A random search for one or more targets in a bounded domain occurs widely in nature, with examples ranging
from animal foraging to the transport of vesicles within cells. Most theoretical studies take a searcher-centric
viewpoint, focusing on the first passage time (FTP) problem to find a target. This single search-and-capture
event then triggers a downstream process or provides the searcher with some resource such as food. In this
paper we take a target-centric viewpoint by considering the accumulation of resources in one or more targets
due to multiple rounds of search-and-capture events combined with resource degradation; whenever a searcher
finds a target it delivers a resource packet to the target, after which it escapes and returns to its initial position.
The searcher is then resupplied with cargo and a new search process is initiated after a random delay. It has
previously been shown how queuing theory can be used to derive general expressions for the steady-state
mean and variance of the resulting resource distributions. Here we apply the theory to some classical FPT
problems involving diffusion in simple geometries with absorbing boundaries, including concentric spheres,
wedge domains, and branching networks. In each case, we determine how the resulting Fano factor depends on
the degradation rate, the delay distribution, and various geometric parameters. We thus establish that the Fano
factor can deviate significantly from Poisson statistics and exhibits a nontrivial dependence on model parameters,
including nonmonotonicity and crossover behavior. This indicates the nontrivial nature of the higher-order
statistics of resource accumulation.