Waiting time analysis in a finite-capacity multi-server systems with dynamic priorities, dynamically evolving customer types, and abandonment

2026-06-30

AI Summary

This study addresses the complex interplay of dynamically evolving customer classes, abandonment behavior, and dynamic prioritization in finite-capacity, multi-server queueing systems. To tackle this challenge, the authors propose a scalable continuous-time Markov chain (CTMC) modeling framework that integrates quasi-birth–death processes, matrix-analytic methods, and Krylov subspace approximations to efficiently compute both conditional and steady-state waiting time distributions for two customer classes. Notably, this work is the first to incorporate dynamic customer-type evolution and reneging into waiting time analysis for such systems. The model’s validity is demonstrated using real-world data from a tertiary referral hospital in Australia, where it successfully quantifies the disparity in waiting times between complex and routine patients, thereby offering actionable, quantitative insights for healthcare operational decision-making.

Abstract

In many service systems, an estimation of customers' waiting times for the service can assist in decision making focused on enhancing the operational efficiency, improving the customers' experience, and ensuring efficient resource allocation. In this paper, we study the customers' waiting times in a finite-capacity service system with a finite number of parallel servers and a shared waiting area. We consider two customer types, Type 1 and Type 2, with dynamic admission priorities, dynamically evolving customer type, and abandonment. We model the system under such assumptions using a continuous-time Markov chain (CTMC) and develop a methodology based on Krylov subspace approximation methods to evaluate the conditional waiting time distributions of Type 1 and Type 2 customers in the system. This methodology (CTMC-Krylov) offers a scalable computational approach that is well suited for analysing large complex systems. Next, we model this system using a quasi-birth-and-death (QBD) process and derive analytical expressions building on matrix-analytic methods to evaluate the conditional and long-run waiting time distributions using recursion. We illustrate the practical applicability of our models in a hospital system through a suite of numerical examples based on a large dataset obtained from a tertiary referral hospital in Australia, considering two types of patients, complex (Type 1) and other (Type 2).

Problem

• waiting time
• dynamic priorities
• customer abandonment
• finite-capacity queue
• multi-server system

Innovation

• Krylov subspace approximation
• continuous-time Markov chain
• quasi-birth-and-death process
• dynamic priorities
• waiting time distribution