Markov Chains: Gibbs fields, Monte Carlo simulation and queues
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Format: | Buch |
Sprache: | English |
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Springer
[2020]
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Ausgabe: | Second edition |
Schriftenreihe: | Texts in applied mathematics
Volume 31 |
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Online-Zugang: | Inhaltsverzeichnis |
Beschreibung: | xvi, 557 Seiten Illustrationen, Diagramme |
ISBN: | 9783030459819 9783030459840 |
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Datensatz im Suchindex
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adam_text | Contents Preface 1 Probability Review 1.1 1.2 1.3 1.4 1.5 Basic Concepts...................................................................................... 1.1.1 Events, Random Variables andProbability.............................. 1.1.2 Independence and ConditionalProbability.............................. 1.1.3 Expectation................................................................................. 1.1.4 Conditional Expectation........................................................... Transforms of Probability Distributions.............................................. 1.2.1 Generating Functions.................................................................. 1.2.2 Characteristic Functions........................................................... Transformations of Random Vectors.................................................... 1.3.1 Smooth Change of Variables..................................................... 1.3.2 Order Statistics........................................................................... Almost-Sure Convergence ................................................................... 1.4.1 Two Basic Tools ........................................................................ 1.4.2 The Strong Law of Large Numbers......................................... Exercises................................................................................................... 2 Discrete-Time Markov Chains 2.1 The Transition Matrix.......................................................................... 2.1.1 The Markov Property
............................................................. 2.1.2 The Distribution of anHMC..................................................... 2.2 Markov Recurrences ............................................................................. 2.2.1 A Canonical Representation.................................................... 2.2.2 First-Step Analysis.................................................................... 2.3 Topology of the TransitionMatrix......................................................... 2.3.1 Communication.......................................................................... 2.3.2 Period......................................................................................... 2.4 Steady State............................................................................................. 2.4.1 Stationarity................................................................................ vii 1 1 1 8 15 33 37 37 42 45 45 47 48 48 52 58 63 63 63 65 67 67 73 80 80 81 85 85 xi
Contents xii 2.4.2 Time Reversal............................................................................. 91 2.5 Regeneration ......................................................................................... 96 2.5.1 The Strong Markov Property ................................................. 96 2.5.2 Regenerative Cycles ................................................................ 99 2.6 Exercises..................................................................................................... 102 3 Recurrence and Ergo dicity 111 3.1 The Potential Matrix Criterion ........................................................... Ill 3.1.1 Recurrent and TransientStates.................................................. Ill 3.1.2 A Criterion of Recurrence........................................................... 113 3.1.3 Structure of the Transition Matrix ........................................... 116 3.2 Recurrence.................................................................................................. 117 3.2.1 Invariant Measures....................................................................... 117 3.2.2 A Positive Recurrence Criterion..................................................120 3.3 Empirical Averages................................................................................... 128 3.3.1 The Ergodic Theorem ................................................................. 128 3.3.2 The Renewal Reward Theorem .................................................. 135 3.4
