Markov Models & Optimization

Routledge (Verlag)
  • erschienen am 19. Februar 2018
  • |
  • 308 Seiten
E-Book | PDF mit Adobe DRM | Systemvoraussetzungen
978-1-351-43349-5 (ISBN)
This book presents a radically new approach to problems of evaluating and optimizing the performance of continuous-time stochastic systems. This approach is based on the use of a family of Markov processes called Piecewise-Deterministic Processes (PDPs) as a general class of stochastic system models. A PDP is a Markov process that follows deterministic trajectories between random jumps, the latter occurring either spontaneously, in a Poisson-like fashion, or when the process hits the boundary of its state space. This formulation includes an enormous variety of applied problems in engineering, operations research, management science and economics as special cases; examples include queueing systems, stochastic scheduling, inventory control, resource allocation problems, optimal planning of production or exploitation of renewable or non-renewable resources, insurance analysis, fault detection in process systems, and tracking of maneuvering targets, among many others.

The first part of the book shows how these applications lead to the PDP as a system model, and the main properties of PDPs are derived. There is particular emphasis on the so-called extended generator of the process, which gives a general method for calculating expectations and distributions of system performance functions. The second half of the book is devoted to control theory for PDPs, with a view to controlling PDP models for optimal performance: characterizations are obtained of optimal strategies both for continuously-acting controllers and for control by intervention (impulse control). Throughout the book, modern methods of stochastic analysis are used, but all the necessary theory is developed from scratch and presented in a self-contained way. The book will be useful to engineers and scientists in the application areas as well as to mathematicians interested in applications of stochastic analysis.
  • Englisch
  • Boca Raton
  • |
  • USA
Taylor & Francis Ltd
  • Für höhere Schule und Studium
  • |
  • Für Beruf und Forschung
978-1-351-43349-5 (9781351433495)
weitere Ausgaben werden ermittelt
  • Cover
  • Title Page
  • Copyright Page
  • Dedication
  • Table of Contents
  • Preface
  • 1: Analysis, probability and stochastic processes
  • 11 Analysis
  • 12 Probability theory
  • 13 Stochastic processes
  • 14 Markov processes
  • 15 Notes and references
  • 2: Piecewise-deterministic Markov processes
  • 21 Markov models and supplementary variables
  • 22 Ordinary differential equations and vector fields
  • 23 Simulation
  • 24 Definition of the PDP
  • 25 The strong Markov property
  • 26 The extended generator of the PDP
  • 27 Further Markov properties of the PDP
  • 28 Notes and references
  • 3: Distributions and expectations
  • 31 The differential formula and transformations of PDPs
  • 32 Expectations
  • 33 Applications
  • 34 Stationary distributions
  • 35 Notes and references
  • 4: Control theory
  • 41 Feedback control of PDPs
  • 42 Naïve dynamic programming
  • 43 Relaxed controls
  • 44 Control via discrete-time dynamic programming
  • 45 Non-smooth analysis and deterministic optimal control
  • 46 Necessary and sufficient conditions for optimality: the generalized Bellman equation
  • 47 Discrete-stage Markov decision models
  • 48 Notes and references
  • 5: Control by intervention
  • 51 Optimal stopping
  • 52 Randomized stopping
  • 53 Variational inequalities and the penalty method
  • 54 Impulse control
  • 55 Computational methods
  • 56 Notes and references
  • Appendix: Jump processes and their martingales
  • A1 Definition of the jump process
  • A2 Structure of stopping times and stopped s-fields
  • A3 Predictability
  • A4 The single-jump process
  • A5 Local martingale representation for the general jump process
  • Bibliography
  • Index of notation
  • Subject index

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