Scheduling to Minimize Age of Information in Multi-State Time-Varying Networks with Power Constraints

Haoyue Tang; Jian Song; Jintao Wang; Linqi Song

REVIEW 3 major objections 5 minor 31 references

Scheduling to Minimize Age of Information in Multi-State Time-Varying Networks with Power Constraints

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a simple truncated scheduling policy — transmit when a user's age exceeds a per-channel threshold, and cap simultaneous transmissions at M — achieves average Age of Information within O(1/√N) of the optimal value as…

desk verdict A genuinely useful LP-based framework for AoI scheduling with power constraints, but the advertised O(1/sqrt(N)) optimality is not yet rigorously backed because the threshold-structure theorem is deferred and the proof gaps are non-trivial. read the letter →

arxiv 1908.01334 v6 pith:WZYS3IS7 submitted 2019-08-04 cs.IT cs.NImath.IT

Haoyue Tang , Jintao Wang , Linqi Song , Jian Song This is my paper · ORCID

classification cs.ITcs.NImath.IT

keywords AgeofInformationconstrainedMarkovdecisionprocessthresholdstructureopportunisticschedulingpowerconstraintbandwidthasymptoticoptimalitytime-varyingchannels

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how a central controller should schedule updates from many battery- and bandwidth-limited users over wireless channels that vary randomly from slot to slot, so that the average age of the freshest information stays small. The paper proves that the problem can be broken into per-user subproblems once the hard bandwidth limit is relaxed, that each subproblem's optimal policy has a simple threshold form (transmit only when the information is old enough or the channel is good enough, with thresholds ordered by channel quality), and that a truncated multi-user rule that simply caps the number of simultaneous transmissions is asymptotically optimal: as the number of users grows with the bandwidth fraction held fixed, the average age it achieves approaches the relaxed-problem lower bound at rate O(1/√N). The practical consequence is a computationally tractable way to design near-optimal freshness-aware schedules in large networks, with the scheduler using good channels for power-starved users and keeping well-powered users updated often enough to fill the bandwidth.

What carries the argument

The load-bearing object is the threshold-structured stationary randomized policy for the decoupled single-user CMDP, together with the truncated multi-user policy built from it. In the single-user problem the state is (x, q), the current age and channel state; the threshold property says transmission is optimal exactly when age x is at least τ_q, with τ_1 ≤ ⋯ ≤ τ_Q, and at most one state randomizes. This structure turns the Markov decision process into a finite linear program whose variables y_{x,q} = μ_x η_q ξ_{x,q} are the stationary probabilities of being in each state and transmitting. The multi-user policy then schedules all eager users when at most M qualify and otherwise selects M eager users uniformly; the proof of asymptotic optimality bounds the AoI loss from these truncation events by a concentration bound on the number of eager users, giving the O(1/√N) gap.

What would settle it

Simulate the truncated policy with M/N = 1/5 on the paper's four-state i.i.d. channel model for N = 100, 400, and 1600, measuring (J(π̃) − AoI_LB)/AoI_LB; Theorem 3 predicts this normalized gap decays like 1/√N, so a plateau or growth in the gap would refute asymptotic optimality.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3: for a network of N users with M = θN schedulable slots per time step, the expected average Age of Information under the proposed truncated policy — schedule every user whose individual policy says 'transmit', and if too many qualify, pick M of them uniformly at random — differs from the optimal value of the original bandwidth- and power-constrained problem by at most O(1/√N). Hence, as N grows, the truncated policy is asymptotically optimal. The path to this result is a chain of reductions: relax the hard bandwidth constraint to a time-average constraint, decouple the users through a Lagrange multiplier W, show the single-user constrained Markov decision process has an optimal stationary randomized policy with threshold structure (for each channel state q there is a threshold τ_q such that the user transmits whenever its age is at least τ_q, and τ_q is smaller for better channels), solve the single-user problem exactly as a linear program, and then use concentration of the number of 'eager' users to control the loss from truncation.

Load-bearing premise

The entire chain — decoupling, threshold structure, and the O(1/√N) concentration bound — assumes each user's channel state is independent and identically distributed from slot to slot with a known distribution; if channels are temporally correlated, the age-and-channel state no longer summarizes the user and the theorem's scope ends.

