## The least variable phase type distribution is Erlang.(English)Zbl 0635.60086

Let $$(X_ t,t\geq 0)$$ be a continuous-time Markov process with state space $$\{$$ 0,1,...,n$$\}$$ where 0 is an absorbing state, and let $$T_{n0}$$ denote the time to reach 0 if $$X_ 0=n$$. It is shown that $$var(T_{n0})/(E T_{n0})$$ $$2\geq 1/n$$ with equality if and only if $$(X_ t,t\geq 0)$$ is a pure death process with constant death rate. In other words, the coefficient of variation of a random variable with phase-type distribution (with states 0,1,...,n, absorbing state 0 and initial state n) is minimal and equal to 1/n if and only if this phase- type distribution is an Erlang distribution.
Reviewer: E.A.v.Doorn

### MSC:

 60J27 Continuous-time Markov processes on discrete state spaces 60K15 Markov renewal processes, semi-Markov processes 60K05 Renewal theory
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