# 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!

No, the holding times are not necessarily independent, even for a continuous-time Markov chain.

Think about the following chain, with jump rates as shown:

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.

• Thanks! How did you draw that nice graph?
– Tim
Apr 29, 2011 at 19:38
• Graphviz: graphviz.org Apr 29, 2011 at 20:00
• Thanks! For any two holding times, (1) if given their beginning states, will they be conditionally independent? (2) if given their beginning and end states, will they be conditionally independent? If yes to both (1) and (2), is it because of Markov property and how to explain that based on that?
– Tim
May 9, 2011 at 6:20