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Asymptotics of the occupation measure defined on a nonnegative matrix of two-dimensional quasi-birth-and-death type

Toshihisa Ozawa

TL;DR

This work analyzes the occupation measure and its dual hitting measure for a nonnegative matrix with the same block structure as a 2d-QBD transition matrix. By constructing potential matrices, compensation equations, and matrix-valued generating functions, it derives convergence domains and exact asymptotic decay rates of the occupation measure in arbitrary directions, with the hitting measure results obtained via a duality through transposition. The key contributions include a precise characterization of the convergence domain $\mathcal{D} = [\Gamma]^{ex}$ and explicit formulas for the decay rates $\xi_{\boldsymbol{c}}$, along with a framework that extends to transient and higher-dimensional queueing-perturbed random walks using G-matrix techniques. The methodology bridges MAP-type and QBD-type analyses, enabling exact tail asymptotics for occupation and hitting measures and offering a pathway to 3D generalizations in reflected random walks.

Abstract

We consider a nonnegative matrix having the same block structure as that of the transition probability matrix of a two-dimensional quasi-birth-and-death process (2d-QBD process for short) and define two kinds of measure for the nonnegative matrix. One corresponds to the mean number of visits to each state before the 2d-QBD process starting from the level zero returns to the level zero for the first time. The other corresponds to the probabilities that the 2d-QBD process starting from each state visits the level zero. We call the former the occupation measure and the latter the hitting measure. We obtain asymptotic properties of the occupation measure such as the asymptotic decay rate in an arbitrary direction. Those of the hitting measure can be obtained from the results for the occupation measure by using a kind of duality between the two measures.

Asymptotics of the occupation measure defined on a nonnegative matrix of two-dimensional quasi-birth-and-death type

TL;DR

This work analyzes the occupation measure and its dual hitting measure for a nonnegative matrix with the same block structure as a 2d-QBD transition matrix. By constructing potential matrices, compensation equations, and matrix-valued generating functions, it derives convergence domains and exact asymptotic decay rates of the occupation measure in arbitrary directions, with the hitting measure results obtained via a duality through transposition. The key contributions include a precise characterization of the convergence domain and explicit formulas for the decay rates , along with a framework that extends to transient and higher-dimensional queueing-perturbed random walks using G-matrix techniques. The methodology bridges MAP-type and QBD-type analyses, enabling exact tail asymptotics for occupation and hitting measures and offering a pathway to 3D generalizations in reflected random walks.

Abstract

We consider a nonnegative matrix having the same block structure as that of the transition probability matrix of a two-dimensional quasi-birth-and-death process (2d-QBD process for short) and define two kinds of measure for the nonnegative matrix. One corresponds to the mean number of visits to each state before the 2d-QBD process starting from the level zero returns to the level zero for the first time. The other corresponds to the probabilities that the 2d-QBD process starting from each state visits the level zero. We call the former the occupation measure and the latter the hitting measure. We obtain asymptotic properties of the occupation measure such as the asymptotic decay rate in an arbitrary direction. Those of the hitting measure can be obtained from the results for the occupation measure by using a kind of duality between the two measures.

Paper Structure

This paper contains 14 sections, 13 theorems, 119 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Potential matrices
  4. Compensation equations
  5. Asymptotic decay rates
  6. Convergence domains
  7. Matrix-valued generating functions of the occupation measure
  8. Matrix-valued generating functions of the potential matrices
  9. Convergence domains of the generating functions
  10. Asymptotics of the occupation measure
  11. Identifying the convergence domains
  12. Asymptotic decay rates and functions
  13. Asymptotics of the hitting measure
  14. Concluding remark

Key Result

Lemma 2.1

For $\alpha\in\{\{1\},\{2\},\{1,2\}\}$,

Figures (4)

  • Figure 1: Transition probability blocks
  • Figure 2: Convergence domain
  • Figure 3: Function $f_1(s)$
  • Figure 4: An example of case (C4: $s_1^*=\bar{\theta}_1^*$, $s_2^*=\bar{\theta}_2^{\{2\}}$)

Theorems & Definitions (24)

  • Remark 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • Lemma 2.2
  • Remark 2.1
  • Remark 2.2
  • Proposition 3.1
  • Proposition 3.2
  • ...and 14 more