Studies in Probability, Optimization and Statistics (IOS Press) – serie

This book presents a unified theory of dynamic programming and Markov decision processes and its application to a major field of operations research and operations management: inventory control. Models are developed in discrete time as well as in continuous time. For continuous time, this book concentrates only on models of interest to inventory control. For discrete time, the focus is mainly on infinite horizon models.The book also covers the difference between impulse control and continuous control. Ergodic control is considered in the context of impulse control, and some simple rules currently used in practice are justified. Chapter 2 introduces some of the classical static problems which are preliminary to the dynamic models of interest in inventory control.This book is not a general text on control theory and dynamic programming, in that the systems dynamics are mostly limited to inventory models. For these models, however, it seeks to be as comprehensive as possible, although finite horizon models in discrete time are not developed, since they are largely described in existing literature. On the other hand, the ergodic control problem is considered in detail, and probabilistic proofs as well as analytical proofs are provided.The techniques developed in this work can be extended to more complex models, covering additional aspects of inventory control.

This book is a survey of work on passage times in stable Markov chains with a discrete state space and a continuous time. Passage times have been investigated since early days of probability theory and its applications. The best known example is the first entrance time to a set, which embraces waiting times, busy periods, absorption problems, extinction phenomena, etc. Another example of great interest is the last exit time from a set. The book presents a unifying treatment of passage times, written in a systematic manner and based on modern developments. The appropriate unifying framework is provided by probabilistic potential theory, and the results presented in the text are interpreted from this point of view. In particular, the crucial role of the Dirichlet problem and the Poisson equation is stressed. The work is addressed to applied probalilists, and to those who are interested in applications of probabilistic methods in their own areas of interest. The level of presentation is that of a graduate text in applied stochastic processes. Hence, clarity of presentation takes precedence over secondary mathematical details whenever no serious harm may be expected. Advanced concepts described in the text gain nowadays growing acceptance in applied fields, and it is hoped that this work will serve as an useful introduction.