An interpolation approximation for queueing systems with Poisson input
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Open queueing systems in light traffic
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Wide area traffic: the failure of Poisson modeling
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Tail probabilities for M/G/\infty input processes (I): Preliminary asymptotics
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Heavy-traffic analysis for the $GI/G/1$ queue with heavy-tailed distributions
Heavy-traffic analysis for the $GI/G/1$ queue with heavy-tailed distributions
Tail probabilities for a multiplexer with self-similar traffic
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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 &agr;-stable, 1/&agr; 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-&agr;. 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.