1
Update the Unseen Only: Minimizing AoI for Collaborative Perception through Online Learning
arXiv:2607.20967v1 [cs.NI] 23 Jul 2026
Yanan Ma, Zhuoyi Zhao, Zhengru Fang, Haonan An, Xianhao Chen, Member, IEEE, and Yuguang Fang, Fellow, IEEE
Abstract—While collaborative perception (CP) enhances the safety of autonomous driving, limited bandwidth can cause severe shared data staleness in CP systems. Existing age-of-information (AoI) minimization policies are not well suited for CP, as they overlook the fact that a vehicle’s AoI decreases not only through updates from the source (i.e., a base station) but also through the vehicle’s local sensing. To address this issue, we propose a mobility-aware AoI minimization framework for CP that explicitly accounts for vehicles’ dynamic sensing ranges. We first derive a closed-form expression for the long-term time average sum AoI within a considered region, accommodating an ever-changing vehicle population and their dynamic sensed areas. Based on this characterization, we develop Local-sensing-aware Max-Weight Scheduling (LocMW), an online learning algorithm designed for sensor information broadcast from a source to vehicles under unknown environmental statistics and delayed observations. We provide performance guarantees demonstrating that LocMW achieves a sublinear cumulative excess AoI compared to the optimal stationary randomized benchmark. Extensive simulations using vehicular trajectory datasets and 3D perception tasks demonstrate that our LocMW policy substantially outperforms competing baselines, reducing the time-averaged sum AoI by up to 31.6% and improving mAP detection accuracy by up to 16.3%. Index Terms—Collaborative perception, age of information (AoI), restless multi-armed bandit (RMAB), vehicular networks, online learning
I. I NTRODUCTION
T
HE realization of Level-5 autonomous driving requires vehicles to perceive dynamic traffic environments with high precision. However, a single autonomous vehicle is vulnerable to various occlusions in dense urban scenarios, as it is fundamentally limited by the line-of-sight nature of onboard geometric sensors, such as cameras and LiDAR. To overcome this limitation, collaborative perception (CP) has emerged as a promising solution [1]–[3]. For instance, in an infrastructureassisted CP system, a base station (BS) or roadside unit equipped with camera/LiDAR sensors can broadcast sensor information to surrounding vehicles to complement their local The work was supported in part by the JC STEM Lab of Smart City funded by The Hong Kong Jockey Club Charities Trust under Contract 2023-0108, in part by the Research Grants Council of the Hong Kong SAR, China (Project No. CityU 11216324), and in part by the Hong Kong SAR Government under the Global STEM Professorship. The work of X. Chen was supported in part by the Research Grants Council of Hong Kong under Grant 27213824 and CRS HKU702/24. Y. Ma, Z. Fang, H. An, and Y. Fang are with Hong Kong JC STEM Lab of Smart City and the Department of Computer Science, City University of Hong Kong, Hong Kong, China. (e-mail: [email protected], [email protected], [email protected], [email protected].) Z. Zhao and X. Chen are with the Department of Electrical and Computer Engineering, The University of Hong Kong, Hong Kong, China. (e-mail: [email protected], [email protected].)
Traditional System Source
Update
Receiver
CP System Source
Update
Receiver in CP
Local Update!
Fig. 1: Comparison of information update mechanisms in traditional systems versus CP systems. In a traditional system, AoI decreases for the receiver that relies exclusively on direct transmissions from the source. In a CP system, the receiver is equipped with onboard sensors that allow it to actively perceive its surroundings, which also leads to AoI decrease.
views [4]. By integrating observations from vehicles and road infrastructure, CP eliminates the blind spots of individual vehicles, thus improving the safety of autonomous driving [1]– [3], [5], [6]. Despite its perception benefits, CP is constrained by the limited bandwidth available for sensor data transmissions. Prior works have addressed the communication bottleneck in CP by optimizing resource allocation based on traditional quality-of-service (QoS) metrics, such as network throughput [1], spectral efficiency, and communication volume [7]. Nevertheless, these metrics focus on transmission efficiency rather than on the staleness of information from the vehicles’ perspectives, though the latter is a more important objective for real-time sensing systems. To maintain information freshness, the age of information (AoI) metric has been widely adopted in sensor networks, which measures information freshness from the destination’s perspective [8]–[15]. Optimizing AoI for CP ensures the delivery of fresh sensor data in autonomous driving, thereby enhancing driving safety. Yet, previous research efforts on AoI are ill-suited to CP systems. Specifically, prior work implicitly assumes that receivers lack sensing capabilities, because their AoI decreases only upon receiving updates from a transmitter. In contrast, in CP systems, vehicles (or drones or robots) actively perceive their local environments (see Fig. 1). For instance, if a vehicle cannot see an occluded area, a traditional AoI-minimization scheduling policy may transmit information
2
about this area. However, due to vehicle mobility and environmental changes, the vehicle may see this area directly, implying that data transmissions, if scheduled, are redundant. Conventional AoI scheduling policies that ignore receivers’ local sensing capabilities can therefore lead to significantly suboptimal performance in CP systems. The mobility and sensing dynamics in CP introduce unique challenges for AoI-minimization scheduling policies. First, BS update scheduling depends not only on the current AoI, but also on the set of currently present vehicles and their local sensing coverage. Jointly characterizing and optimizing these time-varying states is challenging. This is because vehicles may enter or leave a target region over time, and their local sensing ranges change dynamically due to mobility, viewpoints, and occlusions. Second, the BS must make scheduling decisions with delayed knowledge of each vehicle’s sensing conditions, because occlusion information is available only after perception and uplink feedback, both of which incur nonnegligible delays. These challenges lead to two key research questions: Q1. How can we characterize a tractable AoI objective that captures both the time-varying receiver population and their dynamic local sensing ranges? • Q2. How can we minimize the formulated AoI under unknown environmental parameters and delayed network status? •
To answer these questions, we develop a unified localsensing-aware online scheduling framework for infrastructureassisted CP. On the modeling side, we characterize the population of vehicles that are interested in a specific area but unable to directly observe it as a temporally correlated stochastic process. By exploiting the dynamics of this unobserving population, we derive a closed-form expression for the timeaverage sum AoI, accounting for the time-varying receiver population and dynamic local sensing ranges. On the optimization side, delayed state observations and unknown environmental parameters render the scheduling problem a partially observable restless multi-armed bandit (PO-RMAB) problem, which is generally PSPACE-hard. To address this challenge, we propose the Local-sensing-aware Max-Weight scheduling (LocMW) policy for real-time update scheduling, a learningaided online policy that combines projected ridge estimation of environmental parameters, certainty-equivalent prediction over the observation-delay window, and Lyapunov-drift-based MaxWeight scheduling. We show that when the system parameters are known, LocMW achieves a time-average sum AoI no older than the optimal stationary randomized benchmark and within a factor of two of the mean-field lower bound. With online learning, LocMW incurs a sublinear cumulative excess AoI relative to the optimal stationary randomized benchmark. This is the first work to address the AoI minimization problem by explicitly accounting for receivers’ dynamic local sensing capabilities. While we specifically focus on vehicular CP systems in this paper, the proposed methodology can be broadly extended to AoI minimization problems in other types of freshness-critical CP systems, such as drone swarms and robotic networks.
