Optimal control of two interacting service stations

Optimal controls described by switching curves in the two-dimensional state space are shown to exist for the optimal control of a Markov network with two service stations and linear cost. The controls govern routing and service priorities. Finite horizon and long run average cost problems are considered and value iteration is a key tool. Nonconvex value functions are shown to exist for slightly more general networks. Nonconvex value functions are also shown to arise for a simple single station control problem in which the instantaneous cost is convex but not monotone. Nevertheless, optimality of threshold policies is established for the single station problem. The proof is based on a novel use of stochastic coupling and policy iteration.