Computational approximations of renewal process relating to a warranty problem: The case of phase-type lifetimes

This paper considers two product warranty policies proposed originally by Nguyen and Murthy in 1984. With the exception of exponential lifetimes, computing expected costs under the two policies is in general complicated. In 1988, Frees and Nam proposed an approach based on a straight line approximation of the related renewal function and showed that their approach yielded satisfactory results. In this paper we study the case in which the lifetime distribution is of phase type. We obtain simplifications by exploiting various properties of phase-type distributions. For lifetime distribution of moderate variability, our empirical observations support those of Frees and Nam. By examining the behavior of the renewal function, we offer an explanation as to why their approach works. However, when the variability of product lifetime is large and the life cycle is small, we demonstrate that the results obtained from straight line approximation is inaccurate and one of the two more accurate approaches should be used instead.