General QBD processes with applications to overload control

In this paper we introduce a general class of level-dependent quasi-birth-and-death (QBD) processes and their stationary solution. We obtain the complete characterization of their fundamental matrices in terms of minimal non-negative solution of the number of matrix quadratic equations. Our results provides a mixed-geometric solution for the stationary solution of level-dependent chains. Applications in overload control in communication networks are also discussed