Notes on conservation laws for preemptive priority queues

A generalized conservation law for mixed preemptive and nonpreemptive discipline is presented. In this discipline there are N job classes which are assigned preemptive resume priority and each job class split into Nj (j=0,...,N) job subclasses which are assigned head-of-the-line priority. The discipline is used, e.g., in communication control in a switching system. The authors give a proof for the conservation law by an intuitively comprehensive direct method, which is also applied for the Kleinrock’s conservation law to make his proof elaborated. An alternative proof for Takahashi’s conservation law on preemptive-resume M GI/G1 G2/1 systems is also presented to clarify the physical meaning in the connection with completion times.