Heavy Traffic Limits Associated with M|GI|Input Processes

We study the heavy traffic regime of a discrete-time queue driven by correlated inputs, namely the M|GI|input processes of Cox. We distinguish between M|GI|processes with short- and long- range dependence, identifying for each case the appropriate heavy traffic scaling that results in non-degenerate limits. As expected, the limits we obtain for short-range dependent input 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 a-stable, 1/a self-similar independent increments levy process. The resulting buffer asymptotics in heavy traffic display a hyperbolic decay, of power 1 - a. Thus M|GI|processes already demonstrate that, within long-range dependence, fractional Brownian motion does not necessarily assume the ubliquitous role that standard Brownian motion plays in the short-range dependence setup