The study of the quasi-stationary behavior was initiated by Yaglom [33]. Since then, significant advances have been made through the efforts of many researchers, e.g., see a continuously updating overview by Pollett [2011]. For the quasi-stationary distributions of block-structured Markov chains, some preliminary results for Markov chains of GI/M/1 type and of M/G/1 type were obtained by Quan-Lin Li [1997], while a complete work were obtained in Li and Zhao [2002, 2003]. The quasi-stationarity distributions of the QBD processes can be found in Makimoto [1993], Bean, Bright, Latouche, Pearce, Pollett and Taylor [1997], and Bean, Pollett and Taylor [1998, 2000].