The main contributions of this paper are summarized as follows: • We propose the first AoI minimization framework that explicitly accounts for receivers’ local sensing capabilities and the time-varying unobserving populations in CP systems. Based on this framework, we derive a closedform characterization of the time-average sum AoI over a target region, revealing how mobility-driven local sensing reshapes the information-staleness dynamics. • Through a mean-field relaxation, we derive an analytical lower bound on the time-average sum AoI. We further characterize the optimal stationary randomized benchmark and show that it is within a constant factor of this lower bound. • We design LocMW scheduling, an online learning policy for sensor information broadcasting to navigate observation delays and unknown environmental dynamics. We also show that it incurs a sublinear cumulative excess AoI against the optimal stationary randomized benchmark. • We conduct extensive simulations using real-world vehicular trajectory datasets (pNEUMA and FLUID) and a V2X CP dataset (V2X-Sim) to evaluate our framework. The results demonstrate that LocMW consistently outperforms other baselines by reducing information staleness and improving perception performance, achieving up to a 37.5% reduction in AoI and a 6.79% gain in mean average precision (mAP). The remainder of this paper is organized as follows. Section II reviews related work on AoI scheduling and resource management in CP. Section III presents the system model and problem formulation. Section IV characterizes the timeaverage sum AoI in closed form. Sections V and VI detail the analytical lower bound and the optimal stationary randomized benchmark, respectively. Section VII develops the LocMW algorithm and establishes its performance guarantee. Finally, Section VIII reports the simulation results, and Section IX concludes the paper. II. R ELATED W ORK A. Age of Information AoI has become a standard metric for quantifying information freshness from the destination’s perspective [8], [15]– [18]. A substantial body of research has investigated AoI minimization under various communication, scheduling, and system constraints. Kadota et al. [8] formulated a discretetime scheduling problem over unreliable channels and developed randomized, Max-Weight, and Whittle-index policies for minimizing the expected weighted-sum AoI. Sun et al. [16] demonstrated that the zero-wait policy is not necessarily ageoptimal and formulated average age-penalty minimization as a constrained semi-Markov decision process, based on which optimal causal update policies were derived. Ji et al. [19] considered age-optimal packet scheduling under long-term resource constraints and delayed feedback, and proposed a low-complexity greedy policy that minimizes the immediate expected Lagrangian cost. Under constrained transmission rates and imperfect feedback, Zhu et al. [20] developed
3
a Lyapunov-optimization-based drift-plus-penalty policy for systems with Bernoulli traffic. Tsai et al. [21] further studied update-through-queue systems with random, unknown delays and subsequently designed online algorithms to adaptively learn the optimal waiting time. In addition to traditional AoI formulations, new AoI metrics have also been introduced to capture the usefulness of status updates. This includes age of changed information (AoCI), which accounts for changes in the underlying source state [22], and age of incorrect information (AoII) [23], which incorporates the discrepancy between the receiver’s estimate and the true system state. However, these studies fail to account for users’ local sensing capabilities and dynamic mobility, in which a time-varying set of vehicles refresh their information through onboard sensing. Beyond traditional scheduling, recent studies have applied advanced learning and optimization techniques to manage AoI in highly dynamic environments. Chen et al. [24] optimized long-term AoI performance in a Manhattan grid vehicle-tovehicle (V2V) network and proposed a decentralized deep reinforcement learning algorithm. Emami et al. [25] studied the trade-off between UAV mobility and data freshness as a mean-field game and developed a hybrid proximal policy optimization scheme to jointly optimize trajectories and communication schedules. Nevertheless, these schemes still do not consider users’ local sensing capabilities. In a nutshell, existing AoI-minimization frameworks cannot be directly applied to CP systems for two reasons: 1) they assume that AoI can only decrease via network transmissions without considering local sensing, and 2) they assume that the set of receivers is fixed, which does not hold in dynamic mobile networks such as vehicular networks. B. Collaborative Perception The transmission of high-dimensional sensory data over bandwidth-limited V2X links remains a primary bottleneck for CP. To alleviate this, a growing body of literature has investigated dynamic resource allocation, user selection, and task-oriented scheduling for communication-efficient CP [1], [2], [7], [26]–[28]. To enhance communication efficiency, recent studies focus on selective transmission strategies that filter redundant data. For example, Where2comm [7] and How2comm [29] reduce communication overhead by transmitting high-uncertainty spatial regions. Other frameworks, such as PACP [1] and Directed-CP [28], further improve CP performance by dynamically prioritizing data based on perception correlation or directional interests under varying wireless conditions. To address the downlink communication bottleneck in vehicle-to-infrastructure (V2I) systems, Ma et al. [2] developed Birdcast, which maximizes network-wide utility by jointly optimizing bird’s-eye-view (BEV) feature selection and multicast grouping. Recent studies have incorporated AoI metrics to manage data freshness in CP systems [12], [13], [30]. To mitigate feature misalignment caused by spatiotemporal heterogeneity, Han et al. [13] proposed a fusion framework that exploits network synchronization and AoI to compensate for clock drifts and communication delays. Wu et al. [26] developed
Which areas to broadcast?
Delayed Feedback AoI: 13
BS AoI: 16
AoI: 20
Fig. 2: Illustration of our framework for an infrastructureassisted CP system. Users, i.e., vehicles, perceive their immediate surroundings via onboard sensors, instantaneously resetting their AoI to 0. Operating on delayed feedback, the BS dynamically schedules broadcast updates for up to K areas per slot to complement the users’ ongoing local sensing and minimize the network’s total AoI.
an AoI-driven optimization framework that jointly controls computing and communication delay. Fang et al. [12] formulated an Age of Perceived Targets (AoPT) minimization problem to prioritize high-quality and task-relevant data for critical targets. To handle asynchronous multi-source updates in CP systems, Wang et al. [27] designed a scheduling policy that minimizes channel utilization while satisfying AoI constraints. Zhu et al. [14] introduced a timeliness-aware prioritized scheduling algorithm for multi-region CP systems, leveraging Lyapunov optimization to balance AoI reduction and communication costs. However, the aforementioned works have not incorporated users’ local sensing capabilities for AoI minimization, which are salient characteristics of CP systems. III. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model We consider a real-time CP system comprising a BS and a dynamic set of users (e.g., vehicles), as illustrated in Fig. 2. The BS coverage region is partitioned into B distinct areas, denoted by the set B = {1, 2, . . . , B}, and time is discretized into a slotted horizon T = {1, 2, . . . , T }. Both the BS and users are equipped with sensing capabilities (e.g., cameras or LiDAR) to perceive the environment. We assume that the BS continuously monitors the entire region of interest, whereas each user observes only a time-varying subset of the region due to mobility and physical occlusions. By adopting appropriate modulation and coding schemes, we assume a broadcast update from the BS can reach all users interested in
𝑎𝑏,𝑖 (AoI of user 𝑖 for area 𝑏)
4
TABLE I: Summary of important notations.
Receive update from BS Sense it locally OR is no longer interested
0
1
2
3
4
5
6
Notation Description
7 Time slot
Fig. 3: Evolution of the AoI ab,i of user i for area b over time slots. The AoI increases by 1 in each time slot if the user is interested in area b but cannot sense it. AoI reduces to 1 upon receiving a new update from the BS and resets to 0 if the user senses the area or loses interest in it.
the corresponding area [2]. As illustrated in Fig. 3, the AoI evolution of user i for area b at time slot t, ab,i (t), comprises the following cases. • Decreases to 1: If user i cannot directly observe area b but receives a broadcast update from BS at slot t, its AoI for area b becomes 1 at slot t + 1. • Resets to 0: If user i directly observes area b or loses interest in area b, the AoI is set to 0 at slot t + 1. • Increases by 1: Otherwise, the AoI increases by 1 in each time slot. User Demand Modeling. AoI evolution depends on user demands, i.e., when they can see the area and when they lose interest. Let Nb (t) denote the instantaneous user demand, defined as the number of users in the network unable to directly observe but interested in area b at time t. Nb (t) changes according to user mobility and environmental dynamics. To capture the temporal correlation and the non-negative integer nature of this population, we model Nb (t) as a first-order integer-valued auto-regressive (INAR(1)) process [31], [32]: Nb (t) = ρb ◦c Nb (t − 1) + ωb (t),
(1)
vis where ρb = ρint b (1 − ρb ) ∈ (0, 1) represents the probability that a user remains in the interested-but-unobserving state for area b. Here, ρint and ρvis denote the probabilities of the user b b maintaining interest and possessing direct observation of area b, respectively. The symbol ◦c is a dependent binomial thinPNb (t−1) Xi (t−1) ning operator, defined as ρb ◦c Nb (t−1) = i=1 with Xi (t − 1) ∼ Bernoulli(ρb ) being identically distributed but potentially correlated Bernoulli random variables (this correlation accounts for the spatial dependencies typically exhibited by proximate users), and ωb (t) is an independent, integer-valued random variable representing the influx of users newly transitioning into the interested-but-unobserving state for area b at slot t due to mobility or physical occlusions. Considering physical road capacity constraints, there exist finite constants Nmax and µmax such that Nb (t) ≤ Nmax and ωb (t) ≤ µmax for all b ∈ B.
