Long-tail buffer-content distributions in broadband networks

We identify conditions under which relatively large buffers will be required in broadband communication networks. For this purpose, we analyze an infinite-capacity stochastic fluid model with a general stationary environment process (without the usual independence or Markov assumptions). With that level of generality, we are unable to establish asymptotic results, but by a very simple argument we are able to obtain a revealing lower bound on the steady-state buffer-content tail probability. The bounding argument shows that the steady-state buffer content will have a long-tail distribution when the sojourn time in a set of states with positive net input rate itself has a long-tail distribution. If a set of independent sources, each with a general stationary environment process, produces a positive net flow when all are in high-activity states, and if each of these sources has a high-activity sojourn-time distribution with a long tail, then the steady-state buffer-content distribution will have a long tail, but possibly one that decays faster than the tail for any single component source. The full buffer-content distribution can be derived in the special case of a two-state fluid model with general high- and low-activity-time distributions, assuming that successive high- and low-activity times come from independent sequences of i.i.d. random variables. In that case the buffer-content distribution will have a long tail when the high-activity-time distribution has a long tail. We illustrate by giving numerical examples of the two-state model based on numerical transform inversion.