An integral equation for the second moment function of a geometric process and its numerical solution

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Abstract
In this article, an integral equation satisfied by the second moment function M2(t) of a geometric process is obtained. The numerical method based on the trapezoidal integration rule proposed by Tang and Lam for the geometric function M(t) is adapted to solve this integral equation. To illustrate the numerical method, the first interarrival time is assumed to be one of four common lifetime distributions, namely, exponential, gamma, Weibull, and lognormal. In addition to this method, a power series expansion is derived using the integral equation for the second moment function M2(t), when the first interarrival time has an exponential distribution.

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