Exercises..................................................................................................... 138 4 Long-Run Behavior 145 4.1 Coupling..................................................................................................... 145 4.1.1 Convergence in Variation.............................................................. 145 4.1.2 The Coupling Method................................................................. 149 4.2 Convergence toSteady State..................................................................... 150 4.2.1 The Positive Recurrent Case........................................................150 4.2.2 The Null Recurrent Case.............................................................. 152 4.3 Convergence Rates,a First Look.................................................................157 4.3.1 Convergence Rates via Coupling..................................................157 4.3.2 The Perron-Frobenius Theorem..................................................158 4.3.3 Quasi-stationary Distributions.....................................................162 4.3.4 Dobrushin’s Ergodic Coefficient..................................................163 4.4 Exercises..................................................................................................... 166 5 Discrete-Time Renewal Theory 171 5.1 The Renewal Process................................................................................ 171 5.1.1 The Renewal Equation................................................................. 171 5.1.2 The Renewal
Theorem................................................................. 175 5.1.3 Defective Renewal Sequences........................................................177 5.2 Regenerative Processes............................................................................. 180 5.2.1 Renewal Equations for Regenerative Processes......................... 180 5.2.2 The Regenerative Theorem........................................................... 182
Contents 5.3 xiii Exercises.....................................................................................................185 6 Absorption and Passage Times 187 6.1 Life Before Absorption............................................................................ 187 6.1.1 Infinite Sojourns ......................................................................... 187 6.1.2 Time to Absorption...................................................................... 192 6.2 Absorption Probabilities.......................................................................... 193 6.2.1 The First Fundamental Matrix ................................................. 193 6.2.2 The Absorption Matrix................................................................. 196 6.2.3 Hitting Times Formula............................................................... 202 6.3 The Second Fundamental Matrix.......................................................... 204 6.3.1 Definition...................................................................................... 204 6.3.2 The Mutual Time-Distance Matrix........................................... 208 6.3.3 Variance of Ergodic Estimates.................................................... 211 6.4 The Branching Process............................................................................ 214 6.4.1 The Galton-Watson Model.......................................................... 214 6.4.2 Tail Distributions......................................................................... 218 6.4.3 Conditioning by Extinction
