The least variable phase type distribution is erlang

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Abstract
Let Xt, t ≥ 0 be any continuous-time Markov process on states 0,1, …, n where Xo = n and To is the time to reach 0 which is absorbing. We prove that To is most nearly constant in the sense of minimizing the coefficient of variation var(T0)/(ET0)2 over all transition matrices Pij and exponential delay parameters λi- in each state when Pii = 1, i = n, n - l,…, 1 and λi, ≡ constant. The latter chain is Erlang's process on n fictitious states and has been used to show that an arbitrary semi-Markov process can be approximated by a Markov process. It has been a long-open problem since the work of Kendall, Cox, and others to try to improve on Erlang's scheme by generalizing the transition structure of X, i.e. adding loops, twists, and turns in order to make the overall waiting time have smaller coefficient of variation. We destroy this hope by showing at last that Erlang's original method is not improvable. Our proof is simple and elegant and is a nice example of the power of martingales; it seems intractible without them.

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