Chapter 7 - Renewal Phenomena

Abstract

This chapter reviews renewal phenomena. Renewal theory began with the study of stochastic systems whose evolution through time was interspersed with renewals or regeneration times when, in a statistical sense, the process began anew. A renewal (counting) process {N(t), t ≥ 0} is a nonnegative integer-valued stochastic process that registers the successive occurrences of an event during the time interval (0, t), where the time durations between consecutive events are positive, independent, identically distributed random variables. Presently, the subject is viewed as the study of general functions of independent, identically distributed, nonnegative random variables representing the successive intervals between renewals. The results are applicable in a wide variety of both theoretical and practical probability models. The Poisson process with parameter λ is a renewal process; the Poisson process is the only renewal process (in continuous time) whose renewal function is exactly linear. A large number of the functionals that have explicit expressions for Poisson renewal processes are far more difficult to compute for other renewal processes. There are, however, many simple formulas that describe the asymptotic behavior. Thereafter, a delayed renewal process will arise when the component in operation at time t = 0 is not new, but all subsequent replacements are new. The renewal theorem provides conditions under which the solution {Vn} to a renewal equation will converge as n grows large; certain periodic behavior must, thus, be precluded.