I place two spherical particles, $P_1$ and $P_2$ (with radii $r_1$ & $r_2$), into a cylindrical container of radius $r_c$ ($r_1$ & $r_2$ $\leq \frac{1}{2}r_c$) and height $h$. While $P_1$ is immobilized at the centerpoint of the cylindrical container, $P_2$ has a coefficient of diffusion $D$, and can freely diffuse throughout the container and across its walls (i.e. the boundaries of the cylindrical container are non-reflecting).

We randomly position $P_1$ and $P_2$ somewhere inside the cylindrical container. Can we derive an expression for the mean residency time of $P_2$ as a function of the relative sizes of the particles and the cylindrical container? Is it a fair approximation to simply estimate the residency time of $P_2$ in a cylindrical container resized to subtract the volume of $P_1$?

Update - $P_1$ is now fixed at the centerpoint of the cylindrical container, and $r_1$ & $r_2$ are defined to be $\leq \frac{1}{2}r_c$, s.t. $P_1$ cannot block access, say, to half the cylindrical container.

Update 2 - In practice, $r_1$, $r_2$, and $r_c$ will be within one or two orders of magnitude of one another, i.e. it is not the case that $r_1$ & $r_2$ $<< \frac{1}{2}r_c$). Also, all collisions between particles, like the walls of the container, are fully reflecting.

Update 3 - I'd be perfectly happy calling the cylindrical container a "spherical container" with the same radius, $r_c$. Similarly, I'd be happy to not pin $P_1$ at any particular point, and instead to simply make the walls of the sphere reflecting specifically for this particle.