Given $N$ nodes randomly placed in a $D\times D$ area, i.e., the position of each node is randomly chosen. Assume that both $N$ and $D$ are sufficiantly large.

An agent can move in the area at constant unit speed. When moving, it marks all nodes within distance $d$ from it ($d$ is Eucliean distance rather than number of hops). It aims at finding an itinerary such that given the time duration $T$, it can mark as many nodes as possible. In other words, the agent wants to find a path of length $l=T$ along which it can mark as many nodes as possile.

Under such setting, if the path is chosen randomly (e.g., the agent goes straightly towards a random direction for the entire time duration $T$), which is a natural strategy, then the number of nodes covered is $\frac{NT}{D^2}$. This gives a reference for comparison. Now I want to derive the expected number of covered nodes (or an upper-bound) for an optimal path, i.e., if the agent strategically chooses an optimal path, what is the number of nodes covered in average?