 The stochastic matrix describing the Markov chain has block structure

where each of "A"0, "A"1 and "A"2 are matrices and "A"*0, "A"*1 and "A"*2 are irregular matrices for the first and second levels.The transition rate matrix for a quasi-birth-death process has a tridiagonal block structure

where each of "B"00, "B"01, "B"10, "A"0, "A"1 and "A"2 are matrices.The process can be viewed as a two dimensional chain where the block structure are called "levels" and the intra-block structure "phases".