Computing overlapping intervals in parallel Poisson processes

Question

Let (X'_t), (X'_t) be two independent Poisson processes, let N,N' respectively be the number of arrivals before time $\beta$, let $0<T_1< T_2,...<T_N$ and $0<T'_1< T'_2...<T'_{N'}$ be the time of each arrival before time $\beta$. Is it possible to compute for each pair of positive integers $i, j$ the following quantity, for any $r$? $$ P( | [T_i, T_{i+1}] \cap [T'_j,T'_{j+1} ] | > r\} \cap \{N \geq i\} \cap \{N' \geq j\} ) $$