Reliability Engineering and Services

6Renewal Theory and Superimposed Renewal

6.1 Introduction

The number of renewals provides valuable information for spares inventory management, queuing analysis, maintenance planning, and warranty services. In this chapter, we introduce some basic theories and models about the renewal process and its applications. There are two different approaches to solving renewal function (RF): the analytical method and numerical computation. In Sections 6.2–6.4, we adopt Laplace transform to search the closed‐form solutions when the inter‐renewal times follow exponential, generalized exponential, or Erlang distribution. In Sections 6.5 and 6.6, we propose several approximation methods to compute the Weibull and gamma renewals based on the mixture of exponential functions and the sinc function. In Sections 6.7 and 6.8 we extend the renewal function to situations where the installed base of field products varies and are time‐dependent. Both the aggregate fleet renewal function and the lead‐time renewal model are explicitly derived under deterministic and stochastic fleet expansion. These models are built upon the superimposed renewal processes. In Section 6.9 we present a case study from the wind industry to illustrate the application of the superimposed renewal process.

6.2 Renewal Integral Equation

6.2.1 Overview of Renewal Solution Methods

The renewal theory was originally proposed by Feller (1941) in 1940s. Later it was further developed by Smith (1958) and Cox (1962). Since then, the ...