Estimation of the expected value of the arrival epochs given a bounded period of observation for the homogeneous and non-homogeneous Poisson processes

In this paper, we derive analytical expressions of the expected value of the arrival epochs given a bounded period of observation for the homogeneous Poisson process (HPP) and for the general case of a non-homogeneous Poisson process (NHPP). These expressions are exact, but their applicability require some computational adaptations. We illustrate the results obtained through the proposed expression by comparing them with Monte Carlo sampling. The results show that these expressions outperform Monte Carlo in precision and computational effort.