Dynamic vehicle routing with moving demands - Part II: High speed demands or low arrival rates

In the companion paper we introduced a vehicle routing problem in which service demands arrive stochastically on a line segment. Upon arrival, the demands translate perpendicular to the line with a fixed speed. A vehicle, with speed greater than that of the demands, seeks to provide service by reaching each mobile demand. In this paper we study a first-come-first-served (FCFS) policy in which the service vehicle serves demands in the order in which they arrive. When the demand arrival rate is very low, we show that the FCFS policy can be used to minimize the expected time, or the worst-case time, to service a demand. We determine necessary and sufficient conditions on the arrival rate of the demands (as a function of the problem parameters) for the stability of the FCFS policy. When the demands are much slower than the service vehicle, the necessary and sufficient conditions become equal. We also show that in the limiting regime when the demands move nearly as fast as the service vehicle; (i) the demand arrival rate must tend to zero; (ii) every stabilizing policy must service the demands in the order in which they arrive, and; (iii) the FCFS policy minimizes the expected time to service a demand.