....................................................... 221 6.5 Exercises.................................................................................................... 223 7 Lyapunov Functions and Martingales 7.1 7.2 7.3 7.4 8 Random Walks on Graphs 8.1 8.2 227 Lyapunov Functions .............................................................................. 227 7.1.1 Foster’s Theorem......................................................................... 227 7.1.2 Queueing Applications................................................................ 231 Martingales and Potentials..................................................................... 237 7.2.1 Harmonic Functions and Martingales........................................237 7.2.2 The Maximum Principle............................................................. 239 Martingales and hmcs........................................................................... 244 7.3.1 The Two Pillars of Martingale Theory.................................... 244 7.3.2 Transience and Recurrence viaMartingales............................... 246 7.3.3 Absorption via Martingales ....................................................... 249 Exercises.................................................................................................... 252 255 Pure Random Walks............................................................................. 255 8.1.1 The Symmetric Random Walks on TL andZ3............................ 255 8.1.2 The Pure Random Walk on a Graph....................................... 259 8.1.3 Spanning Trees and
Cover Times............................................. 260 Symmetric Walks on a Graph................................................................265 8.2.1 Reversible Chains as Symmetric Walks.................................... 265 8.2.2 The Electrical Network Analogy................................................ 268
Contents xiv 8.3 8.4 9 Effective Resistance and Escape Probability........................................ 273 8.3.1 Computation of the Effective Resistance.................................. 273 8.3.2 Thompson’s and Rayleigh’s Principles ..................................... 277 8.3.3 Infinite Networks..........................................................................280 Exercises.................................................................................................... 283 Convergence Rates 289 9.1 Reversible Transition Matrices................................................................ 289 9.1.1 A Characterization of Reversibility........................................... 289 9.1.2 Convergence Rates in Terms of the SLEM.................................. 292 9.1.3 Rayleigh’s Spectral Theorem....................................................... 297 9.2 Bounds for the slem............................................................................... 299 9.2.1 Bounds via Rayleigh’s Characterization..................................... 299 9.2.2 Strong Stationary Times............................................................. 308 9.2.3 Reversibilization ..........................................................................313 9.3 Mixing Times........................................................................................... 315 9.3.1 Basic Definitions..........................................................................315 9.3.2 Upper Bounds via Coupling....................................................... 317 9.3.3 Lower
Bounds................................................................................ 319 9.4 Exercises..................................................................................................... 323 10 Markov Fields on Graphs 331 10.1 The Gibbs-Markov Equivalence.............................................................331 10.1.1 Local Characteristics..................................................................331 10.1.2 Gibbs Distributions..................................................................... 334 10.2 Specific Models........................................................................................ 343 10.2.1 Random Points........................................................................... 343 10.2.2 The Auto-binomial Texture Model ....................... 345 10.2.3 The Pixel-Edge Model...............................................................348 10.2.4 Markov Fields in Image Processing......................................... 351 10.3 Phase Transition in the Ising Model...................................................... 356 10.3.1 Experimental Results.................................................................. 356 10.3.2 The Peierls Argument ............................................................... 359 10.4 Exercises.................................................................................................... 362 11 Monte Carlo Markov Chains 369 11.1 General Principles of Simulation.............................................................369 11.1.1 Simulation via the Law of Large
