Optimal Server Assignment in Queues with Flexible Servers and Abandonments

We study a Markovian system of two stations in tandem and two flexible servers who are capable of working at both stations. The novelty of our work is that jobs waiting in line to be processed by the second station may leave the system (without being processed at the second station) after an exponential amount of time. We show that the dynamic server assignment policy that maximizes the long-run average throughput assigns one server to each station unless station 2 is starved (i.e., has no work) or the number of jobs in the buffer reaches a certain threshold (both servers work together at station 1 when station 2 is starved and at station 2 when the threshold is reached). We specify the criterion for the assignment of the servers to the stations and characterize the threshold. Finally, we investigate how the threshold depends on the service and abandonment rates.