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In this paper we consider the problem of providing statistical Quality of Service (QoS) guarantees defined in terms of packet loss when independent heterogeneous traffic streams access a network router of high capacity. By using a scaling technique we show how this problem becomes tractable when the server capacity is large and many traffic streams are present. In particular we show that we can define an effective bandwidth for the sources that allows us to map the model onto a multirate loss model. In particular we show several insights on the multiplexing problem as the capacity becomes large. We also provide numerical and simulation evidence to show how the largeness of networks can be used to advantage in providing very simple admission control schemes. The techniques are based on large deviations, local limit theorems, and the product-form associated with co-ordinate convex policies.