Rational arrival processes with strictly positive densities need not be Markovian

TL;DR

The paper provides a counterexample showing that rational arrival processes with positive densities are not necessarily equivalent to Markovian arrival processes, challenging a previous conjecture.

Contribution

It constructs a specific counterexample demonstrating the failure of the conjecture even under stronger normalization conditions.

Findings

Counterexample of order 3 disproves the conjecture.

Strict positivity of densities achieved via exponential damping.

Obstruction to MAP realizability linked to irrational rotation poles.

Abstract

Telek (2022) asked whether a rational arrival process (RAP), specified by matrices ${G}_0$ and ${G}_1$ and an initial row vector ${\nu}$, with strictly positive joint densities and a unique dominant real eigenvalue of ${G}_0$ must admit an equivalent Markovian arrival process (MAP). A counterexample of order $3$ is given, showing the answer is no, and that the conjecture fails even under the stronger condition of exact normalisation $({G}_0+{G}_1){1}={0}$. The construction combines a strictly positive exponential baseline with a two-dimensional correction driven by an irrational rotation. Strict positivity of all joint densities follows from the continuous-time damping of the correction block; the obstruction to MAP realisability comes from the poles of the boundary generating function at $e^{\pm i\varphi}$, which cannot be peripheral eigenvalues of any finite nonnegative matrix when $\varphi/\pi$ is irrational.