Throughput-AoI Capacity Region

• Throughput-AoI capacity region is the set of achievable throughput and freshness pairs defined by network scheduling policies.
• It quantifies the trade-off between long-term successful delivery rates and information freshness using mean and variance metrics.
• The region underpins the design of scheduling policies such as Variance-Weighted Deficit to optimize performance in unreliable wireless networks.

Searching arXiv for the focal paper and closely related work on throughput–AoI trade-offs and second-order capacity regions. Throughput-AoI capacity region denotes the set of throughput–Age of Information tuples that can be achieved simultaneously by some admissible scheduling policy, and is used to formalize the trade-off between delivery rate and information freshness. In unreliable wireless networks with $N$ devices, per-device transmission success probabilities $p_i$, and a per-slot scheduling limit of $M$ transmitters, the region is defined over tuples $\{(m_i,h_i)\}_{i=1}^N$, where $m_i$ is long-term average throughput and $h_i$ is long-term average AoI (Wang et al., 17 Aug 2025). Related literature studies analogous feasible sets, frontiers, or higher-dimensional regions in stochastic-geometry IoT systems, fading channels, random access protocols, wired networks, and mixed AoI/timely-throughput settings, even when the term “throughput-AoI capacity region” is not used explicitly (Mankar et al., 2020, Bhat et al., 2019, Munari et al., 2023, Tseng et al., 2021, Fountoulakis et al., 2021).

1. Formal definition and basic objects

In the multi-device unreliable wireless setting, the throughput-AoI capacity region is the set of tuples $\{(m_i,h_i)\}_{i=1}^N$ for which there exists a scheduling policy satisfying

$$\liminf_{T \to \infty} \frac{\sum_{t=1}^T Z_i(t)}{T} \geq m_i, \qquad \limsup_{T \to \infty} \frac{\sum_{t=1}^{T} a_i(t)}{T} \leq h_i,$$

where $Z_i(t)\in\{0,1\}$ indicates whether device $i$'s transmission at time $p_i$0 succeeds, and $p_i$1 is the AoI of device $p_i$2 at time $p_i$3 (Wang et al., 17 Aug 2025). The definition is inherently multi-dimensional: it concerns all users jointly rather than a single scalar trade-off curve.

This formulation makes explicit that throughput and AoI are not interchangeable objectives. Throughput is a long-term average successful delivery rate, whereas AoI is a stateful freshness metric driven by the elapsed time since the freshest received update. In cellular IoT, for example, the AoI at a base station for device $p_i$4 is written as

$p_i$5

where $p_i$6 is the generation time of the latest received update (Mankar et al., 2020). In 5G uplink scheduling, AoI for user equipment $p_i$7 is similarly written as

$p_i$8

(Wu et al., 2021). Across these models, the region concept collects feasible operating points rather than optimizing only one metric in isolation.

A recurrent distinction in the literature is between a formally characterized region and an empirically traced trade-off frontier. Some works provide explicit region definitions or feasibility sets, while others obtain the set of achievable throughput–AoI pairs implicitly by sweeping protocol parameters or policy weights. This suggests that “capacity region” is used both as a rigorous mathematical object and as a practical design abstraction for operating-point selection.

2. Second-order characterization: mean, temporal variance, and AoI

A central development in the area is the use of second-order statistics—mean and temporal variance—to characterize freshness-sensitive performance. In the unreliable wireless formulation, for each device $p_i$9,

$M$0

and

$M$1

The AoI is then approximated by

$M$2

Here, $M$3 is the time-average successful delivery rate, while $M$4 captures temporal delivery fluctuations (Wang et al., 17 Aug 2025).

This mean–variance viewpoint is consistent with the broader second-order wireless optimization program, where the feasible object is a second-order capacity region over $M$5 rather than a classical first-order throughput region. In that framework, feasible tuples satisfy subset mean constraints such as

$M$6

together with a variance constraint of the form

$M$7

which sharply characterizes the second-order capacity region of wireless access networks (Guo et al., 2022, Guo et al., 2024). In those papers, AoI is modeled as a function of both mean and temporal variance, and temporal variance is essential for capturing Markovian fading wireless channels and emerging network performance metrics such as AoI and timely-throughput.

The unreliable-wireless throughput-AoI region specializes this second-order perspective by introducing a system-wide projected martingale process,

$M$8

whose mean remains zero for all policies, and whose system-wide variance is

$M$9

This process links per-device statistics to network-wide constraints and is central to the derivation of outer and inner bounds (Wang et al., 17 Aug 2025). This suggests that the throughput-AoI capacity region can be viewed as an AoI-facing projection of a second-order mean/variance feasibility region.

3. Outer bounds, inner bounds, and optimization inside the region

The region in the unreliable wireless model is characterized through necessary and sufficient conditions on $\{(m_i,h_i)\}_{i=1}^N$0. An outer bound requires, for every device $\{(m_i,h_i)\}_{i=1}^N$1, that $\{(m_i,h_i)\}_{i=1}^N$2 and

$\{(m_i,h_i)\}_{i=1}^N$3

along with the packing and feasibility conditions

$\{(m_i,h_i)\}_{i=1}^N$4

The remaining condition couples individual temporal variances to the system-wide variance through the Cauchy-Schwarz inequality, and the equality on the sum is tight for policies that make instantaneous allocations optimally (Wang et al., 17 Aug 2025).

