stochastic petri nets. modelling, stability, simulation - haas p.

Stochastic Petri Nets: Modelling, Stability, Simulation Peter J. Haas Springer Springer Series in Operations Research Editors: Peter W. Glynn Stephen M. Robinson Peter J. Haas Stochastic Petri Nets Modelling, Stability, Simulation With 71 Illustrations Peter J. Haas IBM Research Division San Jose, CA 95120-6099 USA peterh@almaden.ibm.com Series Editors: Peter W. Glynn Stephen M. Robinson Department of Management Science Department of Industrial Engineering and Engineering University of Wisconsin–Madison Terman Engineering Center Madison, WI 53706-1572 Stanford University USA Stanford, CA 94305-4026 USA Library of Congress Cataloging-in-Publication Data Haas, Peter J. (Peter Jay) Stochastic Petri nets : modelling, stability, simulation / Peter J. Haas. p. cm. — (Springer series in operations research) Includes bibliographical references and index. ISBN 0-387-95445-7 (alk. paper) 1. Petri nets. 2. Stochastic analysis. I. Title. II. Series. QA267 .H3 2002 511.3—dc21 2002019559 Printed on acid-free paper.  2002 Springer-Verlag New York, Inc. 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 hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Manufacturing supervised by Jerome Basma. Camera-ready copy prepared from the author’s LaTeX2e files using Springer’s macros. Printed and bound by Maple-Vail Book Manufacturing Co., York, PA. Printed in the United States of America. 987654321 ISBN 0-387-95445-7 SPIN 10867072 Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH Preface This book was motivated by a desire to bridge the gap between two impor- tant areas of research related to the design and operation of engineering and information systems. The first area concerns the development of mathe- matical tools for formal specification of complex probabilistic systems, with an eye toward subsequent simulation of the resulting stochastic model on a computer. The second area concerns the development of methods for analysis of simulation output. Research on modelling techniques has been driven by the ever-increasing size and complexity of computer, manufacturing, transportation, workflow, and communication systems. Many engineers and systems designers now recognize that the use of formal models has a number of advantages over simply writing complicated simulation programs from scratch. Not only is it much easier to generate software that is free of logical errors, but various qualitative system properties—absence of deadlock, impossibility of reaching catastrophic states, and so forth—can be verified far more easily for a formal model than for an ad-hoc computer program. Indeed, certain system properties can sometimes be verified automatically. Our focus is on systems that can be viewed as making state transitions when events associated with the occupied state occur. More specifically, we consider discrete-event systems in which the stochastic state transi- tions occur only at an increasing sequence of random times. The “Bedi- enungsprozess” (service process) framework, developed by K¨onig, Matthes, and Nawrotzki in the 1960s and early 1970s, provided the first set of build- ing blocks for formal modelling of general discrete-event systems. The mod- ern incarnation of the Bedienungsprozess is the “generalized semi-Markov viii Preface process” (gsmp). Although useful for a unified theoretical treatment of discrete-event stochastic systems, the gsmp framework is not always well suited to practical modelling tasks. In particular, the modeller is forced to specify the “state of the system” directly as an abstract vector of random variables. Such a specification can be highly nontrivial: the system state definition must be as concise as possible for reasons of efficiency, but must also contain enough information so that (1) a sequence of state transitions and transition times can be generated during a simulation run and (2) the system characteristics of interest can be determined from such a sequence. Stochastic Petri nets (spns), introduced in the 1980s, are very appealing in that they not only have the same modelling power as gsmps (see Chapter 4) but also admit a graphical representation that is well suited to top-down and bottom-up modelling of complex systems. In parallel to these advances in modelling, a rigorous theory of simulation output analysis has been developed over the past 25 years. Much of this theory pertains to the problem of obtaining point estimates