  The fractional Poisson process is a renewal process with Mittag-Leffler
waiting times. Its distributions solve a time-fractional analogue of the
Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent
inverse stable subordinator, is also a fractional Poisson process. This result
unifies the two main approaches in the stochastic theory of time-fractional
diffusion equations. The equivalence extends to a broad class of renewal
processes that include models for tempered fractional diffusion, and
distributed-order (e.g., ultraslow) fractional diffusion. The paper also
establishes an interesting connection between the fractional Poisson process
and Brownian time.
