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In this paper we analyse the fractional Poisson process where the state probabilities (t), ... (t), t≥ = 0, are governed by time-fractional equations of order 0 < vk ≤ 1 depending on the number of events that have occurred up to time t. We are able to obtain explicitly the Laplace transform of ... and various representations of state probabilities. We show that the Poisson process with intermediate waiting times depending on vk differs from that constructed from the fractional state equations (in the case of vk = v, for all k, they coincide with the time-fractional Poisson process). We also introduce a different form of fractional state-dependent Poisson process as a weighted sum of homogeneous Poisson processes. Finally, we consider the fractional birth process governed by equations with state-dependent fractionality. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a fractional version of non-linear [InlineEquation not available: see fulltext.], t>0, linear M( t), t>0 and sublinear $\mathfrak{M}^{\nu}(t)$, t>0 death processes. Fractionality is introduced by replacing the usual integer-order derivative in the difference-differential equations governing the state probabilities, with the fractional derivative understood in the sense of Dzhrbashyan–Caputo. We derive explicitly the state probabilities of the three death processes and examine the related probability generating functions and mean values. A useful subordination relation is also proved, allowing us to express the death processes as compositions of their classical counterparts with the random time process T( t), t>0. This random time has one-dimensional distribution which is the folded solution to a Cauchy problem of the fractional diffusion equation. [ABSTRACT FROM AUTHOR]