A general theory on spectral properties of state-homogeneous finite-state quasi-birth–death processes

In this paper a spectral theory pertaining to Quasi-Birth–Death Processes (QBDs) is presented. The QBD, which is a generalization of the birth–death process, is a powerful tool that can be utilized in modeling many stochastic phenomena. Our theory is based on the application of a matrix polynomial method to obtain the steady-state probabilities in state-homogeneous finite-state QBDs. The method is based on finding the eigenvalue–eigenvector pairs that solve a matrix polynomial equation. Since the computational effort in the solution procedure is independent of the cardinality of the counting set, it has an immediate advantage over other solution procedures. We present and prove different properties relating the quantities that arise in the solution procedure. By also compiling and formalizing the previously known properties, we present a formal unified theory on the spectral properties of QBDs, which furnishes a formal framework to embody much of the previous work. This framework carries the prospect of furthering our understanding of the behavior the modeled systems manifest.