Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-24T04:49:33.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Extinction Probability in A Birth-Death Process with Killing

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Erik A. Van Doorn  and
Alexander I. Zeifman
Erik A. Van Doorn*
Affiliation:
University of Twente
Alexander I. Zeifman*
Affiliation:
Vologda State Pedagogical University and Vologda Scientific Coordinate Centre of CEMI RAS
*
∗Postal address: Department of Applied Mathematics, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands. Email address: [email protected]
∗∗Postal address: Vologda State Pedagogical University, S. Orlova 6, Vologda, Russia. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

Keywords

Absorptiondecay parameterextinction timepersistence timerate of convergencelogarithmic norm

MSC classification

Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 42 , Issue 1 , March 2005 , pp. 185 - 198
Copyright
© Applied Probability Trust 2005 

References

Anderson, W. J. (1991). Continuous-Time Markov Chains. Springer, New York.Google Scholar
Bobrowski, A. (2004). Quasi-stationary distributions of a pair of Markov chains related to time evolution of a DNA locus. Adv. Appl. Prob. 36, 56–77.Google Scholar
Brockwell, P. J. (1986). The extinction time of a general birth and death process with catastrophes. J. Appl. Prob. 23, 851–858.Google Scholar
Chen, A., Pollett, P., Zhang, H. and Cairns, B. (2004). Uniqueness criteria for continuous-time Markov chains with general transition structure. Preprint, University of Queensland.Google Scholar
Chen, M. F. (1991). Exponential L2-convergence and L2-spectral gap for Markov processes. Acta Math. Sinica (N.S.) 7, 19–37.Google Scholar
Chen, M. F. (1996). Estimation of the spectral gap for Markov chains. Acta Math. Sinica (N.S.) 12, 337–360.Google Scholar
Chihara, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.Google Scholar
Cohen, J. W. (1982). The Single Server Queue (North-Holland Ser. Appl. Math. Mech. 8), 2nd edn. North-Holland, Amsterdam.Google Scholar
Elmes, S., Pollett, P. and Walker, D. (2000). Further results on the relationship between μ-invariant measures and quasi-stationary distributions for absorbing continuous-time Markov chains. Math. Comput. Modelling 31, 107–113.Google Scholar
Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd edn. John Wiley, New York.Google Scholar
Good, P. (1968). The limiting behavior of transient birth and death processes conditioned on survival. J. Austral. Math. Soc. 8, 716–722.Google Scholar
Granovsky, B. L. and Zeifman, A. I. (1997). The decay function of nonhomogeneous birth–death processes, with applications to mean-field models. Stoch. Process. Appl. 72, 105–120.Google Scholar
Granovsky, B. L. and Zeifman, A. I. (2000). The N-limit of spectral gap of a class of birth–death Markov chains. Appl. Stoch. Models Business Industry 16, 235–248.3.0.CO;2-S>CrossRefGoogle Scholar
Granovsky, B. L. and Zeifman, A. I. (2004). Nonstationary queues: estimation of the rate of convergence. Queueing Systems 46, 363–388.Google Scholar
Jacka, S. D. and Roberts, G. O. (1995). Weak convergence of conditioned processes on a countable state space. J. Appl. Prob. 32, 902–916.CrossRefGoogle Scholar
Karlin, S. and McGregor, J. L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 589–646.Google Scholar
Karlin, S. and Tavaré, S. (1982). Linear birth and death processes with killing. J. Appl. Prob. 19, 477–487.CrossRefGoogle Scholar
Kartashov, N. V. (1998). Calculation of the exponential ergodicity exponent for birth–death processes. Theory Prob. Math. Statist. 57, 53–60.Google Scholar
Kijima, M. (1992). Evaluation of the decay parameter for some specialized birth–death processes. J. Appl. Prob. 29, 781–791.CrossRefGoogle Scholar
Kingman, J. F. C. (1963). The exponential decay of Markov transition probabilities. Proc. London Math. Soc. 13, 337–358.Google Scholar
Lenin, R. B., Parthasarathy, P. R., Scheinhardt, W. R. W. and van Doorn, E. A. (2000). Families of birth–death processes with similar time-dependent behaviour. J. Appl. Prob. 37, 835–849.Google Scholar
Pakes, A. G. (1995). Quasi-stationary laws for Markov processes: examples of an always proximate absorbing state. Adv. Appl. Prob. 27, 120–145.Google Scholar
Van Doorn, E. A. (1985). Conditions for exponential ergodicity and bounds for the decay parameter of a birth–death process. Adv. Appl. Prob. 17, 514–530.Google Scholar
Van Doorn, E. A. (1991). Quasi-stationary distributions and convergence to quasi-stationarity of birth–death processes. Adv. Appl. Prob. 23, 683–700.Google Scholar
Van Doorn, E. A. (2002). Representations for the rate of convergence of birth–death processes. Theory Prob. Math. Statist. 65, 37–43.Google Scholar
Van Doorn, E. A. and Zeifman, A. I. (2005). Birth–death processes with killing. To appear in Statist. Prob. Lett. Google Scholar
Zeifman, A. I. (1991). Some estimates of the rate of convergence for birth and death processes. J. Appl. Prob. 28, 268–277.Google Scholar
Zeifman, A. I. (1995). On the estimation of probabilities for birth and death processes. J. Appl. Prob. 32, 623–634.Google Scholar
Zeifman, A. I. (1995). Upper and lower bounds on the rate of convergence for nonhomogeneous birth and death processes. Stoch. Process. Appl. 59, 157–173.Google Scholar