Synchronization and Linearity An Algebra for Discrete Event Systems Franc¸ois Baccelli INRIA and ´ Ecole Normale Sup´erieure, D´epartement d’Informatique Paris, France Guy Cohen ´ Ecole Nationale des Ponts et Chauss´ees-CERMICS Marne la Vall´ee, France and INRIA Geert Jan Olsder Delft University of Technology Faculty of Technical Mathematics Delft, the Netherlands Jean-Pierre Quadrat Institut National de la Recherche en Informatique et en Automatique INRIA-Rocquencourt Le Chesnay, France Preface to the Web Edition Thefirst edition of this book was published in 1992 by Wiley (ISBN 0 471 93609 X). Since this book is now out of print, and to answer the request of several colleagues, the authors have decided to make it available freely on the Web, while retaining the copyright, for the benefit of the scientific community. Copyright Statement This electronic document is in PDF format. 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These more important modificationswillinitiallybelisted explicitlyand provided separately fromthe original i ii Synchronization and Linearity text. In a more remote future, we may consider providing a true “second edition” in which these changes will be incorporated in the main text itself, sometimes removing the obsolete or wrong corresponding portions. Franc¸ois Baccelli francois.baccelli@ens.fr Guy Cohen guy.cohen@mail.enpc.fr Geert Jan Olsder g.j.olsder@math.tudelft.nl Jean-Pierre Quadrat jean-pierre.quadrat@inria.fr October 2001 Contents Preface ix IDiscrete Event Systems and Petri Nets 1 1Introduction and Motivation 3 1.1PreliminaryRemarks andSomeNotation 3 1.2Miscellaneous Examples 8 1.2.1Planning 9 1.2.2Communication 14 1.2.3 Production 15 1.2.4Queuing System with Finite Capacity 18 1.2.5ParallelComputation 20 1.2.6Traffic. 22 1.2.7 Continuous System Subject to Flow Bounds and Mixing . . . 25 1.3Issues andProblemsinPerformance Evaluation 28 1.4Notes 32 2GraphTheory and Petri Nets 35 2.1 Introduction . . 35 2.2Directed Graphs 35 2.3Graphs andMatrices 38 2.3.1Composition of Matrices and Graphs 41 2.3.2Maximum CycleMean 46 2.3.3The Cayley-Hamilton Theorem . . 48 2.4Petri Nets 53 2.4.1Definition 53 2.4.2Subclassesand PropertiesofPetri Nets 59 2.5Timed EventGraphs 62 2.5.1SimpleExamples 63 2.5.2 The Basic Autonomous Equation . 68 2.5.3Constructiveness of theEvolutionEquations 77 2.5.4 Standard Autonomous Equations . . 81 2.5.5 The Nonautonomous Case 83 2.5.6Constructionofthe Marking 87 2.5.7Stochastic EventGraphs 87 2.6ModelingIssues 88 iii iv Synchronization and Linearity 2.6.1Multigraphs 88 2.6.2Places with Finite Capacity 89 2.6.3SynthesisofEvent Graphs fromInteractingResources 90 2.7Notes 97 II Algebra 99 3Max-Plus Algebra 101 3.1 Introduction . 101 3.1.1Definitions 101 3.1.2Notation 103 3.1.3The minOperationinthe Max-Plus Algebra 103 3.2 Matrices in R max 104 3.2.1Linear andAffine Scalar Functions 105 3.2.2Structures 106 3.2.3 Systems of Linear Equations in (R max ) n 108 3.2.4SpectralTheoryofMatrices 111 3.2.5ApplicationtoEvent Graphs 114 3.3 Scalar Functions in R max 116 3.3.1 Polynomial Functions P(R max ) . 116 3.3.2RationalFunctions . 124 3.3.3Algebraic Equations 127 3.4Symmetrizationofthe Max-Plus Algebra 129 3.4.1 The Algebraic Structure S 129 3.4.2Linear Balances 131 3.5 Linear Systems in S 133 3.5.1Determinant 134 3.5.2 Solving Systems of Linear Balances by the Cramer Rule . . . 135 3.6 Polynomials with Coefficients in S 138 3.6.1 Some Polynomial Functions . . . 138 3.6.2 Factorization of Polynomial Functions 139 3.7 Asymptotic Behavior of A k . 143 3.7.1 Critical Graph of a Matrix A 143 3.7.2Eigenspace Associated with theMaximum Eigenvalue 145 3.7.3SpectralProjector 147 3.7.4 Convergence of A k with k 148 3.7.5Cyclic Matrices 150 3.8Notes 151 4Dioids 153 4.1 Introduction . 