Renewal processes based on generalized Mittag--Leffler waiting times

TL;DR

This paper introduces generalized renewal processes with Mittag--Leffler waiting times, expanding the fractional Poisson process framework for better modeling of real-world phenomena, and provides algorithms for simulation and parameter estimation.

Contribution

It generalizes the fractional Poisson process using Mittag--Leffler distributions and develops algorithms for simulation and parameter estimation, enhancing practical applicability.

Findings

Derived algorithms for simulation and parameter estimation.

Analyzed state probabilities and qualitative features.

Established relations to integral operators with Mittag--Leffler kernels.

Abstract

The fractional Poisson process has recently attracted experts from several fields of study. Its natural generalization of the ordinary Poisson process made the model more appealing for real-world applications. In this paper, we generalized the standard and fractional Poisson processes through the waiting time distribution, and showed their relations to an integral operator with a generalized Mittag--Leffler function in the kernel. The waiting times of the proposed renewal processes have the generalized Mittag--Leffler and stretched-squashed Mittag--Leffler distributions. Note that the generalizations naturally provide greater flexibility in modeling real-life renewal processes. Algorithms to simulate sample paths and to estimate the model parameters are derived. Note also that these procedures are necessary to make these models more usable in practice. State probabilities and other qualitative or quantitative features of the models are also discussed.