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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.
