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We consider a random access-based Internet of Things (IoT) system with periodic status updates. Multiple IoT devices transmit time-stamped status updates, which are generated periodically and synchronously, to the access point (AP) according to a state-dependent slotted Aloha access. The AP aims to maintain the latest possible status information of each device with respect to the age of information (AoI). By considering the characteristics of periodic update generation and state-dependent slotted Aloha, we propose an analytical model to derive the average AoI, peak AoI, and system throughput. Using the analytical model, the impact of the update interval on the average AoI and system throughput is investigated. For a state-dependent slotted Aloha with heuristically set transmission probability, the optimal update interval that minimizes the average AoI is evaluated.