and confidence intervals for long-run performance measures of interest. Such point and in- terval estimates are typically used to compare alternative system designs or operating policies. These estimates also form the basis for simulation- based optimization procedures. Confidence intervals can be particularly difficult to obtain, but are necessary to distinguish true differences in sys- tem behavior from mere random fluctuations. The basic idea is to view each simulation run as the sample path of a precisely defined stochastic process. Point estimates and confidence intervals are then established by appealing to limit theorems for such processes. Unfortunately, many of the results in the output-analysis literature have not been provided in a form that is directly useful to practicing simula- tion analysts. Typically, a specified estimation or optimization procedure is shown to produce valid results if the output process of the simulation has specified stochastic properties—for example, obeys specified limit the- orems or has a sequence of regeneration points. Verification of the required properties for a specific (and usually complicated) simulation model often turns out to be a formidable task. Indeed, when studying the long-run per- formance of a specified system, it is often hard even to establish that the simulation problem at hand is well posed in that the system is stable and long-run performance measures actually exist. This book is largely concerned with making a connection between mod- elling practice and output-analysis theory. We illustrate the use of the spn building blocks for modelling and discuss the basic principles that underlie estimation procedures such as the regenerative method and the method of batch means. Tying these topics together are verifiable conditions on the building blocks of an spn under which the net is stable over time and spec- ified estimation procedures are valid. Our treatment highlights perhaps the most appealing aspect of spns: the formalism is powerful enough to permit Preface ix accurate modelling of a wide range of real-world systems and yet simple enough to be amenable to stability and convergence analysis. When studying the literature related to spns, one quickly encounters a multitude of spn variants as well as a variety of other frameworks for modelling discrete-event systems. Partly for this reason, we provide—in addition to our other results—methods for comparing the modelling power of different discrete-event formalisms. Although we emphasize the compari- son of spns with gsmps, our general approach provides a means for making principled choices between alternative modelling frameworks. Our method- ology can also be used to extend recurrence results and limit theorems from one framework to another. This latter application of our modelling- power theorems both simplifies the proofs of certain results for spns and makes the material in this book relevant not only to spns but also to the general study of discrete-event systems. Indeed, this book can be viewed as a survey of some fundamental stability, convergence, and estimation is- sues for discrete-event systems, using spns as a convenient and appealing framework for the discussion. Our view of spns differs from many in the literature in that we focus on the close relationship between spns and gsmps. To some extent this viewpoint is necessary: because we allow completely arbitrary clock-setting distributions, the underlying marking process of an spn is not, in general, a Markov or semi-Markov process. Our viewpoint also is advantageous, in that it lets us exploit the many powerful results that have been es- tablished for both gsmps and their underlying general state-space Markov chains. We emphasize, however, that spns have unique features that require extension—rather than straightforward adaptation—of results for gsmps. The prime example is given by “immediate transitions,” which have no counterpart in the gsmp model and lead to a variety of mathematical com- plications. The presentation is self-contained. Knowledge of basic probability theory, statistics, and stochastic processes at a first-year graduate level is needed to understand the theory and examples. We occasionally use results from the theory of Markov chains on a general state space—most of the techni- cal complexities for such chains can safely be glossed over in our setting, and the results we use are directly analogous to classical results for chains with finite