Many-Server Queueing Systems with Heterogeneous Strategic Servers in Heavy Traffic

In most service systems, the servers are humans who desire to experience a certain level of idleness. In call centers, this manifests itself as call-avoidance behavior when servers strategically adjust their service rate to strike a balance between the idleness they receive and effort to work harder. Moreover, being human, each server values this trade-off differently and has different capabilities. We develop a novel framework, relying on measure-valued processes and mean-field game theory, to simultaneously address strategic server behavior and inherent server heterogeneity in service systems. This framework significantly extends the literature on strategic servers in four directions by (i) incorporating individual choices of servers, (ii) incorporating individual abilities of servers, (iii) modeling the discomfort experienced because of low levels of idleness, and (iv) considering more general routing policies. It also enables us to move beyond symmetric equilibria by asymptotically characterizing asymmetric Nash equilibria in many-server systems with strategic servers. In simpler cases, it is shown that the purely quality-driven regime is asymptotically optimal when the servers are strategic. However, we show that, if the discomfort increases quickly enough as the idleness approaches zero, the quality- and efficiency-driven regime can be optimal even under strategic behavior. To our knowledge, this is the first time this conclusion appears in the literature.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2022.0608.