Expectiles for subordinated Gaussian processes with applications

In this paper, we introduce a new class of estimators of the Hurst exponent of the fractional Brownian motion (fBm) process. These estimators are based on sample expectiles of discrete variations of a sample path of the fBm process. In order to derive the statistical properties of the proposed estimators, we establish asymptotic results for sample expectiles of subordinated stationary Gaussian processes with unit variance and correlation function satisfying $\rho(i)\sim \kappa|i|^{-\alpha}$ (\kappa\in \RR) with $\alpha>0$. Via a simulation study, we demonstrate the relevance of the expectile-based estimation method and show that the suggested estimators are more robust to data rounding than their sample quantile-based counterparts

Estimation of Scale and Hurst Parameters of Semi-Selfsimilar Processes

The characteristic feature of semi-selfsimilar process is the invariance of its finite dimensional distributions by certain dilation for specific scaling factor. Estimating the scale parameter $\lambda$ and the Hurst index of such processes is one of the fundamental problem in the literature. We present some iterative method for estimation of the scale and Hurst parameters which is addressed for semi-selfsimilar processes with stationary increments. This method is based on some flexible sampling scheme and evaluating sample variance of increments in each scale intervals $[\lambda^{n-1}, \lambda^n)$, $n\in \mathbb{N}$. For such iterative method we find the initial estimation for the scale parameter by evaluating cumulative sum of moving sample variances and also by evaluating sample variance of preceding and succeeding moving sample variance of increments. We also present a new efficient method for estimation of Hurst parameter of selfsimilar processes. As an example we introduce simple fractional Brownian motion (sfBm) which is semi-selfsimilar with stationary increments. We present some simulations and numerical evaluation to illustrate the results and to estimate the scale for sfBm as a semi-selfsimilar process. We also present another simulation and show the efficiency of our method in estimation of Hurst parameter by comparing its performance with some previous methods.Comment: 15 page

Fast, Approximate Synthesis of Fractional Gaussian Noise for Generating Self-Similar Network Traffic

Recent network traffic studies argue that network arrival processes are much more faithfully modeled using statistically self-similar processes instead of traditional Poisson processes [LTWW94,PF95]. One difficulty in dealing with self-similar models is how to efficiently synthesize traces (sample paths) corresponding to self-similar traffic. We present a fast Fourier transform method for synthesizing approximate self-similar sample paths for one type of self-similar process, Fractional Gaussian Noise, and assess its performance and validity. We find that the method is as fast or faster than existing methods and appears to generate close approximations to true self-similar sample paths. We also discuss issues in using such synthesized sample paths for simulating network traffic, and how an approximation used by our method can dramatically speed up evaluation of Whittle's estimator for H, the Hurst parameter giving the strength of long-range dependence present in a self-similar time series.Comment: 14 page

Modeling the Impact of Protocols on Traffic Burstiness At Large Timescales in Wireless Multi-Hop Networks

We investigate the impact of the protocol stack on traffic burstiness at large time-scales in wireless multi-hop network traffic. Origins of traffic burstiness at large scales (like its LRD nature) have been mostly attributed to the heavy-tails in traffic sources. In wired networks, protocol dynamics have little impact on large time-scale dynamics. However, given the nature of wireless networks, the MAC and routing layers together can lead to route flapping or oscillations even in a static network. Hence, we explore whether these dynamics can lead to traffic burstiness and LRD. Using network simulations, we analyze traffic for two MANET routing protocols - OLSR and AODV. By varying the routing protocol parameters, we analyze their role in inducing or preventing route oscillations, and study their impact on traffic LRD. We find that, losses in OLSR control packets, due to congestion at the MAC, can lead to route oscillations and traffic burstiness at large timescales. By tuning the parameters, route oscillations and traffic LRD can be avoided. AODV dynamics show little evidence for traffic LRD, even though we cannot rule out this possibility. We also show that the route oscillations can have heavier body and tail than exponential distribution, and that the Markovian framework for route oscillations is inadequate to explain the observed traffic scaling. Lastly, we give a model that captures the MAC and OLSR routing protocol interactions and depending upon chosen protocol parameters and input load, correctly predicts the presence of traffic LRD. Thus, we use this model to design appropriate choice of protocol parameters to mitigate traffic burstiness at large-timescales.Research supported by the Army Research Office under MURI award W911NF-08-1-0238 and by the National Science Foundation under grant CNS1018346

Variable bit rate video time-series and scene modeling using discrete-time statistically self-similar systems

This thesis investigates the application of discrete-time statistically self-similar (DTSS) systems to modeling of variable bit rate (VBR) video traffic data. The work is motivated by the fact that while VBR video has been characterized as self-similar by various researchers, models based on self-similarity considerations have not been previously studied. Given the relationship between self-similarity and long-range dependence the potential for using DTSS model in applications involving modeling of VBR MPEG video traffic data is presented. This thesis initially explores the characteristic properties of the model and then establishes relationships between the discrete-time self-similar model and fractional order transfer function systems. Using white noise as the input, the modeling approach is presented using least-square fitting technique of the output autocorrelations to the correlations of various VBR video trace sequences. This measure is used to compare the model performance with the performance of other existing models such as Markovian, long-range dependent and M/G/(infinity) . The study shows that using heavy-tailed inputs the output of these models can be used to match both the scene time-series correlations as well as scene density functions. Furthermore, the discrete-time self-similar model is applied to scene classification in VBR MPEG video to provide a demonstration of potential application of discrete-time self-similar models in modeling self-similar and long-range dependent data. Simulation results have shown that the proposed modeling technique is indeed a better approach than several earlier approaches and finds application is areas such as automatic scene classification, estimation of motion intensity and metadata generation for MPEG-7 applications

PREDICTING INTERNET TRAFFIC BURSTS USING EXTREME VALUE THEORY

Computer networks play an important role in today’s organization and people life. These interconnected devices share a common medium and they tend to compete for it. Quality of Service (QoS) comes into play as to define what level of services users get. Accurately defining the QoS metrics is thus important. Bursts and serious deteriorations are omnipresent in Internet and considered as an important aspects of it. This thesis examines bursts and serious deteriorations in Internet traffic and applies Extreme Value Theory (EVT) to their prediction and modelling. EVT itself is a field of statistics that has been in application in fields like hydrology and finance, with only a recent introduction to the field of telecommunications. Model fitting is based on real traces from Belcore laboratory along with some simulated traces based on fractional Gaussian noise and linear fractional alpha stable motion. QoS traces from University of Napoli are also used in the prediction stage. Three methods from EVT are successfully used for the bursts prediction problem. They are Block Maxima (BM) method, Peaks Over Threshold (POT) method, and RLargest Order Statistics (RLOS) method. Bursts in internet traffic are predicted using the above three methods. A clear methodology was developed for the bursts prediction problem. New metrics for QoS are suggested based on Return Level and Return Period. Thus, robust QoS metrics can be defined. In turn, a superior QoS will be obtained that would support mission critical applications