State space expansions and the limiting behavior of quasi-birth-and-death processes

The notion of complete level crossing information, or LCI-completeness, is introduced for quasi-birth-death (QBD) processes. It is shown that state space expansions allow any QBD-process to be modified so that it is LCI-complete. For any LCI-complete, QBD-process, there exists a matrix W such that , where is the vector of limiting probabilities for all states on level n of the process. When W cannot be found in closed form, it can be found via an algorithm requiring fewer than m steps, where m is the number of states on each level of the process. The result of this algorithm is always a linear matrix equation for which W is the solution. In essentially all cases considered in this paper, the matrix W is a solution of the matrix quadratic X2A2 + XA1 + A0 = 0. Despite this fact, W is never equal to Neuts' rate matrix R, although the non-zero eigenvalues and the corresponding left eigenvectors of R are a subset of the eigenvalues and left eigenvectors of W. This fact leads to two methods for determining R from W. If the transition rates of the QBD-process are level-dependent, then it is also shown that matrices W(n) exist such that https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:bina ry:20180209072233157-0291:S0001867800018553:S0001867800018553_inline3.gi f?pub-status=live