This seems to be a simple problem to me, but am not getting right results.
As attached in the figure (not drawn to scale). A circle centered at $X$ (X in Blue) with initial position at time $t=t_0$ moves to next location (X in red) at $t=t_1$ (all within a square of sides $a$).
Some items (called 'nodes' henceforth) are distributed uniformly with density $\lambda$ in the square (not shown in fig.). $X$ has to discover each item within its communication range. Hence every circle has $N = \lambda\pi r^2$ nodes, where $r$ is the radius around $X$. ($X$ is called master).
If $P_{iN} = NP$ (not to be confused by NP-hard etc.) are the nodes that $X$ discover within in communication range at the time $t=i$. Where $P$ is the probability that $X$ discovers a single node.
Then After time $T = t_0+t_1$ what is the probability of finding all discovered nodes.
My solution is: $$P_T = P_{0N} + P_{1N} - P'_{01N'}$$
where $P'_{01}$ is the probability of nodes that might repeat in the shaded area.
I find this shaded area as $A = A_{circle} - A_{Cresent} = \pi r^2 - \frac{\pi r w}{2}$.
I just want to know if my strategy is right. If someone is interested, I can share the formulas for $P$ as well. So $N' = \lambda \times (\pi r^2 - \frac{\pi r w}{2})$
