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
such that , where is the vector of limiting probabilities for all states on level n of the process. When
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
is the solution.
In essentially all cases considered in this paper, the matrix
is a solution of the matrix quadratic
2 + XA
1 + A
0 = 0.
Despite this fact,
is never equal to Neuts' rate matrix
, although the non-zero eigenvalues and the corresponding left eigenvectors of R are a subset of the eigenvalues and left eigenvectors of
. This fact leads to two methods for determining
If the transition rates of the QBD-process are level-dependent, then it is also shown that matrices
W(n) exist such that .