Heavy traffic analysis for a multiplexer driven by M∨GI∨∞ input processes

Long-range dependence has been recently asserted to be an important characteristic in modeling telecommunications traffic. Inspired by the integral relationship between the fractional Brownian motion and the standard Brownian motion, we model a process with long-range dependence, Y, as a fractional integral of Riemann-Liouville type applied to a more standard process X—one that does not have long-range dependence. When X takes the form of a sample path process with bounded stationary increments, we provide a criterion for X to satisfy a moderate deviations principle (MDP). Based on the MDP of X, we then establish the MDP for Y. Furthermore, we characterize, in terms of the MDP, the transient behavior of queues when fed with the long-range dependent input process Y. In particular, we identify the most likely path that leads to a large queue, and demonstrate that unlike the case where the input has short-range dependence, the path here is nonlinear.

Intuition suggests that a finite buffer limits the effect of traffic auto-correlations on the queue length. We investigate the extent to which finite buffers can, therefore, be expected to mitigate the effects of long-range dependence (LRD). With traffic sequences generated by the fractional autoregressive integrated moving average (f-ARIMA) models for LRD, and by AR models for short range dependence (SRD), we investigate the traffic performance for a range of finite buffers, both for single and multiplexed streams. For design, the aim is to 'match' a given LRD auto-correlation function with a suitable SRD function whose performance provides, for a wide range of traffic intensities and buffer sizes, a conservative bound on the performance associated with the LRD function. The results suggest that by suitably 'dominating' an LRD auto-correlation function by an SRD function, one can obtain conservative performance bounds for a realistic range of traffic intensities and buffer sizes (delays). Also, in several cases, a 'cross-over' phenomenon is observed for cell-loss probabilities as the buffer size increases, i.e., the loss probabilities are smaller for LRD than for SRD for small buffers, with the converse true for large buffers. This suggests that finite buffers, can in some cases, counteract the effects of LRD in traffic arrivals, and thus enable conservative designs to be based on Markovian traffic models.

Solutions to BSDEs driven by both fractional Brownian motions and the underlying standard Brownian Motions

The sub-fractional CEV model

Network traffic exhibits fractal properties, such as self- similarity and impulsiveness, and posseses a complex correlation structure, i. e., both long and short-range dependence are present. Models such as Fractional Gaussian Noise can capture the long-range dependence but not the short-range dependence. Therefore, a key issue that must be adressed when using synthesized traffic for network simulations is the incorporation of short-range dependence into the generated trace. This work develops a filter-based method for the introduction of short- range dependence into long-range dependent sample paths that are synthesized via wavelets and studies a method of mapping from a given time series to a packet stream. The generator does not employ training sequences, i. e., it is not based on the use of real traffic traces as training sequences.

Strong approximation of fractional Brownian motion by moving averages of simple random walks

<p>Since the fractional $ G $-Brownian motion (fGBm) generalizes the concepts of the standard Brownian motion, fractional Brownian motion, and $ G $-Brownian motion, while it can exhibit long-range dependence or antipersistence and feature the volatility uncertainty simultaneously, it can be a better alternative stochastic process in the financial applications. Thus, in this paper, some empirical studies for the financial applications of the fGBm were carried out, where the recent high-frequency data for some selected assets in the financial market are from the Oxford-Man Institute of Quantitative Finance Realized Library. There are two main empirical findings. One was that the H-$ G $-normal distributions associated with the fGBm are more suitable in describing the dynamics of daily returns and increments of log-volatility for these assets than the usual distributions, since they not only characterize the properties of skewness, excess kurtosis, and long-range dependence $ \left(\frac{1}{2} < H < 1\right) $ or antipersistence $ \left(0 < H < \frac{1}{2}\right) $, but also feature the volatility uncertainty. The other one was that the daily return and log-volatility both behave essentially as fGBm with different $ \underline{\sigma}^2 $ and $ \overline{\sigma}^2 $, but Hurst parameters $ H < \frac{1}{2} $, at any reasonable time scale. Then a generalized stochastic model for the dynamics of the assets called rough fractional stochastic volatility model driven by fGBm (RFSV-fGBm) was developed. Finally, some parameter estimates and numerical experiments for the RFSV-fGBm model were investigated and carried out.</p>

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

The long-range dependence and self-similarity of fractional Brownian motion make it an attractive model for traffic in many data transfer networks. Reflected fractional Brownian Motion appears in the storage process of such a network. In this paper, we focus on the simulation of reflected fractional Brownian motion using a straightforward discretization scheme and we show that its strong error is of order hH, where h is the discretization step and H ∈ (0, 1) is the Hurst index.

