Fractional Poisson Process on Quantum Time Scale with Applications to Practical Data

Abstract

A q−fractional generalization of the Poisson processes have been provided by replacing the first time derivative in the relaxation equation of the survival probability by a q−fractional derivative of order (0 < ≤ 1). For 0 < < 1, 1/q−renewal processes have been obtained where, the 1/q −exponential probability densities, typical for the 1/q−Poisson processes, are replaced by functions of 1/q−Mittag–Leffler type, that decay in a power law manner with an exponent related to . The distributions obtained by considering the 1/q−sum of k independent identically distributed random variables distributed according to the 1/q−Mittag-Leffler law provide the q−fractional generalization of the corresponding 1/q−Erlang distributions.
Two fitting scenarios are built on a data set including the records of serious earthquakes in Turkey. The first fitting scenario compares the 1/q−Poisson distribution with the Poisson and the negative binomial distributions. In the second fitting scenario, the 1/q−Erlang and the Erlang distributions are discussed. The presented results suggest that the 1/q−Poisson and the 1/q−Erlang are the more adequate choice for the observed data than the other discussed distributions. The parameters of the 1/q−Poisson and the 1/q−Erlang distributions are estimated by maximum likelihood method.