Routing and Staffing when Servers are Strategic

Traditionally, research focusing on the design of routing and staffing policies for service systems has modeled servers as having fixed (possibly heterogeneous) service rates. However, service systems are generally staffed by people, who respond to workload incentives: how hard a person works can depend both on how much work there is, and how the work is divided between the people responsible for it. This observation has consequences when modeling performance of service systems where routing and staffing policies control such workload incentives, and our objective in this work is to investigate those consequences. Under a simple model for strategic servers (that choose their service rate in order to maximize a trade-off between a “cost of effort” and a “value of idleness”), and in the context of the M/M/N queuing model, we characterize the symmetric Nash equilibrium service rate under any routing policy that routes based on the server idle time, e.g., Longest-Idle-Server-First.

We find that the system must operate in a Quality-Driven regime, in which servers have idle time, in order for an equilibrium to exist. Then, within the class of admissible policies, we (asymptotically) solve the problem of minimizing the total cost, when the staffing and waiting costs are linear. Finally, we end by exploring the question of whether routing policies that are based on the service rate, instead of the server idle time, can improve system performance.