T
The set of time slots
B
The set of distinct areas
Nb (t)
The instantaneous user demand for area b at slot t
ρb
The probability of a user remaining in the interested-butunobservable state for area b
ωb (t)
The number of new interested-but-unobservable users for area b at slot t
λb
The steady-state mean user demand for area b
µb
The mean arrival rate of interested-but-unobservable users for area b
d
The observation delay (in time slots)
H(t)
The delayed information available to the BS
ab,i (t)
AoI of user i for area b at slot t
Ab (t)
The aggregate AoI for area b at slot t
ub (t)
The binary scheduling decision for area b at slot t
K
The maximum number of areas can be scheduled for updates
Ab
The long-term time-average AoI for area b
N̂b (t)
The predicted user demand for area b at slot t
Âb (t)
The predicted aggregate AoI of area b at slot t
Delayed Status Upload. To track information staleness, each user reports its location, a perception-quality or visibility indicator1 , and its local AoI for each target area to the BS. Due to uplink latency, however, the BS observes these reports with a constant delay of d ≥ 1 time slots and must operate on delayed observations when making scheduling decisions. Accordingly, the information available to the BS at the beginning of slot t is H(t) = {Nb (τ ), Ab (τ ) : b ∈ B, τ ≤ t − d}. The BS constructs Nb (τ ) by counting the number of users that are interested in, but unable to directly observe area b andP aggregating their individual AoI reports, Nb (τ ) such that Ab (τ ) = i=1 ab,i (τ ). The evolution of ab,i will be detailed in Section IV. Note that Nb (τ ) and Ab (τ ) are derived from uploaded binary matrices and scalar values from users, respectively, which are communication-efficient. The key notations are summarized in Table I. B. Problem Formulation At the beginning of each time slot t, the BS generates fresh information for all B areas via its equipped sensors and then selects a subset of areas to broadcast. Let ub (t) ∈ {0, 1} denote the binary scheduling variable, where ub (t) = 1 if the BS broadcasts an update for area b in slot t, and ub (t) = 0 otherwise. Our objective is to find the optimal scheduling policy π ∗ , i.e., the sequence of scheduling actions {ub (t)}b∈B , that minimizes the long-term time-average sum AoI across the network subject to PTa bandwidth constraint. By defining Ab = lim supT →∞ T1 t=1 E [Ab (t)], the corresponding optimization problem is formulated as min π
B X
Ab
(2a)
b=1
1 This indicator can be directly derived from the user’s perception confidence [7], which is generated by onboard perception modules without incurring extra computing overhead.
5
s.t.
B X
ub (t) ≤ K,
∀t ≥ 1,
(2b)
∀b ∈ B, ∀t ≥ 1,
(2c)
b=1
ub (t) ∈ {0, 1},
where Constraint (2b) indicates that at most K areas can be scheduled for broadcast in any single time slot. As previously discussed, the true system state, specifically, the current demand and aggregate AoI, {Nb (t), Ab (t)}b∈B , is not observable by the BS when making decisions. Instead, the BS must rely on delayed information and historical actions to schedule updates. This naturally leads to a partially observable restless multi-armed bandit (PO-RMAB) problem. In general, PO-RMAB problems are PSPACE-hard, and determining an optimal policy entails exponential complexity and memory requirements [33], [34]. Moreover, different from traditional AoI scheduling problems, even evaluating the objective function (2a) is challenging in our case. ❶ First, information freshness in CP is intertwined with users’ time-varying perception ranges: whenever a user can directly sense a particular area, the AoI of that area is effectively reset to zero even without receiving an update. ❷ Second, the user population in the network evolves over time, which further complicates the analysis of the time-average sum AoI. Consequently, before solving problem (2), we must first develop a tractable expression for the objective (2a). IV. C HARACTERIZATION OF T IME - AVERAGE S UM AO I In this section, we derive a tractable expression for the time-average sum AoI in (2a), which forms the basis for the subsequent development of our solution. Based on this expression, we transform the original AoI minimization into an equivalent weighted AoI reduction maximization problem.
representing a batch of ωb (t+1) users newly transitioning into this state. Since these users could either observe area b locally at slot t or have just entered the network, their initial AoI at slot t is 0. Let λb ≜ limt→∞ E[Nb (t)] denote the steady-state mean demand for area b and let µb ≜ E[ωb (t)] denote the mean influx of users, where µb ≤ µmax . Under the INAR(1) mobility model, we have µb = (1 − ρb )λb . We now establish the onestep evolution and the stability of Ab (t) and then characterize the tractable expression of the time-average AoI Ab . Lemma 1. Given the system filtration Ft containing all system states up to time slot t, the conditional one-step evolution of the aggregate AoI for area b is given by E[Ab (t + 1)|Ft ] = ρb 1 − ub (t) Ab (t) + ρb Nb (t) + µb . (4) Proof: The proof can be found in Appendix A Lemma 2. Under any admissible scheduling policy, the aggre+1)] gate AoI Ab (t) is mean-rate stable, i.e., limT →∞ E[Ab (T = T 0, which implies 1 lim E[Ab (T + 1)] − E[Ab (1)] = 0. (5) T →∞ T Proof: The proof can be found in Appendix B. Proposition 1. Based on Lemma 1 and 2, the time-average aggregate AoI for area b is characterized by λb ρb ub Ab , − 1 − ρb 1 − ρb PT where ub Ab ≜ limT →∞ T1 t=1 E [ub (t)Ab (t)]. Ab =
T
A. Characterization of Time-average AoI Recall that ab,i (t) denotes the AoI of user i for area b at slot PNb (t) t , and Ab (t) = i=1 ab,i (t) represents the instantaneous aggregate AoI of all interested-but-unobserving users. Since our objective is to minimize the time-average sum AoI Ab , we will show that it is sufficient to directly characterize Ab (t), without explicitly tracking the evolution of each individual ab,i (t). Given the scheduling decision ub (t), the aggregate AoI Ab (t) evolves as 2
Nb (t)
Ab (t + 1) =
X
Xi (t) (1 − ub (t))ab,i (t) + 1 + ωb (t + 1) . | {z } i=1 {z } new arrivals | staying users
(3) It comprises two components based on user population dynamics. ❶ The first term accounts for staying users, who remain in the interested-but-unobserving state. The indicator Xi (t) ∼ Bernoulli(ρb ) denotes whether user i persists in this state at slot t + 1. For these users, if BS schedules an update, i.e., ub (t) = 1, their AoI resets to 1 at slot t + 1; otherwise, it increments by 1. ❷ The second term accounts for new arrivals, 2 The index i identifies users only within slot t and is not tracked consis-
tently over time.
1X E[Ab (t)], Āb,T = T t=1
(6)
T
1X Ūb,T = E[ub (t)Ab (t)]. T t=1
Proof: Applying the law of iterated expectations to the one-step conditional drift established in Lemma 1 yields E[Ab (t+1)] = ρb E[Ab (t)]−ρb E[ub (t)Ab (t)]+ρb E[Nb (t)]+µb . (7) Subtracting E[Ab (t)] from both sides and average over the time horizon from t = 1 to T , we obtain T 1 1X E[Ab (T + 1)] − E[Ab (1)] = (ρb − 1) E[Ab (t)] T T t=1 T T 1X 1X E[ub (t)Ab (t)] + ρb E[Nb (t)] + µb . T t=1 T t=1 (8) Taking the limit superior as T → ∞, Lemma 2 guarantees that the boundary difference on the left-hand side vanishes. Concurrently, applying the definition of the long-term time average to the right-hand side yields
− ρb
0 = (ρb − 1)Ab − ρb ub Ab + ρb λb + µb .
(9)
Substituting µb = (1 − ρb )λb simplifies the equation to 0 = (ρb − 1)Ab − ρb ub Ab + λb .
(10)
6
Rearranging these terms to solve for Ab yields (6), which completes the proof. Insights: The expression for Ab in (6) contains two comλb ponents: ❶ A policy-independent baseline ( 1−ρ ), which repb resents the accumulation of information staleness in area b only depending on exogenous factors, i.e., λb and ρb . This term acts as the upper bound on the time-average AoI when there are no updates from the BS. ❷ A policy-dependent term ρb ub Ab ), which captures the AoI reduction achieved via ( 1−ρ b ρb BS updates and is weighted by the coefficient 1−ρ related to b vehicular mobility/occlusion factors. B. Problem Transformation PB Minimizing the time-average sum AoI b=1 Ab in (2a) is mathematically equivalent to maximizing the sum of the weighted AoI reductions. According to Proposition 1, the optimization problem can thus be reformulated as follows max π
B X b=1
ρb ub Ab 1 − ρb
s.t. (2b), (2c).
