On a fractional alternating Poisson process

We propose a generalization of the alternating Poisson process from the point of view of fractional calculus. We consider the system of differential equations governing the state probabilities of the alternating Poisson process and replace the ordinary derivative with the fractional derivative (in the Caputo sense). This produces a fractional 2-state point process. We obtain the probability mass function of this process in terms of the (two-parameter) Mittag-Leffler function. Then we show that it can be recovered also by means of renewal theory. We study the limit state probability, and certain proportions involving the fractional moments of the sub-renewal periods of the process. In conclusion, in order to derive new Mittag-Leffler-like distributions related to the considered process, we exploit a transformation acting on pairs of stochastically ordered random variables, which is an extension of the equilibrium operator and deserves interest in the analysis of alternating stochastic processes.