Table Of Content

Theory of Modeling and Simulation Discrete Event and Iterative System Computational Foundations Theory of Modeling and Simulation Discrete Event and Iterative System Computational Foundations Third Edition Bernard P. Zeigler University of Arizona Tucson, USA Alexandre Muzy CNRS, I3S Laboratory Universté Côte d’Azur Nice Sophia Antipolis, France Ernesto Kofman FCEIA – Universidad Nacional de Rosario CIFASIS – CONICET Rosario, Argentina AcademicPressisanimprintofElsevier 125LondonWall,LondonEC2Y5AS,UnitedKingdom 525BStreet,Suite1650,SanDiego,CA92101,UnitedStates 50HampshireStreet,5thFloor,Cambridge,MA02139,UnitedStates TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UnitedKingdom Copyright©2019ElsevierInc.Allrightsreserved. ThisisthethirdandrevisededitionofTheoryofModelingandSimulation,BernardP.Zeigler,publishedbyWileyInterscience,1976 withreissuebyKriegerPub.1984 Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronicormechanical,including photocopying,recording,oranyinformationstorageandretrievalsystem,withoutpermissioninwritingfromthepublisher.Detailson howtoseekpermission,furtherinformationaboutthePublisher’spermissionspoliciesandourarrangementswithorganizationssuchas theCopyrightClearanceCenterandtheCopyrightLicensingAgency,canbefoundatourwebsite:www.elsevier.com/permissions. ThisbookandtheindividualcontributionscontainedinitareprotectedundercopyrightbythePublisher(otherthanasmaybenoted herein). Notices Knowledgeandbestpracticeinthisfieldareconstantlychanging.Asnewresearchandexperiencebroadenourunderstanding,changes inresearchmethods,professionalpractices,ormedicaltreatmentmaybecomenecessary. Practitionersandresearchersmustalwaysrelyontheirownexperienceandknowledgeinevaluatingandusinganyinformation, methods,compounds,orexperimentsdescribedherein.Inusingsuchinformationormethodstheyshouldbemindfuloftheirownsafety andthesafetyofothers,includingpartiesforwhomtheyhaveaprofessionalresponsibility. Tothefullestextentofthelaw,neitherthePublishernortheauthors,contributors,oreditors,assumeanyliabilityforanyinjuryand/or damagetopersonsorpropertyasamatterofproductsliability,negligenceorotherwise,orfromanyuseoroperationofanymethods, products,instructions,orideascontainedinthematerialherein. LibraryofCongressCataloging-in-PublicationData AcatalogrecordforthisbookisavailablefromtheLibraryofCongress BritishLibraryCataloguing-in-PublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-813370-5 ForinformationonallAcademicPresspublications visitourwebsiteathttps://www.elsevier.com/books-and-journals Publisher:KateyBirtcher AcquisitionEditor:KateyBirtcher EditorialProjectManager:KarenMiller ProductionProjectManager:NileshKumarShah Designer:MarkRogers TypesetbyVTeX Contents Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx1 Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii PART 1 BASICS: MODELING FORMALISMS AND SIMULATION ALGORITHMS CHAPTER 1 Introduction to Systems Modeling Concepts ........................ . 3 1.1 Systems Specification Formalisms ... .. .... ...... .. ........... ... .. . 4 1. 1.1 Relation to Object Orientation .............................. . 5 1.1.2 Evolution of Systems Formalisms ..... ............... .. .. ... . 6 1.1.3 Continuous and Discrete Formalisms ........................ . 7 1.1.4 Quantized Systems .......... .. ........... .. ............. . 8 1.1.5 Extensions of DEVS .. ......... .... ...................... . 9 1.2 Levels of System Knowledge ..................................... . 10 1.3 Introduction to the Hierarchy of Systems Specifications .. ........ ... .... . 12 1.4 The Specification Levels Informally Presented .... ..... . .. ....... ..... . 14 1.4.1 Observation Frame ...... . ........... .. .. ........... .. ... . 14 1.4.2 1/0 Behavior and 1/0 Function ... ............... ........... . 15 1.4.3 State Transition System Specification .... ..... ............... . 16 1. 4.4 Coupled Component System Specification ... .... .. ...... ... .. . 16 1.5 System Specification Morphisms: Basic Concepts ..................... . 17 1.6 Evolution of DEVS ............................................ . 20 1.7 Summary .................................................... . 23 1.8 Sources ... .... ........... .. ................................. . 23 Definitions, Acronyms, Abbreviations ........ .. ......... ........... . 24 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 CHAPTER 2 Framework for Modeling and Simulation .......................... . 27 2.1 The Entities of the Framework .................................... . 28 2.1.1 Source System . .... ... . . .. .. .. ...... . .. .. .. ....... .. .. . . 28 2.1.2 Experimental Frame .... ... . ..... ...... ... . ......... ... . . . 29 Objectives and Experimental Frames ......................... . 29 2.1.3 Model .............................. .................. . 31 2.1 .4 Simulator .... .......... ............. .................. . 32 2.2 Primary Relations Among Entities .. .... ... .... .. .. ................ . 32 2.2. I Modeling Relation: Validity .... ....... ............... ..... . 32 2.2.2 Simulation Relation: Simulator Correctness .......... ......... . 33 2.3 Other Important Relationships ... . . ... .... ...... .. ....... .... .. ... . 34 V vi Contents Modeling as Valid Simplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Experimental Frame - Model Relationships . . . . . . . . . . . . . . . . . . . . 35 2.4 Time 36 2.5 Historical Trace of V & V Streams ................................. . 36 2.5. l Informal V & V Concepts and Processes ....................... . 37 2.5.2 Theory-Based and Model-Driven Developments ................ . 38 2.5.3 Generic Methodology Processes and Best Practice Guides ......... . 38 2.6 Summary ............................. ........... ............ . 39 2.7 Sources ..................................................... . 39 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 40 CHAPTER 3 Modeling Formalisms and Their Simulators .. ...................... . 43 Introduction .................................................. . 44 3.1 Discrete Time Models and Their Simulators .......................... . 44 3.1.1 Discrete Time Simulation ................................. . 46 3.1.2 Cellular Automata ....................................... . 