Mean value function of a discrete-time Markov-modulated Poisson process

Question

By "discrete-time Markov-modulated Poisson process" I mean a semi-Markov process $\{(X_n,T_n):n=0,1,\ldots\} $ which satisfies $$T_{n+1}-T_n\mid X_n\sim\operatorname{Exp}(\lambda_{X_n}), $$ with $\lambda_j>0$ and $$N(t) = \sum_{n=0}^\infty \mathsf 1_{(0,t]}(T_n) $$ the number of arrivals up to time $t$. Let $\alpha$ be the distribution of $X_0$ and assume that $\{X_n\}$ is irreducible and positive recurrent, so a unique stationary distribution $\pi$ exists. Let $\Lambda(t)=\mathbb E[N(t)]$. It is known that in a nonhomogeneous Poisson process with intensity function $\lambda(t)$ that $\Lambda(t)=\int_0^t \lambda(s)\ \mathsf ds$, but here $\lambda(t)=\lambda_{X_{N(t)}}$ is itself a stochastic process, so it is not clear how to compute $\Lambda(t)$.