Optimization-Free Concentrated Matrix-Exponentials

Abstract

Near-deterministic positive delays require highly concentrated distributions, but phase-type models are constrained by the Erlang variance limit. While matrix-exponential distributions can empirically bypass this barrier, prior low-variance constructions relied entirely on numerical optimization. We propose an explicit family of concentrated matrix-exponential densities for the unit delay, obtained by raising the trigonometric Fej\'er kernel to logarithmic power. With exact moments and closed-form parameters, this gives the first analytical proof of a matrix-exponential class that asymptotically surpasses the Erlang bound.