Renewal processes of Mittag - Leffler and Wright type

After sketching the basic principles of renewal theory we discuss the classical Poisson process and offer two other processes, namely the renewal process of Mittag-Leffler type and the renewal process of Wright type, so named by us because special functions of Mittag-Leffler and of Wright type appear in the definition of the relevant waiting times. We compare these three processes with each other, furthermore consider corresponding renewal processes with reward and numerically their long-time behaviour. It is well known that the Poisson process (with and without reward) plays a fundamental role in renewal theory. In this paper, by means of functions of Mittag-Leffler and Wright type we provide a generalization of and a variant to this classical process and construct interesting subordinated stochastic processes of fractional diffusion. The plan of the paper is as follows.