An Optimization Model to Minimize Expected Excess Waiting Time that Employs Accumulating Priority Queues

David Stanford, Statistical & Actuarial Sciences, Western University, London, Canada

joint work with

Peter Taylor, Univ. of Melbourne,

Ilze Ziedins, Univ. of Auckland,

Azaz Sharif, Western University, and

Rick Caron, Univ. of Windsor

## Abstract:

Key Performance Indicators (KPIs) are a measure of service system performance comprising a target delay and compliance probability (the chance a customer starts service by the target). While our motivating KPIs arise in the field of health care, other applications arise in call centre settings, telecommunication messaging systems, and elsewhere. The primary flaw of a pure KPI approach is that no consideration is given for the consequences of customers whose waiting time exceeds the target.

We (Sharif, Stanford, Caron 2014) present an optimization model for the such systems which seeks to minimize the weighted average of expected excess waiting time for the various classes. We test the model extensively in an Accumulating Priority Queue (APQ) setting (Stanford, Taylor and Ziedins, 2013). The APQ selects patients for treatment according to a linear priority accumulation function at a rate that depends upon the acuity class.

The first major focus of the talk is to review key insights gained about the probabilistic behavior of the APQ. We then move on to its use to address the minimization of excess delay. We establish what we can regarding convexity of the problem: in terms of its mathematical properties, the expected excess objective function cannot be readily shown to be convex in the priority accumulation rates.

When numerical inversion is used to assess the results, the computation of the expected excess is a simple extension to the computation of the compliance probabilities.

The presentation concludes with the insights gained from extensive numerical testing. One such result is anticipated: most notably, that the optimal priority accumulation rates are in inverse proportion to the delay targets when the value of the excess waiting time is deemed to be the same for all patient classes.