Coordinated production and inspection in a tandem system

We study the coordination of production and quality control in a tandem-queue system. There are two stages, with a single server at stage one that can engage in processing an item, or inspecting the produced item, or staying idle; whereas the second stage represents the aggregate of the rest of the production facility. We focus on the optimal control of the first stage, where both the production and inspection times follow general distributions. We formulate a semi-Markov decision program with a long-run average objective, and derive the stationary optimal policy to control and coordinate the production, inspection, and idling processes. We show that there exists a threshold value i^*, such that under the optimal policy, once the threshold is reached, production should be suspended at the first stage; and this leads naturally to i^*= 1 being the required buffer capacity between the two stages.