Matrix-Exponential Distributions in Applied Probability

1. Front Matter

   Pages i-xvii

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2. Preliminaries on Stochastic Processes

   • Mogens Bladt, Bo Friis Nielsen

   Pages 1-71

3. Martingales and More General Markov Processes

   • Mogens Bladt, Bo Friis Nielsen

   Pages 73-124

4. Phase-Type Distributions

   • Mogens Bladt, Bo Friis Nielsen

   Pages 125-197

5. Matrix-Exponential Distributions

   • Mogens Bladt, Bo Friis Nielsen

   Pages 199-296

6. Renewal Theory

   • Mogens Bladt, Bo Friis Nielsen

   Pages 297-359

7. Random Walks

   • Mogens Bladt, Bo Friis Nielsen

   Pages 361-386

8. Regeneration and Harris Chains

   • Mogens Bladt, Bo Friis Nielsen

   Pages 387-435

9. Multivariate Distributions

   • Mogens Bladt, Bo Friis Nielsen

   Pages 437-480

10. Markov Additive Processes

    • Mogens Bladt, Bo Friis Nielsen

    Pages 481-516

11. Markovian Point Processes

    • Mogens Bladt, Bo Friis Nielsen

    Pages 517-580

12. Some Applications to Risk Theory

    • Mogens Bladt, Bo Friis Nielsen

    Pages 581-626

13. Statistical Methods for Markov Processes

    • Mogens Bladt, Bo Friis Nielsen

    Pages 627-670

14. Estimation of Phase-Type Distributions

    • Mogens Bladt, Bo Friis Nielsen

    Pages 671-701

15. Back Matter

    Pages 703-736

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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 treatmenton 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

“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)

• Institute for Applied Mathematics (IIMAS), Universidad Nacional Autónoma de México, Coyoacan, Mexico

  Mogens Bladt

• Department of Applied Mathematics and Computer Science, Technical University of Denmark, Kgs. Lyngby, Denmark

  Bo Friis Nielsen

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).

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