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Part Ⅰ MODELING THEORY 1

1 DISTRIBUTION FUNCTIONS 5

1.1 Basic Definitions 5

1.2 The Exponential Distribution 9

1.3 Operations on Random Variables 10

1.4 Exponential Polynomial Distributions 17

1.5 Mixture Distributions 18

1.6 EP and Other Classes of Distributions 21

1.7 Approximating non-EP Distributions with EP Distributions 22

1.8 Operations on Exponential Polynomials 23

2 RELIABILITY AND AVAILABILITY MODELS 27

2.1 Reliability 27

2.2 Availability 30

2.3 Series-Parallel Reliability Block Diagrams 35

2.4 Fault Trees 39

2.5 Reliability Graphs 42

2.6 Analysis of Network Reliability Models 45

3 SERIES-PARALLEL ACYCLIC DIRECTED GRAPHS 47

3.1 A Simple Task Graph Example 48

3.2 Running Example: Performance from a Program's Point of View 49

3.3 Definition of a Series-Parallel Acyclic Directed Graph Model 50

3.4 Series-Parallel Acyclic Directed Graph Analysis 53

4 MARKOV MODELS 55

4.1 Stochastic Processes 55

4.2 Markov Chains 57

4.3 Basic Equations 58

4.4 Classification of States and Chains 61

4.5 Examples of Markov Chain Analysis 63

4.6 Steady-state Solution Techniques 72

4.7 Transient Analysis Methods 73

4.8 Examples 80

5 PRODUCT-FORM QUEUEING NETWORKS 85

5.1 Queueing Terminology 85

5.2 Queueing Network Analysis 89

5.3 Examples 100

6 PERFORMABILITY MODELS 103

6.1 Introduction 104

6.2 Degradable Systems 106

6.3 Largeness and stiffness: the decomposition approach 108

6.4 The Markov Reward Model 109

6.5 Measures of interest 110

6.6 Reward Assignment and Reward Computation 116

7 STOCHASTIC PETRI NET MODELS 119

7.1 Introduction to Petri Net Models 120

7.2 Petri Net Model Definitions 123

7.3 Petri Net Extensions 126

7.4 SPN and GSPN Analysis 133

7.5 GSPN EXAMPLES 137

7.6 Non-Markovian SPN Model Extensions 141

8 SEMI-MARKOV CHAINS 143

8.1 Describing Semi-Markov chains 143

8.2 Analysis of Irreducible Semi-Markov Chains 145

8.3 A Semi-Symbolic Analysis for Acyclic Semi-Markov Chains 147

Part Ⅱ MODELING EXAMPLES 151

9 RELIABILITY AND AVAILABILITY MODELING 155

9.1 Modeling with Block Diagrams 155

9.2 Modeling Reliability and Availability with Fault Trees 172

9.3 Modeling With A Reliability Graph 180

9.4 Modeling Using Markov Chains 183

9.5 Ring Network Reliability Models 193

10 PERFORMANCE MODELING 203

10.1 Program Performance Analysis Using Task Graphs 204

10.2 System Performance Analysis 222

11 HIERARCHICAL MODELS 261

11.1 A Non-Series-Parallel Block Diagram 262

11.2 A Non-Series-Parallel Task Precedence Graph 271

11.3 A Task Graph Containing a Cycle 274

11.4 A Queueing Model with Resource Constraints 277

11.5 A Queueing Model with Simultaneous Resource Possession 280

11.6 A Queueing Model with Job Priorities 284

11.7 Parallel Processing of Task Systems with Resource Con-straints 288

11.8 A Queue Subject to Failure and Repair 294

11.9 Modeling Repair Dependence 295

11.10 Intermittent and Near-coincident Faults 301

12 PERFORMABILITY MODELS 313

12.1 An Acyclic Markov Reward Model 313

12.2 An Irreducible Markov Reward Model 318

12.3 A Hierarchical Markov Reward Model 320

12.4 A Multiprocessor Performability Model 324

13 HANDLING ALGORITHMIC AND NUMERICAL LIMITATIONS 329

13.1 Distributions with Very Large Coefficients 330

13.2 A Phase-type Markov Chain 334

13.3 An Irreducible Markov Chain 337

13.4 An Example Where the Order of States Matters 339

Part Ⅲ APPENDICES 343

A SHARPE COMMAND LINE SYNTAX 345

B SHARPE LANGUAGE DESCRIPTION 347

B.1 Conventions 347

B.2 Basic Language Components 347

B.3 Specification of Exponential Polynomial Functions 352

B.4 Specification of Models 354

B.5 Asking for Results 367

B.6 Built-in Functions 371

B.7 Controlling the Analysis Process 375

B.8 Program Constants 377

B.9 Summary of Top-level Input Statements 378

C USING SHARPE INTERACTIVELY 381

D ALGORITHM CHOICES FOR PHASE-TYPE MARKOV CHAINS 387

REFERENCES 389

INDEX 401