# Extinction probability in a birth-death process with killing

Erik A. van Doorn, A.I. Zeifman

Research output: Book/ReportReportProfessional

## 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 language Undefined Enschede University of Twente, Faculty of Mathematical Sciences 20 Published - 2004

### Publication series

Name Memorandum Faculty of Mathematical Sciences Department of Applied Mathematics, University of Twente 1723 0169-2690

## Keywords

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