Customers 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 routing problem of minimizing a weighted sum of the expected time in the system and the throughput of unsatisfied customers. Using a Markov decision process approach, we prove that the optimal routing of customers in service is of threshold type on the number of customers in the queue. This result is an extension of the known one when only the heterogeneity in speed of the servers is considered.