.Google Scholar
[7]Van Doorn, E. (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. Appl. Prob. 17, 514530.10.2307/1427118Google Scholar
[8]Van Doorn, E. (1987) Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices. J. Approx. Theory 51, 254266.10.1016/0021-9045(87)90038-4Google Scholar
[9]Zeifman, A. I. (1988) Qualitative investigation of nonhomogeneous birth and death processes. In Stability Problems for Stochastic Models. Institute for System Studies, Moscow, 3240 (in Russian).Google Scholar
[10]Zeifman, A. I. (1989) Some properties of loss system in the case of varying intensities. Autom. Remote Control N 1, 107113 (in Russian).Google Scholar
[11]Zeifman, A. I. (1989) On quasi-ergodicity and stability for some nonhomogeneous Markov processes. Siberian Math. J. 30 N 2, 8589 (in Russian).Google Scholar
[12]Zeifman, A. I. (1989) Quasi-ergodicity for non-homogeneous continuous-time Markov chains. J. Appl. Prob. 26, 643648.10.2307/3214422Google Scholar