A statistical network service which allows a certain fraction of traffic to not meet its QoS guarantees can extract additional capacity from a network by exploiting the statistical properties of the traffic. Here we consider a statistical service which assumes statistical independence of flows, but does not make any assumptions on the statistics of traffic sources, other than that they are regulated, e.g., by a leaky bucket. Under these conditions, we present functions, so-called local effective envelopes and global effective envelopes, which are, with high certainty, upper bounds of multiplexed traffic. We show that these envelopes can be used to obtain bounds on the amount of traffic on a link that can be provisioned with statistical QoS. A key advantage of our bounds is that they can be applied with a variety of scheduling algorithms. In fact, we show that one can reuse existing admission control functions that are available for scheduling algorithms with a deterministic service. We present numerical examples which compare the number of flows with statistical QoS guarantees that can be admitted with our effective envelope approach to those achieved with existing methods.