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We explore the tail of various waiting time datasets of processes that follow a nonstationary Poisson distribution with a sinusoidal driver. Analytically, we find that the distribution of large waiting times of such processes can be described using a power law slope of -2.5. We show that this result applies more broadly to any nonstationary Poisson process driven periodically. Examples of such processes include solar flares, coronal mass ejections, geomagnetic storms, and substorms. We also discuss how the power law specifically relates to the behavior of driver near its minima.