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Concentrated matrix exponential distributions with real eigenvalues

Published online by Cambridge University Press:  26 August 2021

András Mészáros  and
Miklós Telek Open the ORCID record for Miklós Telek [Opens in a new window]
András Mészáros
Affiliation:
Department of Networked Systems and Services, Technical University of Budapest, Budapest, Hungary. E-mail: [email protected]
Miklós Telek
Affiliation:
Department of Networked Systems and Services, Technical University of Budapest, Budapest, Hungary. E-mail: [email protected] MTA-BME Information Systems Research Group, Budapest, Hungary. E-mail: [email protected]
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Abstract

Concentrated random variables are frequently used in representing deterministic delays in stochastic models. The squared coefficient of variation ($\mathrm {SCV}$) of the most concentrated phase-type distribution of order $N$ is $1/N$. To further reduce the $\mathrm {SCV}$, concentrated matrix exponential (CME) distributions with complex eigenvalues were investigated recently. It was obtained that the $\mathrm {SCV}$ of an order $N$ CME distribution can be less than $n^{-2.1}$ for odd $N=2n+1$ orders, and the matrix exponential distribution, which exhibits such a low $\mathrm {SCV}$ has complex eigenvalues. In this paper, we consider CME distributions with real eigenvalues (CME-R). We present efficient numerical methods for identifying a CME-R distribution with smallest SCV for a given order $n$. Our investigations show that the $\mathrm {SCV}$ of the most concentrated CME-R of order $N=2n+1$ is less than $n^{-1.85}$. We also discuss how CME-R can be used for numerical inverse Laplace transformation, which is beneficial when the Laplace transform function is impossible to evaluate at complex points.

Keywords

Concentrated matrix exponential distributionsEigenvaluesMatrix exponential distributionPhase-type distributionSquared coefficient of variation
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 4 , October 2022 , pp. 1171 - 1187
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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