In this paper, we study a production system that operates under a lead-time performance constraint which guarantees the completion of an order before a pre-determined lead-time with a certain probability. The demand arrival times and the service requirements for the orders are random. To reduce the capacity-related operational costs, the production system under study has the option to use flexible capacity. We focus on periodic capacity policies and model the production system as a queuing system that can change its capacity periodically and choose to operate in one of the two levels: a permanent capacity level and a permanent plus contingent capacity level. Contingent capacity is supplied if needed at the start of a period, and is available during that period, at a cost rate that is decreasing in period length in different functional forms. Next, we propose a search algorithm that finds the capacity levels and the switching point that minimizes the capacity-related costs for a given period length. The behaviour of the capacity-related costs changes drastically under different period lengths and cost structures. In our computational study, we observe that the periodic capacity flexibility can reduce the capacity-related operational costs significantly (up to 35%). However, in order to achieve these savings, the period length must be chosen carefully depending on ambition level and capacity-related costs. We also observe that the percentage savings are higher for more ambitious lead-time performance constraints. Moreover, we observe that the use of contingent capacity makes the total system costs more insensitive to the lead-time performance requirements.