{'fraction_non_alphanumeric': 0.02242152466367713, 'fraction_numerical': 0.0, 'mean_word_length': 4.826086956521739, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
An introduction to, wait for it...the Renewal Process

A renewal process counts the number of times that a system returns to its initial state, called a renewal. Renewal theory is the study of the waiting times and long-term behaviour of how these renewals occur. In this paper we begin by discussing the general theory of renewal processes and explore the distribution renewal counting process. We discuss the key renewal theorem and its implications. We also give variants of the renewal processes, including renewal reward, alternating renewal, and delayed renewal processes. Finally we explore in detail the inspection paradox and the distribution of waiting times.