  In this paper we study a dynamic vehicle routing problem in which there are
multiple vehicles and multiple classes of demands. Demands of each class arrive
in the environment randomly over time and require a random amount of on-site
service that is characteristic of the class. To service a demand, one of the
vehicles must travel to the demand location and remain there for the required
on-site service time. The quality of service provided to each class is given by
the expected delay between the arrival of a demand in the class, and that
demand's service completion. The goal is to design a routing policy for the
service vehicles which minimizes a convex combination of the delays for each
class. First, we provide a lower bound on the achievable values of the convex
combination of delays. Then, we propose a novel routing policy and analyze its
performance under heavy load conditions (i.e., when the fraction of time the
service vehicles spend performing on-site service approaches one). The policy
performs within a constant factor of the lower bound (and thus the optimal),
where the constant depends only on the number of classes, and is independent of
the number of vehicles, the arrival rates of demands, the on-site service
times, and the convex combination coefficients.
