Peter C. Kiessler

The M/GI/ 1 Bernoulli feedback queue with vacations

Feedback may be introduced as a mechanism for scheduling customer service (for example in systems in which customers bring work that is divided into a random number of stages). A model is developed that characterizes the queue length distribution as seen following vacations and service stage completions. We demonstrate the relationship that exists between these distributions. The ergodic waiting time distribution is formulated in such a way as to reveal the effects of server vacations when feedback is introduced.

Testing for reversibility in markov chain data

This paper introduces two statistics that assess whether (or not) a sequence sampled from a stationary time-homogeneous Markov chain on a finite state space is reversible. The test statistics are based on observed deviations of transition sample counts between each pair of states in the chain. First, the joint asymptotic normality of these sample counts is established. This result is then used to construct two chi-squared-based tests for reversibility. Simulations assess the power and type one error of the proposed tests.

Estimation in Reversible Markov Chains

This article examines estimation of the one-step-ahead transition probabilities in a reversible Markov chain on a countable state space. A symmetrized moment estimator is proposed that exploits the reversible structure. Examples are given where the symmetrized estimator has superior asymptotic properties to those of a naive estimator, implying that knowledge of reversibility can sometimes improve estimation. The asymptotic mean and variance of the estimators are quantified. The results are proven using only elementary results such as the law of large numbers and the central limit theorem.

ON THE CONVEXITY OF VALUE FUNCTIONS FOR A CERTAIN CLASS OF STOCHASTIC DYNAMIC PROGRAMMING PROBLEM

It is a common practice in stochastic dynamic programming problems to assume a priori that the value function is either concave or convex and later verify this assumption after the value function has been identified. It is often a difficult task to establish the concavity or convexity of the value function directly. In this paper, we prove that the value function of a certain type of infinite horizon stochastic dynamic programming problem is convex. This type of value function arises frequently in economic modeling.

A Prediction‐Residual Approach for Identifying Rare Events in Periodic Time Series

Many environmental time series have seasonal structures. This article presents an approach to identifying the rare events of such series based on time-series residuals. The methods justify the application of classical peaks over threshold methods to estimated versions of the one-step-ahead prediction errors of the series. Such methods enable the seasonal means, variances and autocorrelations of the series to be taken into account. Even in stationary settings, the proposed strategy is useful as it bypasses the need for blocking runs of extremes. The mathematics are justified via a limit theorem for a periodic autoregressive moving-average time series. A detailed application to a daily temperature series from Griffin, Georgia, is pursued.

Statistical Inference for Ergodic Diffusion Processes. Yury A. Kutoyants

Lévy Processes” (eight papers), “III. Empirical Processes” (four papers), and “IV. Stochastic Differential Equations” (four papers). Here are some comments about the individual papers: In I.2 (paper 2 of Part I) the covariance representation method, which relies on Clark’s formula on path spaces, is used to obtain concentration inequalities for functionals of Brownian motion on a manifold, allowing one to obtain tail estimates for this Brownian motion. In I.4 a transportation inequality for the canonical Gaussian measure in R is obtained and applied to Khintchine–Kahane inequalities for norms of random series with nonsymmetric Bernoulli coefficients. In II.1 exponential inequalities for U -statistics of order two are presented; these rely upon the Talagrand inequality for empirical processes but also use martingale type inequalities. In II.2 the unconditional convergence of a Gaussian [and, more generally, independent, identically distributed (iid)] series in a Banach space is studied. Applications to Karhunen–Love representations of Gaussian processes are given. In II.3 estimates of tail properties and moments of multidimensional chaos generated by positive random variables with log concave tails are given. In II.4 a quantitative technique for studying the asymptotic distribution of sequences of Markov processes in infinite dimensions is proposed. The proof relies on the properties of an associated sequence of exponential martingales. In II.5 it is shown that a moving average process driven by a symmetric Lévy process and with a kernel with finite total 2-variation admits an almost surely bounded version. In II.6 a Markovian approach to the entropic convergence in the central limit theorem is presented. The emphasis is on the speed of convergence, as well as relaxing a spectral gap assumption. In II.7 a new version of the Khintchine–Kahane inequality for general Bernoulli random variables is presented with the help of hypercontractive methods. In III.1 necessary and sufficient conditions for the moderate deviations of empirical processes and sums of iid random vectors on a separable Banach space are given. In III.2 exponential concentration inequalities for subadditive functions of independent random variables are obtained. As a consequence, Talagrand’s inequality for empirical processes is refined thanks to further developments of the entropy method introduced by M. Ledoux. In III.3 ratio limit theorems for empirical processes are obtained with the help of concentration inequalities. In III.4 asymptotic distributions of trimmed Wasserstein distances between the true and the empirical distribution function are obtained via weighted approximation results for uniform empirical processes. In IV.1 sharp rates of convergence for splitting-up approximations of stochastic partial differential equations are obtained. The error is estimated in terms of Sobolev’s norm. In IV.4 the existence and uniqueness of a strong solution for a stochastic differential equation driven by a fractional Brownian motion with Hurst index H < 1/2 and with a possibly time-dependent drift which satisfies a suitable integrability condition is obtained. In short, the book presents a wide variety of inequalities which are established for either stochastic processes or sequences of random variables. In 2006 this is still an active domain of research in transportation problems.

Rapid training of GIL neural networks

Applying generalized inverse learning to a feedforward neural network has been shown to be an effective tool in pattern recognition. The difficult computational step is finding the pseudo-inverse of a matrix. In this paper, we develop an efficient method using differential equations to calculate the pseudo-inverse.

Workload in queues having priorities assigned according to service time

System designers often implement priority queueing disciplines in order to improve overall system performance; however, improvement is often gained at the expense of lower priority cystomers. Shortest Processing Time is an example of a priority discipline wherein lower priority customers may suffer very long waiting times when compared to their waiting times under a democratic service discipline. In what follows, we shall investigate a queueing system where customers are divided into a finitie number of priority classes according to their service times.We develop the multivariate generating function characterizing the joint workload among the priority classes. First moments obtained from the generating function yield traffic intensities for each priority class. Second moments address expected workloads, in particular, we obtain simple Pollaczek-Khinchine type formulae for the classes. Higher moments address variance and covariance among the workloads of the priority classes.