We study the large buffer asymptotics of a multiplexer under two different self-similar traffic inputs, namely the so-called M G  model of Cox and the fractional Gaussian noise input model. In the former case we show that the tail probabilities for the buffer content (in steady-state) decay at most hyperbolically. This is contrasted with the situation where the input traffic is fractional Gaussian noise, in which case the tail probabilities display a Weibullian character. Therefore, for a given input rate rin and Hurst parameter H, these dissimilar asymptotics would result in vastly differing buffer engineering practices, which points somewhat to the inadequacy of using H as the sole parameter to characterize long-range dependence.