Heavy traffic analysis of multi-class bipartite queueing systems under FCFS

This paper examines the performance of multi-class, multi-server bipartite queueing systems, where each arriving customer is compatible with only a subset of servers. We focus on the system's performance under a first-come, first-served-assign longest idle server service discipline. In this discipline, an idle server is matched with the compatible customer who has been waiting the longest, and a customer who can be served by multiple idle servers is routed to the server that has been idle for the longest period. We analyse the system under conventional heavy-traffic conditions, where the traffic intensity approaches one from below. Building upon the formulation and results of Afeche et al. (Oper Res 70(1):363-401, 2022), we generalize the model by allowing the vector of arrival rates to approach the heavy-traffic limit from an arbitrary direction. We characterize the steady-state waiting times of the various customer classes and demonstrate that a much wider range of waiting time outcomes is achievable. Furthermore, we establish that the matching probabilities, i.e. the probabilities of different customer classes being served by different servers, do not depend on the direction along which the system approaches heavy traffic. We also investigate the design of compatibility between customer classes and servers, finding that a service provider who has complete control over the matching can design a delay-minimizing matching by considering only the limiting arrival rates. When some constraints on the compatibility structure exist, the direction of convergence to heavy-traffic affects which compatibility structure minimizes delay. Additionally, we discover that the bipartite matching queueing system exhibits a form of Braess's paradox, where adding more connectivity to an existing system can lead to higher average waiting times, despite the fact that neither customers nor servers act strategically.