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This paper is principally concerned with resource allocation for connections tolerating statistical quality of service (QoS) guarantees in a public wide-area ATM network. Our aim is to sketch a framework, based on effective bandwidths, for call admission schemes that are sensitive to individual QoS requirements and account for statistical multiplexing. Results approximating the effective bandwidth required by heterogeneous streams sharing buffered links, including results for the packetized generalized processor sharing service discipline, are described. Extensions to networks follow via the concept of decoupling bandwidths, motivated by a study of the input-output properties of queues. Based on these results we claim that networks with sufficient routing diversity will inherently satisfy nodal decoupling. We then discuss on-line methods for estimating the effective bandwidth of connection. Using this type of traffic monitoring we propose an approach to usage parameter control (i.e., policing) for effective bandwidth descriptors. Finally, we suggest how on-line monitoring might be combined with admission control to exploit unknown statistical multiplexing gains and thus increase utilization