We provide the first analytical expressions for the expected waiting time of high- priority customers in the delayed APQ by exploiting a classical conservation law for work-conserving queues. Additionally, we describe an efficient algorithm for exactly computing the expected waiting times of both low-priority and high-priority customers, requiring only the truncation of sums that converge exponentially quickly. These insights are used to demonstrate how the accumulation rate and delay level should be chosen by health care practitioners to optimize common key performance indicators (KPIs). In particular, we demonstrate that for certain nontrivial KPIs, an accumulating priority queue with a delay of zero is always preferable. Finally, we present a detailed investigation of the quality of an exponential approximation to the high-priority waiting time distribution, which we use to optimize the choice of queueing parameters with respect to both classes’ waiting time distributions.