We study the heavy traffic regime of a multiplexer driven by correlated inputs, namely the M|GI|input processes of Cox. We distinguish between M|GI|processes exhibiting short or 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 inputs involve the standard Brownian motion. Of particular interest though are our 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 some other stable non -- Gaussian self-similar process. Thus, the M|GI|processes serve as an example demonstrating that, within long-range dependence, fractional Brownian motion does not assume the ubiquitous role that its counterpart, standard Brownian motion, plays in the short-range dependence setup, and that modeling possibilities attracted to non -- Gaussian limits are not so hard to come by.