On a fractional linear birth--death process

In this paper, we introduce and examine a fractional linear birth–death process $N_{\nu }(t)$, $t>0$, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^{\nu }(t)$, $t>0$, $k\ge 0$. We present a subordination relationship connecting $N_{\nu }(t)$, $t>0$, with the classical birth–death process $N(t)$, $t>0$, by means of the time process $T_{2\nu }(t)$, $t>0$, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^{\nu }(t)$ and the state probabilities $p_k^{\nu }(t)$, $t>0$, $k\ge 1$, 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 $\mathbb{E}{N}_{\nu }(t)$ and $\mathrm{Var}{N}_{\nu }(t)$ are derived and analyzed.