Extinction probability in a birth-death process with killing

Erik A. van Doorn, A.I. Zeifman

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    Abstract

    We study birth-death processes on the non-negative integers where $\{1,2,\ldots\}$ 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 $t\to\infty$ of the probability of absorption at time $t$, and relate it to the common rate of convergence of the transition probabilities which do not involve state $0$. Finally, we derive upper and lower bounds for the probability of absorption at time $t$ by applying a technique which involves the logarithmic norm of an appropriately defined operator.
    Original languageUndefined
    Place of PublicationEnschede
    PublisherUniversity of Twente, Faculty of Mathematical Sciences
    Number of pages20
    Publication statusPublished - 2004

    Publication series

    NameMemorandum Faculty of Mathematical Sciences
    PublisherDepartment of Applied Mathematics, University of Twente
    No.1723
    ISSN (Print)0169-2690

    Keywords

    • MSC-60J27
    • MSC-60J80
    • IR-65907
    • METIS-218214
    • EWI-3543

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