On Transient Probabilities of Fractional Birth-Death Process

TL;DR

This paper introduces a fractional birth-death process with state-dependent rates, deriving explicit transient probabilities using fractional differential equations and the Adomian decomposition method, extending classical results and analyzing distributional properties.

Contribution

It provides closed-form expressions for transient probabilities of a fractional birth-death process, generalizing classical models and exploring their distributional characteristics.

Findings

Derived explicit transient probabilities using Adomian decomposition

Verified results for linear birth-death rates against existing literature

Analyzed the asymptotic behavior of extinction times in time-changed processes

Abstract

We study a fractional birth-death process with state dependent birth and death rates. It is defined using a system of fractional differential equations that generalizes the classical birth-death process introduced by Feller (1939). We obtain the closed form expressions for its transient probabilities using Adomian decomposition method. In this way, we obtain the unknown transient probabilities for the classical birth-death process (see Feller (1968), p. 454). Its various distributional properties are studied. For the case of linear birth and death rates, the obtained results are verified with the existing results. Also, we discuss the cumulative births in the fractional linear birth-death process. Later, we consider a time-changed linear birth-death process where we discuss the asymptotic behaviour of the distribution function of its extinction time at zero.