We consider the problem of quality-of-service (QoS) provisioning in modern high-speed, multimedia, communication networks. We quantify QoS by the probabilities of loss and excessive delay of an arbitrary packet, and introduce the model of a multiclass node (switch) which provides network access to users that may belong to multiple service classes. We treat such a node as a stochastic system which we analyze and control. In particular, we develop an analytical approach to estimate both the delay and the buffer overflow probability per service class, based on ideas from large deviations and optimal control. We exploit these performance analysis results by devising a call admission control algorithm which can provide per class QoS guarantees. We compare the proposed approach to alternative worst-case and effective bandwidth-based schemes and argue that it leads to increased efficiency. Finally, we discuss extensions to the network case in order to provide end-to-end QoS guarantees.