Editorial extensions

If this is right

For large N with fixed bandwidth fraction θ, network operators can implement the truncated policy and be guaranteed average AoI within O(1/√N) of the relaxed-problem lower bound.
Per-user thresholds can be computed offline by solving a small linear program for each distinct channel-distribution and power-budget type, so the online scheduler is simple: compare current age to a precomputed threshold and pick a subset of eager users.
The scheduler automatically differentiates users: power-limited users transmit mainly in good channel states, while power-rich users transmit often and at low age, filling the bandwidth.
The asymptotic-optimality result holds for heterogeneous users because the decoupling is per-user; the proof does not require identical channel distributions or power budgets across users.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper: if the channel process is Markovian rather than i.i.d., the same decomposition could in principle be carried out with the age augmented by channel memory, but the thresholds would become functions of the channel-state history and the O(1/√N) concentration argument would need a new bound; the paper's result does not cover this case.
Beyond the paper: the proof of Theorem 3 bounds the gap using the worst-case threshold span and the worst-case channel probability; a tighter bound is likely available by exploiting the actual distribution of the eager set, which would shrink the constant while preserving the 1/√N rate.
Beyond the paper: the threshold-based policy suggests a natural online-learning extension when channel distributions are unknown — estimate η and adapt the LP thresholds periodically; this paper assumes known distributions.
Beyond the paper: the same LP-with-truncation construction should apply to other freshness metrics, such as peak age, as long as the single-user Bellman equation retains monotonic structure; the paper only treats average AoI.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper does something real: it takes AoI scheduling out of the two-state, identical-user world and gives a tractable LP reformulation for Q-state i.i.d. fading with per-user average power budgets and a hard bandwidth limit. The decoupling into single-user CMDPs and the linear programming in Theorem 1 are clean and, as far as I can tell, correct. The dual search and the truncated policy are natural. Simulations are consistent with the theory. If those were the whole story, this would be a solid journal paper.

The soft spots are in the proofs, and they are not cosmetic. Lemma 1 has a typo in inequality (37): the discount factors are misplaced. The intended inequality is true, so that one is repairable. The bigger issue is Theorem 2. The tight-power case is dismissed with a pointer to [21, Theorem 5] without checking that its queueing-model assumptions transfer to this AoI CMDP. Corollary 2 gives monotonicity of the randomized scheduling probabilities, but monotonicity alone does not imply a single threshold per channel state; there could be many fractional entries. That matters because Appendix C leans directly on the single-threshold structure to bound the probability that an eager user stays unscheduled and to define Γ_n. The claim that τ_{n,Q} stays bounded as N grows is also asserted rather than proved. And the round-robin lower bound in (48) is simply wrong as stated—round robin isn't feasible under power constraints—though you can replace it with the trivial bound J≥1 and the asymptotic argument still goes through.

None of this makes me think the central idea is wrong. The framework is plausible and likely fixable. But the current version overstates what has been proved: the O(1/sqrt(N)) gap is not yet supported, and the threshold structure in the tight-power case is an unverified borrowing. The i.i.d. channel assumption is a scope condition rather than a flaw; it should just be stated more prominently.

Who gets value from this: people working on AoI scheduling, CMDP-based cross-layer design, and asymptotic analysis of constrained schedulers. It advances an active subfield rather than opening a new one. I would send it to peer review, but with a request for major revisions, specifically a self-contained proof of Theorem 2 (or a verified reduction to [21]) and a corrected Appendix C. The LP framework alone justifies refereeing it.

Referee Report

3 major / 5 minor

Summary. The paper considers a slotted wireless network in which a central controller collects status updates from N users over multi-state fading channels. Each user has an average power constraint, and at most M users can be scheduled per slot. The objective is to minimize the long-term average Age of Information per user. The authors relax the hard per-slot bandwidth constraint to a time-average constraint, form a Lagrangian with multiplier W, and decouple the problem into N single-user constrained Markov decision processes. They claim a dual threshold structure for the optimal single-user policy, reformulate each single-user CMDP as a linear program, and then construct a truncated scheduling policy that respects the hard bandwidth constraint. The main theoretical result is Theorem 3: with M/N = θ held constant, the per-user AoI gap between the truncated policy and the relaxed lower bound is O(1/√N), so the truncated policy is asymptotically optimal. Simulation results compare the proposed policy with a greedy baseline and illustrate threshold-type scheduling behavior.

Significance. If the main claims are correct, the paper offers a tractable LP-based solution to a meaningful cross-layer AoI problem and an asymptotically optimal scheduling rule under a hard bandwidth constraint. The LP reformulation in Theorem 1 and the steady-state balance equations appear sound, and the asymptotic optimality statement is concrete and falsifiable. The simulations support the qualitative claims about channel-aware scheduling for power-limited users. The main weakness is the incomplete proof of the threshold structure used in the asymptotic argument: Theorem 2 is not established for the tight-power case, and Appendix C relies on that structure in several places. The paper also contains a repairable algebraic error in the proof of Lemma 1. The central idea is promising, but the current version is not self-contained enough for the claimed Theorem 3.

major comments (3)

