Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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

Economic Control of Quality of Manufactured Product.

Simulation modelling involves the development of models that imitate realworld operations, and statistical analysis of their performance with a view to improving efficiency and effectiveness This nontechnical textbook is focused towards the needs of business, engineering and computer science students

Simulation: The Practice of Model Development and Use

Probably one of the most successful interfaces between operations research and computer science has been the development of discrete-event simulation software. The recent integration of optimization techniques into simulation practice, specically into commercial software, has become nearly ubiquitous, as most discrete-event simulation packages now include some form of ?optimization? routine. The main thesis of this article, how-ever,is that there is a disconnect between research in simulation optimization--which has addressed the stochastic nature of discrete-event simulation by concentratingon theoretical results of convergence and specialized algorithms that are mathematically elegant--and the recent software developments, which implement very general algorithms adopted from techniques in the deterministic optimization metaheuristic literature (e.g., genetic algorithms, tabu search, artificial neural networks). A tutorial exposition that summarizes the approaches found in the research literature is included, as well as a discussion contrasting these approaches with the algorithms implemented in commercial software. The article concludes with the author's speculations on promising research areas and possible future directions in practice.

Feature Article: Optimization for simulation: Theory vs. Practice

We present procedures for selecting the best or near-best of a finite number of simulated systems when best is defined by maximum or minimum expected performance. The procedures are appropriate when it is possible to repeatedly obtain small, incremental samples from each simulated system. The goal of such a sequential procedure is to eliminate, at an early stage of experimentation, those simulated systems that are apparently inferior, and thereby reduce the overall computational effort required to find the best. The procedures we present accommodate unequal variances across systems and the use of common random numbers. However, they are based on the assumption of normally distributed data, so we analyze the impact of batching (to achieve approximate normality or independence) on the performance of the procedures. Comparisons with some existing indifference-zone procedures are also provided.

A fully sequential procedure for indifference-zone selection in simulation

In this paper we address the problem of finding the simulated system with the best (maximum or minimum) expected performance when the number of systems is large and initial samples from each system have already been taken This problem may be encountered when a heuristic search procedure--perhaps one originally designed for use in a deterministic environment--has been applied in a simulation-optimization context Because of stochastic variation, the system with the best sample mean at the end of the search procedure may not coincide with the true best system encountered during the search This paper develops statistical procedures that return the best system encountered by the search (or one near the best) with a prespecified probability We approach this problem using combinations of statistical subset selection and indifference-zone ranking procedures The subset-selection procedures, which use only the data already collected, screen out the obviously inferior systems, while the indifference-zone procedures, which require additional simulation effort, distinguish the best from the less obviously inferior systems

http://users.iems.northwestern.edu/~nelsonb/Publications/BoeselNelsonKim.pdf

Using Ranking and Selection to Clean Up after Simulation Optimization

We present fully sequential procedures for steady-state simulation that are designed to select the best of a finite number of simulated systems when best is defined by the largest or smallest long-run average performance. We also provide a framework for establishing the asymptotic validity of such procedures and prove the validity of our procedures. An example based on the M/M/1 queue is given.

http://users.iems.northwestern.edu/~nelsonb/Publications/KimNelsonOR.pdf

On the Asymptotic Validity of Fully Sequential Selection Procedures for Steady-State Simulation

This tutorial provides an overview on recent advances made in ranking and selection (R&S) for selecting the best simulated system and discusses challenges that still exist in the field. We focus on indifference-zone R&S procedures that provide a guaranteed probability of correct selection when the best system is at least a user-specified amount better than the other systems.

/pdf/recent-advances-in-ranking-and-selection-3c11z1xpvu.pdf

Recent advances in ranking and selection

We consider the problem of finding the system with the best primary performance measure among a finite number of simulated systems in the presence of a stochastic constraint on a single real-valued secondary performance measure. Solving this problem requires the identification and removal from consideration of infeasible systems (Phase I) and of systems whose primary performance measure is dominated by that of other feasible systems (Phase II). We use indifference zones in both phases and consider two approaches, namely, carrying out Phases I and II sequentially and carrying out Phases I and II simultaneously, and we provide specific example procedures of each type. We present theoretical results guaranteeing that our approaches (general and specific, sequential and simultaneous) yield the best system with at least a prespecified probability, and we provide a portion of an extensive numerical study aimed at evaluating and comparing the performance of our approaches. The experimental results show that both new procedures are useful for constrained ranking and selection, with neither procedure showing uniform superiority over the other. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 57: 403-421, 2010

Seong-Hee Kim

Papers

A fully sequential procedure for indifference-zone selection in simulation

Using Ranking and Selection to Clean Up after Simulation Optimization

On the Asymptotic Validity of Fully Sequential Selection Procedures for Steady-State Simulation

Recent advances in ranking and selection

Fully sequential procedures for comparing constrained systems via simulation