The purpose of this text is to study abstract settings that generalize many of the basic concepts in classical reliability theory. Generalized graphs (consisting of a set and a binary relation) are the natural home for generalizations of the reliability function, failure rate function, and particularly constant rate distributions. Certain semigroups are the natural home for generalizations of the memoryless and exponential distributions. The two algebraic structures are closely related since there is a natural graph associated with a semigroup. Please read the Preface for mathematical prerequisites, notational conventions, and some basic assumptions, and read the Introduction for a review of the classical theory that motivates this work.
The text is divided into two basic parts. The first part gives the basic definitions and properties of the two fundamental structures, graphs and semigroups, basic probability and random walks on the two structures, and exponential, memoryless, and constant rate distributions. Part I also gives some basic constructions for creating new graphs and semigroups from existing ones. These include various product structures, a quotient structure, and graphs partitioned by an underlying graph. The second part studies a number of applications and examples in detail. The applications include the standard discrete and continuous spaces, rooted trees, free semigroups on countable alphabets, arithmetic semigroups including the positive integers under mulitplication, and semigroup and graph structures on finite subsets of positive integers.
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