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Potentials of skip-free Markov chains

Wendi Li, Jinpeng Liu, Yuanyuan Liu

TL;DR

The paper develops truncation-approximation methods for potential theory in discrete-time and continuous-time skip-free Markov chains. It defines and analyzes potentials as minimal nonnegative solutions to Poisson-type equations and connects them to Green matrices, providing convergence results for truncated systems to their full counterparts. For upward and downward skip-free chains, it derives explicit representations of the potentials, establishes finiteness criteria via phi(0), and gives Green-matrix formulas along with transience/recurrence conditions. The results are illustrated with GI/M/1 and M/G/1 queue examples and extended to CTMCs, offering practical tools for computing potentials and assessing long-run behavior in queueing and related systems.

Abstract

Potential theory has important applications in various fields such as physics, finance, and biology. In this paper, we investigate the potentials of two classic types of discrete-time skip-free Markov chains: upward skip-free and downward skip-free Markov chains. The key to deriving these potentials lies in the use of truncation approximation techniques. The results are then applied to GI/M/1 queues and M/G/1 queues, and further extended to continuous-time skip-free Markov chains.

Potentials of skip-free Markov chains

TL;DR

The paper develops truncation-approximation methods for potential theory in discrete-time and continuous-time skip-free Markov chains. It defines and analyzes potentials as minimal nonnegative solutions to Poisson-type equations and connects them to Green matrices, providing convergence results for truncated systems to their full counterparts. For upward and downward skip-free chains, it derives explicit representations of the potentials, establishes finiteness criteria via phi(0), and gives Green-matrix formulas along with transience/recurrence conditions. The results are illustrated with GI/M/1 and M/G/1 queue examples and extended to CTMCs, offering practical tools for computing potentials and assessing long-run behavior in queueing and related systems.

Abstract

Potential theory has important applications in various fields such as physics, finance, and biology. In this paper, we investigate the potentials of two classic types of discrete-time skip-free Markov chains: upward skip-free and downward skip-free Markov chains. The key to deriving these potentials lies in the use of truncation approximation techniques. The results are then applied to GI/M/1 queues and M/G/1 queues, and further extended to continuous-time skip-free Markov chains.

Paper Structure

This paper contains 7 sections, 11 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. General Markov chains
  4. Upward skip-free Markov chains
  5. Downward skip-free Markov chains
  6. Examples
  7. Continuous-time skip-free Markov chains

Key Result

Theorem 2.1

Assume that the DTMC $\boldsymbol{X}$ is irreducible and the nonnegative function $\boldsymbol{c}$ on $\mathbb{S}$ is finite. Then, we have

Figures (2)

  • Figure 1: The value of the potential $\boldsymbol{\phi}$ with $z=3, 5, 10$ of Example \ref{['exp1']}.
  • Figure 2: The value of the potential $\boldsymbol{\phi}$ with $z=1.2, 1.5, 1.8$ of Example \ref{['exp2']}.

Theorems & Definitions (22)

  • Theorem 2.1
  • proof
  • Corollary 2.1
  • proof
  • Lemma 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.2
  • Remark 2.1
  • Lemma 2.2
  • ...and 12 more