A c μ rule for parallel servers with two-tiered c μ preferences

The cμ rule is known to be optimal in many queueing systems with memoryless service times. We establish a cμ rule for a parallel flexible server system with two tiers. Multiple job classes arrive and wait in separate queues. The first tier contains classes that can only be served by one server and the second tier contains a class that can be served by all servers. The cμ preferences give priority to the first tier. Sequencing decisions are made to minimize linear holding costs. The proofs employ stochastic coupling sample path arguments.