© 2017

Matrix-Exponential Distributions in Applied Probability


Part of the Probability Theory and Stochastic Modelling book series (PTSM, volume 81)

Table of contents

  1. Front Matter
    Pages i-xvii
  2. Mogens Bladt, Bo Friis Nielsen
    Pages 1-71
  3. Mogens Bladt, Bo Friis Nielsen
    Pages 73-124
  4. Mogens Bladt, Bo Friis Nielsen
    Pages 125-197
  5. Mogens Bladt, Bo Friis Nielsen
    Pages 199-296
  6. Mogens Bladt, Bo Friis Nielsen
    Pages 297-359
  7. Mogens Bladt, Bo Friis Nielsen
    Pages 361-386
  8. Mogens Bladt, Bo Friis Nielsen
    Pages 387-435
  9. Mogens Bladt, Bo Friis Nielsen
    Pages 437-480
  10. Mogens Bladt, Bo Friis Nielsen
    Pages 481-516
  11. Mogens Bladt, Bo Friis Nielsen
    Pages 517-580
  12. Mogens Bladt, Bo Friis Nielsen
    Pages 581-626
  13. Mogens Bladt, Bo Friis Nielsen
    Pages 627-670
  14. Mogens Bladt, Bo Friis Nielsen
    Pages 671-701
  15. Back Matter
    Pages 703-736

About this book


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. 


Applied probability Markov Processes Matrix--exponential distributions Numerical methods Stochastic modeling Uncertainty quantification Phase-type distributions Renewal theory Random walks Ladder Processes Regenerative methods Probability theory and stochastic processes Operations Research Management Science

Authors and affiliations

  1. 1.Institute for Applied Mathematics (IIMAS)Universidad Nacional Autónoma de MéxicoCoyoacanMexico
  2. 2.Department of Applied Mathematics and Computer ScienceTechnical University of DenmarkKgs. LyngbyDenmark

About the authors

Bo Friis Nielsen is an associate professor in the Department of Applied Mathematics and Computer Science at the Technical University of Denmark. 

Mogens Bladt is a researcher in the Department of Probability  and Statistics at the Institute for Applied Mathematics and Systems, National University of Mexico (UNAM).

Bibliographic information

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“The book is useful not only for those who want to learn about ME distributions but also about renewal theory, random walks, Markov chains … The book should be accessible to a beginning graduate student and many portions of it to undergraduates as well. ... Although this is a book geared towards practitioners, it also does a very good job in explaining some of the finest points of the theory.” (Takis Konstantopoulos, Mathematical Reviews, May, 2018)

“This book may be used as a graduate-level textbook, and the authors provide outlines of several possible courses based on it, as well as exercises at the end of each chapter. ... this book is a very good introduction to phase-type and matrix-exponential distributions, which manages to effectively convey the scope of their applications across probability and statistics, and seems well suited to its intended graduate-level audience.” (Fraser Daly, zbMATH 1375.60002, 2018)