Context: I saw the following question in the past papers for my university's probability and stochastic processes course. Our university does not provide worked solutions, and I am having difficulty getting started with this question. I would greatly appreciate any pointers on how to solve this question, or even just on how to get started.
Question: Let $W_1, W_2, \ldots$ be exponentially distributed iid random variables with mean $\frac{1}{\lambda}$ and define $T_n = W_1 + \dots + W_n$ $(T_0 = 0)$. Let $(N_t, t \geq 0)$ be a stochastic process defined as $N_0 = 0$ and for $t \in \mathbb R^+$
$$ N_t = \max\{n \in \mathbb N \;:\; W_1 + \dots + W_n \leq t\}. $$
Define the process $(R_t)$ via $R_t = T_{N_t + 1} - t$. Find the complementary cdf $\mathbb P(R_t \geq u)$ of $R_t$. Having computed the cdf of $R_t$ find the cdf of the random variable $M_t := T_{N_t + 1} - T_{N_t}$. What is the limiting distribution of $M_t$ as $t \rightarrow \infty$?