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Principles of Performance and Reliability Modeling and Evaluation pp 49–75Cite as

Fitting Phase-Type Distributions and Markovian Arrival Processes: Algorithms and Tools

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Part of the book series: Springer Series in Reliability Engineering ((RELIABILITY))

Abstract

This chapter provides a comprehensive survey of PH (phase-type) distribution and MAP (Markovian arrival process) fitting. The PH distribution and MAP are widely used in analytical model-based performance evaluation because they can approximate non-Markovian models with arbitrary accuracy as Markovian models. Among a number of past research results on PH/MAP fitting, we present the mathematical definition of the PH distribution and MAP, and summarize the most recent state-of-the-art results on the fitting methods. We also offer an overview of the software tools for PH/MAP fitting.

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Notes

  1. 1.

    The PH parameters can be defined by \((\varvec{\alpha }, \varvec{T})\). In this chapter, since the diagonals of \(\varvec{T}\) are determined after estimating nondiagonals of \(\varvec{T}\) and \(\varvec{\tau }\), the PH parameters include \(\varvec{\tau }\).

  2. 2.

    When the number of samples increases, the log-likelihood function can be approximated by a multivariate normal distribution.

  3. 3.

    The MCMC is a sample-based approximation method for posterior distributions.

  4. 4.

    In the implementation, to avoid underflow in \(\hat{\varvec{f}}_k\) and \(\hat{\varvec{b}}_k\), the scaling technique should be applied.

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Acknowledgments

The authors (H.O. and T.D.) are grateful to Prof. Kishor S. Trivedi who stimulated their research interests in model-based performance evaluation through his many papers.

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Authors and Affiliations

  1. Department of Information Engineering, Graduate School of Engineering, Hiroshima University, 1-4-1 Kagamiyama, Higashihiroshima, 739-8527, Japan

    Hiroyuki Okamura & Tadashi Dohi

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  1. Hiroyuki Okamura
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  2. Tadashi Dohi
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Correspondence to Hiroyuki Okamura .

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Editors and Affiliations

  1. University of Massachusetts Dartmou, Dartmouth, Massachusetts, USA

    Lance Fiondella

  2. Università degli studi di Messina, Messina, Italy

    Antonio Puliafito

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Okamura, H., Dohi, T. (2016). Fitting Phase-Type Distributions and Markovian Arrival Processes: Algorithms and Tools. In: Fiondella, L., Puliafito, A. (eds) Principles of Performance and Reliability Modeling and Evaluation. Springer Series in Reliability Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-30599-8_3

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  • DOI: https://doi.org/10.1007/978-3-319-30599-8_3

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