The Markov modulated Poisson process (MMPP) extends the classical Poisson process by allowing the arrival intensity to evolve according to an underlying continuous–time Markov chain, thus capturing regime-switching behavior and temporal dependence. We study a 2-state Markov modulated Poisson process Nt and provide explicit expressions for the probability distribution by making use of probability generating function techniques. The limiting and asymptotic behavior of the state probabilities are also analyzed, providing insight into the role of switching intensities and transition rates. In particular, limiting regimes are examined, revealing connections with the standard Poisson process and highlighting structural transitions in the distributional behavior. We address stationary and interval-stationary versions of the process, and introduce time-changed versions of Nt obtained through different subordinators, including Poisson, Gamma and a-stable. These extensions are consistent with broader Markov-modulated Poisson modeling frameworks. Shock models driven by a MMP process provide a natural and effective framework for applications in which systems accumulate damage or experience failures at rates influenced by an unobservable or fluctuating environmental regime. We investigate both extreme and cumulative shock models driven by Nt, in line with recent developments on shock processes governed by mixed Poisson dynamics. In particular, in the cumulative shock model, system failure is assumed to occur when the total damage produced by successive shocks exceeds a threshold, which is assumed to be either deterministic or exponentially distributed. We provide analytical formulas for the failure rate function, whose monotonic decreasing behavior is discussed, as well as closed-form expressions of the mean and variance of the lifetime distribution.