Geometrical time-changed Poisson processes and their extensions

In this paper, we study generalized versions of Poisson-type processes obtained by time-changing them with the geometric counting process (GCP). Our work generalizes the results of Meoli (2023). We introduce the geometric time-changed Poisson processes of Type I (GTCPP-I) and Type II (GTCPP-II), the geometric generalized compound, and multiplicative Poisson processes for both types. We derive several distributional properties for the above-mentioned processes and study special cases, including the geometric space-fractional Poisson process, the geometric tempered space-fractional Poisson process, and the geometric time-fractional Poisson process. We also analyze the asymptotic behavior of the correlation structure. Finally, we discuss applications in shock models in reliability theory, deriving explicit reliability functions and failure rates under both the extreme shock model and the cumulative shock model.