The rapid development of the Internet of Things has led to the rise of many real-time applications, which are highly dependent on timely information delivery to respond correctly to changes in the surrounding environment. Therefore, age of information (AoI) has aroused widespread attention as a new metric to evaluate information freshness at the destination. In this paper, we investigate the effect of the retransmission mechanism and updating patterns on the AoI performance. Different from the existing work that considers the average value of AoI, we adopt violation probability, the probability that AoI exceeds a certain threshold, as the metric for meeting the strict timeliness requirements of different applications. We obtain closed-form expressions of AoI violation probability for active sources and buffered sources under last-come-first-served discipline with packet discarding. The accuracy of the analyses is verified through simulations. Numerical results show that the buffered sources with periodic arrivals have better performance in terms of information freshness given service rate and updating rate.