Numerical Methods for Simulation and Optimization of Piecewise Deterministic Markov Processes

Application to Reliability

Benoîte de Saporta, François Dufour, Huilong Zhang

• Publisher: Wiley-ISTE
• Year: 2015
• Print ISBN: 9781848218390
• eBook ISBN: 9781119145141
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Mark H.A. Davis introduced the Piecewise-Deterministic Markov Process (PDMP) class of stochastic hybrid models in an article in 1984. Today it is used to model a variety of complex systems in the fields of engineering, economics, management sciences, biology, Internet traffic, networks and many more. Yet, despite this, there is very little in the way of literature devoted to the development of numerical methods for PDMDs to solve problems of practical importance, or the computational control of PDMPs.

This book therefore presents a collection of mathematical tools that have been recently developed to tackle such problems. It begins by doing so through examples in several application domains such as reliability. The second part is devoted to the study and simulation of expectations of functionals of PDMPs. Finally, the third part introduces the development of numerical techniques for optimal control problems such as stopping and impulse control problems.

Part 1
Piecewise Deterministic Markov Processes and Quantization

1
Piecewise Deterministic Markov Processes

1.1. Introduction

Davis [DAV 93] introduced in probability theory the class of piecewise deterministic Markov processes (PDMPs) as a general class of models suitable for formulating optimization problems in queuing and inventory systems, maintenance-replacement models, investment scheduling and in many other areas of operation research. These models are described by two variables. To the usual Euclidean variable representative of the state of the process, we add a discrete variable, called regime or mode that takes values in a finite or countable set. In this context, the state variable represents the physical parameters of the system under consideration. For example, it can be the position or the orientation of a satellite or the pressure in a tank. The mode characterizes the regimes of operation of the physical system from nominal to failure regime.

From a mathematical point of view, the notion of a PDMP is very intuitive and simple to describe. Starting from a point of the state space, the process follows a deterministic trajectory, namely a flow indexed by the mode, until the first jump time, which occurs either spontaneously in a random manner or when the trajectory hits the boundary of the state space. Between two jumps, the mode is assumed to be constant. In both cases, a new point and a new regime are selected by a random operator and the process restarts from this new point under this new mode. There exist two types of jumps. The first type is deterministic. From the mathematical point of view, it is given by the trajectory hitting the boundary of the state space. From the physical point of view, it can be seen as a mo...

Table of contents

• Cover
• Table of Contents
• Title
• Copyright
• Preface
• Introduction
• Part 1: Piecewise Deterministic MarkovProcesses and Quantization
• Part 2: Simulation of Functionals.
• Part 3: Optimization
• Bibliography
• Index
• End User License Agreement