Average time of permanence in a state of a Markov-chain

Question

I know that in a Markov-chain the mean permanence time in a state is a random variable distributed accordingly to:

• Geometric distribution for Time Discrete Markov Chains
• Exponential distribution for Time Continuos Markov Chains

This happens because these two distributions are the only ones having no memory, and thus Markov-chains have no memory, that makes perfect sense.

But, and here is my question, how to get the parameters for these two distributions from the chain?

Answer 1

How is the chain you are working with defined?

In a discrete time Markov chain the parameter for the holding time distribution in state $i$ is $p_{ii}$ (where $P$ is the stochastic matrix). If $p_{ii}=0$ then the holding time is 1 with probability 1.

In a continuous time Markov chain the diagonal elements in the transition rate matrix are the (negative) rates for the exponential distribution holding time in each state $i$, so the parameter you want for state $i$ is $-q_{ii}$.