# Performance of the smallest-variance-first rule in appointment sequencing

## TL;DR

This paper analyzes the performance of the smallest-variance-first (SVF) rule in appointment scheduling, providing theoretical bounds and showing asymptotic optimality as the number of patients increases.

## Contribution

It offers the first theoretical bounds on SVF's worst-case performance and proves its asymptotic optimality in appointment sequencing.

## Key findings

- SVF has bounded worst-case ratio to optimal in various settings.
- SVF's ratio approaches 1 as the number of patients grows large.
- This is the first application of approximation ratio analysis in appointment scheduling.

## Abstract

A classical problem in appointment scheduling, with applications in health care, concerns the determination of the patients' arrival times that minimize a cost function that is a weighted sum of mean waiting times and mean idle times. One aspect of this problem is the sequencing problem, which focuses on ordering the patients. We assess the performance of the smallest-variance-first (SVF) rule, which sequences patients in order of increasing variance of their service durations. While it was known that SVF is not always optimal, it has been widely observed that it performs well in practice and simulation. We provide a theoretical justification for this observation by proving, in various settings, quantitative worst-case bounds on the ratio between the cost incurred by the SVF rule and the minimum attainable cost. We also show that, in great generality, SVF is asymptotically optimal, i.e., the ratio approaches 1 as the number of patients grows large. While evaluating policies by considering an approximation ratio is a standard approach in many algorithmic settings, our results appear to be the first of this type in the appointment scheduling literature.