The author's preface gives an outline: "This book is about weakconvergence methods in metric spaces, with applications sufficient to show their power and utility. The Introduction motivates the definitions and indicates how the theory will yield solutions to problems arising outside it. Chapter 1 sets out the basic general theorems, which are then specialized in Chapter 2 to the space C[0, l ] of continuous functions on the unit interval and in Chapter 3 to the space D [0, 1 ] of functions with discontinuities of the first kind. The results of the first three chapters are used in Chapter 4 to derive a variety of limit theorems for dependent sequences of random variables. " The book develops and expands on Donsker's 1951 and 1952 papers on the invariance principle and empirical distributions. The basic random variables remain real-valued although, of course, measures on C[0, l ] and D[0, l ] are vitally used. Within this framework, there are various possibilities for a different and apparently better treatment of the material. More of the general theory of weak convergence of probabilities on separable metric spaces would be useful. Metrizability of the convergence is not brought up until late in the Appendix. The close relation of the Prokhorov metric and a metric for convergence in probability is (hence) not mentioned (see V. Strassen, Ann. Math. Statist. 36 (1965), 423-439; the reviewer, ibid. 39 (1968), 1563-1572). This relation would illuminate and organize such results as Theorems 4.1, 4.2 and 4.4 which give isolated, ad hoc connections between weak convergence of measures and nearness in probability. In the middle of p. 16, it should be noted that C*(S) consists of signed measures which need only be finitely additive if 5 is not compact. On p. 239, where the author twice speaks of separable subsets having nonmeasurable cardinal, he means "discrete" rather than "separable." Theorem 1.4 is Ulam's theorem that a Borel probability on a complete separable metric space is tight. Theorem 1 of Appendix 3 weakens completeness to topological completeness. After mentioning that probabilities on the rationals are tight, the author says it is an

Convergence of probability measures

Praise for the Third Edition: "This is one of the best books available. Its excellent organizational structure allows quick reference to specific models and its clear presentation . . . solidifies the understanding of the concepts being presented."IIE Transactions on Operations EngineeringThoroughly revised and expanded to reflect the latest developments in the field, Fundamentals of Queueing Theory, Fourth Edition continues to present the basic statistical principles that are necessary to analyze the probabilistic nature of queues. Rather than presenting a narrow focus on the subject, this update illustrates the wide-reaching, fundamental concepts in queueing theory and its applications to diverse areas such as computer science, engineering, business, and operations research.This update takes a numerical approach to understanding and making probable estimations relating to queues, with a comprehensive outline of simple and more advanced queueing models. Newly featured topics of the Fourth Edition include:Retrial queuesApproximations for queueing networksNumerical inversion of transformsDetermining the appropriate number of servers to balance quality and cost of serviceEach chapter provides a self-contained presentation of key concepts and formulae, allowing readers to work with each section independently, while a summary table at the end of the book outlines the types of queues that have been discussed and their results. In addition, two new appendices have been added, discussing transforms and generating functions as well as the fundamentals of differential and difference equations. New examples are now included along with problems that incorporate QtsPlus software, which is freely available via the book's related Web site.With its accessible style and wealth of real-world examples, Fundamentals of Queueing Theory, Fourth Edition is an ideal book for courses on queueing theory at the upper-undergraduate and graduate levels. It is also a valuable resource for researchers and practitioners who analyze congestion in the fields of telecommunications, transportation, aviation, and management science.

Fundamentals of Queueing Theory

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

https://www.cs.ubc.ca/~murphyk/Teaching/CS540-Spring10/projects/Yu-hsmm09.pdf

Hidden semi-Markov models

The age of big data is now coming. But the traditional data 
analytics may not be able to handle such large quantities of data. The question that arises now is, how to develop a high performance platform to efficiently analyze big data and 
how to design an appropriate mining algorithm to find the useful things from big data. To deeply discuss this issue, this paper begins with a brief introduction to data analytics, followed by the discussions of big data analytics. 
Some important open issues and further research directions will also be presented for the next step of big data analytics.

