Let $\{N_1 (t); t> 0\}$ a Poisson process of rate $\lambda$. These arrivals are registered by an alternating process: the $n$th arrival of the process $\{N_1 (t); t> 0\}$ turns off the alternating process during a random time $X_n$ distributed according to $G$, after which the alternating process reignites. (The $\{X_n\}$ times are i.i.d.) Suppose the first arrival of the process $\{N1 (t); t> 0\}$ occur at $t = 0$, and define $L$ as the elapsed time until the alternating process turns on again. I want to write the renewal equation for $Z(t)=\mathbb{P}(L>t)$. But first, i need an expression for $L$.
Everytime an $N_1$ renewal occurs, the the alternating process (let' say Y(t)) will be of for a random time. So, eventually, this time will be so short that the alternating process will turn back on. So, when: \begin{align*} S_{n+1}-S_{n}>X_{n} \end{align*} The alternating process will be switched on again, until another renewal for $N_{1}$ occurs. So L must be somthing like: \begin{align*} L= (S_{n+1}-S_n-X_n)1_{S_{n+1}-S_n>X_n} \end{align*}. So, is there a better expression to get the renewal equation for $Z(t)$ using this L?