We have two urns (blue and red) that are connected, and two particles, $p_1$ and $p_2$, are traveling between these urns independently. The amount of time $Z_1$ that $p_1$ spends in blue urn is iid with distribution $\exp{(\lambda_1)}$, and the amount of time it spends in red urn $Y_1$ is iid with distribution $F$. The amount of time $Z_2$ that $p_2$ spends in blue urn is iid with distribution $\exp(\lambda_2)$, and the amount of time it spends in red urn $Y_2$ is iid with distribution $G$. Both $F$ and $G$ are nonlattice.
I look at the urns at time $t$ and see $p_1$ is in the blue urn and $p_2$ is in the red urn. What is the expected time $E[T]$ until for the first time, both particles are in the blue urn ($t\to\infty$).
If we had just $p_2$, the mean time until for the first time it is in the blue urn is: $$E[T']=\frac{E[Y_2^2]}{2E[Y_2]}$$ but here we have two renewal processes.