{"id":"e7901470-5102-475c-9412-f803d867c147","arxiv_id":"2509.10475","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"A Lyapunov-based greedy offloading algorithm is claimed to reduce long-term energy-and-delay cost while stabilizing multi-edge-server queues, but the convergence proof is not sound as written.","lead":"An edge-computing paper proposes LDSO, a Lyapunov-optimization rule for deciding which nearby server should process each service request, balancing cost against queue stability. It reports 10% lower cost than two baselines, but the proof of the optimality guarantee contains a reversed inequality, so the result is not yet established.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The optimality guarantee depends on a stationarity assumption the paper never states, and Theorem 2's proof uses Eq. (38) in the wrong direction, so the O(ε) claim is unsupported even under ideal assumptions.","rationale":"The reader's REJECT verdict is well supported, and I agree that the unstated stationarity assumption is load-bearing: the Lyapunov drift-plus-penalty framework's optimality guarantees require a stationary randomized policy with cost C*, and the paper's time-varying request-probability model never provides that foundation. I mark agreement as partial rather than full because the most directly checkable flaw is internal: Theorem 2's proof reverses an inequality, and Theorem 1's per-service decomposition changes the objective. Even if one granted stationarity, the derivation as printed does not establish the advertised O(ε)/O(1/ε²) convergence. The experimental section gives real evidence for good empirical behavior, but it does not repair the theoretical argument, and no code, data, or confidence intervals are provided to test the 10%/18.75% claims independently. I would therefore keep the reader's REJECT rather than upgrade to a stronger or weaker verdict: the central contribution is a theoretical guarantee, and that guarantee is not established by the submitted proof.","tokens_in":17133,"tokens_out":9067,"duration_ms":109387,"concrete_test":"Analytically re-derive Theorem 2 while tracking inequality directions: from Eq. (38) derive only C* − O(ε) ≤ \\bar C(t), not \\bar C(t) ≤ C* + O(ε); then check whether replacing the invalid step with the standard telescoping inequality from Eq. (39) yields the claimed convergence time. Separately, instantiate the Section 4.1 model with a two-state periodic P_k(t) and two servers, compute LDSO's limiting time-average cost for both phases, and compare against C* from the time-averaged stationary problem: if the phase matters, the bound in Theorem 3 fails without a stationarity/ergodicity assumption.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that LDSO solves (1) with an O(ε) approximation, O(1/ε²) convergence, and average cost ≤ C*+B/V. These guarantees come from Neely's drift-plus-penalty framework, which requires the controlled random processes—A_i(t), μ_i(t), r(t)—to be stationary/ergodic so that a stationary randomized policy with optimal cost C* and strong stability exists. Section 4.1 instead adopts a time-varying request-probability model P_k(t), and no stationarity or ergodicity condition is stated or proved. Without it, C* in Theorems 2 and 3 is not well-defined, and the O(1/ε²) convergence argument lacks a target. The proof trouble is not merely an omitted assumption. In Theorem 2, Eq. (38) states C* ≤ \\bar C(t) + Σ_i δ_i E{Q_i(T)}/T, which bounds \\bar C from below, yet Eq. (49) concludes \\bar C(t) ≤ C* + O(ε), an upper bound. That reversal is invalid. The standard telescoping argument around Eq. (39) can give a B/V-style cost bound, but it does not establish the O(ε) approximation or the O(1/ε²) convergence time claimed via the supporting-hyperplane argument. Additionally, Theorem 1 replaces the aggregate drift penalty Q_i(t)A_i(t) with Σ_k Q_i^k(t)A_i^k(t), weighting each arrival by its own per-service queue rather than the server's total backlog; the two objectives are not generally equal, so the per-slot minimization analyzed is not exactly the drift-plus-penalty minimization used in the proofs. The