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CHAPTER 1．PRELIMINARIES：DISCRETE INDEX SETS AND/OR DISCRETE STATE SPACES1

1.1．Non-negative integer valued random variables1

1.2．Convolution5

1.3．Generating functions7

1.3.1．Differentiation of generating functions9

1.3.2．Generating functions and moments10

1.3.3．Generating functions and convolution12

1.3.4．Generating functions,compounding and random sums15

1.4．The simple branching process18

1.5．Limit distributions and the continuity theorem27

1.5.1．The law of rare events30

1.6．The simple random walk33

1.7．The distribution of a process40

1.8．Stopping times44

1.8.1．Wald's identity47

1.8.2．Splitting an iid sequence at a stopping time48

Exercises for Chapter 151

CHAPTER 2．MARKOV CHAINS61

2.1．Construction and first properties61

2.2．Examples66

2.3．Higher order transition probabilities72

2.4．Decomposition of the state space77

2.5．The dissection principle81

2.6．Transience and recurrence85

2.7．Periodicity91

2.8．Solidarity properties92

2.9．Examples94

2.10．Canonical decomposition98

2.11．Absorption probabilities102

2.12．Invariant measures and stationary distributions116

2.12.1．Time averages122

2.13．Limit distributions126

2.13.1 More on null recurrence and transience134

2.14．Computation of the stationary distribution137

2.15．Classification techniques142

Exercises for Chapter 2147

CHAPTER 3．RENEWAL THEORY174

3.1．Basics174

3.2．Analytic interlude176

3.2.1．Integration176

3.2.2．Convolution178

3.2.3．Laplace transforms181

3.3．Counting renewals185

3.4．Renewal reward processes192

3.5．The renewal equation197

3.5.1．Risk processes205

3.6．The Poisson process as a renewal process211

3.7．Informal discussion of renewal limit theorems;regenerative processes212

3.7.1 An informal discussion of regenerative processes215

3.8．Discrete renewal theory221

3.9．Stationary renewal processes224

3.10．Blackwell and key renewal theorems230

3.10.1．Direct Riemann integrability231

3.10.2．Equivalent forms of the renewal theorems237

3.10.3．Proof of the renewal theorem243

3.11．Improper renewal equations253

3.12．More regenerative processes259

3.12.1．Definitions and examples259

3.12.2．The renewal equation and Smith's theorem263

3.12.3．Queueing examples269

Exercises for Chapter 3280

CHAPTER 4．POINT PROCESSES300

4.1．Basics300

4.2．The Poisson process303

4.3．Transforming Poisson processes308

4.3.1．Max-stable and stable random variables313

4.4．More transformation theory;marking and thinning316

4.5．The order statistic property321

4.6．Variants of the Poisson process327

4.7．Technical basics333

4.7.1．The Laplace functional336

4.8．More on the Poisson process337

4.9．A general construction of the Poisson process;a simple derivation of the order statistic property341

4.10．More transformation theory;location dependent thinning343

4.11．Records346

Exercises for Chapter 4349

CHAPTER 5．CONTINUOUS TIME MARKOV CHAINS367

5.1．Definitions and construction367

5.2．Stability and explosions375

5.2.1．The Markov property377

5.3．Dissection380

5.3.1．More detail on dissection380

5.4．The backward equation and the generator matrix382

5.5．Stationary and limiting distributions392

5.5.1．More on invariant measures398

5.6．Laplace transform methods402

5.7．Calculations and examples406

5.7.1．Queueing networks415

5.8．Time dependent solutions426

5.9．Reversibility431

5.10．Uniformizability436

5.11．The linear birth process as a point process439

Exercises for Chapter 5446

CHAPTER 6．BROWNIAN MOTION482

6.1．Introduction482

6.2．Preliminaries487

6.3．Construction of Brownian motion489

6.4．Simple properties of standard Brownian motion494

6.5．The reflection principle and the distribution of the maximum497

6.6．The strong independent increment property and reflection504

6.7．Escape from a strip508

6.8．Brownian motion with drift511

6.9．Heavy traffic approximations in queueing theory514

6.10．The Brownian bridge and the Kolmogorov-Smirnov statistic524

6.11．Path properties539

6.12．Quadratic variation542

6.13．Khintchine's law of the iterated logarithm for Brownian motion546

Exercises for Chapter 6551

CHAPTER 7．THE GENERAL RANDOM WALK559

7.1．Stopping times559

7.2．Global properties561

7.3．Prelude to Wiener-Hopf：Probabilistic interpretations of transforms564

7.4．Dual pairs of stopping times568

7.5．Wiener-Hopf decompositions573

7.6．Consequences of the Wiener-Hopf factorization581

7.7．The maximum of a random walk587

7.8．Random walks and the G/G/1 queue591

7.8.1．Exponential right tail595

7.8.2．Application to G/M/1 queueing model599

7.8.3．Exponential left tail602

7.8.4．The M/G/1 queue605

7.8.5．Queue lengths607

References613

Index617