The inner bound retains the same structural conditions but replaces the non-strict admission fraction constraints with

$\{(m_i,h_i)\}_{i=1}^N$5

and tightens the variance relation to equality (Wang et al., 17 Aug 2025). The paper states that every interior point of this tight inner bound is achievable.

The region is typically visualized as a surface or volume in the multi-dimensional space of throughput/AoI pairs for all users (Wang et al., 17 Aug 2025). This geometric viewpoint supports several optimization problems that are mapped to finding suitable points within the region. Examples given explicitly are: $\{(m_i,h_i)\}_{i=1}^N$6

$\{(m_i,h_i)\}_{i=1}^N$7

$\{(m_i,h_i)\}_{i=1}^N$8

and the admission-control feasibility question of whether $\{(m_i,h_i)\}_{i=1}^N$9 is achievable (Wang et al., 17 Aug 2025).

The significance of these formulations is that the region is not merely descriptive. It functions as a design domain for resource allocation, fairness, and feasibility analysis. In this sense, the capacity region is both a converse object, via outer bounds, and an optimization substrate, via inner-bound achievability.

4. Achievability and scheduling policies

The principal achievability result in the unreliable wireless setting is the Variance-Weighted Deficit (VWD) policy. For each device $m_i$0 at time $m_i$1,

$m_i$2

and at each slot the policy schedules the $m_i$3 devices with largest $m_i$4 (Wang et al., 17 Aug 2025). The policy is analytically shown to achieve every interior point of the inner bound.

Two analytical properties are emphasized. First, VWD drives the time-averaged throughput and error fluctuations to the exact targets $m_i$5 whenever the target lies in the inner bound. Second, the vector-of-deficits process is shown, by Lyapunov/Foster's theorem, to be positive recurrent, ensuring stabilizability and steady-state performance (Wang et al., 17 Aug 2025). The policy is also described as simple and low complexity.

VWD has clear antecedents in the second-order wireless optimization literature. There, the scheduler maintains a deficit

$m_i$6

normalizes it by $m_i$7, and schedules the client with the largest normalized deficit among currently ON clients; that policy achieves every point in the strict interior of the second-order capacity region (Guo et al., 2022, Guo et al., 2024). The unreliable-wireless throughput-AoI capacity region therefore inherits a policy architecture that is explicitly variance-aware rather than throughput-only.

Other models adopt different structural policies. In single-user fading channels, age-independent stationary randomized policies allocate powers based only on channel state and/or distribution information, without any knowledge of the AoI, and achieve at least half the respective optimal long-term average throughput under average AoI and power constraints, both with perfect and no CSIT (Bhat et al., 2019). In a mixed-traffic two-user wireless system, the AoI/timely-throughput problem is cast as a CMDP, relaxed via Lyapunov optimization, and solved per frame by backward dynamic programming, with a low-complexity algorithm that guarantees that the timely-throughput constraint is satisfied (Fountoulakis et al., 2021). These policies do not define the same region as VWD, but they illustrate the broader methodological point that achievability proofs are inseparable from scheduler structure.

5. Related formulations across network models

The same underlying trade-off appears under substantially different mathematical models. Some papers derive an explicit region, while others characterize an implicit achievable set through closed-form metrics and numerical sweeps.

Setting	Region or frontier	Characterization
Unreliable multi-device wireless (Wang et al., 17 Aug 2025)	Throughput-AoI capacity region	Tuples $m_i$8 with outer/inner bounds from $m_i$9
Cellular IoT with PPP and JM cells (Mankar et al., 2020)	Implicit achievable $h_i$0 set	$h_i$1
Single-user fading (Bhat et al., 2019)	Throughput-AoI-power region	Achievable triples $h_i$2
Two-user AoI/timely-throughput scheduling (Fountoulakis et al., 2021)	Feasible pair region	$h_i$3
Frameless ALOHA (Munari et al., 2023)	Fundamental frontier	Achievable $h_i$4 pairs as $h_i$5 and $h_i$6 vary
Wired mixed-flow networks (Tseng et al., 2021)	Throughput–freshness frontier	Tunable frontier via FATE and IFC

In the stochastic-geometry cellular IoT model, network throughput over D2D links is

$h_i$7

while the temporal mean AoI of a status-update link is

$h_i$8

and the spatial mean AoI is

$h_i$9

The achievable region is given implicitly by varying system design parameters such as device and base-station densities, Johnson-Mehl cell radius, access probabilities, and power control fraction (Mankar et al., 2020).

