Applications of renewal theory in analysis of the free‐replacement warranty

TLDR

A free‑replacement warranty of duration W allows a customer to receive replacements for an initial cost C, with the expected number of payments over the item’s life cycle given by the renewal function M_Y(L) for the random cycle length Y = W + γ(W), which depends on the item’s failure distribution. The study analytically and numerically investigates the renewal function and compares it with known asymptotic results. Payments occur in random cycles of length Y = W + γ(W), where γ(W) is the remaining lifetime after W time units, and the analysis considers exponential, uniform, gamma, and Weibull failure distributions.

Abstract

Abstract Under a free‐replacement warranty of duration W , the customer is provided, for an initial cost of C , as many replacement items as needed to provide service for a period W . Payments of C are not made at fixed intervals of length W , but in random cycles of length Y = W + γ(W) , where γ(W) is the (random) remaining life‐time of the item in service W time units after the beginning of a cycle. The expected number of payments over the life cycle, L , of the item is given by M Y (L) , the renewal function for the random variable Y . We investigate this renewal function analytically and numerically and compare the latter with known asymptotic results. The distribution of Y , and hence the renewal function, depends on the underlying failure distribution of the items. Several choices for this distribution, including the exponential, uniform, gamma and Weibull, are considered.