On the first-visit-time problem for birth and death processes with catastrophes

TL;DR

This paper analyzes the first-visit-time problem in birth-death processes with catastrophes, deriving Laplace transforms, moments, and extensions to non-homogeneous cases, with applications to specific processes.

Contribution

It provides explicit Laplace transforms and moments for first-visit times and catastrophe occurrences in birth-death processes, including non-homogeneous extensions.

Findings

Laplace transform of first-visit-time density derived

Mean and variance of first-visit times calculated

Extensions to time-non-homogeneous processes included

Abstract

For a birth-death process subject to catastrophes, defined on the state-space $S=\{r,r+1,r+2,...\}$, with $r$ a positive integer or zero, the first-visit time to a state $k\in S$ is considered and the Laplace transform of its probability density function is determined, use of which is then made to obtain mean and variance. The Laplace transform of the probability density function of the first effective catastrophe occurrence time and its expected value are also obtained. Some extensions to time-non-homogeneous processes are then provided. Finally, certain additional results concerning the determination of the steady-state distribution and the representation of the transition probabilities are worked out, while some applications to particular birth-death processes are shown in the Appendix.