©This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible..
1
SAOITHE: Sustainable Age-of-Information-Based Timely Status Updating for Hardware-constrained Edge networks
arXiv:2605.21328v1 [cs.NI] 20 May 2026
Shih-Kai Chou, Maice Costa, Mihael Mohorčič, and Jernej Hribar
Abstract—In future large-scale deployments of 6G and beyond networks, collecting timely information, as measured by the Age of Information (AoI) metric, is becoming increasingly important. At the same time, the environmental impact, often characterized by the resulting Carbon Footprint (CF), depends on both the amount of consumed energy and the Carbon Intensity (CI), i.e., the amount of CO2 -equivalent emissions produced per unit of consumed energy. Since CI varies over time, minimizing energy is not equivalent to minimizing CF, as a status update with the same energy demand may result in a different carbon cost depending on when it is transmitted. This makes timely status updating a nontrivial scheduling problem. To address this challenge, we formulate carbon-aware status updating as a constrained Markov Decision Process (MDP) that minimizes AoI subject to CF budget, transmission duty-cycle, and channel-capacity constraints. We then propose Sustainable Age-of-InformationBased Timely Status Updating for Hardware-constrained Edge networks (SAOITHE), a Whittle-index-based scheduling solution that enables scalable real-time scheduling. Using real-world CI traces across low-, medium-, and high-CI regions, the results show that SAOITHE remains within the allocated CF budget while achieving lower AoI than baseline policies. Moreover, the gains are around 25% and 20% in low- and medium-CI regions, respectively, and up to 75% in high-CI settings, while preserving scalability. Index Terms—Age of Information, Sustainability, Carbon Footprint
I. I NTRODUCTION With the rise of data-driven and latency-sensitive applications, such as Connected Autonomous Vehicles (CAV), Industrial Internet of Things (IIoT), remote healthcare, and Integrated Sensing and Communication (ISAC), transmitting fresh and up-to-date information is critical [1]. For such systems, the Age of Information (AoI) metric [2] has been adopted not only to measure the timeliness of collected information, but also to support the design of scheduling policies for real-time systems [3]. Scheduling data collection can have a crucial impact on how sustainable, i.e., environmentally friendly, future networks can become, as the Carbon Footprint This work was performed while S. -K. Chou was at Jožef Stefan Institute. S. -K. Chou is currently with Ericsson AB, Kista, Sweden. The views expressed in this paper are those of the author and do not necessarily reflect the views of Ericsson AB. M. Costa is with Nexcepta Inc., Gaithersburg MD 20878, USA, M. Mohorčič and J. Hribar are with Jožef Stefan Institute, Ljubljana, Slovenia The corresponding author is Jernej Hribar ([email protected]). This work was supported by the Slovenian Research Agency under grants P2-0016, MN-0009, and J2-50071.
CI(gCO2eq/kWh) 400 300
Region with low CI Region with medium CI Region with high CI
200 100
t(h) 2
4
6
8
10
12
14
16
18
20
22
Fig. 1: Daily variation in CI for regions with low, medium, and high CI over a day.
(CF) depends not only on the amount of energy required to collect information, but also on how the consumed energy was generated [4], [5]. The greater the share of renewable energy sources used for electricity generation, the lower the CF of the consumed energy. Therefore, when considering the sustainability of 6G and beyond networks, where international standardization bodies have explicitly identified environmental sustainability as a core Key Performance Indicator (KPI) [6], scheduling, i.e., deciding when information is transmitted, captured, and processed, becomes crucial. Typically, the CF of an Information and Communications Technology (ICT) system is estimated by associating the energy consumed for data transmission with the corresponding Carbon Intensity (CI) per unit of energy [7]. CI is defined as the amount of CO2 -equivalent (CO2 eq) emissions produced per unit of consumed energy and varies over time and geographical region. For example, Fig. 1 shows the variation of CI over a day for three different CI regions using CI traces [8]. Consequently, the same amount of energy required to transmit information can result in a relatively low CF if data transmission and processing occur during periods of low CI, whereas the CF can be three to four times higher when the CI is high. Moreover, CI cannot be directly controlled because it depends on the time-varying electricity generation mix and is influenced by factors such as demand, pricing, dispatch rules, and imports. Therefore, reducing the CF of data collection requires deciding when information is generated and processed to minimize its environmental impact. For many systems, especially large-scale deployments using Low-Power Wide-Area Network (LPWAN) networks such as Long Range Wide Area Network (LoRaWAN), the AoI has
2
been shown to be a suitable metric for characterizing the freshness of information [9]. In such systems, maintaining fresh information is fundamentally a scheduling problem, since status updates cannot be transmitted arbitrarily often. This is due to additional constraints, such as duty-cycle regulations and restricted channel access at the gateway. As a result, the scheduler must decide carefully when information sources should transmit. Moreover, as the number of information sources increases, the complexity of the scheduling problem also grows, which makes it increasingly difficult to find a suitable solution. Typically, the scheduling problem is formulated as a Markov Decision Process (MDP), but the state space grows rapidly with the number of sources. To overcome this, we employ the Whittle Index, which was originally proposed to reduce the complexity of restless bandit problems [10]. This approach is widely adopted in AoI literature to decompose intractable multi-source problems into manageable subproblems and prioritize transmissions based on their urgency [11]–[13]. In short, this paper has the following contributions: • We formulate the carbon-aware status-updating problem as a constrained MDP that minimizes AoI subject to CF budget, LPWAN transmission duty-cycle, and channelcapacity constraints. • We propose Sustainable Age-of-Information-Based Timely Status Updating for Hardware-constrained Edge networks (SAOITHE) framework, a Whittle-index-based scheduling solution, to solve the formulated optimization problem in a scalable manner, and further provide a complexity analysis showing that the computational complexity of the proposed solution is logarithmic in the number of nodes, enabling real-time scheduling. • We evaluate the proposed framework using real-world CI traces across low-, medium-, and high-CI regions. The results show that SAOITHE remains within the allocated CF budget while achieving a lower average AoI compared with baseline solutions. Furthermore, we demonstrate that SAOITHE performs near the system optimum. Specifically, the AoI reduction over the Round Robin baseline under the same CF budget is approximately 25% and 20% in low- and medium-CI regions, respectively, and increases to up to 75% in high-CI settings, where blind schedulers incur an inadequately higher carbon cost. Next, we discuss the related work. II. R ELATED W ORK This work is related to studies that extend the concept of AoI by incorporating additional network conditions and application requirements [14]–[17]. Metrics such as Effective AoI [14] extend the baseline definition by incorporating semantic and correctness aspects of received data, while others adapt AoI to various queuing models, scheduling policies, and update mechanisms. Metrics like Age-Quality Information (AQI) [15] and Value of Information (VoI) [16] jointly optimize timeliness, data quality, and energy efficiency, often employing a utilitybased function that also encompasses AoI. Similarly, the age of incorrect information (AoII) metric [17] accounts for the
