We consider a base station supporting two types of traffics, i.e., status update traffic and timely throughput traffic. The goal is to improve the information freshness of status update traffic while satisfying timely throughput constraints. Age of Information (AoI) is adopted as a metric for information freshness. We first propose an age-aware policy that makes scheduling decisions based on the current value of AoI directly. Given timely throughput constraint, an upper bound of the weighted average AoI under this policy is provided. To evaluate policy performance, it is important to obtain the minimum weighted average AoI achievable given timely throughput constraint. A low complexity method is proposed to estimate a lower bound of this value. Furthermore, inspired by the estimation procedure, we design an age-oblivious policy that does not rely on the current AoI to make scheduling decisions. Surprisingly, simulation results show that the weighted average AoI of the age-oblivious policy is comparable to that of the age-aware policy, and both are close to the lower bound.