Stationary Distribution Convergence of the Offered Waiting Processes for GI/GI/1+GI Queues in Heavy Traffic

A result of Ward and Glynn (2005) asserts that the sequence of scaled offered waiting time processes of the $GI/GI/1+GI$ queue converges weakly to a reflected Ornstein-Uhlenbeck process (ROU) in the positive real line, as the traffic intensity approaches one. As a consequence, the stationary distribution of a ROU process, which is a truncated normal, should approximate the scaled stationary distribution of the offered waiting time in a $GI/GI/1+GI$ queue; however, no such result has been proved. We prove the aforementioned convergence, and the convergence of the moments, in heavy traffic, thus resolving a question left open in Ward and Glynn (2005). In comparison to Kingman's classical result in Kingman (1961) showing that an exponential distribution approximates the scaled stationary offered waiting time distribution in a $GI/GI/1$ queue in heavy traffic, our result confirms that the addition of customer abandonment has a non-trivial effect on the queue stationary behavior.Comment: 29 page

A Method for Detecting Murmurous Heart Sounds based on Self-similar Properties

A heart murmur is an atypical sound produced by the flow of blood through the heart. It can be a sign of a serious heart condition, so detecting heart murmurs is critical for identifying and managing cardiovascular diseases. However, current methods for identifying murmurous heart sounds do not fully utilize the valuable insights that can be gained by exploring intrinsic properties of heart sound signals. To address this issue, this study proposes a new discriminatory set of multiscale features based on the self-similarity and complexity properties of heart sounds, as derived in the wavelet domain. Self-similarity is characterized by assessing fractal behaviors, while complexity is explored by calculating wavelet entropy. We evaluated the diagnostic performance of these proposed features for detecting murmurs using a set of standard classifiers. When applied to a publicly available heart sound dataset, our proposed wavelet-based multiscale features achieved comparable performance to existing methods with fewer features. This suggests that self-similarity and complexity properties in heart sounds could be potential biomarkers for improving the accuracy of murmur detection

Long time stability and control problems for stochastic networks in heavy traffic

Stochastic processing networks arise commonly from applications in computers, telecommunications, and large manufacturing systems. Study of stability and control for such networks is an active and important area of research. In general the networks are too complex for direct analysis and therefore one seeks tractable approximate models. Heavy traffic limit theory yields one of the most useful collection of such approximate models. Typical results in the theory say that, when the network processing resources are roughly balanced with the system load, one can approximate such systems by suitable diffiusion processes that are constrained to live within certain polyhedral domains (e.g., positive orthants). Stability and control problems for such diffusion models are easier to analyze and, once these are resolved, one can then infer stability properties and construct good control policies for the original physical networks. In my dissertation we consider three related problems concerning stability and long time control for such networks and their diffusion approximations. In the first part of the dissertation we establish results on long time asymptotic properties, in particular geometric ergodicity, for limit diffusion models obtained from heavy traffic analysis of stochastic networks. The results provide the rate of convergence to steady state, moment estimates for steady state, uniform in time moment estimates for the process and central limit type results for time averages of such processes. In the second part of the dissertation we consider invariant distributions of an important subclass of stochastic networks, namely the generalized Jackson networks (GJN). It is shown that, under natural stability and heavy traffic conditions, the invariant distributions of GJN converge to unique invariant probability distribution of the corresponding constrained diffusion model. The result leads to natural methodologies for approximation and simulation of steady state behavior of such networks. In the final part of the dissertation we consider a rate control problem for stochastic processing networks with an ergodic cost criterion. It is shown that value functions and near optimal controls for limit diffusion models serve as good approximations for the same quantities for the underlying physical queueing networks that are heavily loaded

Direction-Projection-Permutation for High Dimensional Hypothesis Tests

Motivated by the prevalence of high dimensional low sample size datasets in modern statistical applications, we propose a general nonparametric framework, Direction-Projection-Permutation (DiProPerm), for testing high dimensional hypotheses. The method is aimed at rigorous testing of whether lower dimensional visual differences are statistically significant. Theoretical analysis under the non-classical asymptotic regime of dimension going to infinity for fixed sample size reveals that certain natural variations of DiProPerm can have very different behaviors. An empirical power study both confirms the theoretical results and suggests DiProPerm is a powerful test in many settings. Finally DiProPerm is applied to a high dimensional gene expression dataset

Direction-Projection-Permutation for High-Dimensional Hypothesis Tests

High-dimensional low sample size (HDLSS) data are becoming increasingly common in statistical applications. When the data can be partitioned into two classes, a basic task is to construct a classifier that can assign objects to the correct class. Binary linear classifiers have been shown to be especially useful in HDLSS settings and preferable to more complicated classifiers because of their ease of interpretability. We propose a computational tool called direction-projection-permutation (DiProPerm), which rigorously assesses whether a binary linear classifier is detecting statistically significant differences between two high-dimensional distributions. The basic idea behind DiProPerm involves working directly with the one-dimensional projections of the data induced by binary linear classifier. Theoretical properties of DiProPerm are studied under the HDLSS asymptotic regime whereby dimension diverges to infinity while sample size remains fixed. We show that certain variations of DiProPerm are consistent and that consistency is a nontrivial property of tests in the HDLSS asymptotic regime. The practical utility of DiProPerm is demonstrated on HDLSS gene expression microarray datasets. Finally, an empirical power study is conducted comparing DiProPerm to several alternative two-sample HDLSS tests to understand the advantages and disadvantages of each method. Susan Wei, Department of Statistics and Operations Research, University of North Carolina - Chapel Hill, NC 27599-3260 (E-mail: [email protected] ). Chihoon Lee, Assistant Professor, Department of Statistics and Operations Research, University of North Carolina - Chapel Hill, NC 27599-3260 and currently at Department of Statistics, Colorado State University, Fort Collins, CO 80523-1877 (E-mail: [email protected] ). Lindsay Wichers, Department of Environmental Sciences and Engineering, School of Public Health, University of North Carolina - Chapel Hill, NC 27599-3260 and currently at Environmental Media Assessment Group, MD B243-01, National Center for Environmental Assessment, Office of Research and Development, U.S. Environmental Protection Agency, Research Triangle Park, NC 27711 (E-mail: [email protected] ), J. S. Marron, Department of Statistics and Operations Research, University of North Carolina - Chapel Hill, NC 27599-3260 (E-mail: [email protected] )