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
I don't understand why if the two are dependent, the Markov property is violated.
Thanks and regards!