Potentials of skip-free Markov chains

TL;DR

This paper explores the potentials of upward and downward skip-free Markov chains using truncation techniques, with applications to queue models and extensions to continuous-time chains.

Contribution

It introduces a novel approach to compute potentials of skip-free Markov chains and applies these results to queueing systems and continuous-time models.

Findings

Derived explicit potential formulas for discrete-time skip-free chains.

Extended potential analysis to continuous-time skip-free Markov chains.

Applied potential theory to queueing models like GI/M/1 and M/G/1.

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.