Switching processes, invented by the author in 1977, is the main tool used in the investigation of traffic problems from automotive to telecommunications. The title provides a new approach to low traffic problems based on the analysis of flows of rare events and queuing models. In the case of fast switching, averaging principle and diffusion approximation results are proved and applied to the investigation of transient phenomena for wide classes of overloading queuing networks. The book is devoted to developing the asymptotic theory for the class of switching queuing models which covers models in a Markov or semi-Markov environment, models under the influence of flows of external or internal perturbations, unreliable and hierarchic networks, etc.
Vladimir V. Anisimov is currently Director of the Research Statistics Unit at GlaxoSmithKline, UK. He has written about 200 papers, nine books and manuals in this area.
Preface. Definitions. Chapter 1. Switching Stochastic Models. 1.1. Random processes with discrete component. 1.2. Switching processes. 1.3. Switching stochastic models. 1.4.Bibliography. Chapter 2. Switching Queueing Models. 2.1. Introduction. 2.2. Queueing systems. 2.3. Queueing networks. 2.4.Bibliography. Chapter 3. Processes of Sums of Weakly-dependent Variables. 3.1. Limit theorems for processes of sums of conditionally independent random variables. 3.2. Limit theorems for sums with Markov switching. 3.3. Quasi-ergodic Markov processes. 3.4. Limit theorems for non-homogenous Markov processes. 3.5.Bibliography. Chapter 4. Averaging Principle and Diffusion Approximation for Switching Processes. 4.1. Introduction. 4.2. Averaging principle for switching recurrent sequences. 4.3. Averaging principle and diffusion approximation for RPSMs. 4.4. Averaging principle and diffusion approximation for recurrent processes of semi-Markov type (Markov case). 4.5. Averaging principle for RPSM with feedback. 4.6. Averaging principle and diffusion approximation for switching processes. 4.7.Bibliography. Chapter 5. Averaging and Diffusion Approximation in Overloaded Switching Queueing Systems and Networks. 5.1. Introduction. 5.2. Markov queueing models. 5.3. Non-Markov queueing models. 5.4. Retrial queueing systems. 5.5. Queueing networks. 5.6. Non-Markov queueing networks. 5.7.Bibliography. Chapter 6. Systems in Low Traffic Conditions. 6.1. Introduction. 6.2.Analysis of the first exit time fromthe subset of states. 6.3. Markov queueing systems with fast service. 6.4. Single-server retrial queueing model. 6.5. Multiserver retrial queueing models. 6.6.Bibliography. Chapter 7. Flows of Rare Events in Low and Heavy Traffic Conditions. 7.1. Introduction. 7.2. Flows of rare events in systemswithmixing. 7.3. Asymptotically connected sets ( Vn - S -sets). 7.4. Heavy traffic conditions. 7.5. Flows of rare events in queueing models. 7.6.Bibliography. Chapter 8. Asymptotic Aggregation of State Space. 8.1. Introduction. 8.2. Aggregation of finite Markov processes (stationary behavior). 8.3. Convergence of switching processes. 8.4. Aggregation of states in Markov models. 8.5. Asymptotic behavior of the first exit time from the subset of states (non-homogenous in time case). 8.6. Aggregation of states of non-homogenous Markov processes. 8.7. Averaging principle for RPSM in the asymptotically aggregated Markov environment. 8.8. Diffusion approximation for RPSM in the asymptotically aggregated Markov environment. 8.9. Aggregation of states in Markov queueing models. 8.10. Aggregation of states in semi-Markov queueing models. 8.11.Analysis of flows of lost calls. 8.12.Bibliography. Chapter 9. Aggregation in Markov Models with Fast Markov Switching. 9.1. Introduction. 9.2. Markov models with fast Markov switching. 9.3. Proofs of theorems. 9.4. Queueing systems with fast Markov type switching. 9.5. Non-homogenous in time queueing models. 9.6.Numerical examples. 9.7.Bibliography. Chapter 10. Aggregation in Markov Models with Fast Semi-Markov Switching. 10.1. Markov processes with fast semi-Markov switches. 10.2. Averaging and aggregation in Markov queueing systems with semi-Markov switching. 10.3.Bibliography. Chapter 11. Other Applications of Switching Processes. 11.1. Self-organization in multicomponent interacting Markov systems. 11.2. Averaging principle and diffusion approximation for dynamic systems with stochastic perturbations. 11.3. Random movements. 11.4.Bibliography. Chapter 12. Simulation Examples. 12.1. Simulation of recurrent sequences. 12.2. Simulation of recurrent point processes. 12.3. Simulation ofRPSM. 12.4. Simulation of state-dependent queueing models. 12.5. Simulation of the exit time from a subset of states of a Markov chain. 12.6. Aggregation of states in Markov models. Index.