Conservation laws for single-server fluid networks

We consider single-server fluid networks with feedback and arbitrary input processes. The server has to be scheduled in order to minimize a linear holding cost. This model is the fluid analogue of the so-called Klimov problem. Using the achievable-region approach, we show that the Gittins index rule is optimal in a strong sense: it minimizes the linear holding cost for arbitrary input processes and for all time points t ⩾ 0.