  We introduce a problem in which a service vehicle seeks to guard a deadline
(boundary) from dynamically arriving mobile targets. The environment is a
rectangle and the deadline is one of its edges. Targets arrive continuously
over time on the edge opposite the deadline, and move towards the deadline at a
fixed speed. The goal for the vehicle is to maximize the fraction of targets
that are captured before reaching the deadline. We consider two cases; when the
service vehicle is faster than the targets, and; when the service vehicle is
slower than the targets. In the first case we develop a novel vehicle policy
based on computing longest paths in a directed acyclic graph. We give a lower
bound on the capture fraction of the policy and show that the policy is optimal
when the distance between the target arrival edge and deadline becomes very
large. We present numerical results which suggest near optimal performance away
from this limiting regime. In the second case, when the targets are slower than
the vehicle, we propose a policy based on servicing fractions of the
translational minimum Hamiltonian path. In the limit of low target speed and
high arrival rate, the capture fraction of this policy is within a small
constant factor of the optimal.