[Appendix A, Eq. (37)] The displayed inequality chain in Eq. (37) is algebraically inconsistent. From Eq. (36) the correct implication for x′ > x is α Σ η_{q′} V_α(x′+1,q′) ≥ α Σ η_{q′} V_α(x+1,q′) ≥ α(λω(q)+W)+α Σ η_{q′} V_α(1,q′). The printed line omits the discount factor from parts of the right-hand side and asserts α Σ η_{q′} V_α(x′+1,q′) > Σ η_{q′} V_α(x+1,q′), which is not a consequence of monotonicity because α<1. Since Lemma 1 is the basis for the threshold-structure claims, this step must be corrected.
[Section III-D, Theorem 2] The proof of Theorem 2 does not establish the claimed single-threshold structure. Corollary 2 supplies only monotonicity of ξ*_{x,q} in x and q. By Corollary 1, the optimal stationary randomized policy is a mixture of two deterministic threshold policies; if those two policies have different threshold vectors, then ξ*_{x,q} takes a constant fractional value on an interval of x, so the assertion that randomization occurs only at a single x per channel state does not follow. The tight-power case is deferred to [21, Theorem 5] without verifying that the queueing and arrival assumptions of [21] transfer to the AoI CMDP in Eqs. (11)-(12). This matters because Appendix C uses the threshold structure both to bound x_n(t) ≤ τ_{n,Q} and to control the probability that an eager user remains eager in subsequent slots.
[Appendix C, Eqs. (44)-(46)] The O(1/√N) bound contains several unproved assertions that are load-bearing for Theorem 3. First, the quantity p in the paragraph before Eq. (42) is never defined, and the subsequent construction of z = (N-M)/N + (M/N)(1-η_{n,1}) is not derived from a precise conditional probability over the event that the relaxed policy would schedule the user. Second, Eq. (46) asserts that τ_{n,Q} and Γ_n do not grow with N for users with fixed power constraints; this is not proved and depends on the behavior of the Lagrange multiplier W as N grows. Third, the concentration bound [31] is applied to ||Ω(t)|−Ω|, but the independence assumptions underlying that bound should be stated explicitly. The asymptotic optimality claim therefore needs a more rigorous proof or clearly stated extra conditions.

minor comments (5)

[Eq. (27)] The subgradient formula d_W g(W(k)) should contain the factor 1/N if g(W) is defined as in Eq. (25); the printed formula (1/N)Σ A_n(W(k)) - M is missing the 1/N. This does not change the zero set but it affects the step-size interpretation in the subgradient update.
[Eq. (16b)] The definition of β_x should be Σ_{q=1}^Q η_q ξ_{x,q}; the subscript x on ξ is missing in the displayed equation.
[Eq. (9) and Fig. 4] The notation WMN in Eq. (9) should be written as W M/N, and in Fig. 4 the caption appears to have a typo: “ρ = {0.2, 0.4.1.4, 1.6}” should be “ρ = {0.2, 0.4, 1.4, 1.6}”.
[Eq. (44c)] The expressions for z and 1/(1-z) should be reconciled: from the displayed z one obtains z = 1 - θ η_{n,1}, whereas the text writes 1/(1-z) = 1/(min_n η_{n,q}(1-(N-M)/N)). The current notation is confusing and should be cleaned up.
[Reference [31]] The concentration step in Eq. (45) would be clearer if it used a standard Berry-Esseen or Lyapunov CLT statement for sums of independent bounded variables, and if the independence of the indicators s_n(t) across users were stated explicitly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the relaxed lower bound is computed independently via LP and the truncated-policy gap is measured against that bound, not fitted or self-referentially defined.

full rationale

The central derivation is self-contained against external benchmarks. The relaxed RB&P-Constrained AoI problem (Problem 2) is obtained by replacing the per-slot hard bandwidth constraint (7b) with a time-average constraint (8b), and its decoupled per-user CMDP (Problem 3) is solved as an LP (Theorem 1). The lower bound AoILB in Eq. (32) is therefore an independent computed benchmark, not a fitted parameter renamed as a prediction. The truncated policy is then compared with that lower bound in Theorem 3/Appendix C; the O(1/sqrt(N)) gap is derived from a concentration argument in [31], which is external to this paper. The threshold-structure results (Lemma 1, Corollary 2, Theorem 2) are established in the paper for the deterministic case and, for the tight-power randomized case, are delegated to the external result [21, Theorem 5]; even if that delegation is a proof gap (monotone xi matrices alone do not obviously force a single randomized threshold), reliance on an external theorem is not circularity. No parameter is fit to the AoI curve being 'predicted,' and no load-bearing self-citation is used. The proof gap in Appendix C is a correctness risk, not a circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard CMDP theory and a concentration bound, plus two domain assumptions (i.i.d. channels and lossless transmission). The most paper-specific axiom is the borrowing of the threshold-randomization structure from [21] in the tight-power case; it is not proved in this preprint. No new physical entities are introduced, and no constants are fitted to the simulation data.

