Poisson-type processes governed by fractional and higher-order recursive differential equations

TL;DR

This paper extends the Poisson process using fractional and higher-order differential equations, revealing new renewal processes with Mittag-Leffler interarrival times that exhibit power-law decay and parameter-dependent behaviors.

Contribution

It introduces fractional and higher-order recursive differential equations for Poisson processes, linking them to generalized Mittag-Leffler functions and analyzing their renewal properties.

Findings

Interarrival times follow Mittag-Leffler distributions with power-law decay.

Behavior near zero varies with fractional parameter in (0,1).

Integer parameter models relate to higher-order Poisson processes.

Abstract

We consider some fractional extensions of the recursive differential equation governing the Poisson process, by introducing combinations of different fractional time-derivatives. We show that the so-called "Generalized Mittag-Leffler functions" (introduced by Prabhakar [20]) arise as solutions of these equations. The corresponding processes are proved to be renewal, with density of the intearrival times (represented by Mittag-Leffler functions) possessing power, instead of exponential, decay, for t tending to infinite. On the other hand, near the origin the behavior of the law of the interarrival times drastically changes for the parameter fractional parameter varying in the interval (0,1). For integer values of the parameter, these models can be viewed as a higher-order Poisson processes, connected with the standard case by simple and explict relationships.