Pull or Wait: How to Optimize Query Age of Information

We study a pull-based status update communication model where a source node submits update packets to a channel with random transmission delay, at times requested by a remote destination node. The objective is to minimize the average query-age-of-information (QAoI), defined as the average age-of-information (AoI) measured at query instants that occur at the destination side according to a stochastic arrival process. In reference to a push-based problem formulation defined in the literature where the source decides to \textit{update or wait} at will, with the objective of minimizing the time average AoI at the destination, we name this problem the \textit{Pull-or-Wait} (PoW) problem. We provide a comparison of the two formulations: (i) Under Poisson query arrivals, an optimal policy that minimizes the time average AoI also minimizes the average QAoI, and these minimum values are equal; and (ii) the optimal average QAoI under periodic query arrivals is always less than or equal to the optimal time average AoI. We identify the PoW problem in the case of a single query as a stochastic shortest path (SSP) problem with uncountable state and action spaces, which has been not solved in previous literature. We derive an optimal solution for this SSP problem and use it as a building block for the solution of the PoW problem under periodic query arrivals.