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. ErgodicTheory: 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

uses the full strength of stochastic calculus methods applied to networks detailed presentation of most classical queueing models new material on scaling methods, fluid limits are thoroughly presented and discussed extensive use of martingale and stochastic calculus methods to investigate transient behaviour of queueing models Includes supplementary material: sn.pub/extras

GPSR Compliance The European Union's (EU) General Product Safety Regulation (GPSR) is a set of rules that requires consumer products to be safe and our obligations to ensure this. If you have any concerns about our products you can contact us on ProductSafety@springernature.com. In case Publisher is established outside the EU, the EU authorized representative is: Springer Nature Customer Service Center GmbH Europaplatz 3 69115 Heidelberg, Germany ProductSafety@springernature.com