The objective of this paper is to demonstrate the limitations of the variance-time (VT) test as a statistical tool for inferring long-range dependence (LRD) in network traffic. Since the early Bellcore studies, LRD has been in the center of a continuous debate within the teletraffic community. The controversy is typically focused on the utility of LRD models to predict the performance at network buffers. Our work here is not meant to advocate one modeling approach over another, but to point out (experimentally and theoretically) to the caveats in using the VT test as a tool for detecting LRD. We do that by deriving simple analytical expressions for the slope of the aggregated variance in three autocorrelated traffic models: M/G//spl infin/ process (short-range dependent (SRD) but non-Markovian), the discrete autoregressive of order one model (SRD Markovian), and the fractional ARIMA process (LRD). Our main result is that the VT test often indicates, falsely, the existence of an LRD structure (i.e., H>0.5) in synthetically generated traces from the two SRD models. The bias in the VT test, however, diminishes monotonically with the length of the trace. We provide some guidelines on selecting the minimum trace length so that the bias is negligible.

We investigate transport layer processing intended to improve the communication of multimedia data over wireless and wireline networks. Specifically, we consider data striping and thinning techniques, which are applicable to the multipath/multiflow transmission of multimedia (e.g., MPEG-4). We also introduce the shuffling procedure, which reorders the data at the network edges. We show that all three techniques break up short-term correlations in data traffic thus improving its queueing performance. We demonstrate that while both long range dependence (LRD) and short range dependence (SRD) influence the queueing performance for some timescale of the queueing system, it is the short-range statistical properties of multimedia traffic within the critical time scale that are dominant in determining the buffer efficiency of the queue. We also show that estimation of LRD for thinned and striped data may lead to a misleading notion of LRD reduction, when none is warranted. We further outline the ideas for a new transport layer protocol that explores the combination of thinning, striping and shuffling approaches to multimedia data transmission.

Time series modeling and analysis provides means of predicting the future and has been widely used in a variety of fields ranging from seismology for predicting earthquake and volcanic eruption, to finance for risk assessment, and to quantum information processing. The conventional integer order models can only capture short-range dependence; for example, Poisson processes, Markov processes, autoregressive (AR), moving average (MA), autoregressive moving average (ARMA) and autoregressive integrated moving average (ARIMA) processes. In time series analysis, one of the conventional assumptions is that the coupling between values at different time instants decreases rapidly as the time difference or distance increases. However, there are situations where strong coupling between values at different times exhibit properties of long range dependence which cannot be processed by the conventional time series analysis. Typical examples of long range dependence signals include financial time series, underwater noise, electroencephalography (EEG) signal, etc. ARFIMA, a fractional order signal processing technique, is the generalization of the conventional integer order techniques, namely, ARIMA and ARMA methods. Hence, it is capable of capturing both short-range dependence and long-range dependence in signals. Compared to conventional integer order models, the ARFIMA model gives a better fit and result when dealing with the data which possess the long range dependence property. In this paper, we investigate the application of the ARFIMA as well as AR methods to model EEG signals obtained from different brain channels. We analyze the resulting correlations for comparison the benefits of ARFIMA over AR on the EEG data exhibiting the long range dependency property. The results showed that the prediction results have a better performance compared to the conventional ARMA models.

In this paper, we focus on a class of stochastic differential equations driven by the standard Brownian motion and fractional Brownian motion with Hurst parameter under a weaker conditions than Lipschitz one. In the sense of the pathwise Riemann–Stieltjes integral, we prove the convergence of solutions for the considered equations. By making use of some transformation techniques and approximation means, we obtain some sufficient conditions on the viability for the stochastic systems under investigation. Subsequently, by using some distance functions, we establish some necessary conditions and some sufficient conditions for the viability property with respect to a given non-empty closed set K, but K can be not smooth. As applications of the results, we explore the comparison theorems about the mixed stochastic differential equations driven by fractional Brownian motion under some non-Lipschitz conditions.

Application of a long range dependent model includes several fields of science and economics as geophysics, hydrology, turbulence, weather and so on. Recently it has been successfully used for modeling network traffic data (see [WTLW]). The basic stochastic process of this kind is the fractional Brownian motion defined in [MvN]. The fractional Brownian motion is given as a particular fractional operator on the standard Brownian motion. The linear or Gaussian parametric models of long range dependent phenomena are both the linear stochastic differential equations with fractional Brownian motion input and the fractional operator on the solution of a linear stochastic differential equation (see [C]). Actually these two types of processes are equivalent. Because most of the observations are not Gaussian there is a need nonlinear modeling of long range dependence. One possibility is to get rid of Gaussianity is the bilinear model started by Subba Rao [SR] in the discrete time case. The easy way to get a long range non-Gaussian process is to apply the fractional operator to the solution of the bilinear SDE. It is more painful to consider a bilinear SDE with fractional Brownian motion input.

Network study finds simultaneous presentation of long-range dependence and short-range dependence in Network traffic., which makes the performance of the traditional model used to describe the short-range dependence for prediction lower. And in different scales of these two characteristics of network traffic have different impact for network performance. In small enough scale the short-range dependence has a greater impact on network performance, while in large enough scale, long-range dependence characteristics play a leading role. Therefore, in order to obtain better prediction this paper makes network traffic sequence more smooth and easy to be fitted by the traditional model, through weakening the dependence not affording a major role in small scale and enhancing the dependence playing a leading role in the impact of network.