Two asymptotic analyses of the queue length distribution at a statistical multiplexor supporting heterogeneous exponential on-off sources are considered. The first analysis is performed by approximating the cell generation rates as a multi-dimensional Ornstein-Uhlenbeck process and then applying the Benes queueing formula. In the second analysis, w state with a system of linear equations derived from the exact expressions of the dominant eigenvalue of the matrix governing the queue length distribution. Assuming that there are a large number of sources, we obtain asymptotic approximations to the dominant eigenvalue. Based on the analyses, we define a traffic descriptor to include the mean and the variance of the cell generation rate and a burstiness measure. A simple expression for the quality of service (QoS) in cell loss rate is derived in terms of the traffic descriptor parameters and the multiplexor parameters (output link capacity and buffer size). The result is then used to quantify the factors determining the required capacity of a call taking the statistical multiplexing gain into consideration. As an application of the analyses, we can use the required capacity calculation for simple yet effective connection admission control(CAC) algorithms.