Routing in a queueing system with two heterogeneous servers in speed and in quality of resolution

Heterogeneous servers, in manufacturing and service systems, may have different speeds and different quality levels for the provided service or good. For a two-server queueing model, we formulate the job routing problem for minimizing the long-run weighted sum of the expected time spent in the system and the throughput of unsatisfied customers. Using a Markov decision process approach, we prove that the optimal routing policy of jobs to service is a threshold policy that depends on the queue length. When the number of waiting jobs in the queue is below a certain threshold, only one server should work and the other one remains idle. At or above this threshold, both servers should serve jobs. This is an extension of the known result where only the heterogeneity in speed is considered.