/pdf/big-data-analytics-a-survey-2m5v4g89fb.pdf

Big data analytics: a survey

Fleet-sizing and service availability for a vehicle rental system via closed queueing networks

The overwhelming success of the Web as a mechanism for facilitating information retrieval and for conducting business transactions has ledto an increase in the deployment of complex enterprise applications. These applications typically run on Web Application Servers, which assume the burden of managing many tasks, such as concurrency, memory management, database access, etc., required by these applications. The performance of an Application Server depends heavily on appropriate configuration. Configuration is a difficult and error-prone task dueto the large number of configuration parameters and complex interactions between them. We formulate the problem of finding an optimal configuration for a given application as a black-box optimization problem. We propose a smart hill-climbing algorithm using ideas of importance sampling and Latin Hypercube Sampling (LHS). The algorithm is efficient in both searching and random sampling. It consists of estimating a local function, and then, hill-climbing in the steepest descent direction. The algorithm also learns from past searches and restarts in a smart and selective fashion using the idea of importance sampling. We have carried out extensive experiments with an on-line brokerage application running in a WebSphere environment. Empirical results demonstrate that our algorithm is more efficient than and superior to traditional heuristic methods.

/pdf/a-smart-hill-climbing-algorithm-for-application-server-36stzffisa.pdf

A smart hill-climbing algorithm for application server configuration

Parameter inference of queueing models for IT systems using end-to-end measurements

Recent advances in computer technology and wireless communications have enabled the emergence of stream-based sensor networks. In such sensor networks, real-time data are generated by a large number of distributed sources. Queries are made that may require sophisticated processing and filtering of the data. A query is represented by a query graph. In order to reduce the data transmission and to better utilize resources, it is desirable to place operators of the query graph inside the network, and thus to perform in-network processing. Moreover, given that various queries occur with different frequencies and that only a subset of sensor data may actually be queried, caching intermediate data objects inside the network can help improve query efficiency. In this paper, we consider the problem of placing both operators and intermediate data objects inside the network for a set of queries so as to minimize the total cost of storage, computation, and data transmission. We propose distributed algorithms that achieve optimal solutions for tree-structured query graph topologies and general network topologies. The algorithms converge in Lmax(.HQ + 1) iterations, where Lmax is the order of the diameter of the sensor network, and Hq represents the depth of the query graph, defined as the maximum number of operations needed for a raw data to become a final data. For a regular grid network and complete binary tree query graph, the complexity is 0(radic(N)log2 M), where N is the number of nodes in the sensor network and M is the number of data objects in a query graph. The most attractive features of these algorithms are that they require only information exchanges between neighbors, can be executed asynchronously, are adaptive to cost change and topology change, and are resilient to node or link failures.

http://inlab.lab.asu.edu/Publications/DQP.pdf

Distributed Operator Placement and Data Caching in Large-Scale Sensor Networks

As Information Technology (IT) matures and expands in the scope of available applications, IT systems increase at a fascinating rate in both size and complexity. For example, today, a typical Web service hosting center may have hundreds of nodes and dozens of different applications simultaneously running on it. Each of the nodes in turn has often multiple processors and layered caches. These nodes make use of both local and shared storage systems. The size and complexity of such systems make performance modeling much more difficult, if at all tractable. Detailed modeling, fine tuning and accurate analysis can be carried out only on very small IT systems or very small components in a system. On the other hand, IT systems have become vital to business. Losses of millions of dollars per minute when an IT system goes down are well-documented, leading users to impose stringent QoS requirements. In IT outsourcing, servicelevel agreements (SLA) stipulate service quality guarantees, often with associated penalties in case of violations. Thus, predictive modeling is vital in the capacity planning and QoS management of IT systems. We propose a method based on queueing networks. While many parameters such as the queueing discipline, number of servers in a queueing station, buffer size, etc., of the queueing network can be easily set by the modeler, the service requirements (i.e. job processing times) of customers are more difficult to estimate. However, without an accurate estimation of the service requirements, performance predictions can vary wildly. In other words, a principal difficulty in building a valid queueing network of an IT system is the fine tuning of the service requirements. This step requires benchmarking of the real system and an adjustment of the

Cathy H. Xia

Papers

Fleet-sizing and service availability for a vehicle rental system via closed queueing networks

A smart hill-climbing algorithm for application server configuration

Parameter inference of queueing models for IT systems using end-to-end measurements

Distributed Operator Placement and Data Caching in Large-Scale Sensor Networks

Parameter inference of queueing models for IT systems using end-to-end measurements