On a fractional linear birth-death process

In this paper, we introduce and examine a fractional linear birth-death process N(v)(t), t > 0, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities p(k)(v)(t) > 0, k >= 0. We present a subordination relationship connecting N(v)(t), t > 0, with the classical birth-death process N(t), t > 0, by means of the time process T(2v)(t), t > 0, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability p(0)(v)(t) and the state probabilities p(k)(v)(t), t > 0, k >= 1, in the three relevant cases lambda > mu, lambda < mu, lambda = mu (where lambda and mu are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth-death process with the fractional pure birth process. Finally, the mean values EN(v)(t) and Var N(v)(t) are derived and analyzed.