Stochastic Networks and Queues

ISBN-13: 9783540006572 / Angielski / Twarda / 2003 / 399 str.

Philippe Robert

Opis

Queues and stochastic networks are analyzed in this book with purely probabilistic methods. The purpose of these lectures is to show that general results from Markov processes, martingales or ergodic theory can be used directly to study the corresponding stochastic processes. Recent developments have shown that, instead of having ad-hoc methods, a better understanding of fundamental results on stochastic processes is crucial to study the complex behavior of stochastic networks. In this book, various aspects of these stochastic models are investigated in depth in an elementary way: Existence of equilibrium, characterization of stationary regimes, transient behaviors (rare events, hitting times) and critical regimes, etc. A simple presentation of stationary point processes and Palm measures is given. Scaling methods and functional limit theorems are a major theme of this book. In particular, a complete chapter is devoted to fluid limits of Markov processes.

Spis treści

1. 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.