A unifying conservation law for single server queues.

textabstractIn this paper we develop a conservation law for a work conserving multi-class $GI/GI/1$ queue operating under a general scheduling discipline. In the context of single-class queues, conservation laws have been obtained for both non-anticipating and anticipating disciplines with general service time distributions. In the context of multi-class queues, conservation laws have been previously obtained for (i) non-anticipating disciplines and exponential service time distribution and (ii) non-preemptive disciplines and general service time distribution. The conservation law we develop generalizes already existing conservation laws, and includes in particular popular multi-class time-sharing disciplines such as Discriminatory Processor Sharing (DPS) and Generalized Processor Sharing (GPS). In the literature, the conservation laws for single-class and multi-class queues are presented as if they were different in nature. The conservation law we develop includes existing conservation laws as special case