The fractional non-homogeneous Poisson process

TL;DR

This paper introduces a non-homogeneous fractional Poisson process by replacing the time variable with a function of time, deriving its governing equation, moments, and arrival time distribution, extending known homogeneous results.

Contribution

It presents a novel non-homogeneous fractional Poisson process and characterizes it through equations, moments, and distributions, generalizing previous homogeneous models.

Findings

Derived the non-local governing equation for the process.

Computed the first and second moments of the process.

Obtained the distribution of arrival times.

Abstract

We introduce a non-homogeneous fractional Poisson process by replacing the time variable in the fractional Poisson process of renewal type with an appropriate function of time. We characterize the resulting process by deriving its non-local governing equation. We further compute the first and second moments of the process. Eventually, we derive the distribution of arrival times. Constant reference is made to previous known results in the homogeneous case and to how they can be derived from the specialization of the non-homogeneous process.