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On Transient Probabilities of Fractional Birth-Death Process

K. K. Kataria, P. Vishwakarma

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.

On Transient Probabilities of Fractional Birth-Death Process

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.

Paper Structure

This paper contains 11 sections, 9 theorems, 146 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Subordinator and its first hitting time
  4. Adomian decomposition method (ADM)
  5. Fractional birth-death process
  6. Application of ADM to FBDP
  7. FBDP with linear rates
  8. Cumulative births in linear FBDP
  9. Asymptotic distribution of cumulative birth process
  10. Extinction time of a time-changed linear BDP
  11. Extinction times of time-changed linear BDP

Key Result

Proposition 3.1

For $t\ge0$ and $n\ge k+2$, the series component $p_k^\alpha(n,t)$ defined by (adm2) vanishes, that is, $p_k^\alpha(n,t)=0$ given $n-k\ge2$.

Figures (1)

  • Figure 1: Correlation coefficient of $\tilde{N}^\alpha(t)$ and $\tilde{B}^\alpha(t)$ for different values of $\alpha$

Theorems & Definitions (23)

  • proof
  • Definition 3.1
  • Proposition 3.1
  • proof
  • proof
  • Proposition 3.2
  • proof
  • proof
  • Corollary 3.1
  • Proposition 3.3
  • ...and 13 more