(11a) (11b)
However, solving the reformulated problem in (11) remains PSPACE-hard. In our optimization, this difficulty is further exacerbated by two main factors. First, due to the d-slot observation delay, the current system state, {Nb (t), Ab (t)}b∈B , is hidden from the BS at the moment the scheduling decision ub (t) must be made. Second, the underlying system dynamics, θ b = [ρb , µb ]⊤ , are unknown a priori and must be learned online. To address these coupled challenges, we propose an efficient online LocMW policy in Section VII. To establish performance benchmarks, we first derive a fundamental lower bound and a stationary randomized scheduling policy in Sections V and VI. V. L OWER B OUND ON T IME - AVERAGE S UM AO I In this section, we derive a mean-field lower bound on the time-average sum AoI in (2) for any admissible scheduling policy. PT We define pb ≜ limT →∞ T1 t=1 E[ub (t)] as the expected PB update rate for area b, which satisfies b=1 pb ≤ K. For each area b, let Ñb (t) ≜ E[Nb (t)|H(t)] = λb + ρdb (Nb (t − d) − λb ) denote the d-slot demand predictor. The corresponding prediction error is defined as eb (t) ≜ Nb (t) − Ñb (t), which satisfies E[eb (t)|H(t)] = 0. Let Zb (t) ≜ E[Ab (t)|H(t)] denote the conditional expected aggregate AoI. By the law of iterated expectations, we have E[Ñb (t)] = E[E[Nb (t)|H(t)]] = E[Nb (t)], E[Zb (t)] = E[E[Ab (t)|H(t)]] = E[Ab (t)].
(12)
We then define the deterministic long-term PT means zb ≜ limT →∞ T1 t=1 E[Zb (t)] and PT xb ≜ limT →∞ T1 t=1 E[ub (t)Zb (t)]. In addition, let M̃b (t + 1) ≜ ρb Ñb (t) + µb represent the predicted one-step input to the AoI dynamics conditioned on H(t). Since µb = (1 − ρb )λb , its asymptotic expectation is given by limt→∞ E[M̃b (t + 1)] = ρb λb + µb = λb . We introduce the
following assumption regarding the asymptotic behavior of the predicted dynamics. Assumption 1 (Mean-field Concentration). For each area b, the predicted one-step input satisfies T
1X Var[M̃b (t + 1)] = 0. T →∞ T t=1
(13)
lim
Assumption 1 imposes a mean-field concentration on the predicted input, which is assumed to converge to its mean. This is a standard assumption in stochastic network analysis, which becomes true as the number of entities grows large [35], [36]. In our system, it becomes asymptotically accurate for large unobserving populations (e.g., at dense urban intersections). Under this premise, we establish the following theorem. Theorem 1. Under Assumption 1, the time-average sum AoI achieved by any admissible scheduling policy satisfies B X b=1
Āb ≥ C
LB
≜
min
B X
0≤p ≤1 PB b b=1 b=1 pb ≤K
Lb (pb ),
(14)
pb +1 . Moreover, the optimal where Lb (pb ) ≜ λb pb (1+ρ b )+1−ρb solution of (14) has the water-filling form "p #1 2λb ρb /γ ∗ − (1 − ρb ) ∗ ∗ pb (γ ) = , (15) 1 + ρb 0
where [x]10 ≜ min{1, max{0,P x}}, and γ ∗ > 0 is a Lagrange B multiplier chosen such that b=1 p∗b (γ ∗ ) = K. The optimal ∗ value of γ can be found using the bisection method. Proof: The proof can be found in Appendix C. Note that C LB is a structural lower bound based on a mean-field assumption with fractional update rates and known statistical parameters, serving as an analytical baseline for performance evaluation. VI. O PTIMAL R ANDOMIZED S CHEDULING P OLICY This section develops an optimal randomized scheduling policy for Problem (2). Unlike state-aware policies, this policy depends only on the statistical parameters {λb , ρb }b∈B and can be implemented without feedback. This randomized policy serves as a strong benchmark for our subsequent development. Let ΠR denote the class of stationary randomized policies. Each policy π R ∈ ΠR is specified by a vector of marginal R ⊤ broadcasting probabilities η R = [η1R , . . . , ηB ] , where ηbR ≜ PB R R Pr{ub (t) = 1} ∈ [0, 1] with b=1 ηb ≤ K. At each slot, the BS samples a feasible subset S R (t) ⊆ B satisfying |S R (t)| ≤ R K, and sets uR b (t) = 1 if and only if b ∈ S (t). This sampling process is independent across time and the system history. Theorem 2. The optimal randomized marginal probability is "p #1 λb ρb /ν ∗ − (1 − ρb ) R,∗ ∗ ηb (ν ) = , (16) ρb 0
7
where the dual variable ν ∗ >P0 is determined via onedimensional bisection to satisfy b∈B ηbR,∗ (ν ∗ ) = K. Consequently, the minimum expected time-average sum AoI achieved by this policy is C R,∗ =
B X
λb
b=1
1 − ρb + ρb ηbR,∗
.
(17)
E[Ab (t + 1)] = ρb (1 − ηbR )E[Ab (t)] + ρb E[Nb (t)] + µb . (18) Averaging over t = 1, . . . , T , letting T → ∞, and invoking the mean-rate stability established in Lemma 2, we obtain 0 = − 1 − ρb + ρb ηbR ĀR (19) b + ρb λb + µb . Substituting µb = (1 − ρb )λb and solving for ĀR b gives λb ≜ Fb (ηbR ). 1 − ρb + ρb ηbR
η∈PK
B X
Fb (pLB b )≤2
Fb (ηb ),
(21)
n o PB where PK ≜ η ∈ [0, 1]B : b=1 ηb ≤ K is the feasible set. It is straightforward to verify that Fb (ηbR ) is convex and monotonically decreasing with respect to ηbR . Let ν ≥ 0 be the Lagrange multiplier associated with the bandwidth constraint. Solving the KKT stationarity conditions yields p λb ρb /ν − (1 − ρb ) . (22) ηb = ρb Projecting this onto [0, 1] results in the formulation in (16). The optimal ν ∗ is efficiently found via bisection. Finally, substituting ηbR,∗ back into the objective provides the performance bound in (17), completing the proof. Theorem 3. The optimal stationary randomized policy satisfies C LB ≤ C R,∗ ≤ 2C LB . (23) Proof: Recall that the per-area term in the mean-field lower bound is defined as Lb (x) ≜ λb x(1+ρx+1 . For any b )+1−ρb x ∈ [0, 1], evaluating the ratio between Fb (x) and Lb (x) yields Fb (x) 1 − ρb + (1 + ρb )x = . Lb (x) (1 − ρb + ρb x)(1 + x)
(24)
We have 2(1 + x)(1 − ρb + ρb x) − 1 − ρb + (1 + ρb )x
B X
LB Lb (pLB , b ) = 2C
(26)
b=1
completing the proof. Theorem 3 establishes that the optimal randomized policy achieves a 2-approximation of the mean-field lower bound. Since this randomized approach does not condition on state history, its performance C R,∗ is independent of the observation delay d. VII. L OCAL - SENSING - AWARE M AX -W EIGHT P OLICY In this section, we develop an efficient online learning approach, termed Local-sensing-aware Max-Weight (LocMW) scheduling, to solve the PO-RMAB problem formulated in Sections III and IV. We first detail the proposed LocMW policy and then derive theoretical performance guarantees. A. The LocMW Framework
b=1
=(1 − ρb )(1 + x) + 2ρb x2 > 0,
B X
(20)
The optimal randomized policy is thus obtained by minimizing the sum of these individual functions: C R,∗ ≜ min
C R,∗ ≤
b=1
Proof: Taking the expectation of the one-step AoI evoR lution in (4), and noting that E[uR b (t)Ab (t)] = ηb E[Ab (t)] as R ub (t) is independent of the current AoI state, yields
R ĀR b (ηb ) =
marginal probability vector for the randomized policy, the optimal performance is bounded by
(25)
which implies Fb (x) ≤ 2Lb (x) for all b ∈ B and x ∈ [0, 1]. LB ⊤ Let pLB = [pLB denote the optimal allocation 1 , . . . , pB ] vector for the lower-bound. As pLB ∈ PK is also a feasible
The proposed LocMW policy employs a three-step learningand-control framework. Specifically, the BS first estimates the system parameters θ b = [ρb , µb ]⊤ from delayed reports, then predicts the current network states, and finally schedules updates via a Max-Weight scheduling approach. Step 1. Online Parameter Estimation: At the beginning of slot t, the BS observes the delayed historical states H(t) = {Nb (τ ), Ab (τ ) : b ∈ B, τ ≤ t − d}. For each area b, define the state regressor xb (τ ) = [Nb (τ −1), 1]⊤ and the target observation yb (τ ) = Nb (τ ). Under the INAR(1) demand model, the expected demand evolves as E[yb (τ ) | xb (τ )] = xb (τ )⊤ θ b . Thus, the BS estimates θ b via an ℓ2 -regularized least squares (ridge regression) estimator [37], [38]: ! t−d X 2 ⊤ 2 θ̃ b (t) = arg min2 yb (τ ) − xb (τ ) θ + ζ∥θ∥2 . θ∈R
τ =2
(27) where ζ > 0 is the regularization parameter. The ℓ2 -penalty prevents the parameter estimates from overfitting to stochastic traffic anomalies. To enhance computational efficiency, this estimator is computed recursively rather than scratch at every slot. By Pfrom t−d maintaining Gb (t) = ζI + τ =2 xb (τ )xb (τ )⊤ and hb (t) = Pt−d τ =2 xb (τ )yb (τ ), the estimate simplifies to θ̃ b (t) = Gb (t)−1 hb (t).