or countably infinite state spaces. The Appendix summarizes the key mathematical results used in the text. To increase accessibility, we suppress measure-theoretic notation whenever possible—the Appendix contains a discussion of basic measure-theoretic concepts and their relation to the terminology used in the text. The more applied reader will wish to focus primarily on the discussion of modelling techniques and on spe- cific estimation methods. These topics are covered primarily in Chapter 1, Chapter 2, Section 3.1.3, Section 6.3, Sections 7.2.2–7.2.4 and 7.3.3–7.3.5, Sections 8.1, 8.2.2–8.2.4, 8.3.2, and 8.3.3, and Sections 9.1 and 9.3. x Preface I am grateful to the IBM Corporation for support of this work and for the resources of the Almaden Research Center. I also wish to thank Thomas Kurtz and the Center for the Mathematical Sciences at the University of Wisconsin–Madison for hospitality during the 1992–1993 academic year. I have benefitted from conversations with many colleagues over the years, including Sigrun Andradottir, James Calvin, Donald Iglehart, Sean Meyn, Joseph Mitchell, William Peterson, Karl Sigman, and Mary Vernon. Thanks also are due to the students of the graduate course on simulation that I taught at Stanford University during the 1998–1999 and 2000–2001 aca- demic years. Shane Henderson provided valuable feedback on an initial version of the manuscript. As is apparent from the notes and references in the text, I am deeply indebted to Gerald Shedler, who introduced me to both spns and stochastic simulation and who has co-authored most of the papers I have written on these topics. Perhaps less apparent, but equally important, are the technical insights and general encouragement that I have received from Peter Glynn. The staff of Springer-Verlag has been exceed- ingly helpful throughout the production of this book—special thanks go to Achi Dosanjh for her help in jump-starting the project and to Kristen Cassereau for her meticulous copyediting. Finally, I wish to thank my wife, Laura, and my children, Joshua and Daniel, for their love, patience, and support. San Jose, California Peter J. Haas March 2002 Contents Preface vii List of Figures xv Selected Notation xix 1 Introduction 1 1.1 Modelling 2 1.2 Stability and Simulation 9 1.3 Overview of Topics 13 2 Modelling with Stochastic Petri Nets 17 2.1 Building Blocks 17 2.2 Illustrative Examples 24 2.2.1 Priorities: Producer–Consumer Systems 24 2.2.2 Marking-dependent Transitions 31 2.2.3 Synchronization: Flexible Manufacturing System . . 41 2.2.4 Resetting Clocks: Particle Counter 45 2.2.5 Compound Events: Slotted Ring 47 2.3 Concise Specification of New-Marking Probabilities 49 2.3.1 Transition Firings That Never Occur 50 2.3.2 Numerical Priorities 51 2.4 Alternative Building Blocks 64 xii Contents 3 The Marking Process 69 3.1 Definition of the Marking Process 70 3.1.1 General State-Space Markov Chains 70 3.1.2 Definition of the Continuous-Time Process 72 3.1.3 Generation of Sample Paths 75 3.2 Performance Measures 77 3.2.1 Simple Time-Average Limits and Ratios 77 3.2.2 Conversion of Limit Results to Continuous Time . . 78 3.2.3 Rewards and Throughput 81 3.2.4 General Functions of Time-Average Limits 86 3.3 The Lifetime of the Marking Process 87 3.3.1 Absorption into the Set of Immediate Markings . . . 87 3.3.2 Explosions 90 3.3.3 Sufficient Conditions for Infinite Lifetimes 91 3.4 Markovian Marking Processes 92 3.4.1 Continuous-Time Markov Chains 93 3.4.2 Conditional Distribution of Clock Readings 95 3.4.3 The Markov Property 102 4 Modelling Power 111 4.1 Generalized Semi-Markov Processes 113 4.2 Mimicry and Strong Mimicry 116 4.2.1 Definitions 116 4.2.2 Sufficient Conditions for Strong Mimicry 120 4.3 Mimicry Theorems for Marking Processes 127 4.3.1 Finite-State Processes 128 4.3.2 Countable-State Processes 132 4.4 Converse Results 136 5 Recurrence 145 5.1 Drift Criteria 146 5.1.1 Harris Recurrence and Drift 146 5.1.2 The Positive Density Condition 150 5.1.3 Proof of Theorem 1.22 157 5.2 The Geometric Trials Technique 164 5.2.1 A Geometric Trials Criterion 165 5.2.2 GNBU Distributions 166 5.2.3 A Simple Recurrence Argument 172 5.2.4 Recurrence Theorems 174 5.2.5 Some Ad-Hoc Recurrence Arguments 182 6 Regenerative Simulation 189 6.1 Regenerative Processes 190 6.1.1 Definition of a Regenerative Process 190 6.1.2 Stability of Regenerative Processes 193 [...]