Queueing Networks and Markov Chains

Transcription

1 Queueing Networks and Markov Chains Modeling and Performance Evaluation with Computer Science Applications Gunter Bolch, Stefan Greiner, Hermann de Meer, and Kishor S. Trivedi A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto

2 Contents Preface xv 1 Introduction \ 1.1 Motivation Basics of Probabihty and Statistics Random Variables Discrete Random Variables Continuous Random, Variables Multiple Random Variables Independence Condiüonal ProbaHUty Irnportant Relations The Central Limit Theorem Tmnsforms z-transform Laplace Tra/asform, Parameter Estimation Method of Moments Maximum-Likelihood Estimation Confidance Intervall Order Statistics 30

3 1.2.6 Distribution of Sums 31 Markov Chains Markov Processes Stochastic and Markov Processes Markov Chains Discrete-Time Markov Chains Continuous-Time Markov Chains Recapitulation The Modeling Process Modeling Life-cycle Phases Performance Measures A Simple Example Markov Reward Models A Gase Study Generation Methods Petri Nets Generalized Stochastic Petri Nets Stochastic Reward Nets GSPN/SRN Analysis A Larger Example 98 Steady-State Solutions of Markov Chains Symbolic Solution: Birth-Death Process Hessenberg Matrix: Non-Markovian Queues Non-Exponential Service Times Server with Vacations Polling Systems Analysis Numerical Solution: Direct Methods Gaussian Elimination The Grassm.ann Algorithm Numerical Solution: Iterative Methods Convergence of Iterative Methods Power Method Jacobi's Method Gauss-Seidel Method 139 3J h 5 The Method of Successive Over-Relaxation 140

4 The Relaxation Parameter The Test of Convergence Overflow and Underflow The Algorithm Comparison of Numerical Solution Methods Gase Studies Steady-State Aggregation/Disaggregation Methods 153 J h l Courtois's Approximate Method Decomposition Applicabüity Analysis of the Substructures Aggregation and Unconditioning The Algorithm Takahashi's Iterative Method The Fundamental Equations Applicabüity The Algorithm Application Aggregation Disaggregation Final Remarks Transient Solution of Markov Chains Transient Analysis Using Exact Methods A Pure Birth Process A Two-State CTMC Solution Using Laplace Transforrns I Numerical Solution Using Uniformization The Instantaneous Gase Stiffness Tolerant Uniformization The Cumulaiive Gase Other Numerical Methods Ordinary Differential Equations Weak Lumpability Aggregation of Stiff Markov Chains Outline and Basic Definitions Aggregation of Fast Recurrent Subsets 192

5 5.2.3 Aggregation of Fast Transient Subsets Aggregation of Initial State Probahüities Disaggregations Fast Transient States Fast Recurrent States The Algorühm An Example: Server Breakdown and Repair Single Station Queueing Systems Kendall's Notation Performance Measures The M/M/l System The M/M/oo System The M/M/m System The M/M/l/K Fimte Capacity System Machine Repairman Model Closed Tandem Network The M/G/l System The GI/M/1 System The GI/M/m System The GI/G/1 System The M/G/m System The Gl/G/m System Priority Queueing System without Preemption Conservation Laws System with Preemption System with Time-Dependent Priorities The Asymmetrie System Approximate Analysis Exact Analysis Analysis of M/M/m Loss Systems Extending to Non-Lossy System Exact Analysis of an Asymmetrie System M/M/ Systems with Batch Service Queueing Networks 263

6 ix 7.1 Definitions and Notation Single Class Networks Multiclass Networks Performance Measures Single Class Networks Multiclass Networks Product-Form Queueing Networks Global Balance Local Balance Product-Form Jackson Networks Gordon/Newell Networks BCMP Networks The Concept of Chains BCMP Theorem 300 Algorühms for Product-Form Networks The Convolution Algorithm Single Class Closed Networks Multiclass Closed Networks The Mean Value Analysis Single Class Closed Networks Multiclass Closed Networks Mixed Networks Networks with Load-Dependent Service Closed Networks Mixed Networks The REGAL Method Flow Equivalent Server Method FES Method for a Single Node FES Method for Multiple Nödes Summary 376 Approximation Algorühms for Product-Form Networks Approximations Based on the MVA Bard-Schweitzer Approximation Self-Correcting Approximation Technique Single Server Nödes 385

7 Multiple Server Nödes Extended SC AT Algorithm Summaüon Method Single Class Networks Multiclass Networks Bottapprox Method Initial Value of X Single Class Networks 405 rr Multiclass Networks Bounds Analysis Asymptotic Bounds Analysis 4H Balanced Job Bounds Analysis Networks with Variabiliiies in Workload Summary 4^0 10 Algorithms for Non-Product-Form Networks Non-Exponential Bistributions 4^ Diffusion Approximation 4% Open Networks 4% Closed Networks Maximum Entropy Method Open Networks Closed Networks Decomposition for Open Networks Methods for Closed Networks 448 IO.I.4.I Robustness for Closed Networks 448 IOA.4.2 Marie 's Method Extended SUM and BOTT Closing Method for Mvxed Networks Different Service Times at FCFS Nodes Priority Networks 4^ Extended MVA (PRIOMVA) The Method of Shadow Server 4^ The Original Shadow Techmque Extensions of the Shadow Technique Extended SUM Sirnultaneous Resource Possession 407 IO.4.I Memory Constraints 498

8 xi I/O Subsystems Method of Surrogate Delays 504 IO.4.4 Serialization Programs with Internal Concurrency Parallel Processing Asynchronous Tasks Fork-Join Systems Modeimg Performance Measures Analysis Networks with Asymmetrie Nodes Closed Networks Asymmetrie SUM (ASUM) Asymmetrie MVA (AMVA) ASymmetrie SCAT (ASCA T) Open Networks Networks with Blocking Different Blocking Types Solution for Networks with Two Nodes Optimization Optimization Problems and Cost Functions OptirnizaMon based on the Summaüon Method Maximization of the Throughput Minimization of Cost Minimization of the Response Tim,e Optimization based on the Convolution Algorithm Maximization of the Throughput Optimal Design of Storage HierareMes Performance Analysis Tools PEPSY Structure of PEPSY The Input File The Output File Control Files Different Programs in PEPSY Example of using PEPSY 575

9 xii CONTENTS Graphieal User Interface XPEPSY SPNP SPNP Features The CSPL Language ispn MOSES The Model Descnption Language MOSEL Examples Central-Server Model Fault Tolerant Multiprocessor System SHARPE 594 I2.4.I Central-Server Queueing Network 594 T2.4.2 M/M/m/K System M/M/l/K System with Server Failure and Repair Charactertstics of Same Tools Applications Case Studies of Queueing Networks Multiprocessor Systems Tightly Coupled Systems Loosely Coupled Systems Client Server Systems Communication Systems Description of the System Queueing Network Model Model Parameters Results UNIX Kernel Model of the UNIX Kernel Analysis Flexible Production Systems An Open Network Model A Clos ed Network Model Case Studies. of Markov Chains Wafer Production System Polling Systems Client Server Systems 645

10 XIII ISDN Channel The Baseline Model Markov Modutated Poisson Processen ATM Network under Overload Case Studies of Hiemrchical Models A Multiprocessor with Different Cache Strategies Performabüity of a Multiprocessor System 674 Glossary ßjg Bibliography 691 Index