correctness of updates and supports the design of optimal transmission policies under unreliable channel conditions and power constraints. In contrast to the aforementioned AoIbased metrics, our work proposes a scheduler that minimizes AoI while accounting for sustainability through CF, thereby considering the environmental aspects that have been largely overlooked in prior studies. This work also builds on research that explored the role of AoI in enhancing energy efficiency [18]–[22]. For example, the authors in [18] applied Reinforcement Learning (RL) and system-level strategies to jointly optimize AoI and energy efficiency for computation offloading in IIoT, supported by queuing models. Freshness in caching-enabled Internet of Things (IoT) networks was addressed in [19], while the approach proposed in [20] leveraged Mobile Edge Computing (MEC) in a Wireless Sensor Network (WSN) for environmental monitoring to manage energy and computational constraints. In [21], the authors focused on energy-harvesting IoT systems with on-demand updates, while in [22] they explored Unmanned Aerial Vehicle (UAV)-assisted data collection, revealing tradeoffs between data freshness and energy cost. However, none of these works investigated the environmental impact and its connection to AoI, as considered in this paper. Moreover, while minimizing energy consumption is a necessary step toward reducing the CF of a system, it is not sufficient on its own. To overcome this gap, in this work, we propose a solution that minimizes the environmental impact of the system by also taking into account the temporal variation of CI. On the other hand, the Whittle index [10] has been widely adopted to design transmission scheduling solutions that minimize AoI in the system [11]–[13], [23]–[25]. For example, the authors in [11] applied Whittle’s framework for restless bandits to design a transmission scheduler that minimizes AoI. In [12], the authors demonstrated that Whittle index scheduling policies result in near-optimal performance for minimizing AoI, while the authors in [23] showed that the Whittle policy has analytically provable optimality in the many-user regime. Similarly, in [13], the authors provided a proof of optimality for a low-complexity Whittle-index scheduling policy for multi-source status updating systems. The low-complexity Whittle index was also employed in [24], where the authors proposed Channel-Aware Age of Information (CA-AoI), which considers the channel conditions at the source in a multi-source system. In addition, the authors in [25] employ the Whittle index method to design an energy-aware scheduling policy for sources that minimize AoI in the system. However, none of these papers consider the CI and the resulting CF in connection with AoI, as proposed in our work. This work also builds on our prior work [26], in which we analyzed a system with one information source. Our results showed that CF grows with the number of status updates from the source, i.e., the update rate, whereas the gain in average AoI is generally nonlinear and may exhibit either diminishing returns or even degradation due to queuing and other delays in the system. Therefore, scheduling updates from sources so as to minimize the average AoI while satisfying a constraint on the total available CF budget becomes, in general, a more complex problem than the energy-minimization problem, since
3
Fig. 2: Considered system model with N sources transmitting status updates through a gateway to a server.
the CI also varies over time. In this paper, we build on our prior findings and propose a scalable scheduling solution, SAOITHE, capable of minimizing AoI while ensuring that the set CF requirements are met in an environment with a dynamic CI. III. S YSTEM M ODEL Let us consider a system of N sensors, i.e., information sources, denoted as Zn , where n ∈ {1, 2, . . . , N }, that transmit status updates through a gateway to a server, as illustrated in Fig. 2. Such a system, in which information is generated by hardware-constrained devices at the edge and transmissions are often scheduled, is typical of LPWAN based on technologies such as LoRaWAN, Sigfox, or Mioty. The system operates over a finite time horizon of length T , which is discretized into T decision intervals (slots) indexed by t ∈ {1, . . . , T }. Each generated status update contains a measurement along with a timestamp indicating when it was generated. Therefore, we define the AoI for the n-th source as follows: ∆n (t) = t − zn (t),
(1)
where zn (t) is the generation time slot of the most recently received status update from source n available at the server at time slot t. To model the sequential decision-making problem of scheduling status updates over time, we formulate the considered system as a MDP. State space: We define the joint state space for the considered system model as follows: St = (∆(t), bDEV (t), τ DEV (t), hGW (t), bGW (t), τ GW (t), hSV (t)) ,
(2)
where ∆(t) = [∆1 (t), . . . , ∆N (t)] is a vector of AoI values for all information sources in the system. Moreover, bDEV (t) and τ DEV (t) are vectors tracking the busy status and the remaining transmission time for each source, respectively. The terms bGW (t) and τ GW (t) represent the corresponding forwarding status and timer for the gateway. Furthermore, hGW (t) and hSV (t) denote the buffered status update ages in the gateway and server buffers, respectively. From the perspective of the n-th source, the binary variables bDEV,n (t) and bGW,n (t) indicate the busy status of the device and gateway, respectively. Specifically, when bDEV,n (t) = 1 or bGW,n (t) = 1, the corresponding entity is in active mode, whereas 0 indicates idle mode. The variables τDEV,n (t) and τGW,n (t) track the remaining time in the current transmission and forwarding cycle, respectively. Finally, hGW,n (t) and hSV,n (t), with values in {0, . . . , δmax }, denote the AoI of the packet currently residing in the gateway and server buffers, respectively. Actions: At each time-step, the system has to coordinate the activity of all network entities, i.e., sources, gateway, and server. As a result, we define the joint action space as At = ADEV × AGW × ASV , with each action a(t) ∈ At given by: a(t) = (aDEV (t), aGW (t), aSV (t)),
(3)
where aDEV (t) is a binary vector indicating which of the N sources initiate a transmission cycle, i.e., aDEV (t) = [aDEV,1 (t), . . . , aDEV,N (t)]T . Similarly, aGW (t) and aSV (t) control the forwarding and processing decisions for the gateway and server, respectively. From the perspective of an n-th source in the system, each action element is a binary decision variable, i.e., aDEV,n (t), aGW,n (t), and aSV,n (t) ∈ {0, 1}. This means that, when aDEV,n (t) = 1, the device initiates sensing and transmission, while aGW,n (t) = 1 and aSV,n (t) = 1 indicate that the gateway forwards and the server processes the status update, respectively. Furthermore, due to LPWAN constraints, an action is feasible only if the corresponding entity is idle, i.e., bn (t) = 0, and, for forwarding or processing, if a packet is available in the buffer, i.e., hn (t) > 0. At the system level, the number of simultaneous transmissions is limited by the available channel capacity, which we denote by M . Consequently, the feasible transmission actions at every time slot t are constrained by: N X
aDEV,n (t) ≤ M.
