Independence of holding time and next state in continuous-time Markov chain 
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*In a continuous-time Markov chain, I was wondering why the holding time and the next state are independent? Are the independence a conditional one given the current state?

*Quoted from Ross's Stochastic processes:

The amount of time the process spends in state $i$, and the next state visited, must be independent random variables. For if the next state visited were dependent on $\tau_i$, then information as to how long the process has already been in state $i$ would be relevant to the prediction of the next state—and this would contradict the Markovian assumption.

One can also find identical claim at another book here with more
context available.
I don't understand why if the two are dependent, the Markov property
is violated.
Thanks and regards!
 A: The next state depends on the recent states, not how long you stayed in them. Notice that this is implicit in the state transition matrix.
A: For question 1, the answer is yes.  You might first consider the case of 2 possible "next states", which corresponds to showing that if $U,V$ are two independent exponential random variables (with possibly different rates), then $\min(U,V)$ is independent of the event $U < V$.  Counterintuitive but true.
For question 2, consider a simple example: suppose $X_t$ is a process that, whenever it enters a certain state $a$, it does one of two things: waits 5 seconds and then transitions to $b$, or waits 10 seconds and then transitions to $c$.  Here the holding time is completely correlated with the next state.  Such a process cannot be Markov.  For instance, we have $P(X_{20} = b | X_{17} = a) > 0$, but $P(X_{20} = b | X_{17} = a, X_{14} \ne a) = 0$.
Essentially, information about the holding time is information about the history of the process (where it was at an earlier time), and in a Markov process this is not allowed to prejudice where it goes next.
