On quadratic matrix equations with infinite size coefficients encountered in QBD stochastic processes

Matrix equations of the kind $A_1 X^2 + A0 X + A_{-1} = X$, where both the matrix coefficients and the unknown are semi-infinite matrices belonging to a Banach algebra, are considered. These equations, where coefficients are quasi-Toeplitz matrices, are encountered in certain quasi-birth-death processes as the tandem Jackson queue or in any other processes that can be modeled as a reflecting random walk in the quarter plane. We provide a numerical framework for approximating the minimal nonnegative solution of these equations that relies on semi-infinite quasi-Toeplitz matrix arithmetic. In particular, we show that the algorithm of cyclic reduction can be effectively applied and can approxi- mate the infinite-dimensional solutions with quadratic convergence at a cost that is comparable to that of the finite case. This way, we may compute a finite approximation of the sought solution and of the invariant probability measure of the associated quasi-birth-death process, within a given accuracy. Numerical experiments, performed on a collection of benchmarks, confirm the theoretical analysis.