(4)
n=1
where aDEV,n (t) represents the binary transmission decision for the n-th source. Transition Function: The system dynamics are governed by the physical constraints of a typical LPWAN system, e.g., store-and-forward logic for status updates. In particular, while a generated status update waits in a buffer before being transmitted, its age continues to increase, which prevents the system from arbitrarily delaying transmissions to reduce CF without incurring an AoI penalty at the destination. Accordingly, the vector of destination-side AoI values, ∆(t), evolves such that the age of the n-th source is reset not to zero, but to the age
4
of the processed status update upon reception. Otherwise, the AoI increases linearly: ∆n (t + 1) = (∆n (t) + 1)(1 − δn (t)) + hSV,n (t)δn (t), (5) where δn (t) = aSV,n (t)·I(hSV,n (t) > 0) indicates that a valid packet from the n-th device has been processed and delivered during time slot t, and hSV,n (t) denotes the AoI of that packet at the time of reception. Reflecting the three-entity architecture of the considered system, the transition function of the n-th source can be decomposed into three distinct stages: 1. Source-to-Gateway (Sensing & Transmission): The source state of n-th source (bDEV,n , τDEV,n ) evolves based on the action aDEV,n (t), i.e., a transmission cycle (bDEV,n = 1) decrements the timer τDEV,n . Here, we first define Iarr,n (t) as the uplink completion indicator, which is expressed as: ( 1, if bDEV,n (t) = 1 ∧ τDEV,n (t) = 1, Iarr,n (t) = 0, otherwise.
freshness of the information. We adopt a quadratic staleness penalty to strongly penalize large peaks in AoI, ensuring the reliability of status updates. As a result, it can be expressed as:
C(St , At ) =
N X
∆2n (t).
(10)
n=1
Simultaneously, we track the environmental impact of the system using CF as a key metric [7]. For the n-th source, the CF depends on the time-varying CI, denoted by ξ(t), and on the energy consumed by that source over time. The CI reflects how carbon-intensive the energy source is at a given time, as discussed in [27]. We adopt the standard measure of CI as the amount of CO2 -equivalent emissions produced per kilowatt-hour of consumed energy.
(6)
2. Gateway-to-Server (Buffering & Forwarding): The gateway buffer state for the n-th source hGW,n (t) represents the AoI of the packet. If a new packet arrives (Iarr,n = 1), it enters the buffer with an initial AoI of 1 (representing the transmission slot). If the gateway holds the packet, its AoI increments. If the gateway forwards (aGW,n = 1), the buffer clears. Therefore, we can express the evolution of gateway buffer state (hGW,n (t + 1)) as follows: hGW,n (t + 1) = 1, if Iarr,n (t) = 1, 0, if aGW,n (t) = 1 ∧ hGW,n (t) > 0, h (t) + 1, if hGW,n (t) > 0, GW,n 0, otherwise. (7) The server buffer hSV (t) evolves similarly. If the gateway completes a forwarding cycle (τGW = 1), the packet arrives at the server, carrying its accumulated AoI from the gateway plus the forwarding time. Therefore, the evolution of hSV (t) can be derived as: hSV,n (t + 1) = hGW,n (t) + Tfwd , 0, hSV,n (t) + 1, 0,
if τGW,n (t) = 1, if aSV,n (t) = 1 ∧ hSV,n (t) > 0, if hSV,n (t) > 0, otherwise. (8) 3. Server Processing (AoI Update): Finally, the global AoI ∆(t) is updated based on the server’s processing actions. More specifically, for each information source n ∈ {1, . . . , N }, the corresponding AoI value ∆n (t) resets to the packet’s current AoI hSV,n (t) plus the processing slot, reflecting the true staleness of the information: ( hSV,n (t) + 1, if δn (t) = 1, ∆n (t + 1) = (9) min(∆n (t) + 1, δmax ), otherwise. Cost Function: To evaluate the system performance, we define the immediate cost function, C(St , At ), primarily based on the
IV. P ROBLEM F ORMULATION Having defined the considered system with a MDP, we now formulate the corresponding optimization problem together with the associated CF constraint. The main objective is to obtain an optimal scheduling policy, π ∗ , that minimizes the cumulative staleness penalty (in the form of AoI) over a finite operational horizon of T time slots, e.g., a day, while strictly satisfying the environmental and regulatory constraints, i.e., a set CF budget. First, we define the total cost over the considered time horizon T as follows:
J∆ (π) =
T X N X
∆2n (t).
(11)
t=1 n=1
Unlike a linear function (∆n (t)), the squared term ensures that the marginal penalty for delaying an update increases with the age itself, thereby forcing the scheduler to prioritize transmissions when information becomes critically stale. Similarly, the total cumulative CF (JC (π)) and the total transmission duty usage (JD (π)) over the horizon are given by:
JC (π) =
T X N X
ξ(t) · Etot,n (St , At ),
(12)
t=1 n=1
and
JD (π) =
T X N X
aDEV,n (t) · Cduty ,
(13)
t=1 n=1
respectively, where Cduty represents the normalized airtime cost of a single transmission relative to the decision time step size t. Finally, the constrained optimization problem is formulated as the minimization of AoI subject to constraints on the cumulative CF, transmission duty usage, and the channel capacity M , and is expressed as follows:
5
P1 :
This structure allows us to solve for π ∗ efficiently by employing a Dynamic Programming (DP) technique and adapt it for the proposed solution, described in the next section.
min J∆ (π) π
s.t.
