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This contribution is concerned with Gumbel limiting results for supremum M-n = sup(t epsilon[0,Tn])X(n)(t)vertical bar with X (n) ,n epsilon N-2 centered Gaussian random fields with continuous trajectories. We show first the convergence of a related point process to a Poisson point process thereby extending previous results obtained in [8] for Gaussian processes. Furthermore, we derive Gumbel limit results for M-n as n -> infinity and show a second-order approximation for E{M-n(p)}(1/p) for any p >= 1.

In large scale data centers, a single fault can lead to correlated failures of several physical machines and the tasks running on them, simultaneously. Such correlated failures can severely damage the reliability of a service or a job running on the failed hardware. This paper models the impact of stochastic and correlated failures on job reliability in a data center. We focus on correlated failures caused by power outages or failures of network components, on jobs running multiple replicas of identical tasks. We present a statistical reliability model and an approximation technique for computing a job’s reliability in the presence of correlated failures. In addition, we address the problem of scheduling a job with reliability constraints.We formulate the scheduling problem as an optimization problem, with the aim being to maintain the desired reliability with the minimum number of extra tasks to resist failures.We present a scheduling algorithm that approximates the minimum number of required tasks and a placement to achieve a desired job reliability. We study the efficiency of our algorithm using an analytical approach and by simulating a cluster with different failure sources and reliabilities. The results show that the algorithm can effectively approximate the minimum number of extra tasks required to achieve the job’s reliability.

Entropy-type integral functionals of densities are widely used in mathematical statistics, information theory, and computer science. Examples include measures of closeness between distributions (e.g., density power divergence) and uncertainty characteristics for a random variable (e.g., Renyi entropy). In this paper, we study U-statistic estimators for a class of such functionals. The estimators are based on ε-close vector observations in the corresponding independent and identically distributed samples. We prove asymptotic properties of the estimators (consistency and asymptotic normality) under mild integrability and smoothness conditions for the densities. The results can be applied in diverse problems in mathematical statistics and computer science (e.g., distribution identication problems, approximate matching for random databases, two-sample problems).

We consider the rate of piecewise constant approximation to a locally stationary process X(t),t∈[0,1] , having a variable smoothness index α(t) . Assuming that α(⋅) attains its unique minimum at zero and satisfies α(t)=α0+btγ+o(tγ) as t→0, we propose a method for construction of observation points (composite dilated design) such that the integrated mean square error ∫10E{(X(t)−Xn(t))2}dt∼Knα0(logn)(α0+1)/γ as n→∞, where a piecewise constant approximation Xn is based on N(n)∼n observations of X . Further, we prove that the suggested approximation rate is optimal, and then show how to find an optimal constant K .

In this paper, we investigate the boundary non-crossing probabilities of a fractional Brownian motion considering some general deterministic trend function. We derive bounds for non-crossing probabilities and discuss the case of a large trend function. As a by-product, we solve a minimization problem related to the norm of the trend function.

parameter θ and where the noise is modeled as fractional Brownian motionwith Hurst index H ∈ (0, 12 ). The solution corresponds to the fractionalOrnstein–Uhlenbeck process. We construct an estimator, based on discreteobservations in time, of the unknown drift parameter, that is similar in formto the maximum likelihood estimator for the drift parameter in Langevinequation with standard Brownian motion. It is assumed that the intervalbetween observations is n−1, i.e. tends to zero (high-frequency data) andthe number of observations increases to infinity as nm with m > 1. It isproved that for strictly positive θ the estimator is strongly consistent forany m > 1, while for θ ≤ 0 it is consistent when m > 12H .

We consider the problem of numerical approximation of integrals of random fields over a unit hypercube. We use a stratified Monte Carlo quadrature and measure the approximation performance by the mean squared error. The quadrature is defined by a finite number of stratified randomly chosen observations with the partition generated by a rectangular grid (or design). We study the class of locally stationary random fields whose local behavior is like a fractional Brownian field in the mean square sense and find the asymptotic approximation accuracy for a sequence of designs for large number of the observations. For the H¨older class of random functions, we provide an upper bound for the approximation error. Additionally, for a certain class of isotropic random functions with an isolated singularity at the origin, we construct a sequence of designs eliminating the effect of the singularity point.

In this chapter, we consider a problem of statistical estimation of an unknown drift parameter for a stochastic differential equation driven by fractional Brownian motion. Two estimators based on discrete observations of solution to the stochastic differential equations are constructed. It is proved that the estimators converge almost surely to the parameter value, as the observation interval expands and the distance between observations vanishes. A bound for the rate of convergence is given and numerical simulations are presented. As an auxilliary result of independent interest we establish global estimates for fractional derivative of fractional Brownian motion.

Accurate understanding of workloads is key to efficient cloud resource management as well as to the design of large-scale applications. We analyze and model the workload of Wikipedia, one of the world's largest web sites. With descriptive statistics, time-series analysis, and polynomial splines, we study the trend and seasonality of the workload, its evolution over the years, and also investigate patterns in page popularity. Our results indicate that the workload is highly predictable with a strong seasonality. Our short term prediction algorithm is able to predict the workload with a Mean Absolute Percentage Error of around 2%.

Workload burstiness and spikes are among the main reasons for service disruptions and decrease in the Quality-of-Service (QoS) of online services. They are hurdles that complicate autonomic resource management of datacenters. In this paper, we review the state-of-the-art in online identification of workload spikes and quantifying burstiness. The applicability of some of the proposed techniques is examined for Cloud systems where various workloads are co-hosted on the same platform. We discuss Sample Entropy (SampEn), a measure used in biomedical signal analysis, as a potential measure for burstiness. A modification to the original measure is introduced to make it more suitable for Cloud workloads.