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On vector-valued functional equations with multiple recursive terms

Ioannis Dimitriou, Ivo J. B. F. Adan

TL;DR

This work develops a comprehensive, matrix-valued framework for vector-valued functional equations arising from a broad class of Markov-modulated and dependent recursions, including Lindley-type autoregressions and reflected processes. It leverages Liouville's theorem and Wiener-Hopf boundary-value theory to obtain explicit Laplace-Stieltjes transforms and joint distributions, addressing both transient and stationary regimes. The contributions cover Markov-modulated queues, fluid flows with consumption, shot-noise queues, tandem networks with consumption, integer-valued and VAR(1) processes, and variations with copula-based dependencies, providing a unified analytic approach for complex dependent systems. The results have potential implications for exact performance analysis in advanced queueing networks and multivariate autoregressive models with Markov modulation and structured dependencies.

Abstract

In this work, we study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. Our work is strongly motivated by a wide range of semi-Markovian queueing, and vector-valued autoregressive processes. In all cases, we provide explicit expressions for the joint distribution in terms of Laplace-Stieltjes transforms/generating functions.

On vector-valued functional equations with multiple recursive terms

TL;DR

This work develops a comprehensive, matrix-valued framework for vector-valued functional equations arising from a broad class of Markov-modulated and dependent recursions, including Lindley-type autoregressions and reflected processes. It leverages Liouville's theorem and Wiener-Hopf boundary-value theory to obtain explicit Laplace-Stieltjes transforms and joint distributions, addressing both transient and stationary regimes. The contributions cover Markov-modulated queues, fluid flows with consumption, shot-noise queues, tandem networks with consumption, integer-valued and VAR(1) processes, and variations with copula-based dependencies, providing a unified analytic approach for complex dependent systems. The results have potential implications for exact performance analysis in advanced queueing networks and multivariate autoregressive models with Markov modulation and structured dependencies.

Abstract

In this work, we study vector-valued functional equations with multiple recursive terms that arise naturally when we are dealing with vector-valued multiplicative Lindley-type recursions. Our work is strongly motivated by a wide range of semi-Markovian queueing, and vector-valued autoregressive processes. In all cases, we provide explicit expressions for the joint distribution in terms of Laplace-Stieltjes transforms/generating functions.

Paper Structure

This paper contains 18 sections, 12 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Related work
  3. Our contribution
  4. Organization of the paper
  5. The Markov modulated $M/G/1$-type reflected autoregressive process
  6. A Markovian fluid flow model with consumption
  7. The Markov dependent case
  8. The case where $A_{n+1}$, $S_{n}$ are conditionally dependent based on the FGM copula
  9. A modulated $D_{N}/G/1$ shot-noise queue
  10. A modulated Markovian queue where service time depends on the waiting time
  11. A modulated ASIP tandem network with consumption
  12. The buffer content of $Q_{1}$
  13. The case $a_{1,i}=0$, $i=1,\ldots,N$
  14. An integer vector-valued reflected autoregressive process
  15. On a reflected VAR(1) process
  16. ...and 3 more sections

Key Result

Theorem 1

For $Re(s)=0$, $Re(\eta)\geq 0$, $|r|<1$, or equivalently, in matrix notation, where $\tilde{P}^{(i)}:=(\tilde{P}^{(i)})_{p,q}$, $i,p,q\in E$ is an $N\times N$ matrix, with the element $\tilde{P}^{(i)}_{i,i}=1$, and all the other elements $\tilde{P}^{(i)}_{p,q}=0$, $p,q\neq i$. Note that $\sum_{i=1}^{N}\tilde{P}^{(i)}=I$.

Theorems & Definitions (25)

  • Theorem 1
  • Lemma 2
  • Remark 3
  • Remark 4
  • Theorem 5
  • Remark 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • Theorem 10
  • ...and 15 more