Fractional multiple birth–death processes with birth probabilities λi(Δt)α+o((Δt)α)

The usual model for (Poissonian) linear birth–death processes is extended to multiple birth–death processes with fractional birth probabilities in the form λi(Δt)α+o((Δt)α, 0<α<1. The probability generating function for the time dependent population size is provided by a fractional partial differential equation. The solution of the latter is obtained and comparison with the usual model is made. The probability of ultimate extinction is obtained. One considers the special case of fractional Poissonian processes with individual arrivals only, and then one outlines basic results for continuous processes defined by fractional Poissonian noises. The key is the Taylor’s series of fractional order f(x+h)=Eα(hαDxα)f(x), where Eα(·) is the Mittag–Leffler function, and Dxα is the modified Riemann–Liouville fractional derivative, as previously introduced by the author.