153 4.2Basic Definitions andExamples 154 4.2.1Axiomatics 154 4.2.2SomeExamples 155 4.2.3 Subdioids 156 4.2.4 Homomorphisms, Isomorphisms and Congruences 157 Contents v 4.3Lattice Properties of Dioids . . 158 4.3.1Basic Notions in Lattice Theory . . 158 4.3.2Order StructureofDioids 160 4.3.3CompleteDioids, ArchimedianDioids 162 4.3.4 Lower Bound 164 4.3.5DistributiveDioids 165 4.4Isotone Mappings andResiduation 167 4.4.1Isotony andContinuity of Mappings 167 4.4.2ElementsofResiduationTheory 171 4.4.3Closure Mappings 177 4.4.4Residuation of Addition and Multiplication 178 4.5 Fixed-Point Equations, Closure of Mappings and Best Approximation 184 4.5.1General Fixed-PointEquations 184 4.5.2 The Case (x) = a ◦ \x ∧ b 188 4.5.3 The Case (x) = ax ⊕ b 189 4.5.4SomeProblemsofBestApproximation 191 4.6MatrixDioids 194 4.6.1From‘Scalars’ to Matrices 194 4.6.2Residuation of Matrices and Invertibility 195 4.7 Dioids of Polynomials and Power Series . . 197 4.7.1 Definitions and Properties of Formal Polynomials and Power Series . 197 4.7.2Subtractionand Division of PowerSeries 201 4.7.3 Polynomial Matrices . 201 4.8RationalClosure andRationalRepresentations 203 4.8.1RationalClosure andRationalCalculus 203 4.8.2RationalRepresentations 205 4.8.3Yet OtherRationalRepresentations 207 4.8.4RationalRepresentations in Commutative Dioids . 208 4.9Notes 210 4.9.1Dioidsand RelatedStructures 210 4.9.2Related Results 211 III Deterministic System Theory 213 5Two-Dimensional Domain Description of Event Graphs 215 5.1 Introduction . . 215 5.2 A Comparison Between Counter and Dater Descriptions . 217 5.3 Daters and their Embedding in Nonmonotonic Functions . 221 5.3.1ADioidofNondecreasingMappings 221 5.3.2 γ -Transforms of Daters and Representation by Power Series in γ 224 5.4Movingtothe Two-DimensionalDescription 230 5.4.1 The Z max Algebra through Another Shift Operator 230 5.4.2 The M ax in [[ γ,δ]] Algebra 232 5.4.3 Algebra of Information about Events 238 5.4.4 M ax in [[ γ,δ]] Equations forEvent Graphs 238 5.5 Counters 244 vi Synchronization and Linearity 5.5.1 A First Derivation of Counters . . 244 5.5.2 Counters Derived from Daters . . 245 5.5.3 Alternative Definition of Counters 247 5.5.4 Dynamic Equations of Counters . 248 5.6Backward Equations 249 5.6.1 M ax in [[γ,δ]] Backward Equations . 249 5.6.2Backward Equations forDaters 251 5.7Rationality, Realizability and Periodicity . 253 5.7.1Preliminaries 253 5.7.2Definitions 254 5.7.3MainTheorem 255 5.7.4Onthe Coding of RationalElements 257 5.7.5 Realizations by γ -andδ-Transforms 259 5.8 Frequency Response of Event Graphs . . 261 5.8.1 Numerical Functions Associated with Elements of B[[ γ,δ]] . . 261 5.8.2 Specialization to M ax in [[ γ,δ]] 263 5.8.3Eigenfunctions of RationalTransferFunctions . 264 5.9Notes 268 6Max-Plus Linear System Theory 271 6.1 Introduction . 271 6.2SystemAlgebra 271 6.2.1Definitions 271 6.2.2SomeElementarySystems 273 6.3 Impulse Responses of Linear Systems . . 276 6.3.1 The Algebra of Impulse Responses 276 6.3.2Shift-InvariantSystems 278 6.3.3 Systems with Nondecreasing Impulse Response . 279 6.4TransferFunctions 280 6.4.1EvaluationHomomorphism 280 6.4.2 Closed Concave Impulse Responses and Inputs . 282 6.4.3 Closed Convex Inputs 284 6.5RationalSystems 286 6.5.1 Polynomial, Rational and Algebraic Systems . . 286 6.5.2 Examples of Polynomial Systems 287 6.5.3CharacterizationofRationalSystems 288 6.5.4Minimal Representationand Realization 292 6.6Correlations andFeedback Stabilization 294 6.6.1SojournTimeand Correlations 294 6.6.2Stability and Stabilization 298 6.6.3Loop Shaping 301 6.7Notes 301 Contents vii IV Stochastic Systems 303 7Ergodic Theory of Event Graphs 305 7.1 Introduction . . 