JC (π) ≤ κ, JD (π) ≤ D, N X
(14)
V. T HE SAOITHE F RAMEWORK
where κ represents the total CF budget allocated for the operational horizon, and D denotes the maximum allowable transmission time, derived from regulatory duty cycle limits, e.g., 1% of the horizon T for LoRaWAN. The problem P1 is a constrained finite-horizon optimization problem. We employ the Lagrange multiplier method to relax the hard cumulative constraints into the objective function. We introduce non-negative Lagrange multipliers, λ and µ, corresponding to the CF budget constraint and the duty cycle constraint, respectively. Therefore, the Lagrangian function L(π, λ, µ) is defined as the sum of the cumulative staleness cost and the penalized constraint violations, which can be expressed as:
In the section, we describe the proposed SAOITHE framework, which is capable of optimizing the trade-off between information freshness, i.e., AoI, and carbon cost, i,e, CF. We employ the Whittle Index approach [10], which decomposes the intractable joint optimization into N independent singlesource subproblems by relaxing the hard per-slot constraint into a time-averaged constraint. While standard numerical Whittle Index methods reduce complexity from exponential O(∆N ) to linear O(N · ∆) by iteratively searching for the index, we derive a closed-form expression of the Whittle Index. This eliminates the iterative search, further reducing the complexity to O(N · log N ) and enabling real-time scheduling for a large-scale network deployment. For an n-th source, the relaxed cost function Ln (∆n , aDEV,n , t) at state ∆n with action aDEV,n ∈ {0, 1} that involves a subsidy ν for passivity, is defined as:
L(π, λ, µ) = J∆ (π)+λ (JC (π) − κ)+µ (JD (π) − D) . (15)
Ln (∆n , u, t) = ∆2n + aDEV,n Cn (t) − (1 − aDEV,n )ν, (19)
By substituting the cumulative definitions from Eqs. (11), (12), and (13) into Eq. (15), we can then rewrite the Lagrangian function as the sum of a modified immediate cost function over the horizon T :
where Cn (t) = λξ(t)Etot,n + µCduty is the corresponding immediate aggregated cost of the update.
aDEV,n (t) ≤ M,
n=1
L(π, λ, µ) =
T X
Cλ,µ (St , At ) − λκ − µD,
(16)
t=1
where the immediate Lagrangian cost, Cλ,µ (St , At ), serves as the per-slot objective. It is constructed by updating the base cost defined in Eq. (10) with the weighted resource penalties:
Cλ,µ (St , At ) =
N X
∆2n (t) | {z } n=1
Base Cost
+ λξ(t)Etot,n (St , At ) + µaDEV,n (t)Cduty . | {z } Resource Penalties
(17) This formulation reveals the physical intuition behind the multipliers. More specifically, λ acts as a dynamic price for CF, inflating the cost of energy-intensive actions during high-CF periods, while µ penalizes channel-access usage. The constant terms λκ and µD do not affect the optimal policy π ∗ for fixed multipliers and can be ignored during the minimization step, which leads to the following simplified unconstrained objective: min π
T X N X
∆2n (t) + λξ(t)Etot,n (St , At )
t=1 n=1
+µaDEV,n (t)Cduty ) .
(18)
Proposition 1 (Closed-Form Whittle Index). Under the relaxed immediate update model, the problem satisfies indexability, and the Whittle Index Wn (∆, t) for source n at state ∆ is given by: Wn (∆, t) =
4∆3 + 9∆2 + 5∆ − (λξ(t)Etot,n + µCduty ). 6 (20)
Proof: The derivation involves evaluating the difference in the long-term average cost between a threshold policy m and m + 1 for a quadratic aging penalty. The detailed proof is provided in Appendix A. The validity of the Whittle Index approach relies on the problem satisfying indexability (i.e., the set of passive states decreases monotonically as the subsidy ν increases). While the indexability of convex AoI cost functions has been structurally analyzed in prior literature [13], we further validated this property numerically for our specific parameters to ensure the robustness of the derived index. Corollary 1 (Asymptotic Stability). For large ∆, the Whittle Index behaves asymptotically as Wn (∆) ≈ 32 ∆3 . Since the transmission cost Cn (t) is bounded, there exists a critical age ∆crit such that for all ∆ > ∆crit , the index Wn (∆) > 0. This guarantees that the scheduling policy prevents infinite staleness. Proof: We evaluate the limit of the index normalized by the cubic term: Wn (∆) 4∆3 + 9∆2 + 5∆ − 6Cn (t) 2 = lim = . 3 ∆→∞ ∆→∞ ∆ 6∆3 3 (21) lim
6
Since the limit is a strictly positive constant, the urgency term Wn (∆) dominates the cost term Cn (t) for sufficiently large ∆, forcing source activation. The closed-form Whittle Index derived in Proposition 1 applies strictly to sources where transmission resets the AoI to 1 (∆ → 1). However, the gateway and server operate under buffered-update dynamics (Eqs. 7 and 8), where forwarding a packet resets the system AoI to the age of the buffered packet (hn = h{GW,n,SV,n} ). To extend the scalable framework to incorporate these entities, we first define the cubic staleness urgency function U (·) identified in Proposition 1 as: 4x3 + 9x2 + 5x . (22) 6 We then generalize the Whittle Index as the differential urgency between the current system age ∆ and the buffered packet age h, which is expressed as: U (x) =
Wbuf,n (∆n , hn ) ≈ [U (∆n ) − U (hn )] − Cn (t),
(23)
where Cn (t) represents the immediate resource cost (energy and duty cycle penalty) for the specific operation. Eq. 23 captures the physical intuition that if the buffered packet is already stale (hn ≈ ∆n ), the differential urgency approaches zero, thereby suppressing non-energy-efficient forwarding actions. We outline the proposed solution in Alg. 1. In short, our solution decouples decision-making into four sequential steps per time slot: 1) Dynamic Cost Assessment (Step 1): At the beginning of time slot t, the scheduler computes the global effective cost C(t). This scalar aggregates the variable carbon penalty ξ(t) and the fixed duty-cycle cost, weighted by the optimal multipliers learned from the previous stage. 2) Parallel Index Computation (Step 2): For every source n ∈ {1, . . . , N }, the algorithm determines the appropriate staleness urgency based on the current state of that source. If the source is idle, the urgency is calculated as U (∆n ) using the cubic polynomial derived in Proposition 1. If a packet for source n is currently residing in the gateway buffer, the scheduler instead evaluates the differential urgency [U (∆n )−U (hGW,n )] as defined in Eq. (23). The Whittle Index Wn (t) is then obtained by subtracting the global effective cost C(t) from the calculated urgency value. Since these calculations rely solely on the local state components (∆n , hGW,n ), this step maintains O(1) complexity per source and is fully parallelizable. 3) Prioritization (Step 3): The scheduler sorts the indices in descending order to identify the top-M subset A∗ . It strictly respects the mechanism capacity M while excluding any sources with negative indices to maintain energy efficiency. 4) Execution and Update (Step 4): Finally, the scheduler generates the activation vector aDEV . The system then updates the local states ∆n and channel variables based on these activation decisions and the current system dynamics. The state space defined in Eq. 2 grows exponentially with the number of sources in the system. As a result, obtaining the optimal scheduling policy through an exact MDP solution quickly becomes computationally intractable as N increases.
