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

We study the heavy traffic regime of a discrete-time queue driven by correl ated inputs, namely the M/G/infinity input processes of Cox. We distinguish between M/G/infinity processes with short- and long-range dependence, iden tifying in each case the appropriate heavy traffic scaling that results in a nondegenerate limit. As expected, the limits we obtain for short-range de pendent inputs involve the standard Brownian motion. Of particular interest are the conclusions for the long-range dependent case: the normalized queu e length can be expressed as a function not of a fractional Brownian motion , but of an alpha-stable, 1/alpha self-similar independent increment Levy p rocess. The resulting buffer content distribution in heavy traffic is expre ssed through a Mittag-Leffler special function and displays a hyperbolic de cay of power 1-alpha. Thus, M/G/infinity processes already demonstrate that under long-range dependence, fractional Brownian motion does not necessari ly assume the ubiquitous role that standard Brownian motion plays in the sh ort-range dependence setup.