  We introduce and study here a renewal process defined by means of a
time-fractional relaxation equation with derivative order $\alpha(t)$ varying
with time $t\geq0$. In particular, we use the operator introduced by Scarpi in
the Seventies and later reformulated in the regularized Caputo sense in
Garrappa et al. (2021), inside the framework of the so-called general
fractional calculus. The model obtained extends the well-known time-fractional
Poisson process of fixed order $\alpha \in (0,1)$ and tries to overcome its
limitation consisting in the constancy of the derivative order (and therefore
of the memory degree of the interarrival times) with respect to time. The
variable order renewal process is proved to fall outside the usual subordinated
representation, since it can not be simply defined as a Poisson process with
random time (as happens in the standard fractional case). Finally a related
continuous-time random walk model is analysed and its limiting behavior
established.
