On a Fractional Variant of Linear Birth-Death Process

TL;DR

This paper introduces a new fractional variant of the linear birth-death process using advanced fractional derivatives, deriving explicit formulas for its probabilities, mean, variance, and exploring its limiting behavior and applications.

Contribution

It proposes the generalized fractional linear birth-death process (GFLBDP) with explicit state probabilities and links to fractional Poisson processes, extending classical models with fractional calculus.

Findings

Derived explicit state probabilities for GFLBDP

Established the relation between extinction probability and fractional Poisson process

Analyzed asymptotic distributional characteristics and applications

Abstract

We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP) which is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed.