Hostname: page-component-848d4c4894-v5vhk Total loading time: 0 Render date: 2024-06-25T06:23:39.824Z Has data issue: false hasContentIssue false
  • English
  • Français

On the estimation of probabilities for birth and death processes

Part of: Special processes General theory in ordinary differential equations Markov processes

Published online by Cambridge University Press:  14 July 2016

A. I. Zeifman
A. I. Zeifman*
Affiliation:
Vologda State Pedagogical Institute
*
∗Postal address: Vologda State Pedagogical Institute, Vologda, S. Orlova, 6, 160600, Russia.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let X(t) be a non-homogeneous birth and death process. In this paper we develop a general method of estimating bounds for the state probabilities for X(t), based on inequalities for the solutions of the forward Kolmogorov equations. Specific examples covered include simple estimates of Pr(X(t) < j | X(0) = k) for the M(t)/M(t)/N/0 and M(t)/M(t)/N queue-length processes.

Keywords

DIFFERENTIAL EQUATIONSESTIMATES OF STATE PROBABILITIES

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60K25: Queueing theory 34A30: Linear equations and systems, general
Type
Research Papers
Information
Journal of Applied Probability , Volume 32 , Issue 3 , September 1995 , pp. 623 - 634
Copyright
Copyright © Applied Probability Trust 1995 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Dalecku, Ju. L. and Krein, M. G. (1974) Stability of solutions of differential equations in Banach space. Amer. Math. Soc. Transl. 43.Google Scholar
[2] Gnedenko, B. V. and Makarov, I. P. (1971) Properties of solutions of loss system in the case of periodic intensities. Diff. Equations 7, 1696–1698 (in Russian).Google Scholar
[3] Losinsku, S. M. (1958) Error estimate of numerical integration of ordinary differential equation. Izv. VUZov. Math. 5, 52–90 (in Russian).Google Scholar
[4] Zeifman, A. I. (1985) Stability for continuous-time nonhomogeneous Markov chains. Lect. Notes Math. 1155, 401–414.Google Scholar
[5] Zeifman, A. I. (1988) Qualitative properties of nonhomogeneous birth and death processes. In Stability Problems for Stochastic Models, pp. 32–40. Institute for Systems Studies, Moscow (in Russian).Google Scholar
[6] Zeifman, A. I. (1989a) Some properties of the loss system in the case of varying intensities. Autom. Remote Control 1, 107–113 (in Russian).Google Scholar
[7] Zeifman, A. I. (1989b) On quasi-ergodicity and stability for some nonhomogeneous Markov processes. Sib. Math. J. 2, 85–89 (in Russian).Google Scholar
[8] Zeifman, A. I. (1991) Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob. 28, 268–277.Google Scholar