Power series expansions for the probability distribution, mean value and variance functions of a geometric process with gamma interarrival times

Abstract

The geometric process is used widely as a stochastic monotone model in many areas since its introduction. In this study, this process is considered when the distribution of the first interarrival time follows a gamma distribution. One dimensional probability distribution of the process is obtained by expanding the convolution of gamma distributions into a power series. Further, its mean value, second moment and variance functions are derived as a power series expansion with the help of the integral equations given for mean value and second moment functions. The illustrative examples are also given. Finally, a real-world data set is considered to see the applicability of the results.