Conditional probability of overlapping intervals of a Poisson process

Question

I am trying to solve the following problem: Suppose you have a Poisson process with rate $\lambda$. Now define $Y_n =\sum_{i=n}^{n+N-1} X_i$ as the sum of N subsequent inter-arrival times where $X_i$s are exponential i.i.d. rv's with mean $1/\lambda$. The $Y_n$s (which can be thought of as the inter-arrival time of job $n$ and job $n+N-1$) are Erlang distributed with parameters $(N,\lambda)$. Clearly, $Y_n$ and $Y_{n+1}=\sum_{i=n+1}^{n+N} X_i$ are not independent (in fact the correlation is given as $corr(Y_n,Y_{n+1}) = 1-1/w, \ \forall w \in \mathbb Z_+\setminus\{0\}$). How can I determine $\mathbb P(Y_{n+1}\leq x|Y_n\leq x)$? Thanks for your help! :)