Strategic Servers in Queuing Systems with Idle-Time Incentives

We study queuing systems in which strategic servers choose service speed while valuing idle time. In a single-server Markovian queue, effort raises both the service rate and the instantaneous cost of working, with costs incurred only while busy. We show that the optimal stationary policy has a threshold form: the server works fast when the queue is short and slows down once it is sufficiently long. The threshold is governed by the marginal value of reducing the queue relative to the cost per additional unit of service speed. We further show that this structure is robust to finite buffers: the threshold weakly decreases with the arrival rate and weakly increases with buffer size. Finally, we study a common-queue system with multiple strategic servers, where race-to-idleness incentives interact with free-riding. We prove existence of a symmetric stationary relaxed-effort equilibrium and show that equilibrium effort is weakly decreasing in the fully congested region.