Consider sequence $A$ containing numbers $0 \leq x_1 \leq x_2 \leq \ldots \leq x_n \leq T$ which represent points on the t-axis, in the interval $[0,T]$. Then, you make sequence $B$ of numbers $0 \leq Y_1 \leq Y_2 \ldots \leq Y_m$ using a stationary Poisson point process with parameter $\lambda$. The superposition of $A$ and $B$ gives you $m+n$ points in the interval $[0,T]$. Find the joint probability distribution (pdf) of the superposition.
I think we can make the pdf using the law of total probability and conditioning based on the relative locations of points of $B$ with respect to sequence $A$. This probably is a very ugly solution and will not yield a clean close form. Any idea?