Heavy traffic limits in a wireless queueing model with long range dependence

High-speed wireless networks carrying multimedia applications are becoming a reality and the transmitted data exhibit long range dependence and heavy-tailed properties. We consider the heavy traffic approach in working towards queue models under these properties, extending the model in R.T. Buche and H.J. Kushner (2002). Our focus is on the scalings used in the heavy traffic approach which are determined by combinations of the source rate of an infinite source Poisson model of the arrival process, the tail distribution of data transmitted by these sources, and the rate of variation of the random process (channel process) modeling the wireless medium. A fundamental inequality between the exponent in the power tail distribution of the data from the source and the parameter specifying the rate of channel variations is obtain. This inequality is important in both the "fast growth" and "slow growth" regimes for the arrival process and along with the source rate is used to define the possible cases for obtaining limit models for the queueing process. Across the cases, the possible limit models include reflected Brownian motion, reflected stable Levy motion, or reflected fractional Brownian motion.