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Age-Minimal Transmission for Energy Harvesting Sensors with Finite Batteries: Online Policies
Ahmed Arafa, Jing Yang, Sennur Ulukus, H. Vincent Poor
TL;DR
The paper asks how an energy-harvesting sensor with a finite battery should schedule updates online to minimize long-term average AoI. It proves renewal optimality and derives energy-dependent threshold policies for RBR and IBR models, including closed-form thresholds tied to the optimal AoI. The thresholds decrease with available energy, and the full-battery threshold equals the optimal AoI.
Problem
The paper addresses how to minimize long-term average AoI when a finite-battery sensor receives energy causally and each update consumes one energy unit.
Method
The paper proves renewal optimality and applies a Lagrangian approach to characterize energy-dependent threshold policies for RBR and IBR recharge models.
Results
For both models, optimal thresholds are explicitly characterized in terms of optimal AoI, decrease with available energy, and the full-battery threshold equals the optimal AoI.
Takeaways & Limitations
The resulting online policies provide explicit age-minimal threshold rules for finite-battery sensors under the two modeled recharge processes.
Takeaways & Limitations
The paper identifies extensions involving erasures, combined recharge amounts, higher-priority updates, and general energy-arrival models.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper studies age-minimal online transmission for energy-harvesting sensors with finite batteries, where energy is revealed causally and updates consume energy. It establishes renewal-based, energy-dependent threshold policies for both RBR and IBR models, with thresholds characterized in closed form.
- Motivation: The objective is to minimize long-term average AoI for energy-constrained sensors whose harvested energy is revealed only causally.AoI measures the time since the latest update reached the destination.
- Models: The paper considers RBR, which fills the battery at each arrival, and IBR, which replenishes it unit by unit.Both models use unit-rate Poisson energy arrivals.
- Approach: Renewal theory shows that optimal update times follow model-specific renewal processes with independent inter-update delays.The renewal structure reduces policy characterization to model-dependent inter-update behavior.
- Results: A Lagrangian analysis yields energy-dependent threshold policies that transmit only when AoI exceeds the threshold associated with available energy.The thresholds are characterized in closed form in terms of the optimal AoI.
- Novelty: The closed-form characterization distinguishes this work from concurrent IBR analysis that finds optimal thresholds numerically.The comparison concerns the IBR model and the form of threshold characterization.
2 System Model and Problem Formulation
The system is an energy-harvesting sensor with a finite battery that sends instantaneous, error-free updates using one energy unit each. The online policy minimizes long-term average AoI under either recharge model.
- System model: The sensor stores harvested energy in a finite battery of size B and consumes one energy unit per measurement update.Updates are transmitted over an error-free link with negligible transmission time.
- Feasibility: The online policy chooses feasible transmission times using only causally revealed energy information.The battery starts empty at time 0.
- Objective: For either recharge model, the objective is to minimize the long-term average AoI at the destination.AoI is the elapsed time since the latest received update.
- AoI measurement: The cumulative AoI is represented by the area under the age curve over time.Figure 1 illustrates an example age evolution with n(t) = 3 updates.
- Special case: When B = 1, the RBR and IBR feasible-policy sets and optimization problems coincide.The paper treats this as a special case before analyzing larger batteries.
3 Unit Battery Case: A Review
For a unit battery, the two recharge models are equivalent and renewal policies provide the tractable baseline. The optimal renewal policy is a threshold rule whose parameter is obtained as the unique root of a decreasing function.
- Unit battery: With B = 1, the RBR and IBR models are equivalent, and renewal policies outperform other uniformly bounded policies.A uniformly bounded policy has inter-update times with bounded second moment as functions of arrival times.
- Renewal structure: Under a renewal policy, each inter-update time depends only on the next exponential energy-arrival time and cannot be shorter than that arrival time.The system resets when an update occurs because both battery and age return to zero.
- Optimization: The analysis introduces a parameterized optimization problem and identifies the optimal parameter λ* as the unique solution of p1(λ*) = 0.The relevant objective relation follows from comparing the parameterized and original problems.
- Threshold policy: The optimal inter-update rule is λ-threshold based: delay an early arrival until age λ, but transmit immediately when the arrival occurs at or after λ.This rule follows from the Lagrangian stationarity and complementary-slackness conditions.
- Baseline result: For the unit-battery case, the threshold equation has a unique solution λ* ≈ 0.9012.The paper then extends the approach to larger batteries under RBR and IBR.
4 Random Battery Recharge (RBR) Model
For the RBR model with B > 1, the paper decomposes operation into epochs and proves that optimal updates have a renewal structure. Lagrangian optimization then produces a multi-threshold policy whose thresholds decrease with available energy.
- Renewal-type policies: An epoch is defined between successive times when the battery level falls to B − 1, ensuring that each epoch contains at most B updates.This choice makes the renewal analysis finite and tractable.
- Renewal-type policies: Within an epoch, update delays before recharge are constants, while the final delay may depend on the next recharge time.The policy distinguishes updates made before and after the battery recharge.
- Renewal-type policies: Theorem 1 proves that the optimal RBR policy is a renewal policy, with constant pre-recharge delays and a final delay depending only on the next arrival time.The renewal process is formed by the epoch-boundary times.
- Renewal-type policies: Renewal-reward theory reduces the long-term AoI problem to optimization over a single epoch and its expected age-curve area and length.The resulting objective repeats the optimized epoch without loss of optimality.