46 3.1.3 Cellular Automaton Simulation Algorithms .................... . 48 3.1.4 Discrete Event Approach to CeUular Automaton Simulation ....... . 50 3.1.5 Switching Automata/Sequential Machines .. ........... .... .... . 51 3.1.6 Linear Discrete Time Networks and Their State Behavior ......... . 53 3.2 Differential Equation Models and Their Simulators .................... . 55 3.2.1 Linear ODE Models ..................................... . 57 3.2.2 Continuous System Simulation ............................. . 63 3.2.3 Euler's Methods ........................................ . 64 3.2.4 Accuracy of the Approximations .......... . .............. . .. . 65 3.2.5 Convergence of the Numerical Scheme ....................... . 68 3.2.6 Numerical Stability ...................................... . 70 3.2.7 One-Step Methods ...................................... . 72 3.2.8 Multi-Step Methods ..................................... . 74 3.2.9 Step Size Control ....................................... . 75 3.2.10 Stiff, Marginally Stable and Discontinuous Systems ............. . 78 3.3 Discrete Event Models and Their Simulators ......................... . 84 3.3.1 Introduction .................... ...... ................. . 84 3.3.2 Discrete Event Cellular Automata ........................... . 84 3.3.3 Discrete Event World Views ............................... . 87 Event Scheduling World View ....... ..... . ............ . .... . 87 3.4 Summary ..... ............................................... . 89 3.5 Sources ......... .. ...... ..... ........ ...... ................. . 90 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 CHAPTER 4 Introduction to Discrete Event System Specification {DEVS) . . . . . . . . . . . . . 93 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2 Classic DEVS System Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.2.1 DEVS Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Passive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 Contents vii Storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 Binary Counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Processor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . l Ol Ramp ................................................. 101 4.2.2 Classic DEVS With Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Switch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.3 Classic DEVS Coupled Models...... ................... . . . . . 104 Simple Pipeline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Switch Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.3 Parallel DEVS System Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Processor With Buffer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3.l Parallel DEVS Coupled Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Simple Pipeline (Parallel DEVS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 4.4 Hierarchical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 4.5 Object-Oriented Implementations of DEVS: an Introduction . . . . . . . . . . . . . . 111 4.5.1 Structural Inheritance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.6 DEVS and Hierarchy of System Specifications: Turing Machine Example . . . . 114 4.6.1 Turing Machine as a Modular Composition . . . . . . . . . . . . . . . . . . . . . 114 4.6.2 Tape System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.6.3 TM Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 TM Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.6.4 Simulation of the TM Coupled Model . . . . . . . . . . . . . . . . . . . . . . . . . 118 4.6.5 Example of Simulation Run . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4.6.6 Turing Machine Example of Hierarchy of System Specifications . . . . . 120 1/0 Frame at Level 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 1/0 Relation Observation at Level 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 121 1/0 Function Observation at Level 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 122 DEVS 1/0 System Specification at Level 4 . . . . . . . . . . . . . . . . . . . . . 122 4.6.7 Empirical Investigation of Halting . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.7 Are DEVS State Sets Essentially Discrete? . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 4.9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 CHAPTER 5 Hierarchy of System Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.1 Time Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Segments and Trajectories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.2. l Piecewise Continuous Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.2 Piecewise Constant Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 5.2.3 Event Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Correspondence Between Piecewise Constant and Event Segments . . . 132 5.2.4 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 5.3 1/0 Observation Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.4 1/0 Relation Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 viii Contents 5.5 1/0 Function Observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.6 1/0 System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6.1 Going From System Structure to Behavior . . . . . . . . . . . . . . . . . . . . . 139 5.6.2 Time Invariant Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Example: Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 5.6.3 Special Cases: Input-Free and Memoryless Systems . . . . . . . . . . . . . . 143 Input-Free Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Memoryless Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 5.7 Multi-Component System Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.8 Network of System Specifications (Coupled Systems) . . . . . . . . . . . . . . . . . . . 146 5.8.1 Coupled System Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.8.2 Coupled System Specification at the Structured System Level . . . . . . . 148 5.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 CHAPTER 6 Basic Formalisms: DEVS, DESS, DTSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1 Basic System Specification Formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.2 Discrete Event System Specification (DEVS) . