Finite‐Horizon Behavior of the Periodically Observed Time‐Homogeneous Poisson Process

ABSTRACT When evaluating whether system arrivals can be characterized as a Poisson process, researchers draw conclusions from observed data. In practice, many systems are observed periodically (e.g., hourly, daily, monthly), which leads to discretization errors in the observed data. The periodically observed time‐homogeneous Poisson process (PTPP) model is an existing framework for capturing the long‐run effect of periodic observation on a Poisson process. This paper extends the PTPP model to capture the behavior of periodically observed Poisson systems over a finite number of events . This paper formulates the ‐step transition probabilities in an infinite‐state PTPP system and uses them to determine the PTPP's mixing time. The PTPP's interarrival distribution and the quality of its arrival rate moment estimator are evaluated as a function of . After extending the PTPP model for finite , the model's applicability is demonstrated through a case study of search‐and‐rescue operations in New York wilderness regions. The results suggest that five regions behave similarly to daily‐observed Poisson processes, which can inform practical resourcing decisions over extended periods (e.g., days, weeks, quarters).