Numbers............................. 369 11.1.2 Two Methods for Sampling a pdf............................................. 370 11.2 Monte Carlo Markov Chain Sampling................................................... 373 11.2.1 The Basic Idea............................................................................ 373
Contents XV 11.2.2 Convergence Rates in MCMC................................................... 375 11.2.3 Variance of Monte Carlo Estimators.......................................... 378 11.3 The Gibbs Sampler.................................................................................. 386 11.3.1 Simulation of Random Fields ................................................... 386 11.3.2 Convergence Rate of the Gibbs Sampler ................................ 388 11.4 Exact sampling........................................................................................ 389 11.4.1 The Propp-Wilson algorithm ................................................... 389 11.4.2 Sandwiching.................................................................................. 393 11.5 Exercises.................................................................................................... 397 12 Non-homogeneous Markov Chains 399 12.1 Weak and Strong Ergodicity...................................................................399 12.1.1 Ergodicity of Non-Homogeneous Markov Chains.....................399 12.1.2 The Block Criterion of Weak Ergodicity ................................. 401 12.1.3 A Sufficient Condition for StrongErgodicity............................ 402 12.2 Simulated Annealing............................................................................... 408 12.2.1 Stochastic Descent and Cooling................................................ 408 12.2.2 Convergence of Simulated Annealing....................................... 416 12.3
Exercises.................................................................................................... 421 13 Continuous-Time Markov Chains 423 13.1 Poisson Processes..................................................................................... 423 13.1.1 Point Processes........................................................................... 423 13.1.2 The Counting Process of an hpp ............................................ 424 13.1.3 Competing Poisson Processes .................................................. 427 13.2 Distribution of a Continuous-Time hmc ...........................................430 13.2.1 The Transition Semi-Group..................................................... 430 13.2.2 The Infinitesimal Generator..................................................... 434 13.2.3 Kolmogorov’s Differential Systems............................................ 440 13.3 The Regenerative Structure.................................................................. 447 13.3.1 The Strong Markov Property .................................................. 447 13.3.2 Embedded Chain and Transition Times................................... 450 13.3.3 Explosions.................................................................................... 453 13.4 Recurrence and Long-Run Behavior...................................................... 460 13.4.1 Stationary Distribution Criterion of Ergodicity....................... 460 13.4.2 Ergodicity.................................................................................... 467 13.4.3 Absorption
................................................................................. 469 13.5 Continuous-Time HMCs from hpps ...................................................... 470 13.5.1 The Strong Markov Property of hpps...................................... 470 13.5.2 Թօա Generator to Markov Chain............................................ 473 13.5.3 Poisson Calculus and Continuous-time HMCs.......................... 477
Contents xvi 13.6 Exercises.................................................................................................... 486 14 Markovian Queueing Theory 491 14.1 Poisson Systems........................................................................................ 491 14.1.1 The Purely Poissonian Description .......................................... 491 14.1.2 Markovian Queues as Poisson Systems.................................... 494 14.2 Isolated Markovian Queues...................................................................... 500 14.2.1 As Birth-and-Death Processes................................................... 500 14.2.2 TheM/GI/1/oo/fifo Queue .................................................... 