Algorithm 1 Proposed SAOITHE approach Require: States ∆(t), ξ(t), Multipliers λ∗ , µ∗ , Capacity M . Ensure: Activation Vector aDEV (t). 1: // Step 1: Compute Dynamic Cost 2: C(t) ← λ∗ ξ(t)Etot + µ∗ Cduty 3: // Step 2: Compute Indices (Parallel) 4: for all n ∈ {1, . . . , N } do 5: if bDEV,n (t) = 0 then 6: Wn (t) ← U (∆n ) − C(t) 7: else if hGW,n (t) > 0 then 8: Wn (t) ← [U (∆n ) − U (hGW,n (t))] − C(t) 9: else 10: Wn (t) ← 0 11: end if 12: end for 13: // Step 3: Prioritization 14: Sort W descending: W(1) ≥ W(2) ≥ . . . ≥ W(N ) 15: A∗ ← ∅ 16: for k = 1 to min(N, M ) do 17: if W(k) > 0 then add (k) to A∗ else break 18: end for 19: // Step 4: Execute & Update States 20: for all n ∈ {1, . . . , N } do 21: aDEV,n ← 1 if n ∈ A∗ else 0 22: Update ∆n and channel states based on aDEV,n 23: end for 24: return aDEV (t)
This motivates the need for scalable scheduling policies whose per-slot computational cost remains manageable even in larger systems. To demonstrate that the proposed SAOITHE solution is practically deployable, we therefore analyze its computational complexity and compare it with that of optimal and existing sub-optimal approaches. Proposition 2 (Computational Complexity). The per-slot computational complexity of the proposed SAOITHE algorithm is O(N ·log N ). This represents a significant reduction compared N to the exponential complexity, i.e., O(δmax ) , of the optimal DP solution and remains scalable with the number of sources. Proof: The complexity analysis is provided in Appendix B. In summary, although the underlying state space St grows exponentially with N , the SAOITHE solution avoids this bottleneck by relying on a policy whose per-slot complexity scales only as O(N · log N ). This makes the approach suitable for practical deployment in real systems with a large number of sources, as we show in the evaluation presented in the next section. VI. VALIDATION AND R ESULTS In this section, we evaluate the performance of the proposed SAOITHE scheduler against two baseline policies, namely a Round Robin policy and a Random policy. The Round Robin policy transmits status updates periodically with a fixed period p, where p is determined from the number of
7
SAOITHE
Round Robin
(a) Low CI Region CI (gCO2 eq/kWh)
(c) High CI Region CI (gCO2 eq/kWh)
CI (gCO2 eq/kWh) 400
140
50
340
120
280
40
100 t(h)
30 4
8
12
16
20
24
4
7 6 5 4 3 2 1
t(h) 8
12
16
8
12
16
20
20
24
4
24
(g) Low CI Region Cumulative CF
7 6 5 4 3 2 1
t(h) 8
12
16
20
24
Cumul. CF (µgCO2 eq)
BUDGET EXCEEDED
100
16
20
24
t(h) 8
12
16
20
24
(i) High CI Region Cumulative CF Cumul. CF (µgCO2 eq)
BUDGET EXCEEDED
300
12
∆n (h)
4
(h) Medium CI Region Cumulative CF
Cumul. CF (µgCO2 eq)
8
(f) High CI Region AoI
∆n (h)
4
t(h)
160
(e) Medium CI Region AoI
∆n (h)
4
220 t(h)
80
(d) Low CI Region AoI
150
Budget
(b) Medium CI Region
60
7 6 5 4 3 2 1
Random
BUDGET EXCEEDED
1, 000 750
200
500
50
100
250 t(h)
t(h) 4
8
12
16
20
24
4
8
12
16
20
24
t(h) 4
8
12
16
20
24
Fig. 3: Comparison of CI, resulting AoI, and cumulative CF for SAOITHE, Round Robin, and Random scheduling in low, medium, and high CI regions. N = 1 TABLE I: Values of experimental parameters. Parameter BW D Etot Ftask L N PGW,tx PIoT,rx PSV,proc T
Value 125 kHz 1% 0.9251 J 50 MFLOP 256 bits {10, 50, 100} 3W 0.010 W 150 W 288 slots (24 h)
Parameter Cduty ∆max Fsv κ M PGW,idle Pidle PIoT,tx SF τslot
Value 1296 J 10 slots 10 GFLOP/s {.5, 2, 5, 21.5}µgCO2 eq 8 1.5 W 101.5 W 0.125 W 12 5 min (300 s)
transmissions permitted by the carbon budget κ under the ¯ average grid CI, ξ(t). In contrast, the Random policy is configured to have the same average inter-transmission interval p, but in each time slot it performs an independent Bernoulli trial with transmission probability ptx = 1/p, which results in geometrically distributed inter-transmission intervals. To reflect a practical LPWAN setting, we adopt LoRaWAN Class A [28] to model all relevant timing parameters and energy
consumption. For example, standard LoRaWAN gateways can demodulate only a limited number of simultaneous transmissions; therefore, the number of gateway channels M is set to M = 8, which is typical for Class-A systems [29], [30]. To model the energy consumption and timing constraints of the system, we define several physical and hardwarespecific parameters as summarized in Table I. The physical layer is configured with a bandwidth (BW ) of 125 kHz and a spreading factor (SF ) of 12, transmitting packets with a length (L) of 256 bits. For the server-side processing, we assume a computational capacity (Fsv ) of 10 GFLOP/s, where each status update requires Ftask = 50 MFLOP for processing and verification. The energy model is derived from the power consumption levels of the network entities: the IoT devices consume PIoT,tx = 0.125 W during transmission and PIoT,rx = 0.01 W during the two mandatory Class-A downlink windows, each lasting 100 ms, which are counted as a marginal per-transmission cost. In contrast, the gateway receive power
8
∆(h) 5
SAOITHE κ = .5µCO2 eq Limit
SAOITHE Round Robin
4
Random κ = 5µCO2 eq
101
102
Random
∆(min)
3 2
500
1 0.5 Low CI Region 2 eq (ξ̄ = 52 gCO ) kW h
Round Robin κ = 2µCO2 eq
ξ̄ Medium CI Region 2 eq (ξ̄ = 111 gCO ) kW h
High CI Region 2 eq (ξ̄ = 374 gCO ) kW h
50
Fig. 4: Average AoI for low, medium, and high CI regions with N = 50, κ = 21.5 µ gCO2 eq.
