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EnglishIn this paper we introduce and examine a fractional linear birth-death process $N_\nu \left( t \right)$, $t>0$, whose fractionality is obtained by replacing the time-derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^\nu \left( t \right)$, $t>0$, $k \geq 0$. We present a subordination relationship connecting $N_\nu \left( t \right)$, $t>0$ with the classical birth-death process $N \left( t \right)$, $t>0$ by means of the time process $T_{2 \nu} \left( t \right)$, $t>0$ whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^\nu \left( t \right)$, and the state probabilities $p_k^\nu \left( t \right)$, $t>0$, $k \geq 1$ in the three relevant cases $\lambda > \mu$, $\lambda < \mu$, $\lambda = \mu$ (where $\lambda$ and $\mu$ are respectively the birth and the 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 $\mathbb{E} N_\nu \left( t \right)$ and $\mathbb{V}\!\text{ar}N_\nu \left( t \right)$ are derived and analysed.
| Titolo: | On a fractional linear birth–death process |
| Autori Riconosciuti: | |
| Autori: | E. ORSINGHER;F. POLITO |
| Data di pubblicazione: | 2011 |
| Abstract: | In this paper we introduce and examine a fractional linear birth-death process $N_\nu \left( t \right)$, $t>0$, whose fractionality is obtained by replacing the time-derivative with a fractional derivative in the system of difference-differential equations governing the state probabilities $p_k^\nu \left( t \right)$, $t>0$, $k \geq 0$. We present a subordination relationship connecting $N_\nu \left( t \right)$, $t>0$ with the classical birth-death process $N \left( t \right)$, $t>0$ by means of the time process $T_{2 \nu} \left( t \right)$, $t>0$ whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_0^\nu \left( t \right)$, and the state probabilities $p_k^\nu \left( t \right)$, $t>0$, $k \geq 1$ in the three relevant cases $\lambda > \mu$, $\lambda < \mu$, $\lambda = \mu$ (where $\lambda$ and $\mu$ are respectively the birth and the 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 $\mathbb{E} N_\nu \left( t \right)$ and $\mathbb{V}\!\text{ar}N_\nu \left( t \right)$ are derived and analysed. |
| Volume: | 17 |
| Fascicolo: | 1 |
| Pagina iniziale: | 114 |
| Pagina finale: | 137 |
| Digital Object Identifier (DOI): | 10.3150/10-BEJ263 |
| URL: | http://arxiv.org/pdf/1102.1620.pdf http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bj/1297173835 |
| Parole Chiave: | Extinction Probabilities; Fractional Derivatives; Fractional Diffusion Equations; Generalised Birth-Death Process; Iterated Brownian Motion; Mittag-Leffler Functions. |
| Rivista: | BERNOULLI |
| Appare nelle tipologie: | 03A-Articolo su Rivista |
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http://hdl.handle.net/2318/92916
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