On a fractional alternating Poisson process

Abstract

We propose a generalization of the alternating Poisson process from the point of view offractional calculus. We consider the system of differential equations governing the state probabilitiesof the alternating Poisson process and replace the ordinary derivative with the fractional derivative (inthe Caputo sense). This produces a fractional 2-state point process. We obtain the probability massfunction of this process in terms of the (two-parameter) Mittag-Leffler function. Then we show thatit can be recovered also by means of renewal theory. We study the limit state probability, and certainproportions 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 atransformation acting on pairs of stochastically ordered random variables, which is an extension of theequilibrium operator and deserves interest in the analysis of alternating stochastic processes.