Renewal Theory and its Ramifications

TLDR

This expository article surveys theoretical and computational aspects of renewal theory, including its generalizations and applications to biology, queues, dams, and electronic counters. The paper develops the theory through a sequence of sections covering foundational results (Blackwell’s theorem, renewal density, asymptotic normality), asymptotic behavior of regenerative and semi‑Markov processes, infinite sums and products arising in electronic particle counters, and various proposed generalizations including infinitesimal renewal processes. Illustrative examples from biology, queueing, dam, and electronic counter models are provided, along with a summary table of related literature on electronic counters.

Abstract

Summary This is an expository article on the theoretical and some computational aspects of renewal theory. §1 describes basic theory, including Blackwell’s theorem; renewal density theorem; cumulants and asymptotic normality of the number of renewals in (0, t); and the integral equations of renewal theory. §2 is concerned with asymptotic behaviour of processes having an embedded renewal process: regenerative stochastic processes; semi-Markov processes; cumulative processes. §3 discusses infinite sums and products connected with a renewal process which arise out of the study of electronic particle counters, and an integral equation connected with the infinite products. §4 describes some generalizations of renewal theory that have been proposed by different writers, and also “infinitesimal” renewal processes. Illustrative examples are drawn from biology and from the theories of queues, dams, and electronic counters; a table summarizing some of the papers on the last subject is given.