  Ergodicity is a fundamental issue for a stochastic process. In this paper, we
refine results on ergodicity for a general type of Markov chain to a specific
type or the $GI/G/1$-type Markov chain, which has many interesting and
important applications in various areas. It is of interest to obtain conditions
in terms of system parameters or the given information about the process, under
which the chain has various ergodic properties. Specifically, we provide
necessary and sufficient conditions for geometric, strong and polynomial
ergodicity, respectively.
