A self-similar point process is developed by embedding a process with bursty behavior over timescales. This embedding is not arbitrary, but achieved through a model which itself has fractal patterns. This model decomposes the self-similar point process over its timescales in a manner that may be tractable for accurate characterization and control of packet traffic. The limiting behavior of the model is shown to possess the properties of a self-similar point process, namely, bursts of arrivals which have no (intrinsic) timescale. Furthermore, this model leads to efficient synthesis of such a process.