Renewal processes of Mittag-Leffler and Wright type

TL;DR

This paper explores renewal processes based on Mittag-Leffler and Wright functions, comparing them to the classical Poisson process and analyzing their long-term behavior through numerical methods.

Contribution

It introduces and studies renewal processes of Mittag-Leffler and Wright types, highlighting their differences from the classical Poisson process and analyzing their long-time dynamics.

Findings

Mittag-Leffler and Wright renewal processes differ from Poisson in waiting time distributions

Numerical analysis reveals distinct long-term behaviors of these processes

Comparison clarifies the role of special functions in renewal theory

Abstract

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.