On a fractional linear birth - Death process

Abstract

In this paper, we introduce and examine a fractional linear birth - death process Nv(t), t > 0, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of differencedifferential equations governing the state probabilities p kv(t), t > 0, k ≥ 0. We present a subordination relationship connecting Nv (t), t > 0, with the classical birth - death process N(t), t > 0, by means of the time process T2v (t), t > 0, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability p0v (t) and the state probabilities pkv (t), t > 0, k ≥ 1, in the three relevant cases λ>μ, λ