Construction of Phase Type Distributions by Bernstein Exponentials.

Analysis of stochastic models is often hurdled by the complexity of probability density functions associated with generally distributed random variables. Among the approximation techniques that can be employed to reduce computational complexity, Bernstein polynomials (BP) exhibit some properties that make them suitable for the needs of that particular context. However, they also show some drawbacks; notably, their application is limited to bounded supports. We introduce Bernstein exponentials (BE) by transforming BP so as to enable approximation over unbounded intervals. We show that BE form a subclass of acyclic phase type (PH) distributions, thus possessing a well defined stochastic interpretation. The characteristics of this subclass allows for efficient analysis in the context of M/PH/1 queues. In particular, we develop a technique to calculate the queue length distribution in case of BE service time with linear time complexity in the number of phases of the service time distribution. Finally, we experiment BE approximations in distribution fitting and in the analysis M/G/1 queues.