305 7.2 A Simple Example in R max 306 7.2.1The EventGraph 306 7.2.2Statistical Assumptions 307 7.2.3Statement of theEigenvalueProblem 309 7.2.4Relationwith theEvent Graph 313 7.2.5Uniqueness andCoupling 315 7.2.6 First-Order and Second-Order Theorems 317 7.3First-Order Theorems 319 7.3.1Notationand Statistical Assumptions 319 7.3.2 Examples in R max 320 7.3.3 Maximal Lyapunov Exponent in R max 321 7.3.4 The Strongly Connected Case . . . 323 7.3.5General Graph 324 7.3.6First-Order Theorems in OtherDioids 328 7.4 Second-Order Theorems; Nonautonomous Case 329 7.4.1Notationand Assumptions 329 7.4.2Ratio EquationinaGeneralDioid . 330 7.4.3Stationary Solutionofthe Ratio Equation 331 7.4.4 Specialization to R max 333 7.4.5 Multiplicative Ergodic Theorems in R max 347 7.5 Second-Order Theorems; Autonomous Case 348 7.5.1Ratio Equation 348 7.5.2Backward Process 350 7.5.3 From Stationary Ratios to Random Eigenpairs . . 352 7.5.4 Finiteness and Coupling in R max ;Positive Case 353 7.5.5 Finiteness and Coupling in R max ;Strongly Connected Case . . 362 7.5.6 Finiteness and Coupling in R max ;General Case 364 7.5.7 Multiplicative Ergodic Theorems in R max 365 7.6Stationary MarkingofStochastic EventGraphs 366 7.7 Appendix on Ergodic Theorems 369 7.8Notes 370 8Computational Issues in Stochastic Event Graphs 373 8.1 Introduction . . 373 8.2 Monotonicity Properties . . . 374 8.2.1Notationfor Stochastic Ordering 374 8.2.2 Monotonicity Table for Stochastic Event Graphs . 374 8.2.3PropertiesofDaters. . 375 8.2.4 Properties of Counters 379 8.2.5PropertiesofCycle Times 383 8.2.6ComparisonofRatios 386 8.3Event Graphs andBranching Processes 388 8.3.1Statistical Assumptions 389 viii Synchronization and Linearity 8.3.2Statistical Properties 389 8.3.3 Simple Bounds on Cycle Times . 390 8.3.4General Case 395 8.4Markovian Analysis 400 8.4.1MarkovProperty 400 8.4.2DiscreteDistributions 400 8.4.3 Continuous DistributionFunctions 409 8.5Appendix 412 8.5.1Stochastic Comparison 412 8.5.2MarkovChains 414 8.6Notes 414 VPostface 417 9Related Topics andOpenEnds 419 9.1 Introduction . 419 9.2 About Realization Theory . . 419 9.2.1 The Exponential as a Tool; Another View on Cayley-Hamilton 419 9.2.2RationalTransferFunctions andARMAModels 422 9.2.3RealizationTheory 423 9.2.4MoreonMinimal Realizations 425 9.3ControlofDiscreteEvent Systems 427 9.4Brownianand DiffusionDecision Processes 429 9.4.1Inf-Convolutions of Quadratic Forms 430 9.4.2DynamicProgramming 430 9.4.3Fenchel andCramerTransforms 432 9.4.4Law of LargeNumbers in DynamicProgramming 433 9.4.5Central LimitTheorem in DynamicProgramming 433 9.4.6The Brownian Decision Process 434 9.4.7Diffusion Decision Process 436 9.5EvolutionEquations of GeneralTimed PetriNets 437 9.5.1FIFOTimed PetriNets 437 9.5.2EvolutionEquations 439 9.5.3EvolutionEquations forSwitching 444 9.5.4Integrationofthe RecursiveEquations 446 9.6Min-Max Systems 447 9.6.1General TimedPetri Nets andDescriptorSystems 449 9.6.2Existence of PeriodicBehavior. . 450 9.6.3Numerical Procedures forthe Eigenvalue 452 9.6.4Stochastic Min-MaxSystems 455 9.7 About Cycle Times in General Petri Nets 456 9.8Notes 458 Bibliography 461 Notation 471 [...]... theory and linear system theory form an ideal background For algebra, [61] for instance provides suitable background material; for probability theory this role is for instance played by [20], and for linear system theory it is [72] or the more recent [122] The heart of the book consists of four main parts, each of which consists of two chapters Part I (Chapters 1 and 2) provides a natural motivation for. .. stochastically For a deterministic and invariant (‘customer invariant’) system, the serving times do not, by definition, depend on the particular customer 20 1.2.5 Synchronization and Linearity Parallel Computation The application of this subsection belongs