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Convex Analysisの二,三の進展について

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

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

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Price experimentation is an important tool for firms to find the optimal selling price of their products. It should be conducted properly, since experimenting with selling prices can be costly. A firm, therefore, needs to find a pricing policy that optimally balances between learning the optimal price and gaining revenue. In this paper, we propose such a pricing policy, called controlled variance pricing CVP. The key idea of the policy is to enhance the certainty equivalent pricing policy with a taboo interval around the average of previously chosen prices. The width of the taboo interval shrinks at an appropriate rate as the amount of data gathered gets large; this guarantees sufficient price dispersion. For a large class of demand models, we show that this procedure is strongly consistent, which means that eventually the value of the optimal price will be learned, and derive upper bounds on the regret, which is the expected amount of money lost due to not using the optimal price. Numerical tests indicate that CVP performs well on different demand models and time scales.

This paper was accepted by Assaf Zeevi, stochastic models and simulation.

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Simultaneously Learning and Optimizing Using Controlled Variance Pricing

Block rate pricing of piped water in Indonesian cities has a progressive structure: the marginal price paid increases with the volume of demand. This paper estimates household water demand in Salatiga city using the Burtless and Hausman model, and finds that its distribution is not unimodal—that data cluster around kinks. The main estimation results are a price elasticity of approximately–1.2 and an income elasticity of 0.05. These elasticities are mutually dependent. The estimated model is used to investigate the social welfare consequences of a shift to uniform pricing. The principal beneficiaries would be large households, which are not necessarily wealthy. While replacing the complex rate structure by a uniform marginal price would have positive effects on average welfare, the equity consequences would be small. To improve equity, water companies could reduce installation fees, giving low-income households access to water connections, or reinvest profits in network expansion to unserviced areas.

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Block Rate Pricing of Water in Indonesia: An Analysis of Welfare Effects

This paper gives a detailed empirical analysis of the relationships between different indicators of costs of commuting trips by car: difference as the crow flies, shortest travel time according to route planner, corresponding travel distance, and reported travel time. Reported travel times are usually rounded in multiples of five minutes. This calls for special statistical techniques. Ignoring the phenomenon of rounding leads to biased estimation results for shorter distances. Rather surprisingly, the distance as the crow flies and the network distance appear to be slightly better proxies of the reported travel time compared with the shortest network travel time as indicated by the route planner. We conclude that where actual driving times are missing in commuting research the other three indicators mentioned may be used as proxies, but that the following problems may emerge: actual travel times may be considerably higher than network times generated by route planners, and the average speed of trips increases considerably with distance, implying an overestimate of travel time for long distance commuters. The only personal feature that contributes significantly to variations in reported travel times is gender: women appear to drive at lower average speeds according to our data. As indicated in the paper this may be explained by the differences in the car types of male and female drivers (females drive older and smaller cars) as well as higher numbers of stops/trip chaining among women. A concise analysis is carried out for carpoolers. Car-pooling leads to an increase in travel time of some 17% compared with solo drivers covering the same distance. In the case of car poolers, the above mentioned measures appear to be very poor proxies for the actual commuting times.

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On the relationship between travel time and travel distance of commuters. Reported versus network travel data in the Netherlands

We apply a new corrected diffusion approximation for the Erlang C formula to determine staffing levels in cost minimization and constraint satisfaction problems. These problems are motivated by large customer contact centers that are modeled as an M/M/s queue with s the number of servers or agents. The proposed staffing levels are refinements of the celebrated square-root safety-staffing rule and have the appealing property that they are as simple as the conventional square-root safety-staffing rule. In addition, we provide theoretical support for the empirical fact that square-root safety-staffing works well for moderate-sized systems.

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Refining Square-Root Safety Staffing by Expanding Erlang C

This paper gives an overview of recent research on the impact of scheduling on the tail behavior of the response time of a job. We cover preemptive and non-preemptive scheduling disciplines, consider light-tailed and heavy-tailed distributions, and discuss optimality properties. The focus is on results, intuition and insight rather than methods and techniques.

Bert Zwart

Papers

Simultaneously Learning and Optimizing Using Controlled Variance Pricing

Block Rate Pricing of Water in Indonesia: An Analysis of Welfare Effects

On the relationship between travel time and travel distance of commuters. Reported versus network travel data in the Netherlands

Refining Square-Root Safety Staffing by Expanding Erlang C

Tails in scheduling