AbstractQueue disciplines studied in the past have not given the system designer sufficient freedom with which to alter the relative waiting times of the various priority groups. In this paper, results are derived for a delay dependent priority system in which a unit's priority is increased, from zero, linearly with time in proportion to a rate assigned to the unit's priority group. The utility of this new priority structure is that it provides a number of degrees of freedom with which to manipulate the relative waiting times for each priority group.