  This paper examines the performance of multi-class multi-server bipartite
queueing systems under a FCFS-ALIS service discipline, where each arriving
customer is only compatible with a subset of servers. We analyze 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. 2021, 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 menu by considering only the limiting arrival rates. When some
constraints on the compatibility structure exist, the direction of convergence
to heavy-traffic affects which menu 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.
