Tests for the Nonergodicity of Multidimensional Markov Chains

A Markov chain is ergodic if it has a proper steady-state distribution. The stability of a queueing system often reduces to the ergodicity of an underlying multidimensional chain. We gave criteria for the nonergodicity of a multidimensional Markov chain and a generalization to continuous time Markov chains. We also give queueing model examples.