The lower-class waiting time distribution in the delayed accumulating priority queue

The Accumulating Priority Queue model, in which customers accumulate priority as a linear function of their time in the queue, was first introduced by Kleinrock in 1964 under another name. All publications addressing the APQ since then have assumed that customers start accumulating priority credits upon arrival. The model we present herein, called the Delayed Accumulating Priority Queue, entails an initial delay prior to priority accumulation for low-priority customers. The waiting time distribution for the lower class of customers in such an APQ is determined, and the impact of the initial delay upon that distribution is assessed. The equivalence to another model, the Affine Accumulating Priority Queue, is established. A motivation for our work is the potential for very long waiting times for low-acuity patients in health care systems operating at very high utilization. We test cases in an idealized setting motivated by access targets which differ by a factor of two (as occurs in the Canadian Triage and Acuity Scale (CTAS)). This work also considers the problem of finding the minimal lower-class priority accumulation rate which allows for the lower-class customers to meet their access target, as a function of the duration of the initial delay involved.