503 14.2.3 TheGI/M/1/oo/fifo Queue .................................................... 508 14.3 Markovian Queueing Networks................................................................ 513 14.3.1 TheTandem Network ................................................................ 513 14.3.2 TheJackson Network................................................................... 515 14.3.3 The Gordon-Newell Network ................................................... 519 14.4 Exercises.....................................................................................................522 A Appendix 527 A.l Number TheoryandCalculus.................................................................. 527 A. 1.1 Greatest Common Divisor......................................................... 527 A. 1.2 Abel’s
Theorem........................................................................... 528 A. 1.3 Cesàro’s Lemma ........................................................................ 529 A. 1.4 Lebesgue’s Theorems forSeries................................................. 530 A. 1.5 Infinite Products........................................................................ 532 A. 1.6 Tychonov’s Theorem.................................................................. 533 A.1.7 Subadditive Functions.............................................................. 533 A.2 Linear Algebra ........................................................................................ 534 A.2.1 Eigenvalues and Eigenvectors .................................................. 534 A.2.2 Exponential of a Matrix........................................................... 536 A.2.3 Gershgorin’s Bound.................................................................... 538 A.3 Probability and Expectation...................................................................539 A.3.1 Expectation Revisited .............................................................. 539 A.3.2 Lebesgue’sTheoremfor Expectation.......................................... 541 Bibliography 545 Index 553
|
adam_txt |
Contents Preface 1 Probability Review 1.1 1.2 1.3 1.4 1.5 Basic Concepts. 1.1.1 Events, Random Variables andProbability. 1.1.2 Independence and ConditionalProbability. 1.1.3 Expectation. 1.1.4 Conditional Expectation. Transforms of Probability Distributions. 1.2.1 Generating Functions. 1.2.2 Characteristic Functions. Transformations of Random Vectors. 1.3.1 Smooth Change of Variables. 1.3.2 Order Statistics. Almost-Sure Convergence . 1.4.1 Two Basic Tools . 1.4.2 The Strong Law of Large Numbers. Exercises. 2 Discrete-Time Markov Chains 2.1 The Transition Matrix. 2.1.1 The Markov Property
. 2.1.2 The Distribution of anHMC. 2.2 Markov Recurrences . 2.2.1 A Canonical Representation. 2.2.2 First-Step Analysis. 2.3 Topology of the TransitionMatrix. 2.3.1 Communication. 2.3.2 Period. 2.4 Steady State. 2.4.1 Stationarity. vii 1 1 1 8 15 33 37 37 42 45 45 47 48 48 52 58 63 63 63 65 67 67 73 80 80 81 85 85 xi
Contents xii 2.4.2 Time Reversal. 91 2.5 Regeneration . 96 2.5.1 The Strong Markov Property . 96 2.5.2 Regenerative Cycles . 99 2.6 Exercises. 102 3 Recurrence and Ergo dicity 111 3.1 The Potential Matrix Criterion . Ill 3.1.1 Recurrent and TransientStates. Ill 3.1.2 A Criterion of Recurrence. 113 3.1.3 Structure of the Transition Matrix . 116 3.2 Recurrence. 117 3.2.1 Invariant Measures. 117 3.2.2 A Positive Recurrence Criterion.120 3.3 Empirical Averages. 128 3.3.1 The Ergodic Theorem . 128 3.3.2 The Renewal Reward Theorem . 135 3.4
Exercises. 138 4 Long-Run Behavior 145 4.1 Coupling. 145 4.1.1 Convergence in Variation. 145 4.1.2 The Coupling Method. 149 4.2 Convergence toSteady State. 150 4.2.1 The Positive Recurrent Case.150 4.2.2 The Null Recurrent Case. 152 4.3 Convergence Rates,a First Look.157 4.3.1 Convergence Rates via Coupling.157 4.3.2 The Perron-Frobenius Theorem.158 4.3.3 Quasi-stationary Distributions.162 4.3.4 Dobrushin’s Ergodic Coefficient.163 4.4 Exercises. 166 5 Discrete-Time Renewal Theory 171 5.1 The Renewal Process. 171 5.1.1 The Renewal Equation. 171 5.1.2 The Renewal
Theorem. 175 5.1.3 Defective Renewal Sequences.177 5.2 Regenerative Processes. 180 5.2.1 Renewal Equations for Regenerative Processes. 180 5.2.2 The Regenerative Theorem. 182
Contents 5.3 xiii Exercises.185 6 Absorption and Passage Times 187 6.1 Life Before Absorption. 187 6.1.1 Infinite Sojourns . 187 6.1.2 Time to Absorption. 192 6.2 Absorption Probabilities. 193 6.2.1 The First Fundamental Matrix . 193 6.2.2 The Absorption Matrix. 196 6.2.3 Hitting Times Formula. 202 6.3 The Second Fundamental Matrix. 204 6.3.1 Definition. 204 6.3.2 The Mutual Time-Distance Matrix. 208 6.3.3 Variance of Ergodic Estimates. 211 6.4 The Branching Process. 214 6.4.1 The Galton-Watson Model. 214 6.4.2 Tail Distributions. 218 6.4.3 Conditioning by Extinction
