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In this paper, we analyze the non-linear age of information (AoI) performance in a point-to-point short packet communication system, where a transmitter generates packets based on status updates and transmits the packets to a receiver. Specifically, we investigate three packet management strategies, namely, the non-preemption with no buffer strategy, the non-preemption with one buffer strategy, and the preemption strategy. To characterize the level of the receiver's dissatisfaction on outdated data, we adopt a generalized α-βAoI penalty function into the analysis and derive closed-form expressions for the average AoI penalty achieved by the three packet management strategies. Simulation results are used to corroborate our analysis and explicitly evaluate the impact of various system parameters, such as the coding rate and status update generation rate, on the AoI performance. Additionally, we find that the value of αreflects the system transmission reliability.