Tunable approximations for the mean and variance of the maximum of heterogeneous geometrically distributed random variables

Analysis of the maximum of n independent geometrically distributed random variables arises in a variety of applications in computer science and engineering. Evaluating the mean and variance of the maximum when n is large presents considerable computational challenges. Although approximate formulas have been proposed in the case where each geometric distribution has the same probability of success, the heterogeneous case has not received any attention. We derive an epsilon-accurate approximation for both the mean and the variance in the heterogeneous case. The approximations also apply to the homogeneous case, and offer something new with their ability to tune the approximation to any desired level of accuracy. We illustrate the formulas with a reliability application where the heterogeneous context arose quite naturally.