Applications to be supported on broadband networks exhibit a wide range of traffic statistics and many of them are sensitive to delay and loss violations. To accurately estimate admissible workload and bandwidth requirement, a detailed traffic model, HAP (Hierarchical Arrival Process) is proposed in this paper. Packets generated from HAP are modulated by processes at user, application, and message levels. This model is a generalization of on-off traffic models and is shown to be equivalent to a special class of MMPP (Markov Modulated Poisson Process). Three algorithmic methods along with simulations are applied to evaluate the queueing performance under HAP traffic. Delay under HAP traffic can be well over tens of times higher than Poisson traffic, depending on parameters and load. Congestion may persist for minutes. HAP's dramatic short-term behavior explains the occasional congestion in the real networks. Conventional traffic models, however, do not exhibit this behavior. With these results, we give implications for broadband network control.