Matrix-multiplicative approach to quasi-birth-and-death processes analysis

Abstract Pure algebraic proofs of some known and some new results on the minimal nonnegative solutions of matrix quadratic equations are given. Then we study linear systems with block tridiagonal matrices . We find a method which generalizes the method based on block triangular factorisation. We show that stationary vector of a Markov process with an irreducible block tridiagonal generator can be expressed as a sum of two matrix-multiplicative terms. In the particular case of Quasi-Birth-and-Death processes the solution is given by a sum of two matrix-geometric terms.