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This study considers a wireless network where multiple nodes transmit status updates to a base station (BS) through a shared bandwidth-limited channel. Considering the random arrival of status updates, we measure the data freshness with the age of synchronization (AoS) metric; specifically, we use the time elapsed since the latest synchronization as a metric. The objective of this study is to minimize the weighted sum of the average AoS of the entire network while meeting the minimum throughput requirement of each node. We consider both the central scheduling scenario and the distributed scheduling scenario. In the central scheduling scenario, we propose the optimal stationary randomized policy when the transmission feedback is unavailable and the max-weight policy when it is available. In the distributed scenario, we propose a distributed policy. The complexity of the three scheduling policies is significantly low. Numerical simulations show that the policies can satisfy the throughput constraint in the central controlling scenario and the AoS performance of the max-weight policy is close to the lower bound.