An Inequality for Query Age of Information and Age of Information

We study a status update system where a destination node requests from a source node to submit update packets to a channel. The update packets incur a constant transmission delay in the channel. The update packets sent from the source node will be utilized by the destination node at certain query instants generated toward a computation. The query instants are modeled as a stochastic arrival process. The Age of Information (AoI) at the query instants is called as the Query Age of Information (QAoI). The pull-or-wait (PoW) problem is defined as an optimal control problem in which the destination node decides when to request new update packets from the source to minimize the average QAoI. We compare the solution of the PoW problem with the optimal time average AoI obtained as a result of the update-or-wait (UoW) problem, where the source node decides on packet transmission times to minimize the time-average AoI. We show that for all query arrival processes the optimal QAoI is always less than or equal to the optimal time average AoI under the same power constraint.