On the Distribution of Age of Information in Time-varying Updating Systems

Age of Information (AoI) is a crucial metric for quantifying information freshness in real-time systems where the sampling rate of data packets is time-varying. Evaluating AoI under such conditions is challenging, as system states become temporally correlated and traditional stationary analysis is inapplicable. We investigate an $M_{t}/G/1/1$ queueing system with a time-varying sampling rate and probabilistic preemption, proposing a novel analytical framework based on multi-dimensional partial differential equations (PDEs) to capture the time evolution of the system's status distribution. To solve the PDEs, we develop a decomposition technique that breaks the high-dimensional PDE into lower-dimensional subsystems. Solving these subsystems allows us to derive the Aol distribution at arbitrary time instances. We show AoI does not exhibit a memoryless property, even with negligible processing times, due to its dependence on the historical sampling process. Our framework extends to the stationary setting, where we derive a closed-form expression for the Laplace-Stieltjes Transform (LST) of the steady-state AoI. Numerical experiments reveal AoI exhibits a non-trivial lag in response to sampling rate changes. Our results also show that no single preemption probability or processing time distribution can minimize Aol violation probability across all thresholds in either time-varying or stationary scenarios. Finally, we formulate an optimization problem and propose a heuristic method to find sampling rates that reduce costs while satisfying AoI constraints.