Matrix-Exponential Distributions in Applied Probability

✍ Scribed by Mogens Bladt, Bo Friis Nielsen (auth.)

• Publisher: Springer US
• Year: 2017
• Tongue: English
• Leaves: 749
• Series: Probability Theory and Stochastic Modelling 81
• Edition: 1
• Category: Library

✦ Synopsis

This book contains an in-depth treatment of matrix-exponential (ME) distributions and their sub-class of phase-type (PH) distributions. Loosely speaking, an ME distribution is obtained through replacing the intensity parameter in an exponential distribution by a matrix. The ME distributions can also be identified as the class of non-negative distributions with rational Laplace transforms. If the matrix has the structure of a sub-intensity matrix for a Markov jump process we obtain a PH distribution which allows for nice probabilistic interpretations facilitating the derivation of exact solutions and closed form formulas.

The full potential of ME and PH unfolds in their use in stochastic modelling. Several chapters on generic applications, like renewal theory, random walks and regenerative processes, are included together with some specific examples from queueing theory and insurance risk. We emphasize our intention towards applications by including an extensive treatment on statistical methods for PH distributions and related processes that will allow practitioners to calibrate models to real data.

Aimed as a textbook for graduate students in applied probability and statistics, the book provides all the necessary background on Poisson processes, Markov chains, jump processes, martingales and re-generative methods. It is our hope that the provided background may encourage researchers and practitioners from other fields, like biology, genetics and medicine, who wish to become acquainted with the matrix-exponential method and its applications.

✦ Table of Contents

Front Matter....Pages i-xvii
Preliminaries on Stochastic Processes....Pages 1-71
Martingales and More General Markov Processes....Pages 73-124
Phase-Type Distributions....Pages 125-197
Matrix-Exponential Distributions....Pages 199-296
Renewal Theory....Pages 297-359
Random Walks....Pages 361-386
Regeneration and Harris Chains....Pages 387-435
Multivariate Distributions....Pages 437-480
Markov Additive Processes....Pages 481-516
Markovian Point Processes....Pages 517-580
Some Applications to Risk Theory....Pages 581-626
Statistical Methods for Markov Processes....Pages 627-670
Estimation of Phase-Type Distributions....Pages 671-701
Back Matter....Pages 703-736

✦ Subjects

Probability Theory and Stochastic Processes;Operations Research, Management Science