free parameters (2)

Xmax (AoI truncation bound) = unspecified (stated as chosen large enough to contain thresholds)
The LP (20) requires a finite AoI state space; the paper states in Section III-D that a large Xmax is chosen with Xmax >= tau_Q, but gives no constructive rule, so the reported optimum may depend on this hand-chosen truncation.
Subgradient stepsize gamma_k = not given (chosen similar to [7])
The dual search (28) requires a stepsize sequence; the paper references [7] instead of specifying gamma_k or convergence criteria beyond |W(k)-W(k+1)| < epsilon, so implementation requires guessing or borrowing another paper's schedule.

assumptions (6)

standard math For the decoupled CMDP, an optimal stationary randomized policy exists and is a mixture of no more than two stationary deterministic policies (Corollary 1 via [28, Theorem 4.4]).
Invoked in Section III-C to justify the structure of the optimal single-user policy; standard CMDP theory, not proved in the paper.
domain assumption The threshold structure for the tight-power-constraint case follows by 'a similar manner' to [21, Lemma 2 and Theorem 5].
Theorem 2's proof, Section III-D, defers the at-most-one-randomized-state property to a prior paper on delay-optimal scheduling without establishing that the AoI cost and power constraint satisfy the same conditions.
standard math Averaging over the alpha-discounted optimal policies and taking a sequence alpha_k to 1 recovers the average-cost optimal policy ([30]).
Used in Appendix A to extend the threshold structure from discounted to average cost; standard but infinite-state MDP conditions are not verified.
standard math The number of eager users |Omega(t)| concentrates as E[1/N ||Omega(t)|-E[|Omega(t)|]|] = O(1/sqrt(N)) ([31]), which requires the per-user scheduling decisions to be nearly independent.
Used in Appendix C Eq. (45); independence holds under the relaxed problem because the single-user policies are decoupled, but the paper does not justify the i.i.d./independence condition explicitly.
domain assumption Each user's channel state is i.i.d. over slots with known distribution eta_{n,q}, independent across users.
Section II-A, Eq. (2); this makes (x,q) a sufficient state for the CMDP and underpins the threshold and concentration proofs.
domain assumption Transmissions are always successful when a scheduled user spends omega(q) power; there is no packet loss or estimate error.
Section II-A states 'The transmitted packet will be successfully received by the central controller at the end of the slot.' This keeps AoI evolution deterministic on success and is not relaxed or discussed.

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Pith. "Pith review of Scheduling to Minimize Age of Information in Multi-State Time-Varying Networks with Power Constraints." pith.science (2026). https://pith.science/paper/WZYS3IS7

@misc{pith2026190801334,
  author       = {Pith},
  title        = {Pith review of: Scheduling to Minimize Age of Information in Multi-State Time-Varying Networks with Power Constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZYS3IS7}},
  note         = {Machine review of arXiv:1908.01334}
}

read the original abstract

In this paper, we study how to collect fresh data in time-varying networks with power constrained users. We measure data freshness from the perspective of the central controller by using the metric Age of Information, namely the time elapsed since the generation time-stamp of the freshest information. We wonder what is the minimum AoI performance the network can achieve and how to design scheduling algorithms to approach it. To answer these questions when scheduling decisions are restricted to bandwidth constraint, we first decouple the multi-user scheduling problem into a single user constrained Markov decision process (CMDP) through relaxation of the hard bandwidth constraint. Next we exploit the threshold structure of the optimal policy for the decoupled single user CMDP and obtain the optimum solution through linear programming (LP). Finally, an asymptotic optimal truncated policy that can satisfy the hard bandwidth constraint is built upon the optimal solution to each of the decoupled single-user sub-problem. The performance is verified through simulations. Our investigation shows that to obtain a small AoI performance, the scheduler exploits good channels to schedule users supported by limited power. Users equipped with enough transmission power are updated in a timely manner such that the bandwidth constraint can be satisfied.

Figures

Figures reproduced from arXiv: 1908.01334 by the authors.

**Figure 1.** Illustrative of the probability transfer graph for a stationary randomized policy. The circles denotes the AoI and the square denotes [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 3.** Asymptotic average AoI performance as a number of users [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 2.** Average AoI performance as a number of users [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Scheduling decisions for users with different power constraint [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Age-minimal online policies for energy harvesting sensors with random battery recharges,

A. Arafa, J. Yang, and S. Ulukus, “Age-minimal online policies for energy harvesting sensors with random battery recharges,” in 2018 IEEE International Conference on Communications (ICC) , May 2018, pp. 1– 6

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Optimal transmission for energy harvesting nodes under battery size and usage constraints,

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