# Are holding times independent in a continuous-time Markov chain and in a semi-Markov process

I was wondering if the holding times are independent in a continuous-time Markov chain?

Similar question in a semi-Markov process?

From what I have read, it is not mentioned that the holding times are independent in both cases, but it is in Poisson processes and in renewal processes. So I would like to know if independent holding times are true in the two more general cases: a continuous-time Markov chain and a semi-Markov process, or if there are some famous counterexamples?

Thanks and regards!

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If $H_n$ is the $n$th holding time (so $X_s = X_{H_n}$ for $H_n \le s < H_{n+1}$), then $H_2$ and $H_3$ are not independent. You can do the computation explicitly if you like, but intuitively, if $H_2$ is large, you're much more likely to be in state $B$ than $E$. This means that your next jump will most likely be to $C$ rather than $F$, and so the next holding time $H_3$ is more likely to have rate $1$ than $100$, so $H_3$ is more likely to be large.