[1][2]: 118 As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.
[4] The transition rate matrix for a quasi-birth-death process has a tridiagonal block structure where each of B00, B01, B10, A0, A1 and A2 are matrices.
[6] When describing the process by both level and phase it is a continuous-time Markov chain, but when considering levels only it is a semi-Markov process (as transition times are then not exponentially distributed).
Usually the blocks have finitely many phases, but models like the Jackson network can be considered as quasi-birth-death processes with infinitely (but countably) many phases.
[6][7] The stationary distribution of a quasi-birth-death process can be computed using the matrix geometric method.