Rational arrival processes with strictly positive densities need not be Markovian

Abstract

Telek (2022) asked whether a rational arrival process (RAP), specified by matrices $G_0$ and $G_1$ and an initial row vector $\hat{I}½$, 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 $e^{\pm i\ddot{I}†}$, which cannot be peripheral eigenvalues of any finite nonnegative matrix when $\ddot{I}†/\ddot{I}€$ is irrational.