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Stochastic Modeling and Analysis of Manufacturing Systems PDF

368 Pages·1994·29.455 MB·English
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Springer Series in Operations Research Editor: Peter Glynn Springer Series in Operations Research Yao (Ed.): Stochastic Modeling and Analysis of Manufacturing Systems David D. Yao Editor Stochastic Modeling and Analysis of Manufacturing Systems With 18 Illustrations Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest David D. Yao Department of Industrial Engineering and Operations Research Columbia University New York, NY 10027-6699 USA Series Editor: Peter Glynn Department of Operations Research Stanford University Stanford, CA 94305 USA Library of Congress Cataloging-in-Publication Data Stochastic modeling and analysis of manufacturing systems / David D. Yao. (ed.). p. cm. - (Springer series in operations research) Includes bibliographical references. 1. Production management-Mathematical models. 2. Stochastic analysis. I. Yao, David D., 1950- II. Series. TS155.S787 1994 658.8'01 '5118-dc20 94-19979 Printed on acid~free paper. © 1994 Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1994 All rights reserved; This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereaf ter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. Production managed by Natalie Johnson; manufacturing supervised by Vincent Scelta Photocomposed using the editor's LaTeX files. 987654321 ISBN-13: 978-1-4612-7628-9 e-ISBN-13: 978-1-4612-2670-3 DOl: 10.1007/978-1-4612-2670-3 Preface This book is a collection of chapters, each focusing on a specific topic. The selection of topics attempts to reflect the development in recent years of those new probabilistic models and methodologies that have been either motivated by manufacturing systems research or demonstrated to have im portant potential in such research. Here, manufacturing systems research is broadly interpreted to include modeling and analysis, design, planning, scheduling and control. Each chapter is written by one or several experts, in a self-contained, expository style, and aimed at the graduate level. The authors who wrote for the book were urged to make a special effort to be both informative and expository in their writing. From the outset, we agreed that no chapter should be a general survey (such as who did what when, etc.). Instead, I would like each chapter to be a detailed exposition on a well-selected subset of topics, with a carefully thought-out theme and focus, and self-contained in coverage (for instance, proofs of main results are included wherever necessary). I suggested to the authors that they should write with their graduate students in mind, and as if they were preparing a polished version of their lecture notes. My main objective is that the book be a useful text or reference, at the graduate level, that can be effectively used both in classroom and for self-study. For instance, I have used some of the chapters for graduate courses at Columbia that teach modeling and analysis of manufacturing systems. Students who take the courses come mostly from Operations Re search, Electrical Engineering, Mathematical Statistics, Industrial Engi neering, Management Science, and occasionally from Applied Mathematics and Economics. It is exactly this type of readership that I would like the book to serve. Alternatively, selected chapters of the book can be used to supplement the teaching and readings of other graduate courses in systems and con trol theory, stochastic processes, queueing theory, decision analysis, and advanced simulation. Used this way, the book offers not only detailed ex position to important methodologies (which might relate closely to the core of the courses in question), but also motivating applications of the method ologies. The book can also be a useful reference for researchers in manufacturing systems who want to learn new tools and new methodologies, as well as for researchers in mathematical sciences who are interested in studying manufacturing problems and applications of stochastic modeling. vi Preface Roughly the book consists of two parts. The first four chapters are ba sically concerned with models, while the remaining four chapters address design and control methodologies. In the first part, Chapter 1 studies Jackson networks, emphasizing the effectiveness of the models in capturing the fundamental qualitative and structural features of batch manufacturing, and