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The study of a system's reliability has played a crucial role in business and industry since the dawn of modern technology. Current graphical models utilized in reliability theory are limited in that no one model or technique allows for a thorough analysis of system reliability. This research introduces a new graphical model and methodology to be used in the field of reliability that addresses this concern. Event Occurrence Networks (EONs) and their solution methodologies provide an all-inclusive graphical model that allows for the manipulation of several important reliability measures. An EON is a probabilistic network that represents the superposition of several terminating counting processes and is an efficient tool in both non-repairable and repairable systems. Current methodologies are also restricted in the distributions that characterize component life and repair times. This concern is alleviated via EONs coupled with piecewise polynomial approximation.