On Time-Changed Linear Birth-Death Process with Immigration at Extinction

TL;DR

This paper introduces a novel time-fractional linear birth-death process with immigration at extinction, deriving its probabilities and moments, and analyzing its distributional properties to understand its behavior.

Contribution

It presents the first analysis of a time-fractional birth-death process with extinction-based immigration, providing explicit transient probabilities and moments.

Findings

Transient probabilities obtained via Adomian decomposition

First two moments derived explicitly

Distributional properties analyzed in special cases

Abstract

We study an immigration effect in the time-changed linear birth-death process where the immigration occurs only if the population goes extinct. We call this process as the time-fractional linear birth-death process with immigration (TFLBDPwI). Its transient probabilities are obtained using the Adomian decomposition method. In a particular case, we discuss the accuracy of approximation for the obtained transient probabilities. Also, the first two moments of TFLBDPwI are derived. Later, we consider some particular cases of the TFLBDPwI and study their distributional properties in detail.