5 is absorbed into the system idle power Pidle = 101.5 W, since the gateway continuously listens for all N sources regardless of scheduling decisions and is therefore treated as a fixed infrastructure overhead rather than a per-packet cost. Similarly, the gateway forwarding and server processing power (PGW,tx and PSV,proc ) contribute to Etot as active costs. Finally, the operational horizon T consists of 288 slots, each with a duration τslot = 5 minutes, ensuring a total evaluation period of 24 hours. In Fig. 3, we illustrate how the SAOITHE, Round Robin, and Random scheduling approaches evolve over time across different CI regions. Fig. 3(a)–(c) show the variation of CI over time, Fig. 3(d)–(f) show the corresponding evolution of AoI, and Fig. 3(g)–(i) show the cumulative CF budget consumption. From the CI traces (Fig. 3(a)–(c)), we observe that, within a given region, the highest CI values can be up to three times larger than the lowest ones, indicating high temporal variation. The AoI results (Fig. 3(d)–(f)) further show that both the Random and Round Robin policies may exceed the available budget, whereas the proposed SAOITHE remains within the allocated budget, as seen in Fig. 3(g)–(i). As a result, toward the end of the day, the AoI under both Round Robin and Random increases significantly, since the system no longer has sufficient budget available to transmit new updates, while SAOITHE still has budget remaining and can transmit new updates, thus keeping the AoI lower. Fig. 4 compares the average AoI achieved by the three scheduling policies across low-, medium-, and high-CI regions under the same budget κ = 21.5µgCO2 eq. In the low-CI region, all policies achieve relatively small AoI values, with SAOITHE attaining an AoI of less than 20 minutes, Round Robin around 25 minutes, and Random 45 minutes. In the medium-CI region, the gap widens, where SAOITHE achieves an AoI of 43 minutes, while Round Robin and Random reach 50 and 100 minutes, respectively. The largest difference appears in the high-CI region, where SAOITHE maintains an average AoI of one and a half hours, whereas Round Robin increases to 2 hours 45 minutes and Random to over five and a half hours. These results show that the performance gap between the proposed SAOITHE and the baseline policies becomes larger as the average CI increases. In particular, while
N Fig. 5: Average AoI over the number of sources (N ) in the system for three different CF budgets (.5, 2, and 5µgCO2 eq) in medium CI region.
SAOITHE N = 10
Round Robin N = 50
Random N = 100
∆(h) 10 8 6 4 2 0
10−3
10−2 κ(gCO2 eq)
Fig. 6: Average AoI as a function of the cumulative CF budget under different numbers of sources in medium CI region. all approaches experience higher AoI in more carbon-intensive regions, SAOITHE degrades much more gracefully, since it explicitly accounts for the carbon cost of transmissions when allocating the available budget. In contrast, Round Robin, and especially Random, fail to adapt their transmission decisions to the carbon conditions, which leads to inefficient budget usage and substantially higher AoI. Overall, the results confirm that the advantage of employing SAOITHE is less advantageous in low- and medium-CI regions, where the gains are around 25% and 20%, respectively, but becomes more advantageous in high-CI settings, where such gains can be almost doubled,
9
50
∆crit (min)
40 30 Wn = 0 boundary Transmit Region (Wn > 0) Wait/Buffer Region (Wn ≤ 0)
20 10
ξ(t) (gCO2 /kWh) 100
200
300
400
Fig. 7: Whittle decision boundary of SAOITHE as a function of carbon intensity ξ(t) ∈ [50, 450] gCO2 /kWh, with Etot = 0.9251 J. The boundary ∆crit (ξ) separates the transmit region (Wn > 0, in red) from the wait region (Wn ≤ 0, in blue). up to 75%, compared with Round Robin. To demonstrate the scalability of the proposed solution, Fig. 5 evaluates the proposed approach and two baselines under three different budgets, namely κ = 0.5 µgCO2 eq, κ = 2 µgCO2 eq, and κ = 5 µgCO2 eq, respectively. As the budget increases, all solutions approach the physical limit imposed by the employed LoRaWAN technology. However, regardless of the budget, the proposed SAOITHE achieves lower AoI in comparison. Furthermore, as the number of sources increases, the difference between the approaches becomes larger, as the AoI also increases. The result is most notable when the budget is high, as the proposed solution performs near the hardware limitation of the system, while the performance of Round Robin deteriorates with the increasing number of sources. Consequently, the SAOITHE solution is scalable, since the total number of transmissions per horizon is capped by the gateway demodulation capacity (M = 8), independently of N . Additionally, this reveals a fundamental scalability paradigm: once the gateway capacity is saturated, the network’s maximum CF reaches a physical plateau, meaning that a higher budget will not result in lower AoI. Fig. 6 illustrates the trade-off between information freshness and the available CF budget κ for systems with different numbers of information sources, i.e., N = 10, N = 50, and N = 100. As the number of sources increases, a larger budget is required to achieve the same AoI, which is expected due to the increased contention among sources. Across all considered system sizes, SAOITHE consistently achieves the lowest AoI for a given budget, thereby providing the most efficient freshness-carbon trade-off. In contrast, Round Robin performs worse, while Random exhibits the weakest performance and requires a substantially larger CF budget to achieve the same AoI levels. The results further show that the overall AoI increases with the number of sources, highlighting the greater scheduling complexity in larger systems, while preserving the relative advantage of SAOITHE. Moreover, as the available budget increases, the gap in AoI among the considered approaches decreases, as expected, since the system becomes less constrained under a larger budget. In Fig. 7, we illustrate the theoretical decision boundary of
