Optimal control of a queueing system with heterogeneous servers and setup costs

This paper considers a queueing model with batch Poisson input and two heterogeneous servers, where the service times are exponentially distributed. The faster server is always on, but the slower server is only used when the queue length exceeds a certain level. Activating the slower server involves fixed set-up costs. Also there are linear operating costs and linear holding costs. The class of two-level hysteretic control rules is considered. Rather than proving the overall average cost optimality of a hysteretic rule, the purpose of this paper is to develop a tailor-made policy iteration algorithm for computing the optimal switch-on and switch-off levels for the slower server. An embedding method is used that is generally applicable to structured Markovian control problems with an infinitely large state space