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Heavy traffic limits associated with >M/G/∞ input processes

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Abstract

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 α-stable, 1/α 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-α. 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.

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  1. Electrical Engineering Department and Institute for Systems Research, University of Maryland, College Park, MD, 20742, USA

    Konstantinos P. Tsoukatos & Armand M. Makowski

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  1. Konstantinos P. Tsoukatos
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  2. Armand M. Makowski
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Tsoukatos, K.P., Makowski, A.M. Heavy traffic limits associated with >M/G/∞ input processes. Queueing Systems 34, 101–130 (2000). https://doi.org/10.1023/A:1019100901908

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