On the conservation law and the performance space of single server systems

We consider a multiclass GI/G/1 queueing system, operating under an arbitrary work-conserving scheduling policy π. We derive an invariance relation for the Cesaro sums of waiting times under π, which does not require the existence of limits of the Cesaro sums. This allows us to include important classes in the set of admissible policies such as time-dependent and adaptive policies. For these classes of policies, ergodicity is not known a priori and may not even exist. Therefore, the classical invariance relations that involve statistical averages do not hold. For an M/G/1 system, we derive inequalities involving the Cesaro sums of waiting times that further characterize the achievable performance region of the system.