(28)
Since Gb (t) ∈ R2×2 , its inversion incurs negligible O(1) complexity. In slot t+1, Gb and hb are incrementally updated with the newly available observation (xb (t + 1 − d), yb (t + 1 − d)). Finally, to ensure system stability, this estimate θ̃ b (t) is projected onto the parameter bounds to yield the final result θ̂ b (t) = [ρ̂b (t), µ̂b (t)]⊤ = ProjΘ θ̃ b (t) , (29) where ProjΘ (·) denotes the Euclidean projection operator onto the compact set Θ ≜ [0, ρmax ] × [0, µmax ] with ρmax < 1.
8
Step 2. Certainty-Equivalent State Prediction: To mitigate the intractability of exact belief-state tracking over the unobservable delay window, we adopt the certainty equivalence principle. Specifically, the BS predicts the current system state. Starting from the most recently received delayed reports, N̂b (t − d) = Nb (t − d) and Âb (t − d) = Ab (t − d), the BS recursively propagates the estimated dynamics over the delay window τ = t − d, . . . , t − 1, according to the following rules:
Algorithm 1: The LocMW Scheduling Policy Input: B, K, d, Θ, ζ Output: uLb (t), ∀b ∈ B 1 Initialize Gb ← ζI2 and hb ← 0 for all b ∈ B; 2 for t = 1, 2, . . . do 3 if t ≤ d then 4 Use any warm-up schedule satisfying (2b); 5 continue; 6 end N̂b (τ + 1) = ρ̂b (t)N̂b (τ ) + µ̂b (t), // Parameter Estimation Âb (τ + 1) = ρ̂b (t) 1 − ub (τ ) Âb (τ ) + ρ̂b (t)N̂b (τ ) + µ̂b (t). 7 for b ∈ B do (30) 8 Construct xb (t − d) = [Nb (t − d − 1), 1]⊤ and This recursive process successfully bridges the observation yb (t − d) = Nb (t − d); gap, yielding the current demand N̂b (t) and AoI Âb (t) re9 Update Gb ← Gb + xb (t − d)xb (t − d)⊤ and quired for executing scheduling decisions. hb ← hb + xb (t − d)yb (t − d); Step 3. Local-sensing-aware Max-Weight Scheduling: 10 Compute θ̂ b (t) based on (28) and (29); Given the estimated parameters θ̂ b (t) = [ρ̂b (t), µ̂b (t)]⊤ from 11 end Step 1 and the predicted states N̂b (t) and Âb (t) from Step Compute η̂bR (t) by the water-filling rule in (16) 2, the BS determines which areas to update subject to the 12 using θ̂ b (t); bandwidth constraint (2b). We now derive the scheduling index // State Prediction from a one-slot Lyapunov drift [8], [39], [40]. 13 for b ∈ B do We first define a linear Lyapunov function as 14 Initialize N̂b (t − d) ← Nb (t − d) and B X Âb (t − d) ← Ab (t − d); L(t) ≜ βb Ab (t), (31) 15 Predict N̂b (t) and Âb (t) via the recursion (30); b=1 // LocMW Index Derivation where βb > 0 is a tunable, area-specific parameter. Based on R 16 Compute ĉ (t) ← 1 − ρ̂b (t) + ρ̂b (t)η̂bR (t); b the estimated AoI dynamics, the one-slot Lyapunov drift is 17 Derive the LocMW index WbL (t) via (35); ∆(t) ≜ E[L(t + 1) − L(t) | H(t)] 18 end B h i 19 Select KL (t) as the K areas with the largest X = βb (ρ̂b (t) − 1)Âb (t) + ρ̂b (t)N̂b (t) + µ̂b (t) WbL (t); b=1 20 Obtain the scheduling decision uLb (t) via (36); B X 21 end − βb ρ̂b (t)Âb (t)ub (t). 22 return ub (t), ∀b ∈ B b=1
(32) Therefore, minimizing the drift is equivalent to maximizing P b βb ρ̂b (t)Âb (t)ub (t), which can be achieved by selecting the K areas with the largest weights Wb (t) = βb ρ̂b (t)Âb (t).
(33)
We define the weighting coefficient by aligning with the stationary randomized benchmark in Section VI 1 cb ≜ 1 − ρ̂b (t) + ρ̂b (t)η̂bR (t), (34) βb ≜ , cb where ηbR (t) is obtained by water-filling in (16) with θ̂ b (t). Substituting this into (33) gives the proposed LocMW index WbL (t) =
ρ̂b (t)Âb (t) . 1 − ρ̂b (t) + ρ̂b (t)η̂bR (t)
(35)
At each slot t, the BS deterministically schedules the subset KL (t) containing the K areas with the largest values of (35). The scheduling decision is therefore ( 1, b ∈ KL (t), uLb (t) = (36) 0, otherwise. As a result, the proposed LocMW algorithm dynamically prioritizes limited communication resources based on information
staleness, underlying local perception capabilities, and user mobility characteristics. The overall procedure is outlined in Algorithm 1. Remark 1. The LocMW index is local-sensing aware because the factor ρ̂b (t) captures the estimated persistence of users in the interested-but-unobserving state. A smaller ρ̂b (t) indicates that users in area b are highly likely to regain local visibility or lose interest, reducing the expected benefit of a BS update. Computational Complexity. At each time slot, parameter estimation across B areas incurs a complexity of O(B), the d-step state prediction requires O(dB) operations, and executing the Max-Weight scheduling takes O(B log B). Consequently, the overall worst-case per-slot computational complexity is O(B log B + dB). This demonstrates that the proposed LocMW policy is highly scalable and computationally efficient, making it well-suited for time-sensitive applications such as CP. B. Performance Guarantee We next establish the performance guarantees of LocMW. We first consider the setting in which the parameters
9
(ρb , µb ), ∀b ∈ B are known. In this case, the BS can exactly compute the conditional predictors Zb (t) ≜ E[Ab (t)|H(t)], Ñb (t) ≜ E[Nb (t)|H(t)]. Theorem 4. Suppose that (ρb , µb ) are known for all b ∈ B and LocMW schedules the K areas with the largest Zb (t) weights Wb (t) = 1−ρρb+ρ R,∗ , then its time-average sum AoI b b ηb P L b∈B Āb satisfies X C LB ≤ ĀLb ≤ C R,∗ ≤ 2C LB . (37) b∈B
Proof: By design, the LocMW policy maximizes the last term in (32). Since P the optimal randomized marginal vector η R,∗ ∈ [0, 1]B with b ηbR,∗ ≤ K lies within the convex hull of the feasible scheduling set, we have B X
B X
βb ρb Zb (t)ηbR,∗ .
