A fractional generalization of the Poisson processes

TL;DR

This paper introduces a fractional calculus-based generalization of Poisson processes, extending renewal theory to include non-Markovian processes with heavy-tailed waiting times modeled by Mittag-Leffler functions.

Contribution

It develops a fractional renewal process framework using Mittag-Leffler waiting times, generalizing classical Poisson and compound Poisson processes with non-Markovian features.

Findings

Mittag-Leffler distribution generalizes exponential waiting times.

Non-Markovian renewal process with power-law tail waiting times.

Framework applicable to renewal processes with rewards and subordinated random walks.

Abstract

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the Mittag-Leffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.