to the field of VLSI array processors (VLSI stands for ‘Very Large Scale Integration’) The theory of discrete events provides a method for analyzing the performances... Baccelli, Sophia Antipolis ¸ Guy Cohen, Fontainebleau Geert Jan Olsder, Delft Jean-Pierre Quadrat, Rocquencourt June 1992 1 GC: I do not agree FB is more prone to that than any of us! xiv Synchronization and Linearity Part I Discrete Event Systems and Petri Nets 1 Chapter 1 Introduction and Motivation 1.1 Preliminary Remarks and Some Notation Probably the most well-known equation in the theory of difference... (1.20) is an implicit equation for the vector x Let us see what we obtain by repeated substitution of the complete right-hand side of (1.20) into x of this same right-hand side After one substitution: x = = A2 x ⊕ ABu ⊕ Bu A2 x ⊕ (A ⊕ e)Bu , and after n substitutions: x = An x ⊕ (An 1 ⊕ An 2 ⊕ · · · ⊕ A ⊕ e)Bu In the formulæ above, e refers to the identity matrix; zeros on the diagonal and ε’s elsewhere... Network with input and output input u(k) only goes to node 2 If one were to replace B by ( 2 1 ) for instance, 8 Synchronization and Linearity where the prime denotes transposition, then each input would ‘spread’ itself over the two nodes In this example from epoch u(k) on, it takes 2 time units for the input to reach node 1 and 1 time unit to reach node 2 In many practical situations an input will enter... introduction and relationships with graph theory and Petri nets Part II (Chapters 3 and 4) is devoted to the underlying algebras Once the reader has gone through this part, he will also appreciate the more abstract approach presented in Parts III and IV Part III (Chapters 5 and 6) deals with deterministic system theory, where the systems are mostly DEDS, but continuous max-plus linear systems also... shorthand symbol ⊕ for the minimum operation, then (1.17) becomes Aik ⊗ Ak j k Note that has been used for both the maximum and the minimum operation It should be clear from the context which is meant The symbol ⊕ will be used similarly The reason for not distinguishing between these two operations is that (R ∪ {−∞}, max, +) and (R ∪ {+∞}, min, +) are isomorphic algebraic structures Chapters 3 and 4... certain node can only start when all preceding (‘directly upstream’) nodes have finished their activities and sent the results of these activities along the arcs to the current node Thus, the arc corresponding to Ai j can be interpreted as an output channel for node j and simultaneously as an input channel for node i Suppose that this node i starts its activity as soon as all preceding nodes have sent their... (lubrication say) and the traveling time of the result (the final product) from node 1 to node 2 is 1 time unit Thus the number A11 = 3 is made up of a production time 1 and a recovery time 2 and the number A21 = 2 is made up of the same production time 1 and a traveling time 1 Similarly, if the production time at node 2 is 4, then this node does not need any time for recovery (because A22 = 4), and the traveling... relevant for some systems which either are continuous or do not involve synchronization Systems which mix Preface xi fluids in certain proportions and which involve flow constraints fall in the former category Recursive ‘optimization processes’, of which dynamic programming is the most immediate example, fall in the latter category All these systems involve max (or min) and + as the basic operations Another . Synchronization and Linearity An Algebra for Discrete Event Systems Franc¸ois Baccelli INRIA and ´ Ecole Normale Sup´erieure, D´epartement d’Informatique Paris,. ix IDiscrete Event Systems and Petri Nets 1 1Introduction and Motivation 3 1.1PreliminaryRemarks andSomeNotation 3 1.2Miscellaneous Examples 8 1.2.1Planning