Heavy traffic analysis for a multiplexer driven by M∨GI∨∞ input processes

Abstract

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 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 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 a-stable, 1/a self-similar, independent increments L~vy process. Thus, the M∨GI∨∞ processes already demonstrate that, within long-range dependence, fractional Brownian motion does not assume the ubiquitous role that its short-range dependent 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 find.