The problem of lot sizing and scheduling of production runs for multiple products to be processed on a machine was studied. The problem is formulated as a dynamic system, for which we show that cyclical schedules with the clearing lot sizing rule leads the system to a steady state. In the steady state we obtained a closed-form formula for the cycle time, the lot size and the maximum inventory level for each product. Finally, we showed that the clearing lot sizing rule is the optimal linear lot sizing rule that minimizes the system's inventory levels in the steady state.