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 ,, = ,,-1W, where i,, 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