I am working on a Markov-modulated Poisson process $\{N_{t}, t \geq 0\}$, which is itself a Poisson but the rates of which are governed by a CTMC. In my case, the CTMC is a one-class, aperiodic and positive recurrent MC.

My questions are the following

Are the interarrival times of $\{N_t, t \geq 0\}$ i.i.d? My guess is NO, and it is because the rate of the second interarrival is dependent on the state of the previous one.

If I want to get the distributions of interarrival times, say $X_1$, can I do \begin{equation} \mathbb{P}\{X_1 > t\} = \sum_{s \in \mathcal{S}}e^{-\mu_{s}t}\pi_{s} \end{equation} where $\pi_{s}$ is a stationary initial distribution for state $s$ in the CTMC. Is this correct?

If I want further to compute the joint distribution of interarrival times, I do the following \begin{equation} \mathbb{P}\{X_1 > t,...,X_n > t\} = \sum_{s_1, s_2,...,s_n}\prod_{k=1}^{n}e^{-\mu_{s_k}t}\prod_{l=1}^{n-1}R_{s_l,s_{l+1}}\pi_{s_1} \end{equation} For instance, \begin{equation} \mathbb{P}\{X_1 > t, X_2 > t, X_3 > t\} = \sum_{s_1 \in \mathcal{S}, s_2 \in S_1, s_3 \in S_2}e^{-\mu_{s_1}t}e^{-\mu_{s_2}t}e^{-\mu_{s_3}t}R_{s_1,s_2}R_{s_2,s_3}\pi_{s_1} \end{equation} where $S_1$ is the possible next states connected to $s_1$; $R_{s_1,s_2}$ is the transition probability from $s_1$ to $s_2$; and $\pi_{s_1}$ is a stationary initial distribution for the CTMC. What I do here is simply conditioning on all possible states for interarrival times and use the independence after conditioning. But I am not quite sure about this.

**Motivation**

Since there has been no reply to the above questions, which may be because they looked like homework questions. But it is not. As a matter of fact, I am trying to solve the following random sum \begin{equation} Q = \sum_{i=1}^{N}X_i \end{equation} wherein $X_i$ is the i-th interarrival time of an MMPP and $N$ is also a random variable defined as \begin{equation} N = \inf\{n : X_1 < P,...,X_n < P, X_{n+1} > P\} \end{equation} namely the first time we have an interarrival time greater than some constant $P$. My goal is to get distributional properties of the random sum (distribution, expectation and so like).