- Threshold policies: The optimal λ* is found by bisection as the solution of pB^rbr(λ*) = 0, with λ* bounded above by the unit-battery value 0.9012.The objective function is decreasing in λ.
- Threshold policies: The optimal policy starts with a threshold λ and uses a cutoff xB−1 when a recharge occurs after λ but before the next scheduled update.It updates at λ for earlier arrivals and immediately for arrivals between λ and the cutoff.
- Threshold policies: The inter-update functions fj(λ) decrease with both update-index progression and λ, yielding progressively shorter delays as available energy increases.The paper gives an intuitive interpretation: low energy makes the sensor less eager to update, while high energy makes it more eager.
5 Incremental Battery Recharge (IBR) Model
For the IBR model, the paper establishes renewal structure and state-dependent scheduling, then derives threshold policies and a general-B objective formulation.
- 5.1 Renewal-Type Policies: The optimal IBR policy is a renewal policy, with visits to any fixed state (k, 0) forming a renewal process.The resulting inter-update times are independent.
- 5.1 Renewal-Type Policies: The next scheduled update depends only on the current battery energy and age state, not on earlier history.This state-based characterization follows from the renewal result and the memoryless property of exponential arrivals.
- 5.1 Renewal-Type Policies: The policy is characterized by constants x1, ..., xB−1 and functions y1(·), ..., yB(·) for zero-age and positive-age states.xk is the scheduled update time after entering (k, 0), while yk(t) is the scheduled time after entering (k, t).
- 5.2 Renewal State Analysis: A novel technique expresses the auxiliary objective explicitly in x and y for general battery size B and simplifies the earlier B = 2 analysis.The analysis uses renewal-state transitions and recursive calculations of age-curve area and epoch duration.
- 5.3 Threshold Policies: The resulting yk policies are threshold policies, with x3, x2, and x1 specified by separate closed-form expressions.For B = 4, the optimal parameter is approximately λ* = 0.6023, within the stated bounds [0.5, 0.72].
6 Numerical Evaluations
Numerical evaluations compare the optimal threshold policy with uniform and battery-aware adaptive updating under RBR and IBR arrivals. The optimal policy outperforms both alternatives; under IBR, policy gaps shrink as battery size grows because all policies approach the infinite-battery optimum.
- Policy comparisons: The experiments compare optimal, best-effort uniform, and battery-aware adaptive updating policies under RBR and IBR energy arrivals.The best-effort policy targets updates every 1/ν when energy is available; ν equals B for RBR and 1 for IBR.
- RBR model: The RBR evaluation shows the optimal policy outperforming both comparison policies, with the performance gap increasing with battery size.Figure 6 reports long term average age versus battery size for the different RBR policies.
- IBR model: The IBR evaluation again shows superiority of the optimal policy, but the gap between policies shrinks as battery size grows.Figure 7 reports long term average age versus battery size for the different IBR policies.
- IBR model: As battery size grows under IBR, all policies converge to 0.5, the optimal policy for the infinite-battery case.
- Markovian arrivals: Under the evaluated Markovian arrivals, threshold policies achieve relatively low age except for the bursty-arrivals case with q = 0.1.For IBR with q = 0.5 and q = 1, results are very close to the 0.5 lower bound.
7 Conclusion and Discussion
The paper establishes optimal online threshold policies for finite-battery energy-harvesting sensors under two Poisson recharging models. It also highlights implementation simplicity and directions for extending the analysis beyond the current assumptions.
- Conclusion: Optimal online threshold policies are established for finite batteries and zero service times under RBR and IBR recharging models.Both models use unit-rate Poisson arrivals revealed causally, with full-battery recharges in RBR and unit energy increments in IBR.
- Conclusion: For both models, optimal update times have renewal structures determined by the recharging model, leading to energy-dependent threshold policies.
- Implementation: Threshold values can be computed offline from the average arrival rate and battery size before communication begins.Online operation still requires only elapsed-time comparisons against the relevant threshold.
- Implementation: Threshold policies are relatively simple to implement because the sensor compares elapsed time with a threshold before transmitting.
- Extensions: Future work includes erasures, mixed recharge amounts, non-Poisson arrivals, prioritized updates, and near-optimality analyses.The paper notes that threshold-policy optimality and analysis become especially involved for the IBR model and more general settings.
8 Appendix
The appendix supports renewal-policy optimality by comparing arbitrary uniformly bounded policies with policies that minimize expected epoch cost, using epoch histories and Poisson energy-arrival patterns. It also records derivative calculations used in the threshold-policy optimization.
- Renewal-policy proof: The proof partitions operation into epochs and evaluates each epoch through its age-curve area and length.Epoch-level quantities are conditioned on histories and energy-arrival patterns.
- Renewal-policy proof: A renewal policy is constructed by repeating the policy that minimizes the epoch cost over all possible histories.For fixed histories, the minimizing policy depends on the epoch’s energy-arrival information rather than earlier history.
- Renewal-policy proof: Poisson arrivals make the relevant inter-arrival times independent exponential variables across epochs, yielding i.i.d. epoch lengths under the repeated policy.
- IBR proof: For the IBR proof, epochs are enumerated by energy-arrival patterns, with each pattern assigned a probability, age-curve area, and epoch length.The proof then averages these pattern-specific quantities to obtain a lower bound for arbitrary policies.
- Threshold optimization: The appendix derives derivatives of nested integrals with respect to threshold boundary functions using changes of variables and Leibniz’s rule.The calculations cover inner, middle, and outer boundary limits, including the B = 4 case.