On the convergence and limits of certain matrix sequences arising in quasi-birth-and-death Markov chains (Q2748444)

Let \(\Omega\) be the set of stochastic and substochastic matrices of order \(m\), \(A_0\), \(A_1\) and \(A_2\) be nonnegative matrices in \(\Omega\) such that \(A=A_0 + A_1 + A_2\) is stochastic and \({\mathbf e}\) be the column vector with all components one. Define a sequence of nonnegative matrices \(\{Z(n), n \geq 0\}\) as follows: \(Z(0) \in \Omega\) and \(Z(n+1)=f(Z(n))\). The sequence \(\{Z(n), n \geq 0\}\) is closely related to the minimal nonegative solution \(G\) to the matrix equation \(G = A_2 + A_1G + A_0G^2\). The second author [``Matrix-geometric solutions in stochastic models: An algorithmic approach'' (1981; Zbl 0469.60002)] introduced this important matrix \(G\), gave its probabilistic interpretation and used it in queueing analysis. Let \(\text{sp}(G)\) denote the Perron-Frobenius eigenvalue of the matrix \(G\) and \({\mathbf g}\) be the left eigenvector of matrix \(G\) corresponding to \(\text{sp}(G)\). The authors prove that under certain conditions, the sequence \(\{Z(n)\), \(n\geq 0\}\) converges either to \(G\) or \(G+(I-G){\mathbf {eg}}\). The matrices \(\{Z(n)\), \(n \geq 0\}\) can be interpreted as the absorption probability matrix of some boundary states of a sequence of finite Markov chains. The authors present numerical examples that clarify some of the technical issues of interest.