Age-Minimal Transmission for Energy Harvesting Sensors with Finite Batteries: Online Policies (1806.07271v2)

Abstract: An energy-harvesting sensor node that is sending status updates to a destination is considered. The sensor is equipped with a battery of finite size to save its incoming energy, and consumes one unit of energy per status update transmission, which is delivered to the destination instantly over an error-free channel. The setting is online in which the harvested energy is revealed to the sensor causally over time, and the goal is to design status update transmission policy such that the long term average age of information (AoI) is minimized. AoI is defined as the time elapsed since the latest update has reached at the destination. Two energy arrival models are considered: a random battery recharge (RBR) model, and an incremental battery recharge (IBR) model. In both models, energy arrives according to a Poisson process with unit rate, with values that completely fill up the battery in the RBR model, and with values that fill up the battery incrementally, unit-by-unit, in the IBR model. The key approach to characterizing the optimal status update policy for both models is showing the optimality of renewal policies, in which the inter-update times follow a specific renewal process that depends on the energy arrival model and the battery size. It is then shown that the optimal renewal policy has an energy-dependent threshold structure, in which the sensor sends a status update only if the AoI grows above a certain threshold that depends on the energy available. For both the RBR and the IBR models, the optimal energy-dependent thresholds are characterized explicitly, i.e., in closed-form, in terms of the optimal long term average AoI. It is also shown that the optimal thresholds are monotonically decreasing in the energy available in the battery, and that the smallest threshold, which comes in effect when the battery is full, is equal to the optimal long term average AoI.

• The paper develops energy-dependent threshold policies that minimize the long-term average age of information (AoI) for sensors with finite batteries.
• It employs renewal theory and fractional programming to design optimal scheduling strategies that adapt based on battery energy levels.
• Numerical results confirm that the proposed policies outperform heuristic alternatives under both random and incremental battery recharge models.

Age-Minimal Transmission Policies for Energy Harvesting Sensors with Finite Batteries in Online Settings

The paper addresses the problem of optimizing status update transmission times in systems with energy harvesting sensors equipped with finite batteries. This research specifically focuses on minimizing the long-term average age of information (AoI), which is a metric that represents the time elapsed since the last update has reached its destination. Two distinct energy arrival models, random battery recharge (RBR) and incremental battery recharge (IBR), are considered under Poisson energy arrival processes with unit rates. The main contribution is the development and proof of optimality for energy-dependent threshold policies in such systems.

Methodology

To solve the problem, the authors utilize renewal theory and fractional programming techniques. The paper begins with simple cases where $B=1$ (battery size equal to one unit) to establish a foundation. In this scenario, the optimal policy is shown to adopt a threshold-based approach where updates are sent only when AoI exceeds a specific threshold decided by the energy dynamics. This simplifies into various closed-form expressions for different battery settings.

Upon transitioning to the general case ($B>1$), where batteries can store more than one energy unit, the paper uses renewal theory to demonstrate that optimal policies have a renewal structure—update times follow an independent and identically distributed (i.i.d.) process. From this starting point, the authors derive step-by-step procedures and perform complex combinatorial analyses to formalize optimal policies in RBR and IBR settings.

The key innovation is a method of expressing expected AoI explicitly in terms of given parameters using nested integral techniques and discrete states tracking in the RBR model. For the IBR model, derivations leverage a carefully crafted hierarchy of Markovian transitions through energy states. This systematic approach uncovers a monotonic relationship between battery energy levels and AoI thresholds, revealing that as battery energy increases, the threshold decreases.

Numerical Insights

The numerical results substantiate the theoretical findings, displaying superior performance of derived threshold policies over two heuristic alternatives. Analyses consider long-term average AoI against varying battery sizes and also evaluate performance under alternative Markovian energy arrival models. Threshold policies maintain effectiveness, showcasing their robustness across diverse settings. Results unify both simple ($B=1$) and complex ($B>1$) battery scenarios with insights into practical implementations.

Implications and Future Directions

The paper significantly advances understanding in the application of information age metrics within energy-constrained environments, crucial for real-time sensing systems. It proposes simplified computations for effective deployment, offering practical solutions and aiding system designers in achieving timely data collection without succumbing to energy limitations.

This paper prompts several promising avenues for future research; exploring the impact of communication channel errors, extending analysis to multi-source systems, and investigating near-optimal policy behaviors under varied stochastic arrival models hold promise. Moreover, mapping theoretical constructs directly onto hardware implementations heightens the potential for real-world applications in IoT devices and Autonomous systems.

In summary, the work rigorously handles the complexities of energy-dependent scheduling within stochastic processes and lays the groundwork for further multifaceted studies in the age of information field.