In this paper we consider Quasi-Birth-and-Death (QBD) processes where the upward (resp. downward) transitions are restricted to occur only from (resp. to) a subset of the phase space. This property is exploited to reduce the computation time to find the matrix R or G of the process. The reduction is done through the definition of a censored process which can be of the M/G/1- or GI/M/1-type. The approach is illustrated through examples that show the applicability and benefits of making use of the additional structure. The examples also show how these special structures arise naturally in the analysis of queuing systems. Even more substantial gains can be realized when we further restrict the class of QBD processes under consideration.