(38)
βb −cR,∗ b Zb (t) + ρb Ñb (t) + µb ,
(39)
βb ρb Zb (t)uLb (t) ≥
b=1
b=1
Therefore, the one-slot drift satisfies ∆(t) ≤
B X b=1
where cR,∗ ≜ 1 − ρb + ρb ηbR,∗ denotes the ideal randomized b benchmark decay factor with known parameters. Applying the law of iterated expectations, averaging over t = 1, . . . , T , and letting T → ∞, the telescoping Lyapunov drift vanishes due to the mean-rate stability established PTin Lemma 2. Applying asymptotic properties limT →∞ T1 t=1 E[Nb (t)] = λb and µb = (1 − ρb )λb yields B X
L βb cR,∗ b Āb ≤
B X
βb λ b .
(40)
= C R,∗ . R,∗ c b=1 b
(41)
b=1
b=1
Since βb cR,∗ = 1, this simplifies to b B X
ĀLb ≤
b=1
B X λb
With the lower bound in Section V and the randomizedpolicy guarantee in Section VI, the proof is completed. Next, we consider the case where (ρb , µb ) are unknown. We introduce two fundamental conditions in the following. Assumption 2 imposes a persistence-of-excitation condition, ensuring that the delayed demand samples exhibit sufficient temporal variation to identify (ρb , µb ) [41], [42]. Assumption 3 establishes a local stochastic Lipschitz condition, which guarantees that small parameter estimation errors translate into bounded perturbations in the scheduling weights [43]. Assumption 2. For each b ∈ B, there exist constants αb > 0 and TbPE < ∞ such that, for all t ≥ TbPE + d, the empirical information matrix satisfies: ! t−d X ⊤ λmin xb (τ )xb (τ ) ≥ αb (t − d), (42) τ =2
where λmin (·) denotes the minimum eigenvalue of a matrix. Let θ ≜ [θ 1 , . . . , θ B ]⊤ ∈ ΘB denote a candidate parameter vector. We define the LocMW index map as Φ(t; θ) ≜
ρb Ab (t;θ b ) , [Φ1 (t; θ), . . . , ΦB (t; θ)]⊤ with Φb (t; θ) ≜ 1−ρ R b +ρb ηb (θ) where Nb (t; θ b ) and Ab (t; θ b ) are recursively propagated via (30) by substituting θ b for θ̂ b (t), ηbR (θ) denotes the randomized marginal probability from (16) under θ. We impose the following regularity condition.
Assumption 3. There exist constants rθ > 0 and LLd < ∞, alongside non-negative H(t)-measurable random variables Γd (t) satisfying supt≥1 E[Γ2d (t)] ≤ (LLd )2 . For any pair θ, θ ′ ∈ ΘB satisfying ∥θ−θ ∗ ∥2,∞ ∨∥θ ′ −θ ∗ ∥2,∞ ≤ rθ , where ∥θ − θ ′ ∥2,∞ ≜ maxb∈B ∥θ b − θ ′b ∥2 and θ ∗ ≜ [θ ∗1 , . . . , θ ∗B ]⊤ denotes the true parameter vector, the following local stochastic Lipschitz condition holds almost surely: Φ(t; θ) − Φ(t; θ ′ ) ∞ ≤ Γd (t) θ − θ ′ 2,∞ ,
(43)
Assumption 3 establishes a local stochastic Lipschitz condition, where the random variable Γd (t) bounds the sensitivity of the LocMW index to parameter estimation errors within a localized neighborhood. PT P Theorem 5. Let RTLocMW ≜ t=1 E b∈B Ab (t) − + T C R,∗ denote the cumulative excess AoI of the proposed LocMW policy relative to the optimal randomized benchmark with known parameters. It satisfies p RTLocMW = O KLLd T log(BT ) + o(T ). (44) set as UK ≜ Proof: Define P the feasible scheduling ∗ u ∈ {0, 1}B : u ≤ K . Let w (t) and u∗ (t) ∈ b∈B b ∗ ⊤ arg maxu∈UK w (t) u denote the ideal Max-Weight index vector and scheduling decision under the true system parameters. Similarly, let wL (t) and uL (t) denote the proposed LocMW index vector and scheduling decision. We define the maximum index error as ϵt ≜ ∥wL (t) − w∗ (t)∥∞ . Since ∥u∥1 ≤ K, the performance of the LocMW decision satisfies w∗ (t)⊤ uL (t) ≥ wL (t)⊤ uL (t) − Kϵt ≥ wL (t)⊤ u∗ (t) − Kϵt ∗
(45)
⊤ ∗
≥ w (t) u (t) − 2Kϵt . Moreover, because the optimal randomized marginal probability vector lies within the convex hull of the feasible set, η R,∗ ∈ conv(UK ), it follows that w∗ (t)⊤ u∗ (t) ≥ w∗ (t)⊤ η R,∗ . Hence w∗ (t)⊤ uL (t) ≥ w∗ (t)⊤ η R,∗ − 2Kϵt .
(46)
Since uL (t) is H(t)-measurable, the conditional one-slot Lyapunov drift can be given as ∆(t) =
B X (ρb − 1)Zb (t) + ρb Ñb (t) + µb
cR,∗ b
b=1
≤
B X −cR,∗ Zb (t) + ρb Ñb (t) + µb b
b=1
cR,∗ b
− w∗ (t)⊤ uL (t)
+ 2Kϵt .
(47) Applying the law of iterated expectations, we obtain X X B B ρb E[Nb (t)] + µb E[∆(t)] ≤ −E Ab (t) + +2KE[ϵt ]. cR,∗ b b=1 b=1 (48)
10
Summing over t = 1, . . . , T , dividing by T , we obtain the upper bound on the average AoI X T B T X X E Ab (t) ≤ E[L(1)] − E[L(T + 1)] + 2K E[ϵt ] t=1
t=1
b=1
+
T X B X ρb E[Nb (t)] + µb t=1 b=1
cR,∗ b
.
(49) Considering L(t) ≥ 0 and recalling the steady-state demand relation µb = (1 − ρb )λb , it yields: X T B T X X E Ab (t) − T C R,∗ ≤ DT + E[L(1)] + 2K E[ϵt ], t=1
t=1
b=1
(50) where DT = b=1 cR,∗ t=1 E[Nb (t)] − λb captures the b transient state-prediction deviation relative to the steady-state demand and DT = O(1) as the INAR(1) dynamics are stable. It remains to bound the expected index error E[ϵt ]. Under Assumption 2 and the bounded martingale-difference noise condition, the projected ridge estimator achieves the following convergence rate: ! r 1/2 1 log(Bt) ∗ 2 +O . E θ̂(t) − θ 2,∞ =O t t (51) Let Et ≜ ∥θ̂(t) − θ ∗ ∥2,∞ ≤ rθ denote the local concentration event. Conditioned on Et , Assumption 3 guarantees that the induced LocMW index is locally stochastic Lipschitz, yielding PB
ρb PT
ϵt = Φ(t; θ̂(t))−Φ(t; θ ∗ ) ∞ ≤ Γd (t) θ̂(t)−θ ∗ 2,∞ . (52) Moreover, the compactness of Θ, the positive denominator 1 − ρb + ρb ηbR (θ) ≥ 1 − ρmax > 0, the finite window d, and the bounded AoI second moment collectively ensure that the index map has a uniformly bounded second moment supt≥1 E supθ∈ΘB ∥Φ(t; θ)∥2∞ < ∞. Consequently, on the complement event Etc , the error is constrained, rendering its expected contribution negligible. Taking the expectation and applying the Cauchy-Schwarz inequality, the expected index error is bounded by ! r log(Bt) 1 L +O . (53) E[ϵt ] ≤ Ld · O t t Summing over T yields T X
VIII. E XPERIMENTS In this section, we present numerical experiments to evaluate the proposed LocMW scheduling framework, comparing it against several baseline methods to demonstrate its superiority in AoI reduction and object detection under communication constraints.
p E[ϵt ] = O LLd T log(BT ) + O(log T ).
