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

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Abstract
Under a free‐replacement warranty of durationW, the customer is provided, for an initial cost ofC, as many replacement items as needed to provide service for a periodW. Payments ofCare not made at fixed intervals of lengthW, but in random cycles of lengthY = W + γ(W), whereγ(W)is the (random) remaining life‐time of the item in serviceWtime units after the beginning of a cycle. The expected number of payments over the life cycle,L, of the item is given byMY(L), the renewal function for the random variableY. We investigate this renewal function analytically and numerically and compare the latter with known asymptotic results. The distribution ofY, 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.

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