Modeling service-time distributions with non-exponential tails: Beta mixtures of exponentials

Motivated by interest in probability density functions (pdfs) with nonexponential tails in queueing and related areas, we introduce and investigate two classes of beta mixtures of exponential pdfs. These classes include distributions introduced by Boxma and Cohen and Gaver and Jacobs to study queues with long-tail service-time distributions. When the standard beta pdf is used as the mixing pdf, we obtain pdfs with an exponentially damped power tail. This pdf decays exponentially, but analysis is complicated by the power term. When the beta pdf of the second kind is used as the mixing pdf, we obtain pdfs with a power tail. We obtain explicit representations for the cumulative distributions functions, Laplace transforms, moments and asymptotics by exploiting connections to the Tricomi function. Properties of the power-tail class can be deduced directly from properties of the other class, because the power-tail pdfs are undamped versions of the other pdfs. The power-tail class can also be represented as gamma mixtures of Pareto pdfs. Both classes of pdfs have simple explicit Lageurre-series expansions.