the SAOITHE solution by tracing ∆crit (ξ) (see Corollary 1) across the CI range ξ ∈ [50, 450]. The boundary partitions the state space into a Transmit Region (Wn > 0, ∆ > ∆crit ), where information staleness outweighs the carbon cost of transmission, and a Wait/Buffer Region (Wn ≤ 0), where the scheduler defers transmissions to preserve the carbon budget. The cube-root non-linearity ∆crit ∝ ξ(t)1/3 confirms that the policy becomes increasingly conservative as the grid becomes more carbon-intensive. More specifically, the critical AoI ∆crit rises from approximately 24 minutes at ξ = 50 gCO2 /kWh to over 50 minutes at ξ = 450 gCO2 /kWh. The boundary is governed by λ∗ Etot , where Etot = 0.9251 J and λ∗ is the Lagrange multiplier that emerges at the convergence of the SAOITHE algorithm, representing the shadow price of the carbon budget κ. Notably, the boundary is defined per device. In other words, the Whittle decomposition reduces the N -device scheduling problem to N independent single-device subproblems. As a result, ∆crit (ξ) applies uniformly to all devices regardless of network size. Furthermore, the curve implicitly encodes a specific κ: a tighter carbon budget drives λ∗ higher, shifting ∆crit upward and forcing the scheduler to defer transmissions more aggressively, while a relaxed budget lowers λ∗ and compresses the curve toward smaller critical ages. Overall, our results show a fundamental trade-off between information freshness and the resulting CF of the system. While all policies achieve lower AoI as the available CF budget increases, the proposed SAOITHE consistently provides the best freshness-carbon trade-off by adapting transmission decisions to the instantaneous CI. In particular, it transmits earlier when the CI is lower and waits with updates when the CI is higher, thereby preserving the available budget more efficiently. The results further show that this advantage is maintained as the number of information sources in the system increases, confirming the scalability and practical viability of the proposed approach. VII. C ONCLUSION In this paper, we proposed the SAOITHE framework, which minimizes AoI subject to an allocated CF budget, transmission duty usage, and channel capacity constraints. We showed that the proposed solution based on Whittle-index-based approach is scalable and outperforms Round Robin baseline solution. Using real-world CI traces, our results show a fundamental trade-off between information freshness and the resulting CF of the system. Furthermore, we demonstrate that SAOITHE consistently provides the best AoI-CF trade-off while remaining scalable and practically viable. Furthermore, the results confirm that the advantage of employing SAOITHE is smaller in low- and medium-CI regions, where the gains are around 25% and 20%, respectively, but becomes more pronounced in high-CI settings, where such gains can be almost doubled, reaching up to 75% compared with the Round Robin baseline. In our future work, we will extend our work by considering systems in which information sources have different priorities. For example, weighted AoI formulations could be used to prioritize critical applications, which is important because not
10
all status updates have the same relevance for the underlying system or service. We will also consider heterogeneity in CI, as different information sources may rely on different energy sources. For example, when two sources transmit at the same time, their updates may result in different CFs due to differences in the energy source used for information generation, transmission, or processing. Therefore, the scheduling mechanism would need to account for these differences when making decisions. We also plan to consider the integration of Simultaneous Wireless Information and Power Transfer (SWIPT), which would further affect how the system accounts for the CF of a status update and would result in additional trade-offs. R EFERENCES [1] I. F. Akyildiz, A. Kak, and S. Nie, “6G and beyond: The future of wireless communications systems,” IEEE access, vol. 8, pp. 133 995– 134 030, 2020. [2] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often should one update?” in Proc. 2012 IEEE INFOCOM. IEEE, 2012, pp. 2731– 2735. [3] R. D. Yates, Y. Sun, D. R. Brown, S. K. Kaul, E. Modiano, and S. Ulukus, “Age of information: An introduction and survey,” IEEE J. Sel. Areas Commun., vol. 39, no. 5, pp. 1183–1210, May 2021. [4] C.-S. Yang, Y.-J. Liao, C.-H. Kuo, C.-H. Hwang, W.-D. Wu, P.-K. Liao, I.-K. Fu, G. Sébire, T. Frost, and N. Tenny, “Toward 6G sustainable mobile communications,” IEEE Wireless Communications, vol. 32, no. 1, pp. 44–50, 2025. [5] S.-K. Chou, J. Hribar, V. Hanžel, M. Mohorčič, and C. Fortuna, “The energy cost of artificial intelligence lifecycle in communication networks,” IEEE Journal on Selected Areas in Communications, vol. 44, pp. 2427–2443, 2026. [6] ITU-R, “Framework and overall objectives of the future development of IMT for 2030 and beyond,” International Telecommunication Union, Recommendation ITU-R M.2160-0, Nov 2023. [Online]. Available: https://www.itu.int/rec/R-REC-M.2160-0-202311-I/en [7] S. Ayers, S. Ballan, V. Gray, and R. McDonald, “Measuring the emissions and energy footprint of the ICT sector: Implications for climate action (English).” http://documents.worldbank.org/curated/en/ 099121223165540890. [8] Electricity Maps, “Electricity map,” https://app.electricitymaps.com/ zone/, accessed: 2025-10-29. [9] J. Hribar, A. Marinescu, A. Chiumento, and L. A. DaSilva, “Energy Aware Deep Reinforcement Learning Scheduling for Sensors Correlated in Time and Space,” IEEE Internet of Things Journal, vol. 9, no. 9, pp. 6732–6744, May 2022. [10] P. Whittle, “Restless bandits: Activity allocation in a changing world,” Journal of Applied Probability, vol. 25, pp. 287–298, 1988. [11] Y.-P. Hsu, “Age of Information: Whittle Index for Scheduling Stochastic Arrivals,” in 2018 IEEE International Symposium on Information Theory (ISIT). IEEE Press, 2018, p. 2634–2638. [Online]. Available: https://doi.org/10.1109/ISIT.2018.8437712 [12] I. Kadota, A. Sinha, E. Uysal-Biyikoglu, R. Singh, and E. Modiano, “Scheduling Policies for Minimizing Age of Information in Broadcast Wireless Networks,” IEEE/ACM Transactions on Networking, vol. 26, no. 6, pp. 2637–2650, 2018. [13] V. Tripathi and E. Modiano, “A Whittle Index Approach to Minimizing Functions of Age of Information,” IEEE/ACM Transactions on Networking, vol. 32, no. 6, pp. 5144–5158, 2024. [14] B. Yin, S. Zhang, Y. Cheng, L. X. Cai, Z. Jiang, S. Zhou, and Z. Niu, “Only those requested count: Proactive scheduling policies for minimizing effective age-of-information,” in Proc. 2019 IEEE INFOCOM. IEEE, 2019, pp. 109–117. [15] N. Rajaraman, R. Vaze, and G. Reddy, “Not just age but age and quality of information,” IEEE J. Sel. Areas Commun., vol. 39, no. 5, pp. 1325– 1338, 2021. [16] P. Zou, O. Ozel, and S. Subramaniam, “On age and value of information in status update systems,” in Proc. 2020 IEEE WCNC. IEEE, 2020, pp. 1–6. [17] A. Maatouk, S. Kriouile, M. Assaad, and A. Ephremides, “The age of incorrect information: A new performance metric for status updates,” IEEE/ACM Trans. Netw., vol. 28, no. 5, pp. 2215–2228, Oct. 2020.
