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Measure-Valued Differentiation for Finite Products of Measures : Theory and Applications

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Measure-valued differentiation for finite products of measures : theory and applications Citation for published version (APA): Leahu, H. (2008). Measure-valued differentiation for finite products of measures : theory and applications. Vrije Universiteit Amsterdam. Document status and date: Published: 01/01/2008 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal. If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement: www.tue.nl/taverne Take down policy If you believe that this document breaches copyright please contact us at: [email protected] providing details and we will investigate your claim. Download date: 25. Sep. 2021 MEASURE{VALUED DIFFERENTIATION FOR FINITE PRODUCTS OF MEASURES: THEORY AND APPLICATIONS ISBN 978 90 5170 905 6 °c Haralambie Leahu, 2008 Cover design: Crasborn Graphic Designers bno, Valkenburg a.d. Geul This book is no. 428 of the Tinbergen Institute Research Series, established through cooperation between Thela Thesis and the Tinbergen Institute. A list of books which already appeared in the series can be found in the back. VRIJE UNIVERSITEIT MEASURE{VALUED DIFFERENTIATION FOR FINITE PRODUCTS OF MEASURES: THEORY AND APPLICATIONS ACADEMISCH PROEFSCHRIFT ter verkrijging van de graad Doctor aan de Vrije Universiteit Amsterdam, op gezag van de rector magni¯cus prof.dr. L. M. Bouter, in het openbaar te verdedigen ten overstaan van de promotiecommissie van de faculteit der Economische Wetenschappen en Bedrijfskunde op maandag 22 september 2008 om 13.45 uur in de aula van de universiteit, De Boelelaan 1105 door Haralambie Leahu geboren te Galat»i, RoemeniÄe promotor: prof.dr. H.C. Tijms copromotor: dr. B.F. Heidergott TO MY PARENTS CONTENTS 1. Measure Theory and Functional Analysis :::::::::::::::::::::: 1 1.1 Introduction . 1 1.2 Elements of Topology and Measure Theory . 2 1.2.1 Topological and Metric Spaces . 2 1.2.2 The Concept of Measure . 5 1.2.3 Cv-spaces . 8 1.2.4 Convergence of Measures . 10 1.3 Norm Linear Spaces . 13 1.3.1 Basic Facts from Functional Analysis . 14 1.3.2 Banach Bases . 17 1.3.3 Spaces of Measures . 19 1.3.4 Banach Bases on Product Spaces . 23 1.4 Concluding Remarks . 25 2. Measure-Valued Di®erentiation ::::::::::::::::::::::::::: 29 2.1 Introduction . 29 2.2 The Concept of Measure-Valued Di®erentiation . 30 2.2.1 Weak, Strong and Regular Di®erentiability . 30 2.2.2 Representation of the Weak Derivatives . 35 2.2.3 Computation of Weak Derivatives and Examples . 40 2.3 Di®erentiability of Product Measures . 45 2.4 Non-Continuous Cost-Functions and Set-Wise Di®erentiation . 48 2.5 Gradient Estimation Examples . 52 2.5.1 The Derivative of a Ruin Probability . 52 2.5.2 Di®erentiation of the Waiting Times in a G/G/1 Queue . 56 2.6 Concluding Remarks . 59 3. Strong Bounds on Perturbations Based on Lipschitz Constants :::::::::: 61 3.1 Introduction . 61 3.2 Bounds on Perturbations . 62 3.2.1 Bounds on Perturbations for Product Measures . 63 3.2.2 Bounds on Perturbations for Markov Chains . 68 3.3 Bounds on Perturbations for the Steady-State Waiting Time . 75 3.3.1 Strong Stability of the Steady-State Waiting Time . 75 3.3.2 Comments and Bound Improvements . 81 3.4 Concluding Remarks . 83 ii Contents 4. Measure-Valued Di®erential Calculus :::::::::::::::::::::::: 85 4.1 Introduction . 85 4.2 Leibnitz-Newton Rule and Weak Analyticity . 86 4.2.1 Leibnitz-Newton Rule and Extensions . 86 4.2.2 Weak Analyticity . 88 4.3 Application: Stochastic Activity Networks (SAN) . 94 4.4 Concluding Remarks . 97 5. A Class of Non-Conventional Algebras with Applications in OR ::::::::: 99 5.1 Introduction . 99 5.2 Topological Algebras of Matrices . 100 5.3 Dp-Di®erentiability . 104 5.3.1 Dp-spaces . 104 5.3.2 Dp-Di®erentiability for Random Matrices . 106 5.4 A Formal Di®erential Calculus for Random Matrices . 108 5.4.1 The Extended Algebra of Matrices . 108 5.4.2 Dp-Di®erential Calculus . 111 5.5 Taylor Series Approximations for Stochastic Max-Plus Systems . 115 5.5.1 A Multi-Server Network with Delays/Breakdowns . 