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| Format: | Preprint |
| Published: |
2026
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| Online Access: | https://arxiv.org/abs/2603.28047 |
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| _version_ | 1866917368597315584 |
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| author | Peralta, Oscar |
| author_facet | Peralta, Oscar |
| contents | Telek (2022) asked whether a rational arrival process (RAP), specified by matrices ${G}_0$ and ${G}_1$ and an initial row vector $ν$, 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 $e^{\pm iφ}$, which cannot be peripheral eigenvalues of any finite nonnegative matrix when $φ/π$ is irrational. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_28047 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rational arrival processes with strictly positive densities need not be Markovian Peralta, Oscar Probability Telek (2022) asked whether a rational arrival process (RAP), specified by matrices ${G}_0$ and ${G}_1$ and an initial row vector $ν$, 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 $e^{\pm iφ}$, which cannot be peripheral eigenvalues of any finite nonnegative matrix when $φ/π$ is irrational. |
| title | Rational arrival processes with strictly positive densities need not be Markovian |
| topic | Probability |
| url | https://arxiv.org/abs/2603.28047 |