An asymptotic solution of the integral equation for the second moment function in geometric processes

In this study, we derive an asymptotic solution of the integral equation satisfied by the second moment function [Formula: see text]. We first find the Laplace transform [Formula: see text] and then obtain [Formula: see text] asymptotically by inversion. Further, we have derived the asymptotic expressions of [Formula: see text] for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Finally, the asymptotic solution is compared with the numerical solution to evaluate its performance.