in providing insight for the design and efficient operation of such systems. Chapters 2 and 3 present a modeling hierarchy that consists of three levels: micro, macro and the intermediate, meso models. In this hierarchy, queueing networks are micro models, fluid networks are macro models, while diffusion and strong ap proximations are meso models. Focusing on the macro and meso models, the two chapters detail the space-time scaling and related limit theorems in a unifying framework. In Chapter 4, the main model is the GSMP (gen eralized semi-Markov process). The treatment is to view the GSMP as a scheme driven by a sequence of clock times (event life times), with a focus on the language - the set of feasible strings (sequences) of events. Struc tural properties of the language lead to important implications in system performance. The theory is illustrated through analyzing in depth a produc tion line operating under a generalized kanban mechanism, which controls at each stage the work-in-process and finished goods inventory, as well as overall buffer content. In the second part, Chapter 5 presents the essentials of the recently developed theory of stochastic convexity and stochastic majorization, em phasizing their interplay and their role in understanding system behavior and supporting system design. Various aspects of the theory are illustrated through numerous applications that include a random yield model, a joint setup problem, a process involving trial runs, a network with constant WIP (work-in-protess) and WIP-dependent production rates, and scheduling in tandem production lines and parallel assembly systems. Chapter 6 is a self contained introduction to the fundamentals of derivative estimation via per turbation analysis and its applications in a variety of production networks, including kanban systems, systems with rework and scrap, with alterna tive sourcing, and with subassemblies. This chapter, along with Chapter 4, also introduces the reader to the newly developed discrete-event sys tems perspective in the modeling and analysis of manufacturing systems. The common theme of the last two chapters, Chapter 7 and Chapter 8, is dynamic scheduling. Chapter 7 presents an approach that is based on modeling the production facilities as Brownian networks. It addresses the objective of minimizing a weighted summation of the sojourn time and the sojourn-time inequity. This treatment unifies and generalizes previous works that focus on single-criterion objectives. Chapter 8 studies scheduling in re-entrant lines, which typically model the configuration of semiconduc tor manufacturing systems. The focus of the chapter is on developing simple but effective scheduling rules, and providing rigorous analytical justification to support the performance of such rules. Preface vii Each chapter was reviewed by one or several independent readers, whose critical reading and constructive comments and suggestions have enhanced the quality of the chapters. For their assistance, I thank the following read ers: Rajeev Agrawal (Wisconsin-Madison), Hong Chen (British Columbia), Dinah Cheng (NYU), Jim Dai (Georgia Tech.), Michael Fu (Maryland), Paul Glasserman (Columbia), Shin-Gang Kou (Columbia), Xiao-Gao Liu (Waterloo), Rajendra Rajan (Wisconsin-Madison), Rhonda Righter (Santa Clara), Aliza Schachter (Columbia), Dequan Shaw (GTE Labs.), and Li Zhang (Columbia). The initial organization of the book and much of my own writing took place during the 1991/92 academic year, when I was on sabbatical leave at Yale. I thank my hosts, Eric Dernardo and Offer Kella, for their hospi tality. During the period, I was also a recipient of a Fellowship from the Guggenheim Foundation, and research grants from the National Science Foundation. I am grateful to these foundations for their support. New York City, March 1994 David D. Yao Contents Preface v Contributors xv 1 Jackson Network Models of Manufacturing Systems John A. Buzacott, J. George Shanthikumar, and David D. Yao 1 1.1 Introduction....... 1 1.2 Jackson Networks. . . . . 4 1.2.1 The Open Model . 5 1.2.2 The Closed Model 8 1.2.3 The Semi-Open Model . 9 1.3 The Throughput Function and Computation 11 1.4 Monotonicity of the Throughput Function 15 1.4.1 Equilibrium Rate. . . . . . 15 1.4.2 P F2 Property . . . . . . . . . . . . 16 1.4.3 Likelihood Ratio Ordering . . . . . 17 1.4.4 Shifted Likelihood Ratio Ordering 20 1.5 Concavity and Convexity 24 1.6 Multiple Servers ........ 27 1. 7 Resource Sharing . . . . . . . . 