  The space-fractional and the time-fractional Poisson processes are two
well-known models of fractional evolution. They can be constructed as standard
Poisson processes with the time variable replaced by a stable subordinator and
its inverse, respectively. The aim of this paper is to study non-homogeneous
versions of such models, which can be defined by means of the so-called
multistable subordinator (a jump process with non-stationary increments),
denoted by H. Firstly, we consider the Poisson process time-changed by H and we
obtain its explicit distribution and governing equation. Then, by using the
right-continuous inverse of H, we define an inhomogeneous analogue of the
time-fractional Poisson process.
