Queues with waiting time dependent service

Motivated by service levels in terms of the waiting time distribution seen in e.g. call centers, we consider two models for systems with a service disciple that depends on the waiting time. The first model deals with a single server that continuously adapts its service rate based on the waiting time of the first customer in line. In the second model, one queue is served by a primary server which is supplemented by a secondary server when the waiting of the first customer in line exceeds a threshold. Using level crossings for the waiting time process of the first customer in line, we derive steady-state waiting time distributions for both models. The results are illustrated with numerical examples.