  This paper presents a new method for deriving the waiting time distribution
of the lowest class in an accumulating priority (AP) queue with positive L\'evy
input. The priority of a non-atomic customer (particle) is a function of their
class and waiting time in the system, and the particles with the highest AP are
the next to be processed. The method relies on the construction of a workload
overtaking process and solving a first-passage time problem using an
appropriate stopping time.
