General solution of the Poisson equation for Quasi-Birth-and-Death processes
This work offers a theoretical tool for analyzing QBD processes, which are important in queueing theory and stochastic modeling, but the result is incremental as it extends known techniques to a specific class of equations.
The paper provides the general solution to the Poisson equation for Quasi-Birth-and-Death processes with infinitely many levels, using resolvent triples to solve matrix difference equations derived from the block tridiagonal and block Toeplitz structure of the transition matrix.
We consider the Poisson equation $(I-P)\boldsymbol{u}=\boldsymbol{g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, $\bm g$ is a given infinite dimensional vector and $\bm u$ is the unknown. Our main result is to provide the general solution of this equation. To this purpose we use the block tridiagonal and block Toeplitz structure of the matrix $P$ to obtain a set of matrix difference equations, which are solved by constructing suitable resolvent triples.