... Interactive video-on-demand system 277 spn representation of video-on-demand system 278 An spn with extremely long cycles 280 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 8.11 Positions of jobs in cyclic queues with feedback spn for measuring delays in cyclic queues with feedback Manufacturing flow-line with shunt bank spn rep of manufacturing flow-line with shunt... number of vehicles on a specified stretch of road exceeds a given threshold spns are conducive to both bottom-up and top-down modelling In bottom-up modelling, a detailed subnet is developed for each component of a system, and then the subnets are combined to form the overall spn model In top-down modelling, a preliminary spn model is developed that captures the main interactions between the components... on the theory of continuous-time Markov chains on a finite or countably infinite state space This book is about stochastic Petri nets (spns), which have proven to be a popular and useful tool for modelling and performance analysis of complex stochastic systems We focus on some fundamental issues that arise when 2 1 Introduction modelling a system as an spn and studying the long-run behavior of the resulting... do various spn building blocks enhance modelling power? • Under what conditions on the building blocks is an spn model stable over time, so that long-run simulation problems are well posed? • What simulation- based methods are available for estimating long-run performance characteristics? How can the validity of a given estimation method be established for a particular spn model? We address the first question... various stochastic processes associated with an spn 1.1 Modelling It is frequently useful to view a complex stochastic system as evolving over continuous time and making state transitions when events associated with the occupied state occur Often the system is a discrete-event system in that the stochastic state transitions occur only at an increasing sequence of random times In a discrete-event system,... transition and each of these events has its own stochastic mechanism for determining the next state At each state transition, new events may be scheduled and previously scheduled events may be cancelled The spn framework provides a powerful set of building blocks for specifying the state-transition mechanism and event-scheduling mechanism of a discrete-event stochastic system An spn is specified by a finite... Collision-free bus network spn representation of collision-free bus network Timeline diagram for collision-free bus network 3.1 3.2 3.3 3.4 3.5 3.6 Supply chain spn representation of supply chain Absorption of the marking process into S Example for proof of Theorem 4.10 Non-Markovian spn with exponential clock-setting... are one such extension of the basic spn model 1.2 Stability and Simulation 9 1.2 Stability and Simulation Engineers and systems designers are often interested in performance characteristics such as the long-run average operating cost for a flexible manufacturing system, the long-run fraction of time a database is accessible, or the long-run utilization of a communications link When the system of interest... continuous-time Markov chain (ctmc) with finite or countably infinite state space; see Section 3.4 A variety of techniques is then available for determining whether the time-average limits of interest exist and, if so, for computing these limits either analytically or numerically In general, however, the stochastic process { X(t) : t ≥ 0 } is not a continuous-time Markov chain or even a semi-Markov process... Determining the existence of time-average limits then becomes a highly nontrivial task and the limits, if they exist, must be estimated using computer simulation. 2 We focus primarily on problems for which simulation is required, 2 Even when the marking process is a ctmc, the chain’s state space may be so large that simulation is the only practical means of assessing long-run behavior Similarly, even when . Stochastic Petri Nets: Modelling, Stability, Simulation Peter J. Haas Springer Springer Series in Operations Research Editors: Peter W. Glynn Stephen M. Robinson Peter J. Haas Stochastic Petri. USA Stanford, CA 9430 5-4 026 USA Library of Congress Cataloging-in-Publication Data Haas, Peter J. (Peter Jay) Stochastic Petri nets : modelling, stability, simulation / Peter J. Haas. p. cm. — (Springer series. rep. of producer–consumer system (nonpreempt.) . . . 25 2.5 Marking changes for spn rep. of producer–consumer sys. . . 28 2.6 spn rep. of producer–consumer sys. (preempt repeat) . . . 29 2.7 spn