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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

While the recent focus on knowledge has undoubtedly benefited organizational studies, the literature still presents a sharply contrasting and even contradictory view of knowledge, which at times is described as "sticky" and at other times "leaky." This paper is written on the premise that there is more than a problem with metaphors at issue here, and more than accounts of different types of knowledge (such as "tacit" and "explicit") can readily explain. Rather, these contrary descriptions of knowledge reflect different, partial, and sometimes "balkanized" perspectives from which knowledge and organization are viewed. Taking the community of practice as a unifying unit of analysis for understanding knowledge in the firm, the paper suggests that often too much attention is paid to the idea of community, too little to the implications of practice. Practice, we suggest, creates epistemic differences among the communities within a firm, and the firm's advantage over the market lies in dynamically coordinating the knowledge produced by these communities despite such differences. In making this argument, we argue that analyses of systemic innovation should be extended to embrace all firms in a knowledge economy, not just the classically innovative. This extension will call for a transformation of conventional ideas coordination and of the trade-off between exploration and exploitation.

Knowledge and Organization: A Social-Practice Perspective

Publisher Summary This chapter reviews the supply chain coordination with contracts. Numerous supply chain models are discussed. In each model, the supply chain optimal actions are identified. The chapter extends the newsvendor model by allowing the retailer to choose the retail price in addition to the stocking quantity. Coordination is more complex in this setting because the incentives provided to align one action might cause distortions with the other action. The newsvendor model is also extended by allowing the retailer to exert costly effort to increase demand. Coordination is challenging because the retailer's effort is noncontractible—that is, the firms cannot write contracts based on the effort chosen. The chapter also discusses an infinite horizon stochastic demand model in which the retailer receives replenishments from a supplier after a constant lead time. Coordination requires that the retailer chooses a large basestock level.

Supply Chain Coordination with Contracts

Those who solve more of a given type of problem tend to get better at it-which suggests that problems of any given type should be brought to specialists for a solution. However, in this paper we argue that agency-related costs and information transfer costs ("sticky" local information) will tend drive the locus of problem-solving in the opposite direction-away from problem-solving by specialist suppliers, and towards those who directly benefit from a solution and who have difficult-to-transfer local information about a particular application being solved, such as the direct users of a product or service. We examine the actual location of design activities in two fields in which custom products are produced by "mass-customization" methods: application-specific integrated circuits (ASICs) and computer telephony integration (CTI) systems. In both, we find that users rather than suppliers are the actual designers of the application-specific portion of the product types examined. We offer anecdotal evidence that the pattern of user-based customization we have documented in these two fields is in fact quite general, and we discuss implications for research and practice.

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Economics of Product Development by Users: the Impact of Sticky Local Information

The most common model to support workforce management of telephone call centers is theM/ M/ N/ B model, in particular its special casesM/ M/ N (Erlang C, which models out busy signals) andM/ M/ N/ N (Erlang B, disallowing waiting). All of these models lack a central prevalent feature, namely, that impatient customers might decide to leave (abandon) before their service begins.In this paper, we analyze the simplest abandonment model, in which customers' patience is exponentially distributed and the system's waiting capacity is unlimited ( M/ M/ N +M). Such a model is both rich and analyzable enough to provide information that is practically important for call-center managers. We first outline a method for exact analysis of theM/ M/ N +M model, that while numerically tractable is not very insightful. We then proceed with an asymptotic analysis of theM/ M/ N +M model, in a regime that is appropriate for large call centers (many agents, high efficiency, high service level). Guided by the asymptotic behavior, we derive approximations for performance measures and propose "rules of thumb" for the design of large call centers. We thus add support to the growing acknowledgment that insights from diffusion approximations are directly applicable to management practice.

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Designing a Call Center with Impatient Customers

This paper presents heavy traffic limit theorems for the queue length and sojourn time processes associated with open queueing networks. These limit theorems state that properly normalized sequences of queue length and sojourn time processes converge weakly to a certain diffusion as the network traffic intensity converges to unity. The limit diffusion is reflected Brownian motion on the nonnegative orthant. This process behaves like Brownian motion on the interior of its state space, and reflects instantaneously on the boundaries. The reflection direction is a constant for each boundary hyperplane.

Open Queueing Networks in Heavy Traffic

We consider a $K$-dimensional diffusion process $Z$ whose state space is the nonnegative orthant. On the interior of the orthant, $Z$ behaves like a $K$-dimensional Brownian motion with arbitrary covariance matrix and drift vector. At each of the (`K-1) dimensional hyperplanes that form the boundary of the orthant, $Z$ reflects instantaneously in a direction that is constant over that hyperplane. There is no extant theory of multidimensional diffusion that applies to this process, because the boundary of its state space is not smooth. We adopt an approach that requires a restriction on the directions of reflection, but Reiman has shown that this restriction is met by all diffusions $Z$ arising as heavy traffic limits in open $K$-station queuing networks. Our process $Z$ is defined as a path-to-path mapping of $K$-dimensional Brownian motion. From this construction it follows that $Z$ is a continuous Markov process and a semimartingale. Using the latter property, we obtain a change of variable formula from which one can develop a complete analytical theory for the process $Z$.

Reflected Brownian Motion on an Orthant

We consider a serial inventory system with N stages. The material flows from an outside supplier to stage N, then to stage N - 1, etc., and finally to stage 1 where random customer demand arises. Each stage replenishes a stage-specific inventory position according to a stage-specific reorder point/order quantity policy. Two variations of this policy are considered. One is based on echelon stock, and the other installation stock. The former requires centralized demand information, while the latter does not. The relative cost difference between the two policies is called the value of centralized demand information. For fixed order quantities, we develop efficient algorithms for computing both the optimal echelon reorder points and the optimal installation reorder points. These algorithms enable us to conduct an extensive computational study to assess the value of centralized demand information and to understand how this value depends on several key system parameters, i.e., the number of stages, lead times, batch sizes, demand variability, and the desired level of customer service.

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Martin I. Reiman

Papers

Economics of Product Development by Users: the Impact of Sticky Local Information

Designing a Call Center with Impatient Customers

Open Queueing Networks in Heavy Traffic

Reflected Brownian Motion on an Orthant

Echelon Reorder Points, Installation Reorder Points, and the Value of Centralized Demand Information