  The aim of this paper is the analysis of the fractional Poisson process where
the state probabilities $p_k^{\nu_k}(t)$, $t\ge 0$, are governed by
time-fractional equations of order $0<\nu_k\leq 1$ depending on the number $k$
of events occurred up to time $t$. We are able to obtain explicitely the
Laplace transform of $p_k^{\nu_k}(t)$ and various representations of state
probabilities. We show that the Poisson process with intermediate waiting times
depending on $\nu_k$ differs from that constructed from the fractional state
equations (in the case $\nu_k = \nu$, 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.
