This paper is concerned with level‐dependent quasi‐birth‐death (LD‐QBD) processes, i.e., multi‐variate Markov chains with a block‐tridiagonal
‐matrix, and a more general class of block‐structured Markov chains, which can be seen as LD‐QBD processes with total catastrophes. Arguments from univariate birth‐death processes are combined with existing matrix‐analytic formulations to obtain sufficient...