In single-user fading channels, the relevant object is a throughput–AoI–power region. The optimization maximizes long-term average throughput subject to average AoI and power constraints, and an upper bound is obtained by reformulating the AoI constraint as a success-frequency requirement of at least $\{(m_i,h_i)\}_{i=1}^N$0 (Bhat et al., 2019). In frameless ALOHA, the achievable $\{(m_i,h_i)\}_{i=1}^N$1 curve is generated by varying transmission probability $\{(m_i,h_i)\}_{i=1}^N$2 and maximum contention-period length $\{(m_i,h_i)\}_{i=1}^N$3, and the paper states that configurations maximizing throughput may result in a degradation of the AoI performance (Munari et al., 2023). In wired networks shared by LDA and AoI flows, the LDA-AoI coscheduling problem uses update frequency as a tractable freshness proxy and produces a tunable throughput–AoI frontier through FATE and IFC (Tseng et al., 2021).

A common misconception is that all such results describe the same mathematical region. They do not. Some are per-user vector regions, some are aggregate two-dimensional frontiers, and some include additional resources such as power or deadlines. The unifying idea is feasibility under simultaneous freshness and delivery-rate objectives.

6. Empirical behavior, learning-based approximations, and interpretation

Simulation evidence in the unreliable wireless formulation serves three roles. First, it validates the second-order approximation. The error between theoretical and empirical results is minimal except at boundaries, and the discrepancy reduces as $\{(m_i,h_i)\}_{i=1}^N$4 grows, so in large, dense systems the continuous approximation is nearly exact (Wang et al., 17 Aug 2025). Second, it tests bound tightness. Plots of achievable and infeasible points confirm the predictive utility of the region. Third, it evaluates the scheduler. VWD nearly matches the theoretical optimum and outperforms Max-Weight and Random policies in both AoI and throughput metrics, especially under hard resource constraints or non-trivial trade-off tuning (Wang et al., 17 Aug 2025).

Learning-based work often traces an achievable throughput–AoI region without deriving a closed-form capacity characterization. In 5G uplink scheduling, the objective

$\{(m_i,h_i)\}_{i=1}^N$5

is optimized by a PPO-based scheduler, and the trade-off curve is controlled by $\{(m_i,h_i)\}_{i=1}^N$6: larger $\{(m_i,h_i)\}_{i=1}^N$7 prioritizes lower AoI, while smaller $\{(m_i,h_i)\}_{i=1}^N$8 emphasizes throughput (Wu et al., 2021). The paper states that the results implicitly outline the scalable and adaptive achievable throughput–AoI region. In NOMA-assisted semi-grant-free systems, hierarchical learning combines beamforming and transmission scheduling; the plots are described as tracing the boundary of the achievable throughput–AoI region, and the reported numerical results include an approximately $\{(m_i,h_i)\}_{i=1}^N$9 throughput gain while maintaining the average AoI of GFUs within $$\liminf_{T \to \infty} \frac{\sum_{t=1}^T Z_i(t)}{T} \geq m_i, \qquad \limsup_{T \to \infty} \frac{\sum_{t=1}^{T} a_i(t)}{T} \leq h_i,$$0 time slots (Liu et al., 15 Aug 2025).

Across models, a recurrent conclusion is that throughput maximization and AoI minimization are generally not simultaneous objectives. In frameless ALOHA, parameter settings that maximize throughput may degrade AoI (Munari et al., 2023). In stochastic-geometry cellular IoT, increasing the power control fraction $$\liminf_{T \to \infty} \frac{\sum_{t=1}^T Z_i(t)}{T} \geq m_i, \qquad \limsup_{T \to \infty} \frac{\sum_{t=1}^{T} a_i(t)}{T} \leq h_i,$$1 decreases AoI but reduces throughput, while increasing the Johnson-Mehl cell radius can improve AoI coverage but eventually increase congestion and interference (Mankar et al., 2020). In wired networks, it is possible to trade a little throughput, specifically $$\liminf_{T \to \infty} \frac{\sum_{t=1}^T Z_i(t)}{T} \geq m_i, \qquad \limsup_{T \to \infty} \frac{\sum_{t=1}^{T} a_i(t)}{T} \leq h_i,$$2 lower, for much shorter AoI, specifically $$\liminf_{T \to \infty} \frac{\sum_{t=1}^T Z_i(t)}{T} \geq m_i, \qquad \limsup_{T \to \infty} \frac{\sum_{t=1}^{T} a_i(t)}{T} \leq h_i,$$3 to $$\liminf_{T \to \infty} \frac{\sum_{t=1}^T Z_i(t)}{T} \geq m_i, \qquad \limsup_{T \to \infty} \frac{\sum_{t=1}^{T} a_i(t)}{T} \leq h_i,$$4 shorter, compared to state-of-the-art traffic engineering (Tseng et al., 2021). The throughput-AoI capacity-region program gives this qualitative observation a precise feasibility language and, in the strongest formulations, a constructive scheduler that reaches every interior operating point (Wang et al., 17 Aug 2025).

The broader significance of the topic lies in this shift from single-metric optimization to structured joint feasibility. Throughput-AoI capacity regions, second-order capacity regions, feasible pair sets, and empirical frontiers all express the same systems question: which combinations of information freshness and delivery performance can actually be sustained under the stochastic, resource-limited dynamics of modern networks.