We study the optimum scheduling of a pushing-based information delivery system, where information is sent from a server to mobile users by using server-initiated pushing actions. Information arrives at the server at random. The server adopts a "hold-then-serve" strategy, where new information is temporarily stored in queue for a later one-time transmission. Based on the current queue status, at any given time instant the server needs to decide whether to push all information in the queue to the mobile user or keep waiting. A shorter waiting time can ensure the timeliness or "freshness" of the information, which can be measured by using age-of-information (AoI). On the other hand, frequent pushing actions will frequently wake up the client thus increase the power consumption of mobile devices. The objective of this paper is to identify the scheduling rule that can optimize the tradeoff between the AoI and energy efficiency measured by energy consumption per bit. With the help of stopping theories, we develop optimum stopping rules that can minimize a weighted combination of AoI and energy efficiency, with the tradeoff between the two metrics determined by the weight coefficient in the cost function. Specifically, if the AoI scales linearly with time, it is proved that the one-step look ahead stopping rule is optimum. The corresponding statistical properties of the random stopping time for push actions are analytically identified, and they are used to obtain the operation parameters of the optimum stopping rule.
