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Survival analysis, also known as failure time analysis or time-to-event analysis, is one of the most significant advancements of mathematical statistics in the last quarter of the 20th century. It has become the de facto standard in biomedical data analysis. Although reliability was conceived as a major application field by the mathematicians who pioneered survival analysis, survival analysis failed to establish itself as a major tool for reliability analysis. In this paper, we attempt to demonstrate, by reviewing and comparing the major mathematical models of both fields, that survival analysis and reliability theory essentially address the same mathematical problems. Therefore, survival analysis should become a major mathematical tool for reliability analysis and related fields such as Prognostics and Health Management (PHM). This paper is the first in a four part series in which we review state-of-the-art studies in survival (univariate) analysis, competing risks analysis, and multivariate survival analysis, with focusing on their applications to reliability and computer science. The present article discusses the univariate survival analysis (survival analysis hereafter).