. 221 6.5 Exercises. 223 7 Lyapunov Functions and Martingales 7.1 7.2 7.3 7.4 8 Random Walks on Graphs 8.1 8.2 227 Lyapunov Functions . 227 7.1.1 Foster’s Theorem. 227 7.1.2 Queueing Applications. 231 Martingales and Potentials. 237 7.2.1 Harmonic Functions and Martingales.237 7.2.2 The Maximum Principle. 239 Martingales and hmcs. 244 7.3.1 The Two Pillars of Martingale Theory. 244 7.3.2 Transience and Recurrence viaMartingales. 246 7.3.3 Absorption via Martingales . 249 Exercises. 252 255 Pure Random Walks. 255 8.1.1 The Symmetric Random Walks on TL andZ3. 255 8.1.2 The Pure Random Walk on a Graph. 259 8.1.3 Spanning Trees and
Cover Times. 260 Symmetric Walks on a Graph.265 8.2.1 Reversible Chains as Symmetric Walks. 265 8.2.2 The Electrical Network Analogy. 268
Contents xiv 8.3 8.4 9 Effective Resistance and Escape Probability. 273 8.3.1 Computation of the Effective Resistance. 273 8.3.2 Thompson’s and Rayleigh’s Principles . 277 8.3.3 Infinite Networks.280 Exercises. 283 Convergence Rates 289 9.1 Reversible Transition Matrices. 289 9.1.1 A Characterization of Reversibility. 289 9.1.2 Convergence Rates in Terms of the SLEM. 292 9.1.3 Rayleigh’s Spectral Theorem. 297 9.2 Bounds for the slem. 299 9.2.1 Bounds via Rayleigh’s Characterization. 299 9.2.2 Strong Stationary Times. 308 9.2.3 Reversibilization .313 9.3 Mixing Times. 315 9.3.1 Basic Definitions.315 9.3.2 Upper Bounds via Coupling. 317 9.3.3 Lower
Bounds. 319 9.4 Exercises. 323 10 Markov Fields on Graphs 331 10.1 The Gibbs-Markov Equivalence.331 10.1.1 Local Characteristics.331 10.1.2 Gibbs Distributions. 334 10.2 Specific Models. 343 10.2.1 Random Points. 343 10.2.2 The Auto-binomial Texture Model . 345 10.2.3 The Pixel-Edge Model.348 10.2.4 Markov Fields in Image Processing. 351 10.3 Phase Transition in the Ising Model. 356 10.3.1 Experimental Results. 356 10.3.2 The Peierls Argument . 359 10.4 Exercises. 362 11 Monte Carlo Markov Chains 369 11.1 General Principles of Simulation.369 11.1.1 Simulation via the Law of Large
Numbers. 369 11.1.2 Two Methods for Sampling a pdf. 370 11.2 Monte Carlo Markov Chain Sampling. 373 11.2.1 The Basic Idea. 373
Contents XV 11.2.2 Convergence Rates in MCMC. 375 11.2.3 Variance of Monte Carlo Estimators. 378 11.3 The Gibbs Sampler. 386 11.3.1 Simulation of Random Fields . 386 11.3.2 Convergence Rate of the Gibbs Sampler . 388 11.4 Exact sampling. 389 11.4.1 The Propp-Wilson algorithm . 389 11.4.2 Sandwiching. 393 11.5 Exercises. 397 12 Non-homogeneous Markov Chains 399 12.1 Weak and Strong Ergodicity.399 12.1.1 Ergodicity of Non-Homogeneous Markov Chains.399 12.1.2 The Block Criterion of Weak Ergodicity . 401 12.1.3 A Sufficient Condition for StrongErgodicity. 402 12.2 Simulated Annealing. 408 12.2.1 Stochastic Descent and Cooling. 408 12.2.2 Convergence of Simulated Annealing. 416 12.3
Exercises. 421 13 Continuous-Time Markov Chains 423 13.1 Poisson Processes. 423 13.1.1 Point Processes. 423 13.1.2 The Counting Process of an hpp . 424 13.1.3 Competing Poisson Processes . 427 13.2 Distribution of a Continuous-Time hmc .430 13.2.1 The Transition Semi-Group. 430 13.2.2 The Infinitesimal Generator. 434 13.2.3 Kolmogorov’s Differential Systems. 440 13.3 The Regenerative Structure. 447 13.3.1 The Strong Markov Property . 447 13.3.2 Embedded Chain and Transition Times. 450 13.3.3 Explosions. 453 13.4 Recurrence and Long-Run Behavior. 460 13.4.1 Stationary Distribution Criterion of Ergodicity. 460 13.4.2 Ergodicity. 467 13.4.3 Absorption
. 469 13.5 Continuous-Time HMCs from hpps . 470 13.5.1 The Strong Markov Property of hpps. 470 13.5.2 Թօա Generator to Markov Chain. 473 13.5.3 Poisson Calculus and Continuous-time HMCs. 477
Contents xvi 13.6 Exercises. 486 14 Markovian Queueing Theory 491 14.1 Poisson Systems. 491 14.1.1 The Purely Poissonian Description . 491 14.1.2 Markovian Queues as Poisson Systems. 494 14.2 Isolated Markovian Queues. 500 14.2.1 As Birth-and-Death Processes. 500 14.2.2 TheM/GI/1/oo/fifo Queue . 503 14.2.3 TheGI/M/1/oo/fifo Queue . 508 14.3 Markovian Queueing Networks. 513 14.3.1 TheTandem Network . 513 14.3.2 TheJackson Network. 515 14.3.3 The Gordon-Newell Network . 519 14.4 Exercises.522 A Appendix 527 A.l Number TheoryandCalculus. 527 A. 1.1 Greatest Common Divisor. 527 A. 1.2 Abel’s