(54)
t=1
Finally, absorbing DT = O(1) and O(log T ) into the o(T ), we obtain p (55) RTLocMW = O KLLd T log(BT ) + o(T ), PT P R,∗ + where RTLocMW ≜ . This t=1 E b∈B Ab (t) − T C completes the proof.
A. Experiment Settings Setting. While the LocMW framework is broadly applicable to diverse wireless systems with local sensing and AoI-aware scheduling, we instantiate and evaluate it in the context of V2I-CP, the primary motivating application of this work. Since standard CP datasets are specifically tailored for egouser object detection, the number of users (vehicles) per scene is inherently limited (e.g., up to five in the V2X-Sim dataset). To assess the effectiveness of our framework in dense traffic, we utilize the pNEUMA [44] and FLUID [45] datasets to simulate dense urban conditions. The pNEUMA dataset provides large-scale, drone-captured vehicle trajectories over congested Athens. We use four days of recordings from drones d6 and d7, concatenating five consecutive 30-minute blocks per day and partitioning the coverage region into 543 areas via OpenStreetMap. The FLUID dataset offers fine-grained trajectories at signalized intersections. We treat each video sequence as a scene, retain only motorized users, and divide the region into B = 201 areas. Time is discretized into 0.1s slots, mapping each vehicle to one area per slot. A vehicle’s region of interest is defined as a radius of 80 m for pNEUMA and 60 m for FLUID. Following distancedependent sensing degradation and stochastic blockage models in vehicular environments [7], [46], We model the visibility of area b to vehicle uh based and local traffic on distance i d (t) nb (t) where du,b (t) + density vu,b (t) = exp −ε u,b RI nmax is the distance, nb (t) is the vehicle count in area b, and nmax is the maximum observed density. We set ζ = 1, ρmax = 0.99, and implement a 500-slot warm-up. In addition, we evaluate the framework under realistic perception conditions using the V2X-Sim dataset [47], a comprehensive V2X CP dataset simulated via SUMO and CARLA, and we randomly select 10 scenes. For the 3D object detection task, we adopt PointPillars [2], [48] as the backbone detector. The feature maps are divided into B = 400 grids, with a data volume of δ = 1 KB per grid. We comprehensively evaluate the system’s performance based on both AoI and mAP. Baselines. We compare the proposed LocMW policy against several baselines. • LocMW (Perfect Est.): This oracle benchmark assumes the BS has perfect knowledge of the true parameters (ρb , µb ), isolating the loss caused by online estimation. • Max-Demand: This policy schedules K areas with the highest predicted demand N̂b (t). It prioritizes areas with the most vehicles requiring BS updates without AoI state. • Traditional Max-Weight: This policy schedules the K R areas that maximize the index ρ̂b ÃT b (t)/(1 − ρ̂b + ρ̂b η̂b ), T where Ãb is the AoI state without accounting for local sensing.
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(a) Time-average sum AoI versus communi- (b) Time-average sum AoI versus observation (c) Time-average sum AoI versus number of cation budget K/B (d = 8). delay d in slot (K/B = 0.03). users (d = 8, K/B = 0.03).
Fig. 4: Performance evaluation of the time-average sum AoI under varying system settings using the pNEUMA dataset. Max-Demand Trad-MW Trad-MaxDemand No-Update
3600
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(a) Time-average sum AoI versus communi- (b) Time-average sum AoI versus observation (c) Time-average sum AoI versus number of cation budget K/B (d = 8). delay d in slot (K/B = 0.06). users (d = 8, K/B = 0.06).
Fig. 5: Performance evaluation of the time-average sum AoI under varying system settings using the FLUID dataset.
Traditional Max-Demand: This policy schedules K areas with the highest traditional demand estimates, ÑbT (t), without accounting for local data sensing, reflecting the behavior of a conventional demand-based scheduler. • Stationary Randomized Policy: The stationary randomized benchmark developed in Section VI. • Mean-field Lower Bound: The analytical lower bound, C LB , derived in Section V. • No Update: This policy schedules no BS transmissions, where AoI depends on users’ local sensing only. •
B. AoI Evaluation Fig. 4a illustrates the time-average sum AoI under various communication budgets K/B. As expected, increasing the communication budget consistently reduces the AoI for all update-based policies by allowing more areas to be refreshed per slot. The proposed LocMW policy achieves the lowest AoI among all online policies, performing comparably to the benchmark with perfect estimation. Notably, it reduces the sum AoI by up to 31.6% compared to the Traditional Max-Demand baseline. This demonstrates that the online parameter estimation incurs negligible performance degradation. Although the stationary randomized policy outperforms the heuristic baselines, it remains inferior to LocMW because it cannot adapt to instantaneous AoI states or dynamic local sensing
events. In contrast, the traditional max-weight policy is less effective because it neglects local-sensing-aware AoI dynamics, wasting limited bandwidth on redundant BS updates for areas already refreshed by onboard vehicular sensors. Furthermore, while Max-Demand effectively prioritizes areas with high user concentrations, its disregard for accumulated freshness degradation leads to suboptimal performance. Finally, the No Update scheme yields the highest AoI, underscoring the importance of BS updates in maintaining system-wide perception freshness. Fig. 4b evaluates the impact of the observation delay d. The AoI of LocMW exhibits only a slight increase as d grows, demonstrating its robustness to delayed state observations. This is because LocMW predicts unobserved system states and schedules areas based on the expected reduction in AoI, rather than relying only on stale observations. Moreover, the performance gap between LocMW and its perfect-estimation counterpart remains marginal across all values of d. The stationary randomized policy is insensitive to d as it relies on long-term statistics. Traditional Max-Weight and Traditional Max-Demand yield significantly higher AoI, indicating that ignoring the local sensing process or the accumulated AoI leads to suboptimal decisions under delayed observations. Fig. 4c shows the performance of the proposed policy versus the average number of users. As expected, the time-average
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LocMW LocMW(Perfect Est.) Trad-MW
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(a) mAP@50 versus the commu- (b) mAP@70 versus the communication budget K/B. nication budget K/B.
(a) mAP@50 versus the observa- (b) mAP@70 versus the observation delay d. tion delay d.
Fig. 6: The perception performance (mAP@50/mAP@70) versus the communication budget K/B with observation delay d = 8 slots.
Fig. 7: The perception performance (mAP@50/mAP@70) versus the observation delay d with communication budget K/B.
LocMW(Perfect Est.)
LocMW
Trad-MW
Max-Demand
Trad-MaxDemand
Fig. 8: Qualitative visualization of 3D object detection on the V2X-Sim dataset, comparing the proposed LocMW policy against baseline scheduling schemes. Red bounding boxes denote predictions, while green ones represent the ground truth (GT).
sum AoI increases for all policies as the user population grows. LocMW consistently achieves the best performance among the online policies and maintains a marginal gap to the theoretical lower bound across the whole range of user densities. Moreover, its performance advantage over Max-Demand and Traditional Max-Weight becomes more pronounced as the number of users increases, because the baselines’ inaccurate modeling of local sensing leads to increasingly inefficient utilization of the limited communication budget.
The simulation results on the FLUID dataset are shown in Fig. 5, which exhibit consistent trends with those observed in the pNEUMA dataset. Across all settings, LocMW consistently attains the lowest AoI among the online policies and performs comparably to its perfect-estimation counterpart. These results demonstrate that the proposed LocMW framework generalizes effectively to diverse real-world traffic traces, delivering robust AoI reduction under varying bandwidth budgets, observation delays, and traffic densities.
