Renewal processes based on generalized Mittag--Leffler waiting times

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.

References in corpus (5)

• Fractional processes: from Poisson to branching one
• A fractional generalization of the Poisson processes
• Renewal processes of Mittag-Leffler and Wright type
• Simulation and estimation for the fractional Yule process
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• On Discrete Time Prabhakar-Generalized Fractional Poisson Processes and Related Stochastic Dynamics
• Biased continuous-time random walks with Mittag-Leffler jumps
• Generalized Nonlinear Yule Models
• Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process
• A fractional generalization of the Dirichlet distribution and related distributions
• Prabhakar discrete-time generalization of the time-fractional Poisson process and related random walks