On the Convergence of MMPP and Fractional ARIMA Processes with Long-Range Dependence to Fractional Brownian Motion

Though the various models proposed in the literature for capturing the long-range dependent nature of network traac are all either exactly or asymptotically second order self-similar, their eeect on network performance can be very diierent. We are thus motivated to characterize the limiting distributions of these models so that they lead to parsimonious modeling and a better understanding of network traac. In this paper we consider long-range dependent arrival processes based on Markovian arrival and fractional ARIMA processes and show that the suitably scaled distributions of these processes converge to fractional Brownian motion in the sense of nite dimensional distributions. Subsequently, we prove that they also converge weakly to fractional Brownian motion in the space of continuous functions. Thus, the behavior of network elements fed with traac from these models has similar characteristics to those fed with fractional Brow-nian motion under suitable limiting conditions. Specii-cally, tails of queues fed with these arrivals have a Weibul-lian shape in sharp contrast with the exponential tails of conventional queues. Also, the weak convergence results allow us to accurately estimate the loss probabilities using the expressions for storage models for fractional Browni-nan motion. The seminal paper of 9] introduced the notion of self-similarity and long-range dependence in network traac. This has spurred research in the area of traac models which account for these second order statistical characteristics of network data. Wavelet models 1], Markovian arrival processes 2], the M=G=1 model 5], chaotic maps 6], Fractional Brownian motion 9], fractional ARIMA processes 9] and superposition of ON/OFF sources 14] are some of the models that have been suggested. Though all these models model the long-range dependence and show either exact or asymptotically second order self-similarity, the performance of network elements fed with these diierent sources diier widely. Studies in 10] and 11] show, for example, that queues fed with self-similar traac from M=G=1 sources have an exponentially distributed queue tail while the tails of queues fed with input sources characterized by fractional Brownian motion and superposition of ON/OFF sources have a Weibullian shape. Thus additional insight into the characterization of the arriving work process is necessary to predict the queue behavior. Motivated by this striking variation in queue performance, we try to model the limiting distribution of some of these self-similar sources. Convergence of the limiting distribution to any given model for which the network element performance has already been characterized then allows us to …