C. Perception Evaluation Fig. 6 presents the 3D object detection accuracy on the V2X-Sim dataset under varying communication budget K/B. As the communication budget increases, both mAP@50 and mAP@70 improve across all update-based schemes, as a greater volume of features can be refreshed in each slot. The proposed LocMW policy consistently achieves the highest mAP-improving mAP@70 by up to 16.3% relative to the Traditional Max-Demand baseline, and performs comparably to the perfect-estimation benchmark. In contrast, traditional Max-Weight and traditional MaxDemand exhibit inferior detection accuracy because they neglect local sensing-induced AoI reductions and tend to allocate bandwidth to grids that have already been refreshed locally. In addition, Max-Demand performs suboptimally as it prioritizes user density without explicitly accounting for accumulated information staleness. The No-Update baseline yields the poorest performance, confirming that BS updates are essential for collaborative perception. Fig. 7 shows the impact of observation delay on perception accuracy. As the delay increases, the mAP of most scheduling
13
policies degrades, given that the BS makes decisions based on increasingly outdated network-state information. Nevertheless, LocMW demonstrates remarkable stability across different delay periods and consistently outperforms the heuristic and traditional baselines. These results show that the proposed local-sensing-aware AoI scheduling framework benefits tasklevel perception quality. Fig. 8 provides qualitative visualization results of object detection on the V2X-Sim dataset. The proposed LocMW produces detection results that are more consistent with ground truth, showing better spatial alignment and fewer missed vehicles. In contrast, Trad-MW, Max-Demand, and TraditionalMaxDemand exhibit more incomplete or less accurate detections, particularly for vehicles located in peripheral or occluded regions. These visualizations confirm that localsensing-aware AoI scheduling improves not only information freshness but also downstream collaborative perception quality. IX. C ONCLUSION In this paper, we have investigated AoI minimization for infrastructure-assisted collaborative perception under limited downlink bandwidth. Unlike conventional AoI scheduling optimizations, the proposed framework explicitly accounts for vehicles’ local sensing capabilities, under which information freshness can be improved not only through BS broadcasts but also through users’ own perception. To capture this distinctive feature, we have modeled the interested-but-unobserving user population as an INAR(1) process and derived a closed-form characterization of the time-average aggregate AoI. Based on this characterization, we have established a mean-field lower bound and an optimal stationary randomized benchmark. We then proposed LocMW, a local-sensing-aware Max-Weight scheduling policy. We have also provided the theoretical analysis, showing that LocMW incurs only sublinear cumulative excess AoI relative to the optimal randomized policy. Extensive experiments on pNEUMA, FLUID, and V2X-Sim datasets have further demonstrated that LocMW consistently reduces information staleness and improves downstream 3D object detection accuracy compared with competing baselines. A PPENDIX A P ROOF OF L EMMA 1 Conditioned on the filtration Ft , the scheduling decision ub (t), the demanding user count Nb (t), and the individual AoI ab,i (t) are deterministic. Conversely, the indicators Xi (t) and the new arrivals ωb (t + 1) are independent of Ft . Thus, taking the conditional expectation E[· | Ft ] of the AoI evolution in (3) yields: b (t) h NX i E[Ab (t + 1)|Ft ] =E Xi (t)((1 − ub (t))ab,i (t) + 1)|Ft
which completes the proof. A PPENDIX B P ROOF OF L EMMA 2 We first analyze the worst-case scenario where the BS never schedules an update for area b, i.e., ub (t) ≡ 0 for all t. Given the physical bounds Nb (t) ≤ Nmax and µb ≤ µmax , applying Lemma 1 and the law of iterated expectations yields E[Ab (t + 1)] ≤ ρb E[Ab (t)] + ρb Nmax + µmax .
(57)
Defining C ≜ ρb Nmax + µmax and unrolling this recursion from t = 1, we obtain E[Ab (t)] ≤ ρt−1 b E[Ab (1)] + C
t−2 X
ρkb (58)
k=0
ρb Nmax + µmax ≜ M < ∞. ≤ E[Ab (1)] + 1 − ρb (a)
where inequality (a) holds because ρb ∈ [0, 1). Since any admissible scheduling policy ub (t) ∈ {0, 1} either maintains or reduces AoI compared to this passive baseline, the upper bound supt≥1 E[Ab (t)] ≤ M holds universally. Consequently, the time-averaged boundary difference satisfies 1
0 ≤ lim
T →∞ T
E[Ab (T + 1)] − E[Ab (1)] ≤ lim
T →∞
2M = 0, T (59)
which directly implies lim
1
T →∞ T
E[Ab (T + 1)] − E[Ab (1)] = 0,
(60)
completing the proof. A PPENDIX C P ROOF OF T HEOREM 1 Conditioning the aggregate AoI recursion in (3) on H(t), and noting that ub (t) is H(t)-measurable, gives Zb+ (t + 1) = E[Ab (t + 1)|H(t)] = ρb (1 − ub (t))Zb (t) + M̃b (t + 1).
(61)
Taking the time average of both sides yields zb = ρb (zb − xb ) + λb ,
xb =
λb − (1 − ρb )zb . ρb
(62)
Squaring both sides of (61) and considering (1 − ub (t))2 = 1 − ub (t) as ub (t) ∈ {0, 1}, we obtain Zb2 (t + 1) =ρ2b (1 − ub (t))Zb2 (t) + M̃b2 (t + 1) + 2ρb (1 − ub (t))Zb (t)M̃b (t + 1).
(63)
i=1
Let Qb (t) ≜ (1 − ub (t))Zb (t), we have
h i + E ωb (t + 1)|Ft
T T 1X 1X E[Qb (t)M̃b (t + 1)] − λb E[Qb (t)] T t=1 T t=1
Nb (t)
=
X
ρb (1 − ub (t))ab,i (t) + 1 + µb
i=1 =ρb 1 − ub (t) Ab (t) + ρb Nb (t) + µb ,
T
(56)
1X = E[Qb (t)(M̃b (t + 1) − λb )]. T t=1
(64)
14
Applying the Cauchy-Schwarz inequality to the right-hand term gives T i 1X h E Qb (t) M̃b (t + 1) − λb T t=1 !1/2 !1/2 T T X 1X 1 2 E[Q2b (t)] E[ M̃b (t + 1) − λb ] . ≤ T t=1 T t=1 (65) PT Under Assumption 1, limT →∞ T1 t=1 E[(M̃b (t+1)−λb )2 ] = 0. By Jensen’s inequality, we have E[Q2b (t)] ≤ E[Zb2 (t)] ≤ E[A2b (t)]. Considering the passive worst-case policy ub (t) ≡ 0 and the physical bounds Nb (t) ≤ Nmax and ωb (t) ≤ µmax , we can bound the conditional second moment of the one-step AoI evolution using Young’s inequality as E[A2b (t + 1)|Ft ] ≤ (1+δ)ρb (Ab (t)+Nmax )2 +(1+1/δ)µ2max for any δ > 0. Since ρb < 1, choosing a sufficiently small δ > 0 yields E[A2b (t + 1)] ≤ αb E[A2b (t)] + Cb for some constants αb < 1 and Cb < ∞. Unrolling this recursion guarantees supt≥1 E[A2b (t)] < ∞, which implies that E[Q2b (t)] remains bounded. Thus, from (64) and setting the left hand side of (65) to zero, we have T i 1X h E (1 − ub (t))Zb (t)M̃b (t + 1) T →∞ T t=1
lim
T
(66)
1X =λb lim E [(1 − ub (t))Zb (t)] = λb (zb − xb ). T →∞ T t=1 PT Define yb ≜ limT →∞ T1 t=1 E[Zb2 (t)] and wb ≜ P T limT →∞ T1 t=1 E[ub (t)Zb2 (t)]. By taking the time-average expectation of (63) and invoking Assumption 1, we obtain yb = ρ2b (yb − wb ) + 2ρb λb (zb − xb ) + λ2b .
(67)
Rearranging the terms and using ρb (zb − xb ) = zb − λb from the first-moment equality, this simplifies to (1 − ρ2b )yb + ρ2b wb = 2ρb λb (zb − xb ) + λ2b = 2λb zb − λ2b . (68) Moreover, Jensen’s inequality gives yb ≥ zb2 and Cauchy– Schwarz inequality gives wb ≥ x2b /pb . Substituting the expression for xb results in 2 λb − (1 − ρb )zb 2 2 (1 − ρb )zb + ≤ 2λb zb − λ2b . (69) pb Solving it for zb establishes the lower bound: zb ≥ Lb (pb ) = λb
pb + 1 . pb (1 + ρb ) + 1 − ρb
(70)
Since E[Zb (t)] = E[Ab (t)], we have zb = Āb in the timeaverage sense. Therefore, every admissible policy satisfies B X b=1
Āb ≥
B X
Lb (pb ).
(71)
b=1
Minimizing the right-hand side over all feasible pb gives C LB . The Karush-Kuhn-Tucker (KKT) stationarity condition is γ=
2λb ρb pb (1 + ρb ) + 1 − ρb
2 .
(72)
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