simulations are suggestive and the empirical comparisons may survive, but the theoretical core—a stated contribution—is not supported as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies cost minimization in a horizontally collaborative edge-computing system. It formulates a long-term stochastic optimization problem (Eq. 1) with buffer-stability constraints, constructs an energy-plus-delay cost model (Eq. 2 and Section 4.2.2), and proposes LDSO, an online Lyapunov-optimization-based algorithm with a greedy matching heuristic (Algorithm 1). The paper claims an O(ε)-approximation with O(1/ε²) convergence (Theorem 2), an average-cost bound C*+B/V (Theorem 3), strong stability of the buffer queues (Theorem 4), and empirical improvements of about 10% lower cost and 18.75% more offloaded data over DSARA and MECNC (Section 6).","tokens_in":17475,"tokens_out":7022,"duration_ms":78590,"significance":"If the theoretical claims were correct, LDSO would be a useful contribution: it would show that a standard drift-plus-penalty construction can be adapted to multi-hop horizontal offloading with a tunable cost/stability trade-off, and the greedy matching would be an implementable per-slot solver. The paper also ships a reasonable level of model detail: multi-hop transmission, per-server queues, energy and delay components, and a Shanghai Telecom trace study. Credit goes to the authors for making the model and algorithm concrete. However, the central theoretical guarantees are not established. The missing stationarity premise, the mismatch between the aggregate and per-service drift terms, the reversed inequality in the proof of Theorem 2, and the non-standard and unproved stability statement in Theorem 4 are load-bearing. Absent these proofs, the paper is an empirical study of a heuristic with a plausible Lyapunov-inspired design, not a validated solution to Eq. (1).","major_comments":[{"comment":"The drift-plus-penalty machinery of [13] requires the controlled stochastic primitives A_i(t), μ_i(t), r(t) to be stationary (typically i.i.d. over slots or at least ergodic) so that a stationary randomized policy with average cost C* exists. This assumption is never stated. Instead, Section 4.1 uses a time-varying service-request probability model P_k(t) from [48], and the simulations use non-stationary scenarios. Without stationarity or ergodicity, C* in Theorems 2 and 3 is not well-defined, and Eq. (29) has no target to converge to. This is a load-bearing gap, not a technicality.","section":"Section 4.1, Theorems 2-3"},{"comment":"The upper bound analyzed in Lemma 2, Eq. (26), contains the aggregate term Σ_i Q_i(t) A_i(t), where Q_i(t) and A_i(t) are the total queue length and total arrival at server i. Theorem 1, Eq. (27), replaces this with Σ_k Q_i^k(t) A_i^k(t), and Algorithm 1 optimizes the per-service expression. In general (Σ_k Q_i^k)(Σ_k A_i^k) ≠ Σ_k Q_i^k A_i^k, so the per-slot problem actually solved is not the minimization of the drift-plus-penalty upper bound used in the proofs. The optimality and stability bounds in Theorems 2 and 3 therefore do not apply to the implemented objective.","section":"Section 5.1, Lemma 2 and Theorem 1"},{"comment":"Eq. (38) states C* ≤ \\bar C(t) + Σ_i δ_i E{Q_i(T)}/T, a lower bound on \\bar C(t). Eq. (49) then concludes \\bar C(t) ≤ C* + O(ε), which is the reverse inequality and does not follow. The intermediate chain in Eqs. (40)-(43) bounds E{L(T)} in terms of ∥E{Q(T)}∥ and Eq. (45) bounds ∥E{Q(T)}∥, but none of these steps yields an upper bound on the average cost \\bar C(t). Thus the O(ε) approximation and the claimed O(1/ε²) convergence time of Theorem 2 are unsupported, even if a stationarity assumption were added.","section":"Theorem 2 proof, Eqs. (38)-(49)"},{"comment":"Strong stability of a queue is standardly defined as limsup_{T→∞} (1/T) Σ_{t=0}^{T-1} E{Q_i(t)} < ∞, as used in constraint (1a). Eq. (54) instead asserts limsup (1/T²) Σ_t Σ_i ½ Q_i(t)² < ∞, which is a different and weaker statement. Moreover, the proof is algebraically invalid: a bound on the endpoint term E{L(T)}−E{L(0)} cannot be converted to Eq. (55)'s bound on the