30 1. 7.1 Aggregation of Servers . 30 1.7.2 Aggregation of Nodes . 32 1.8 Arrangement and Majorization 35 1.9 Conclusiorls 39 1.10 Notes ... 40 1.11 References . 43 2 Hierarchical Modeling of Stochastic Networks, Part I: Fluid Models Hong Chen and Avi Mandelbaum 47 2.1 Introduction........................... 47 2.1.1 Macro, Meso and Microscopic Models for an Li.d. Sequence ...................... 50 2.1.2 Strong Approximations - A Unifying Framework 53 2.1.3 Summary.................. 55 2.2 A Flow Network in Discrete Time ........ 55 2.2.1 The Microscopic Model and Its Dynamics 56 x Contents 2.2.2 Reformulation in Terms of Cumulants and Oblique Reflection . . . . . . . . . . . . . . . . . . . . . . 58 2.2.3 Mesoscopic Models and Strong Approximations . . . 61 2.2.4 Macroscopic Models: FSLLN's ........... . 63 2.2.5 Deviations Between Micro and Macro Models: FCLT 64 2.3 Flow Networks in Continuous Time ............. . 65 2.3.1 Flow Networks with Time Inhomogeneous Dynamics 65 2.3.2 State-Dependent Dynamics ......... . 66 2.4 Linear Fluid Network and Bottleneck Analysis .. . 68 2.4.1 Traffic Equations and Bottleneck Definitions 68 2.4.2 Bottleneck Analysis .......... . 71 2.5 Functional Strong Law of Large Numbers . . . . . . 74 2.5.1 FSLLN's for Nonlinear Fluid Networks .... 74 2.5.2 FSLLN's for Nonparametric Jackson Queueing Networks ................... . 75 2.5.3 FSLLN's for State-Dependent Networks ... . 78 2.6 Applications and Hints at Prospects of Fluid Models . 82 2.6.1 Stochastic Fluid Models for Manufacturing and Communication Systems . . . . . . . . . . . . . 82 2.6.2 Heterogeneous Fluid Networks: Bottleneck Analysis and Scheduling Control .. . . . . . . . . . 85 2.6.3 Transient Analysis of the Mt/Mt/l Queue. 92 2.7 References and Comments 97 2.8 References . . . . . . . . . . . . . . . . . . . . . . 100 3 Hierarchical Modeling of Stochastic Networks, Part II: Strong Approximations Hong Chen and Avi Mandelbaum 107 3.1 Introduction........... 107 3.2 The Model ........... 108 3.2.1 Primitives and Dynamics 108 3.2.2 Underlying Assumptions and Parameters 108 3.2.3 Nonparametric Jackson Networks. 109 3.3 Preliminaries . . . . . . . . . . . . . . . . 111 3.3.1 Traffic Equations and Bottlenecks 111 3.3.2 The Oblique Reflection Mapping 111 3.3.3 Reflected Brownian Motion on the Orthant 112 3.4 The Main Results. . . . . . . . . . . . . . . . . . . 112 3.4.1 Functional Strong Approximations . . . . . 113 3.4.2 Functional Laws of the Iterated Logarithm 114 3.4.3 FSLLN's and Fluid Approximations 114 3.4.4 FCLT's and Diffusion Approximations 115 3.5 Fitting Parametes ............. 116 3.5.1 Nonparametric Jackson Networks. 116 3.5.2 Product Form and Single Station . 117 Contents xi 3.6 Proof of the Main Results . . . . . . . . . . . . . . . 117 3.7 References, Possible Extensions and Future Research 125 3.8 References........................ 128 4 A GSMP Framework for the Analysis of Production Lines Paul Glasserman and David D. Yao 133 4.1 Introduction......... 133 4.2 GSMP and Its Scheme . . . . . 135 4.2.1 The Scheme: GSMS .. 135 4.2.2 Language and Score Space. 136 4.3 Structural Properties of the Scheme 138 4.3.1 Some Useful Properties 138 4.3.2 Condition (M) . . 140 4.3.3 Condition (CX) . . . . 143 4.3.4 Minimal Elements . . . 145 4.3.5 Monotonicity and Convexity 147 4.3.6 Characteristic Function 148 4.3.7 Subschemes...... 150 4.3.8 Synchronized Schemes . 152 4.4 The (a, b, k) Tandem Queue . . 154 4.4.1 Production Lines Under Kanban Control 154 4.4.2 Properties with Respect to Service Times 156 4.5 Properties with Respect to (a, b, k) . . . . . . 160 4.5.1 Monotonicity with Respect to (a, b, k) 160 4.5.2 Concavity with Respect to (a, b, k) 162 4.6 Line Reversal . . . . . . . . . . . . . . . . 164 4.6.1 Reversibility of Departure Epochs 164 4.6.2 Full Reversibility . . . . . . 167 4.7 Subadditivity and Ergodicity . . . . . . . 168 4.7.1 Event-Epoch Vectorization .... 169 4.7.2 The Subadditive Ergodic Theorem 170 4.7.3 More General Matrices. 172 4.8 Cycle Time Limits . . . . . . . 176 4.8.1 Existence of the Limits 176 4.8.2 Rate of Convergence 179 4.9 Notes 182 4.10 References. . . . . . . . . . 185 5 Stochastic Convexity and Stochastic Majorization Cheng-Shang Chang, J. George Shanthikumar, and David D. Yao 189 5.1 Introduction.......................... 189 5.2 Stochastic Order Relations: Functional Characterizations 192 5.3 Second-Order Stochastic Properties. . . . . . . . . . . .. 196

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