Modeling Service-time Distributions with Non-exponential Tails: Beta Mixtures of Exponentials

Motivated by interest in probability density functions (pdf’s) with nonexponential tails in queueing and related areas, we introduce and investigate two classes of beta mixtures of exponential pdf’s. These classes include distributions introduced by Boxma and Cohen (1997) and Gaver and Jacobs (1998) to study queues with long-tail service-time distributions. When the standard beta pdf is used as the mixing pdf, we obtain pdf’s with an exponentially damped power tail, i.e., f(t) ∼ αt−qe−ηt as t→∞. 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 pdf’s with a power tail, i.e., f(t) ∼ αt−q as t→∞. 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 pdf’s are undamped versions of the other pdf’s. The power-tail class can also be represented as gamma mixtures of Pareto pdf’s. Both classes of pdf’s have simple explicit Laguerre-series expansions.