Optimization-Free Concentrated Matrix-Exponentials

TL;DR

This paper introduces an explicit family of matrix-exponential distributions that surpass the Erlang variance limit for positive delays, using a novel analytical approach without numerical optimization.

Contribution

It provides the first analytical construction of concentrated matrix-exponential densities that asymptotically exceed the Erlang bound, using a closed-form, explicit family.

Findings

First analytical proof of a matrix-exponential class surpassing Erlang variance limit.

Explicit family of densities obtained by raising the Fejér kernel to a logarithmic power.

Provides exact moments and closed-form parameters for the new distributions.

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.