Beschreibung
Inhaltsangabe1. Point Processes.- 2. GI/GI/1 FIFO Queues and Random Walks.- 3. Limit Theorems for GI/GI/1 Queues.- 4. Stochastic Networks and Reversibility.- 5. The M/M/1 Queue.- 6. The M/M/? Queue.- 7. Queues with Poisson Arrivals.- 8. Recurrence and Transience of Markov Chains.- 9. Resealed Markov Processes and Fluid Limits.- 10. Ergodic Theory: Basic Results.- 11. Stationary Point Processes.- 12. The G/G/1 FIFO Queue.- A. Martingales.- A.1 Discrete Time Parameter Martingales.- A.2 Continuous Time Martingales.- A.3 The Stochastic Integral for a Poisson Process.- A.4 Stochastic Differential Equations with Jumps.- B. Markovian Jump Processes.- B.2 Global Balance Equations.- B.3 The Associated Martingales.- C. Convergence in Distribution.- C.1 Total Variation Norm on Probability Distributions.- C.2 Convergence of Stochastic Processes.- D. An Introduction to Skorohod Problems.- D.1 Dimension 1.- D.2 Multi-Dimensional Skorohod Problems.- References.- Research Papers.
Inhaltsverzeichnis
1. Point Processes: 1.1 General Definitions; 1.2 Poisson Processes; 1.3 Poisson Point Processes on the Real Line; 1.4 Renewal Point Processes.- 2. The GI/GI/1 FIFO Queue and Random Walks: 2.1 General results on the GI/GI/1 FIFO Queue; 2.2 Wiener-Hopf Factorization; 2.3 Applications to the GI/GI/1 Queue; 2.4 The GI/M/1 and M/GI/1 Queues; 2.5 The H/G/1 Queue; 2.6 A Probabilistic Proof.- 3. Limit Theorems for the GI/GI/1 Queue: 3.1 Introduction; 3.2 The Biased Random Walk; 3.3 The Tail Distribution of W; 3.4 The Maximum of a Busy Period; 3.5 The GI/GI/1 Queue near Saturation; 3.6 The Random Walk Conditioned to Hit Level a.- 4. Stochastic Networks and Reversibility: 4.1 Introduction; 4.2 Reversibility of Markov Processes; 4.3 Local Balance Equations; 4.4 Queueing Networks with Product Form.- 5. The M/M/1 Queue: 5.1 Introduction; 5.2 Exponential Martingales; 5.3 Hitting Times: Downward; 5.4 Convergence to Equilibrium; 5.5 Hitting Times: Upward; 5.6 Rare Events; 5.7 Fluid Limits; 5.8 Large Deviations; 5.9 Appendix.- 6. The M/M/infinity Queue: 6.1 Introduction; 6.2 Positive Martingales; 6.3 Hitting Times: Downward; 6.4 Hitting Times: Upward; 6.5 Fluid Limits; 6.6 A Functional Central Limit Theorem; 6.7 The M/M/N/N Queue; 6.8 Appendix.- 7. Queues with Poisson Arrivals: 7.1 FIFO Queue; 7.2 Infinite Server Queue; 7.3 LIFO Queue with Preemptive Service; 7.4 Processor-Sharing Queue; 7.5 The Insensitivity Property; 7.6 The Distribution Seen by Customers.- 8. Recurrence and Transience of Markov Chains: 8.1 Recurrence of Markov Chains; 8.2 Ergodicity; 8.3 Transience; 8.4 Ergodicity of Markov Processes; 8.5 Some Applications; 8.6 The Classical Version of Lyapunov''s Theorem.- 9. Rescaled Markov Processes and Fluid Limits: 9.1 Introduction; 9.2 Rescaled Markov Processes; 9.3 The Fluid Limits of a Class of Markov Processes; 9.4 Fluid Limits and Skorohod Problems; 9.5 Fluid Limits and Ergodicity Properties; 9.6 Fluid Limits and Local Equilibrium; 9.7 Bibliographical Notes.- 10. Ergodic Theory: Basic Results: 10.1 Discrete Dynamical Systems; 10.2 Ergodic Theorems; 10.3 Continuous Time Dynamical Systems; 10.4 Markovian Endomorphisms.- 11. Stationary Point Processes: 11.1 Introduction; 11.2 The Palm Space of the Arrival Process; 11.3 Construction of a Stationary Point Process; 11.4 Relations Between the Palm Space and Its Extension; 11.5 Joint Distribution of the Points Around t=0; 11.6 Some Properties of Stationary Point Processes; 11.7 Appendix.- 12. The G/G/1 FIFO Queue: 12.1 The Waiting Time; 12.2 Virtual Waiting Time; 12.3 The Number of Customers; 12.4 The Associated Stationary Point Processes; 12.5 The Unstable G/G/1 Queue; 12.6 A Queue with Two Servers, the G/G/2 Queue.- A. Martingales: A.1 Discrete Time Parameter Martingales; A.2 Continuous Time Martingales; A.3 The Stochastic Integral for a Poisson Process; A.4 Stochastic Differential Equations with Jumps.- B. Markovian Jump Processes: B.1 Q-Matrices; B.2 Global Balance Equations; B.3 The Associated Martingales.- C. Convergence in Distribution: C.1 The Total Variation Norm on Probability Distributions; C.2 Convergence of Stochastic Processes.- D. An Introduction to Skorohod Problems: D.1 Dimension 1; D.2 Multi-Dimensional Skorohod Problems