Theorem. 528 A. 1.3 Cesàro’s Lemma . 529 A. 1.4 Lebesgue’s Theorems forSeries. 530 A. 1.5 Infinite Products. 532 A. 1.6 Tychonov’s Theorem. 533 A.1.7 Subadditive Functions. 533 A.2 Linear Algebra . 534 A.2.1 Eigenvalues and Eigenvectors . 534 A.2.2 Exponential of a Matrix. 536 A.2.3 Gershgorin’s Bound. 538 A.3 Probability and Expectation.539 A.3.1 Expectation Revisited . 539 A.3.2 Lebesgue’sTheoremfor Expectation. 541 Bibliography 545 Index 553 |
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any_adam_object_boolean | 1 |
author | Brémaud, Pierre |
author_GND | (DE-588)1060028131 |
author_facet | Brémaud, Pierre |
author_role | aut |
author_sort | Brémaud, Pierre |
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building | Verbundindex |
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classification_rvk | SK 820 |
classification_tum | MAT 629 MAT 607 |
ctrlnum | (OCoLC)1164648628 (DE-599)BVBBV046744732 |
discipline | Mathematik |
discipline_str_mv | Mathematik |
edition | Second edition |
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genre_facet | Lehrbuch |
id | DE-604.BV046744732 |
illustrated | Illustrated |
index_date | 2024-07-03T14:40:22Z |
indexdate | 2024-07-10T08:52:38Z |
institution | BVB |
isbn | 9783030459819 9783030459840 |
language | English |
oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-032154559 |
oclc_num | 1164648628 |
open_access_boolean | |
owner | DE-29T DE-11 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-739 |
owner_facet | DE-29T DE-11 DE-91G DE-BY-TUM DE-19 DE-BY-UBM DE-83 DE-739 |
physical | xvi, 557 Seiten Illustrationen, Diagramme |
publishDate | 2020 |
publishDateSearch | 2020 |
publishDateSort | 2020 |
publisher | Springer |
record_format | marc |
series | Texts in applied mathematics |
series2 | Texts in applied mathematics |
spelling | Brémaud, Pierre (DE-588)1060028131 aut Markov Chains Gibbs fields, Monte Carlo simulation and queues Pierre Brémaud Second edition Cham ; Switzerland Springer [2020] © 2020 xvi, 557 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Texts in applied mathematics Volume 31 bicssc bisacsh Probabilities Operations research Decision making Electrical engineering Mathematics Markov-Kette (DE-588)4037612-6 gnd rswk-swf Stochastischer Prozess (DE-588)4057630-9 gnd rswk-swf Hardcover, Softcover / Mathematik/Wahrscheinlichkeitstheorie, Stochastik, Mathematische Statistik 1\p (DE-588)4123623-3 Lehrbuch gnd-content Markov-Kette (DE-588)4037612-6 s Stochastischer Prozess (DE-588)4057630-9 s 2\p DE-604 DE-604 Erscheint auch als Online-Ausgabe 978-3-030-45982-6 Texts in applied mathematics Volume 31 (DE-604)BV002476038 31 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032154559&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
spellingShingle | Brémaud, Pierre Markov Chains Gibbs fields, Monte Carlo simulation and queues Texts in applied mathematics bicssc bisacsh Probabilities Operations research Decision making Electrical engineering Mathematics Markov-Kette (DE-588)4037612-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
subject_GND | (DE-588)4037612-6 (DE-588)4057630-9 (DE-588)4123623-3 |
title | Markov Chains Gibbs fields, Monte Carlo simulation and queues |
title_auth | Markov Chains Gibbs fields, Monte Carlo simulation and queues |
title_exact_search | Markov Chains Gibbs fields, Monte Carlo simulation and queues |
title_exact_search_txtP | Markov Chains Gibbs fields, Monte Carlo simulation and queues |
title_full | Markov Chains Gibbs fields, Monte Carlo simulation and queues Pierre Brémaud |
title_fullStr | Markov Chains Gibbs fields, Monte Carlo simulation and queues Pierre Brémaud |
title_full_unstemmed | Markov Chains Gibbs fields, Monte Carlo simulation and queues Pierre Brémaud |
title_short | Markov Chains |
title_sort | markov chains gibbs fields monte carlo simulation and queues |
title_sub | Gibbs fields, Monte Carlo simulation and queues |
topic | bicssc bisacsh Probabilities Operations research Decision making Electrical engineering Mathematics Markov-Kette (DE-588)4037612-6 gnd Stochastischer Prozess (DE-588)4057630-9 gnd |
topic_facet | bicssc bisacsh Probabilities Operations research Decision making Electrical engineering Mathematics Markov-Kette Stochastischer Prozess Lehrbuch |
url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=032154559&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
volume_link | (DE-604)BV002476038 |
work_keys_str_mv | AT bremaudpierre markovchainsgibbsfieldsmontecarlosimulationandqueues |