Heavy traffic limits associated with M/G/∞ input processes

We study the heavy traffic regime of a discrete-time queue driven by correlated inputs, namely the M/G/∞ input processes of Cox. We distinguish between M/G/∞ processes with short- and long-range dependence, identifying in each case the appropriate heavy traffic scaling that results in a nondegenerate limit. As expected, the limits we obtain for short-range dependent inputs involve the standard Brownian motion. Of particular interest are the conclusions for the long-range dependent case; the normalized queue length can be expressed as a function not of a fractional Brownian motion, but of an α-stable, 1/α self-similar independent increment Lévy process. The resulting buffer content distribution in heavy traffic is expressed through a Mittag–Leffler special function and displays a hyperbolic decay of power 1–α. Thus, M/G/∞ processes already demonstrate that under long-range dependence, fractional Brownian motion does not necessarily assume the ubiquitous role that standard Brownian motion plays in the short-range dependence setup.