Mind your own customers and ignore the others: Asymptotic optimality of a local policy in multi-class queueing systems with customer feedback

Abstract

This work contributes to the investigation of optimal routing and scheduling policies in multi-class multi-server queueing systems with customer feedback. We propose a new policy, dubbed local policy that requires access to only local queue information. Our new local policy specifies how an idle server chooses the next customer by using the queue length information of not all queues, but only those this server is eligible to serve. To gain useful insights and mathematical tractability, we consider a simple W model with customer feedback, and we establish limit theorems to show that our local policy is asymptotically optimal among all policies that may use the global system information, with the objective of minimizing the cumulative queueing costs measured by convex functions of the queue lengths. Numerical experiments provide convincing engineering confirmations of the effectiveness of our local policy for both W model and a more general non-W model.

KEYWORDS:

• Multi-class queueing systems
• asymptotic optimality
• customer feedback
• local information
• convex delay cost
• $\mathit{c}\mu$-rule

Acknowledgments

The authors would like to thank the associate editor and two anonymous referees for many constructive comments and suggestions. This article has been substantially improved due to their efforts.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under Grants 11971074 and 11871116, and in part by Hong Kong Research Grants Council (Project 14504217).

Notes on contributors

Jiankui Yang

Dr. Jiankui Yang is an associate professor in the School of Science, Beijing University of Posts and Telecommunications. He obtained his BE and PhD degrees from the Mathematics Department at Nanjing University. His current research interests include the stochastic models and quantitative analysis.

Junfei Huang

Dr. Junfei Huang is an associate professor in the Department of Decision Sciences and Managerial Economics at the Chinese University of Hong Kong. His research interests are in asymptotic analysis and optimal control of queueing systems and their applications in manufacturing and services.

Yunan Liu

Dr. Yunan Liu is an associate professor in Department of Industrial and Systems Engineering at North Carolina State University. He obtained his BE degree from the Electrical Engineering Department at Tsinghua University, MS and PhD degrees from the Industrial Engineering and Operations Research Department at Columbia University. His research interests include queueing theory, applied probability, simulations, optimal control, online learning, and their applications to call-center, health-care, transportation, and blockchain. His work was awarded first place in the INFORMS Junior Faculty Interest Group Paper Competition in 2016.