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2.1 Classic DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2.2 Structure Specified by DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 6.2.3 Legitimacy: When is the Structure Specified by a DEVS Really a System? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 6.3 Parallel DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 6.3.1 System Specified by Parallel DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.4 Discrete Time System Specification (DTSS) . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.5 Differential Equation System Specification (DESS) . . . . . . . . . . . . . . . . . . . . . 162 6.6 Example of DESS ....... ...... ...................... .. ......... 163 When is the System Specified by a DESS Well-Defined? . . . . . . . . . . . 164 6.7 Summary..................................................... 164 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 CHAPTER 7 Basic Formalisms: Coupled Multi-Component Systems . . . . . . . . . . . . . . . . . 167 7.1 Discrete Event Specified Network Formalism. . . . . . . . . . . . . . . . . . . . . . . . . . 168 7 .1.1 Classic DEVS Coupled Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 Closure Under Coupling of Classic DEVS . . . . . . . . . . . . . . . . . . . . . . 170 7 .1.2 Parallel DEVS Coupled Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 7 .1.3 Closure Under Coupling of Parallel DEVS . . . . . . . . . . . . . . . . . . . . . 171 Output Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 Internal Transition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 External Transition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.1.4 The Confluent Transition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 7.2 Multi-Component Discrete Event System Formalism . . . . . . . . . . . . . . . . . . . . 174 7 .2.1 Cellular Automata Multi-Component DEVS of GOL Event Model . . . 176 Cellular Automata Multi-Component DEVS of GOL Event Model . . . 177 7.2.2 Event Scheduling Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Implementing Event Scheduling Simulation Systems in Imperative Programming Languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Contents ix 7.2.3 Combined Event Scheduling, Activity Scanning Simulation Strategy.. 179 7.2.4 Process Interaction Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.2.5 Translating Non-Modular Multi-Component DEVS Models Into Modular Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 7.2.6 State Updating in Distributed Simulation. . . . . . . . . . . . . . . . . . . . . . . 182 7.3 Discrete Time Specified Network Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Delay-Free (Algebraic) Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Definition of Discrete Time Coupled Models . . . . . . . . . . . . . . . . . . . . 184 Closure Under Coupling of DTSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 7.4 Multi-Component Discrete Time System Formalism . . . . . . . . . . . . . . . . . . . . 185 Spatial DTSS: Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 7.5 Differential Equation Specified Network Formalism... . . . . . . . . . . . . . . . . . . 187 Closure Under Coupling of DESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 7.6 Multi-Component Differential Equations Specified System Formalism . . . . . . . 188 7.6.1 Spatial DESS: Partial Differential Equation Models. . . . . . . . . . . . . . . 189 7.7 Multi-Component Parallel Discrete Event System Formalism.............. 191 7.8 Summary.................... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 7 .9 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Appendix 7 .A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 CHAPTER 8 Simulators for Basic Formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 8.1 Simulators for DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 8.1.1 Simulator for Basic DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 8.1.2 Simulators for Modular DEVS Networks . . . . . . . . . . . . . . . . . . . . . . 200 8.1.3 The Root-Coordinator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 8.2 DEVS Bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 8.2.1 Simulator for Event Scheduling Multi-Component DEVS . . . . . . . . . . 206 8.2.2 Simulator for Activity Scanning and Process Interaction Multi-Component DEVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 8.3 Simulators for DTSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.3. I Simulator for Atomic DTSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.3.2 Simulator for Instantaneous Functions . . . . . . . . . . . . . . . . . . . . . . . . . 212 8.3.3 Simulator for Non-Modular Multi-Component DTSS . . . . . . . . . . . . . 213 8.3.4 Simulators for Coupled DTSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 8.3.5 The Root-Coordinator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.4 Simulators for DESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.4.1 Causal Simulator for DESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.4.2 Non-Causal Simulator for DESS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 8.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 8.6 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 CHAPTER 9 Multi-Formalism Modeling and Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 223 9.1 Brief Introduction to Specialized Formalisms . . . . . . . . . . . . . . . . . . . . . . . . . . 224

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A consensus on the fundamental status of theory of modeling and simulation is emerging – some recognize the need for a theoretical foundation for M&S as a science. Such a foundation is necessary to foster the development of M&S-specific methods and the use of such methods to solve real world probl
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A consensus on the fundamental status of theory of modeling and simulation is emerging – some recognize the need for a theoretical foundation for M&S as a science. Such a foundation is necessary to foster the development of M&S-specific methods and the use of such methods to solve real world probl

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