Optimal foraging of renewable resources

Consider a team of agents in the plane searching for and visiting target points that appear in a bounded environment, according to a stochastic renewal process with a known absolutely continuous spatial distribution. Agents must detect targets with limited-range onboard sensors. It is desired to minimize the expected waiting time between the appearance of a target point, and the instant it is visited. When the sensing radius is small, the system time is dominated by time spent searching, and it is shown that the optimal policy requires the agents to search a region at a relative frequency proportional to the square root of its renewal rate. On the other hand, when targets appear frequently, the system time is dominated by time spent servicing known targets, and it is shown that the optimal policy requires the agents to service a region at a relative frequency proportional to the cube root of its renewal rate. Furthermore, the presented algorithms in this case recover the optimal performance achieved by agents with full information of the environment. Simulation results verify the theoretical performance of the algorithms.