On a fractional linear birth--death process

Abstract

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 \(p_k^{\nu}(t)\), \(t>0\), \(k\geq0\). 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 \(p_k^{\nu}(t)\), \(t>0\), \(k\geq1\), 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 \(\operatorname {\mathbb{V}ar}N_{\nu}(t)\) are derived and analyzed.