Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rational arrival processes with strictly positive densities need not be Markovian

Oscar Peralta

Abstract

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 $e^{\pm i\varphi}$, which cannot be peripheral eigenvalues of any finite nonnegative matrix when $\varphi/π$ is irrational.

Rational arrival processes with strictly positive densities need not be Markovian

Abstract

Telek (2022) asked whether a rational arrival process (RAP), specified by matrices and and an initial row vector , with strictly positive joint densities and a unique dominant real eigenvalue of must admit an equivalent Markovian arrival process (MAP). A counterexample of order is given, showing the answer is no, and that the conjecture fails even under the stronger condition of exact normalisation . 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 , which cannot be peripheral eigenvalues of any finite nonnegative matrix when is irrational.

Paper Structure

This paper contains 5 sections, 8 theorems, 59 equations.

Table of Contents

  1. Introduction
  2. The counterexample
  3. Normalisation, positivity, and the spectral condition
  4. Evaluation at the origin and its generating function
  5. Non-existence of a MAP representation

Key Result

Lemma 2

For every $k\ge 1$ and every $t_1,\dots,t_k\ge 0$, where $\bm{1}_2=(1,1)^\top$ is the all-ones vector in $\mathbb{R}^2$.

Theorems & Definitions (17)

  • Conjecture 1: Telek Telek2022
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Lemma 6
  • ...and 7 more