A fractal Poisson process and its input queue

An interrupted Poisson process has two states, on-state and off-state. In the onstate, Poisson arrivals occur, and on the other hand, there are no arrivals in the off-state. As the variation becomes larger, the arrival process is changed to a more complex interrupted Poisson process generated by embedding new off-states in each on-state. In such a way, recursively embedding offstates in on-states and taking a limit, a fractal structure can be found in the on-off structure. We name the arrival process fractal Poisson process and study it. The interarrival time density has a heavy tail. In addition, we study queueing models with the fractal Poisson arrivals. Even if the utilization is very low, the waiting time is very long.