  The fractional Poisson process has recently attracted experts from several
fields of study. Its natural generalization of the ordinary Poisson process
made the model more appealing for real-world applications. In this paper, we
generalized the standard and fractional Poisson processes through the waiting
time distribution, and showed their relations to an integral operator with a
generalized Mittag--Leffler function in the kernel. The waiting times of the
proposed renewal processes have the generalized Mittag--Leffler and
stretched-squashed Mittag--Leffler distributions. Note that the generalizations
naturally provide greater flexibility in modeling real-life renewal processes.
Algorithms to simulate sample paths and to estimate the model parameters are
derived. Note also that these procedures are necessary to make these models
more usable in practice. State probabilities and other qualitative or
quantitative features of the models are also discussed.
