ILAS2016 — 11–15 July 2016 — KU Leuven, Belgium

20th Conference of the International Linear Algebra Society (ILAS)

20th ILAS Conference

In minisymposium: Matrix Equations

Wed 12:00–12:30, Room AV 02.17
General solution of the Poisson equation for Quasi-Birth-and-Death processes
Beatrice Meini (University of Pisa)
Joint work with Dario Bini (University of Pisa); Sarah Dendievel (Ghent University); Guy Latouche (ULB Bruxelles)

If $T$ is the transition matrix of a finite irreducible Markov chain and ${b}$ is a given vector in the image of $I-T$, the Poisson equation $(I-T){x}={b}$ has a unique solution, up to an additive constant, given by $ {x}=(I-T)^{\#}{b} +\alpha {1}$, for any $\alpha$ scalar, where ${1}$ is the vector of all ones and $H^{\#}$ denotes the group inverse of $H$. This does not hold when the state space is infinite.

We consider the Poisson equation $(I-P){u}={g}$, where $P$ is the transition matrix of a Quasi-Birth-and-Death (QBD) process with infinitely many levels, ${g}$ is a given infinite–dimensional vector and ${u}$ is the unknown vector.

By using the block tridiagonal and block Toeplitz structure of $P$, we recast the problem in terms of a set of matrix difference equations. We provide an explicit expression of the general solution, relying on the properties of Jordan triples of matrix polynomials and on the solutions of suitable quadratic matrix equations. The uniqueness and boundedness of the solutions are discussed, according to the properties of the right–hand side ${g}$.