We investigate a specific class of irreducible, level-dependent, discrete-time, GI/M/1-type Markov chains. The transition matrices possess a block lower-Hessenberg structure, which shows asymptotic convergence along the rows as the level approaches infinity. Criteria are presented for recurrence, transience, positive recurrence, geometric ergodicity, and geometric transience in terms of elements of the transition matrices. These criteria are established by employing drift functions and matrix-generating functions. Furthermore, we discuss the extension of the main results to the continuous-time case.