115 5.5.2 SAN Modeled as Max-Plus-Linear Systems . 120 5.6 Concluding Remarks . 123 Appendix ::::::::::::::::::::::::::::::::::::::: 125 A. Convergence of In¯nite Series of Real Numbers . 125 B. Interchanging Limits . 126 C. Measure Theory . 127 D. Conditional Expectations . 128 E. Fubini Theorem and Applications . 129 F. Weak Convergence of Measures . 130 G. Functional Analysis . 131 H. Overview of Weakly Di®erentiable Distributions . 132 Summary ::::::::::::::::::::::::::::::::::::::: 133 Samenvatting ::::::::::::::::::::::::::::::::::::: 135 Bibliography ::::::::::::::::::::::::::::::::::::: 137 Index ::::::::::::::::::::::::::::::::::::::::: 141 List of Notations ::::::::::::::::::::::::::::::::::: 143 Acknowledgments ::::::::::::::::::::::::::::::::::: 145 PREFACE A wide range of stochastic systems in the area of manufacturing, transportation, ¯nance and communication can be modeled by studying cost-functions1 over a ¯nite collection of independent random variables, called input variables. From a probabilistic point of view such a system is completely determined by the distributions of the input variables under consideration, which will be called input distributions. Throughout this thesis we consider parameter-dependent stochastic systems, i.e., we assume that the input distributions de- pend on some real-valued parameter denoted by θ. More speci¯cally, let £ ½ R denote an open, connected subset of real numbers and let ¹i,θ, for 1 · i · n, be a ¯nite family of probability measures (input distributions) on some state spaces Si, for 1 · i · n, depending on some parameter θ 2 £, such as, for example, the mean. We consider a stochastic system driven by the above speci¯ed distributions and we call a performance measure of such a system the expression Z Z Pg(θ) := Eθ[g(X1;:::;Xn)] = ::: g(x1; : : : ; xn)¦θ(dx1; : : : ; dxn); (0.1) for an arbitrary cost-function g, where the input variables Xi, for 1 · i · n, are dis- tributed according to ¹i,θ, respectively, and ¦θ denotes the product measure 8θ 2 £ : ¦θ := ¹1,θ £ ::: £ ¹n,θ: (0.2) This thesis is devoted to the analysis of performance measures modeled in (0.1). This class of models covers a wide area of applications such as queueing theory, project eval- uation and review technique (PERT), which provide suitable models for manufacturing or transportation networks, and insurance models. Speci¯cally, the following concrete models will be treated as examples: single-server queueing networks, stochastic activity networks and insurance models over a ¯nite number of claims. Correspondingly, transient waiting times in queueing networks, completion times in stochastic activity networks or ruin probabilities in insurance models are examples of performance measures. The main topic of research put forward in this thesis will be the study of analytical properties of the performance measures Pg(θ) such as continuity, di®erentiability and an- alyticity with respect to the parameter θ, for g belonging to some pre-speci¯ed class of cost-functions D. This allows for a wide range of applications such as gradient estima- tion (which very often is an useful tool for performing stochastic optimization), sensitivity analysis (bounds on perturbations) or Taylor series approximations. To this end, we study the distribution ¦θ of the vector (X1;:::;Xn) rather than investigating each Pg(θ) indi- vidually, i.e., we study weak properties of the probability measure ¦θ. More speci¯cally, if 1 Real-valued functions designed to measure some speci¯c performance of the system. iv Preface D is a set of cost-functions, we say that a property (P) (e.g., continuity, di®erentiability) holds weakly, inR a D-sense, for the measure-valued mapping θ 7! ¹θ if for each g 2 D the mapping θ 7! gd¹θ has the same property (P). It turns out that one can simultaneously handle the whole class of performance measures fPg(θ): g 2 Dg. We propose here a modular approach to the analysis of ¦θ, explained in the following. Let us identify the original stochastic process with the product measure ¦θ de¯ned in (0.2). Assume further that the input distributions ¹i,θ are weakly
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