Let $\mathcal{D}(m)$ be the set of phase-type distributions constructed from $m+1$-state Markov chains. Recall that the *coefficient of variation* of a distribution $D$ is the ratio of the standard deviation to the mean, which we will denote by $C(D)$.
Let
$$ F(m)=\inf_{D\in\mathcal{D}} C(D)$$

What is $F(m)$, and what Markov chains (if any) achieve the infimum?

Note: One candidate distribution for the minimum might be a one-way chain of states with identical transition probabilities. The corresponding phase-type distribution has an Erlang distribution, and the coefficient of variation is $1/\sqrt{m}$. (So, $F(m)\leq 1/\sqrt{m}$.)

Motivation: I'm modeling a stochastic process with a Markov chain and want to match a particular (measured) mean and variance, but for computational reasons I want to use the fewest states possible. It is easy to make the variance unboundedly large (for a fixed mean), but making the variance small seems to require more care.