Equivalences of Subgeometric Ergodicity of Markov Chains and Its Application to M/G/1 Queue

马尔可夫链的遍历性研究在随机过程理论中占有重要地位，本文聚焦于马尔可夫链的次几何遍历性，提出了次几何遍历的六个等价判定条件，这些等价条件对进一步探索一般马尔可夫过程的次几何遍历性提供了理论基础，并将其应用于排队论中经典的M/G/1嵌入过程，得到了易验证的M/G/1嵌入排队过程次几何遍历的判定条件。The study of the ergodicity of Markov chains holds a significant position in the theory of stochastic processes. This paper focuses on the subgeometric ergodicity of Markov chains and proposes six equivalent criteria for subgeometric ergodicity of Markov chains. These equivalent conditions provide a theoretical foundation for further exploration of the subgeometric ergodicity of general Markov processes. In addition, this paper applies these criteria to the classical M/G/1 embedded process in queueing theory, obtaining easily verifiable conditions for determining the subgeometric ergodicity of the M/G/1 embedded queueing process.