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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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