Geodesic
Renewal Processes of Mittag-Leffler and Wright Type
Fractional calculus and applied analysis, Tome 8 (2005) no. 1, pp. 07-38.

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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.
Keywords: Fractional Derivative, Mittag-Leffler Function, Wright Function, Renewal Theory, Poisson Process, Fractional Diffusion, 26A33, 33E12, 33E20, 44A10, 44A35, 60G50, 60J05, 60K05 
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Mainardi, Francesco; Gorenflo, Rudolf; Vivoli, Alessandro. Renewal Processes of Mittag-Leffler and Wright Type. Fractional calculus and applied analysis, Tome 8 (2005) no. 1, pp. 07-38. http://geodesic.mathdoc.fr/item/FCAA_2005_8_1_a0/