We introduce a dynamic vehicle routing problem in which demands arrive via a temporal Poisson process with a certain arrival rate, and uniformly distributed along a line segment. Upon arrival, the demands move in a fixed direction perpendicular to the line with a fixed speed. A service vehicle, modeled as a first-order integrator with speed greater than that of the demands, seeks to serve these mobile demands. For the existence of any stabilizing service policy, we determine a necessary condition on the arrival rate of the demands in terms of the problem parameters; (i) the speed ratio between the demand and service vehicle, and (ii) the length of the line segment on which demands arrive. Next, we propose a novel service policy for the vehicle that involves servicing the outstanding demands as per the traveling salesperson path (t-TSP) through the moving demands. We derive a sufficient condition on the arrival rate of the demands for stability of the TSP-based policy, in terms of the problem parameters. We show that in the limiting case in which the demands move much slower than the service vehicle, the necessary and the sufficient conditions on the arrival rate are within a constant factor. We also provide an upper bound on the steady-state expected time spent by each demand before being served
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