[18] J. Huang, H. Gao, S. Wan, and Y. Chen, “AoI-aware energy control and computation offloading for industrial IoT,” Future Generation Computer Systems, vol. 139, pp. 29–37, 2023. [19] C. Xu, X. Wang, H. H. Yang, H. Sun, and T. Q. S. Quek, “AoI and energy consumption oriented dynamic status updating in caching enabled IoT networks,” in Proc. 2020 IEEE INFOCOM WKSHPS, 2020, pp. 710– 715. [20] G. Zhang, C. Shen, Q. Shi, B. Ai, and Z. Zhong, “AoI minimization for WSN data collection with periodic updating scheme,” IEEE Trans. Wireless Commun., vol. 22, no. 1, pp. 32–46, 2023. [21] M. Hatami, M. Leinonen, and M. Codreanu, “AoI minimization in status update control with energy harvesting sensors,” IEEE Trans. Commun., vol. 69, no. 12, pp. 8335–8351, 2021. [22] X. Zhang, Z. Chang, T. Hämäläinen, and G. Min, “AoI-energy tradeoff for data collection in UAV-assisted wireless networks,” IEEE Trans. Commun., vol. 72, no. 3, pp. 1849–1861, 2024. [23] A. Maatouk, S. Kriouile, M. Assad, and A. Ephremides, “On the Optimality of the Whittle’s Index Policy for Minimizing the Age of Information,” IEEE Transactions on Wireless Communications, vol. 20, no. 2, pp. 1263–1277, 2021. [24] B. Dedhia and S. Moharir, “You Snooze, You Lose: Minimizing Channel-Aware Age of Information,” in 2020 18th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks (WiOPT), 2020, pp. 1–8. [25] P. Lassila and S. Aalto, “Near-optimal uplink scheduling for age-energy tradeoff in wireless systems,” in 2021 33th International Teletraffic Congress (ITC-33), 2021, pp. 1–9. [26] S.-K. Chou, M. Costa, M. Mohorčič, and J. Hribar, “Towards a Sustainable Age of Information Metric: Carbon Footprint of Real-Time Status Updates,” arXiv preprint arXiv:2602.11946, 2026. [27] D. Trihinas, L. Thamsen, J. Beilharz, and M. Symeonides, “Towards energy consumption and carbon footprint testing for AI-driven IoT services,” in Proc. 2022 IEEE IC2E. IEEE, 2022, pp. 29–35. [28] LoRa Alliance, “LoRaWAN 1.0.4 Specification,” LoRa Alliance, Technical Specification, 2020. [Online]. Available: https://lora-alliance. org/resource-hub/lorawan-104-specification-package [29] F. Adelantado, X. Vilajosana, P. Tuset-Peiro, B. Martinez, J. MeliaSegui, and T. Watteyne, “Understanding the limits of LoRaWAN,” IEEE Communications magazine, vol. 55, no. 9, pp. 34–40, 2017. [30] R. B. Sorensen, N. Razmi, J. J. Nielsen, and P. Popovski, “Analysis of LoRaWAN Uplink with Multiple Demodulating Paths and Capture Effect,” in ICC 2019 - 2019 IEEE International Conference on Communications (ICC), 2019, pp. 1–6. [31] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction to Algorithms, 2nd ed. The MIT Press, 2001.
A PPENDICES A PPENDIX A D ERIVATION OF THE C LOSED -F ORM W HITTLE I NDEX We consider the decoupled single-source problem with the relaxed Lagrangian cost function Ln (∆n , a, t) = ∆2n + aCn (t) − (1 − a)ν. Here, ν acts as a subsidy for passivity, making the effective transmission cost Cn (t) + ν. Since the AoI penalty (∆2 ) is convex, the optimal strategy is a threshold policy: transmit if the current age ∆ ≥ H. Under a threshold H, the system evolves in renewal cycles of length H. The cumulative staleness cost over one cycle is J∆ (H) = PH H(H+1)(2H+1) 2 . The long-term average cost Φ(H) h=1 h = 6 is obtained by amortizing the cycle cost (cumulative staleness plus one effective transmission) over the cycle length: 1 H(H + 1)(2H + 1) Φ(H) = + Cn (t) + ν . (24) H 6 The Whittle Index Wn (∆) is defined as the critical subsidy ν that makes the system indifferent between transmitting at age ∆ (policy H = ∆) and waiting one more slot (policy H = ∆ + 1). Setting Φ(∆) = Φ(∆ + 1) implies: J∆ (∆) + Cn (t) + ν J∆ (∆ + 1) + Cn (t) + ν = . ∆ ∆+1
(25)
11
Solving for ν isolates the index: ν = ∆J∆ (∆ + 1) − (∆ + 1)J∆ (∆) − Cn (t).
(26)
Substituting the cubic form of J∆ (·) yields the closed-form Whittle Index: 4∆3 + 9∆2 + 5∆ − Cn (t), 6 where Cn (t) = λξ(t)Etot,n + µCduty . Wn (∆, t) =
(27)
A PPENDIX B C OMPLEXITY A NALYSIS OF SAOITHE-W HITTLE In this appendix, we analyze the asymptotic computational complexity of the proposed scheduling algorithm. Let N = {1, . . . , N } denote the set of devices managed by the gateway. The scheduling decision at each time slot t consists of two sequential phases: Index Computation and Prioritization. The gateway computes the Whittle Index Wn (∆, t) for all n ∈ N using the closed-form expression derived in Eq. (20). Let Top denote the computational cost of a basic floating-point operation (addition, multiplication). The calculation of a cubic polynomial Wn ≈ c3 ∆3 + c2 ∆2 + c1 ∆ + c0 requires a fixed constant number of operations, k, regardless of the network size N . Thus, the total cost for index computation, Tcalc , scales linearly with N : Tcalc (N ) =
N X
k · Top = O(N ).
(28)
n=1
Next, the scheduler must select the subset of devices A∗ ⊂ N with cardinality |A∗ |≤ M that maximizes the sum of indices. This is equivalent to sorting the set {Wn }N n=1 in descending order. According to the information-theoretic lower bound for comparison-based sorting algorithms [31], the worst-case time complexity Tsort is: Tsort (N ) = O(N · log N ).
(29)
As a result, the total time complexity per control slot is the sum of the two phases: Ttotal (N ) = Tcalc (N )+Tsort (N ) = O(N )+O(N ·log N ). (30) Since N log N ≫ N when N → ∞, the asymptotic complexity is: Ttotal (N ) ≈ O(N · log N ).
(31)
This confirms that the algorithm scales pseudo-linearly, making it computationally feasible for resource-constrained gateways even as N → ∞.