This Generalized Processor Sharing (GPS) is the basis for the packet scheduler of choice in IP routers and ATM switches of the future. The currently accepted approach for the design of GPS schedulers is based on deterministic QoS guarantees, which, it is generally accepted, is overly conservative and leads to limitations on capacity. To address this problem we develop a framework for GPS scheduling which is based on statistical QoS guarantees and statistical multiplexing. We give the design of GPS weights which maximize the coverage of operating points, and also the design of the connection admission control (CAC). The general framework is end-to-end, with two heterogeneous QoS classes coexisting with a third, best effort class. Each QoS class has a specified delay bound together with a bound on the probability of its violation. An important objective is to maximize the bandwidth available to best effort traffic while just satisfying the guarantees of the QoS classes. To this end, we consider output regulated GPS scheduling which has the additional feature of limiting each connection's share of the bandwidth to a specified value, a design parameter which is determined by our analysis. The sources are subject to standard dual leaky bucket regulation. For the design of the GPS weights we give procedures based on two key concepts, the realizable set and the critical weights. The realizable set is the union of all admissible sets of connections of both classes over all weights. One of the main contributions is a pragmatic design process by which most of the realizable set is realized by only two critical weights. In the benign case, the system is "effectively homogeneous" and a single GPS weight suffices, while in the complementary "effectively non-homogeneous" case it is necessary to switch between the critical weights. The numerical results, which are for a single node with a wide range of traffic and QoS parameters, validate the design procedure and also show that there are substantial capacity gains from statistical multiplexing.