Fundamentals of Stochastic Networks

Acknowledgments, xii

1 Basic Concepts in Probability 1

1.1 Introduction, 1

1.2 Random Variables, 1

1.3 Transform Methods, 5

1.4 Covariance and Correlation Coefficient, 8

1.5 Sums of Independent Random Variables, 8

1.6 Random Sum of Random Variables, 9

1.7 Some Probability Distributions, 11

1.8 Limit Theorems, 21

2 Overview of Stochastic Processes 26

2.1 Introduction, 26

2.2 Classification of Stochastic Processes, 27

2.3 Stationary Random Processes, 27

2.4 Counting Processes, 28

2.5 Independent Increment Processes, 29

2.6 Stationary Increment Process, 29

2.7 Poisson Processes, 30

2.8 Renewal Processes, 32

2.9 Markov Processes, 37

2.10 Gaussian Processes, 56

3 Elementary Queueing Theory 61

3.1 Introduction, 61

3.2 Description of a Queueing System, 61

3.3 The Kendall Notation, 64

3.4 The Little’s Formula, 65

3.5 The M/M/1 Queueing System, 66

3.6 Examples of Other M/M Queueing Systems, 71

3.7 M/G/1 Queue, 79

4 Advanced Queueing Theory 93

4.1 Introduction, 93

4.2 M/G/1 Queue with Priority, 93

4.3 G/M/1 Queue, 99

4.4 The G/G/1 Queue, 105

4.5 Special Queueing Systems, 109

5 Queueing Networks 124

5.1 Introduction, 124

5.2 Burke’s Output Theorem and Tandem Queues, 126

5.3 Jackson or Open Queueing Networks, 128

5.4 Closed Queueing Networks, 130

5.5 BCMP Networks, 132

5.6 Algorithms for Product-Form Queueing Networks, 138

5.7 Networks with Positive and Negative Customers, 144

6 Approximations of Queueing Systems and Networks 150

6.1 Introduction, 150

6.2 Fluid Approximation, 151

6.3 Diffusion Approximations, 155

7 Elements of Graph Theory 172

7.1 Introduction, 172

7.2 Basic Concepts, 172

7.3 Connected Graphs, 177

7.4 Cut Sets, Bridges, and Cut Vertices, 177

7.5 Euler Graphs, 178

7.6 Hamiltonian Graphs, 178

7.7 Trees and Forests, 179

7.8 Minimum Weight Spanning Trees, 181

7.9 Bipartite Graphs and Matchings, 182

7.10 Independent Set, Domination, and Covering, 186

7.11 Complement of a Graph, 188

7.12 Isomorphic Graphs, 188

7.13 Planar Graphs, 189

7.14 Graph Coloring, 191

7.14.1 Edge Coloring, 191

7.14.2 The Four-Color Problem, 192

7.15 Random Graphs, 192

7.16 Matrix Algebra of Graphs, 195

7.17 Spectral Properties of Graphs, 198

7.18 Graph Entropy, 201

7.19 Directed Acyclic Graphs, 201

7.20 Moral Graphs, 202

7.21 Triangulated Graphs, 202

7.22 Chain Graphs, 203

7.23 Factor Graphs, 204

8 Bayesian Networks 209

8.1 Introduction, 209

8.2 Bayesian Networks, 210

8.3 Classification of BNs, 214

8.4 General Conditional Independence and d-Separation, 215

8.5 Probabilistic Inference in BNs, 215

8.6 Learning BNs, 227

8.7 Dynamic Bayesian Networks, 231

9 Boolean Networks 235

9.1 Introduction, 235

9.2 Introduction to GRNs, 236

9.3 Boolean Network Basics, 236

9.4 Random Boolean Networks, 238

9.5 State Transition Diagram, 239

9.6 Behavior of Boolean Networks, 240

9.7 Petri Net Analysis of Boolean Networks, 242

9.8 Probabilistic Boolean Networks, 250

9.9 Dynamics of a PBN, 251

9.10 Advantages and Disadvantages of Boolean Networks, 252

10 Random Networks 255

10.1 Introduction, 255

10.2 Characterization of Complex Networks, 256

10.3 Models of Complex Networks, 261

10.4 Random Networks, 265

10.5 Random Regular Networks, 267

10.6 Consensus over Random Networks, 268

10.7 Summary, 274

References 276

Index 280