Fluid approximations of many-server queues with delayed feedback and heterogeneous service and patience times

We consider many-server queues with delayed feedback where the service (patience) times of new customers and feedback customers are differentiated, and new and feedback customers are served under the first-come-first-serve (FCFS) discipline in the service station. The arrival process, service, patience and delay times are all general and mutually independent. A two-parameter fluid model for the system dynamics in the many-server regime is investigated, where we use four two-parameter processes to describe the service and queueing processes of the new and feedback customers, two for the service dynamics and two for the queueing dynamics. When the arrival rate is constant, we derive the steady state performance measures and study the impact of impatience differentiation and service differentiation upon them. When the arrival rate is time-varying, we provide an algorithm to compute the fluid processes. Numerical experiments are conducted, and show that the algorithm is very effective, compared with simulations.