time average of L(t) without additional assumptions; Eq. (55) also contains an unexplained factor T on the right-hand side. Theorem 4 does not establish Eq. (1a).","section":"Section 5.3, Theorem 4"},{"comment":"The proofs of Theorems 2 and 3 presuppose that the per-slot decision minimizes the drift-plus-penalty expression. Algorithm 1 is a greedy matching heuristic (lines 12-19), and no lemma shows that it computes the minimizer of {C_i^k(t)} or even a constant-factor approximation to it. Without such a result, the theoretical trade-off is not tied to the implemented LDSO, and the simulations cannot be regarded as verifying Eq. (29) or Eq. (50) for the actual algorithm.","section":"Section 5.2, Algorithm 1 and Theorems 2-3"}],"minor_comments":[{"comment":"Constraint (1c) is printed as KΣ n_i^k(t)b_k ≤ (Qmax_i − Q_i(t)) ≤ μ_i(t). The second inequality appears structurally odd and should be re-derived and restated.","section":"Eq. (1c)"},{"comment":"The denominator 'µi(t)− ∑ nk_i(t) u=1 λk_ubk' requires parentheses; as written, if the arrival sum exceeds μ_i(t), the expression becomes negative and M/M/1 stability is not enforced.","section":"Eq. (13)"},{"comment":"Figure 7(a,b) captions say 'System cost' while the text describes buffered data volume and the y-axis reads 'Data Backlogs'; the labels are inconsistent and should be corrected.","section":"Section 6, Figures 6-7"},{"comment":"The claims 'reduced cost is about an average of 10%' and '18.75 percent more offloaded data' are reported as point values without error bars, confidence intervals, or multiple simulation runs. Statistical support is missing.","section":"Section 6.2"},{"comment":"A_i(t) depends on P_k(t), but the time-varying request-probability model from [48] is not defined in this manuscript. The authors should either define P_k(t) or provide a self-contained reference to the relevant equations.","section":"Eq. (3)"}],"recommendation":"reject","confidential_remarks":"The manuscript has multiple central proof failures (reversed inequality in Theorem 2, invalid conversion in Theorem 4, per-service versus aggregate drift mismatch) and an unstated stationarity premise that undermines the whole DPP theory. These are not local presentation issues; they would require substantial rewriting of the core analysis. The simulation uses the same time-varying request model [48] that is proposed in the model, so the evaluation is not independent at that point. I recommend reject rather than major revision because the load-bearing theoretical guarantees are not merely incomplete; as written, the proofs contain direct logical inversions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a standard Lyapunov drift-plus-penalty offloading work with a legitimately new wrapper: multi-hop horizontal server cooperation, an energy-plus-delay cost, and a greedy matching rule that selects the cheapest per-service Lyapunov cost. That combination is not in the cited baselines, and the simulation setup looks real—Shanghai Telecom deployment, comparison against DSARA and MECNC, about 10% lower cost and 18.75% more offloaded data. There is a genuine engineering idea here, and the empirical story is worth taking seriously.\n\nThe theory, though, is not in publishable shape. The load-bearing O(ε) convergence claim in Theorem 2 is unsupported. Equation (38) states C* ≤ \\bar C(t) + Σ_i δ_i E{Q_i(T)}/T. That is a lower bound on \\bar C(t) given C*, not an upper bound. The proof then uses it to conclude \\bar C(t) ≤ C* + O(ε) in Eq. (49). That is reading the inequality in the wrong direction. The standard telescoping argument gives the B/V cost bound in Theorem 3, but it does not give the claimed O(ε) approximation or the O(1/ε²) convergence time.\n\nTheorem 4 has a different but related problem: it defines “strong stability” as a bound on (1/T²)Σ E{Q_i²}. The constraint the problem actually needs, and the standard notion in queueing theory, is a bound on the (1/T) time-average queue length. The stated bound does not imply that. Theorem 1 also drops cross terms: the drift-plus-penalty objective contains Σ_i Q_i(t)A_i(t), and replacing it with Σ_k Q_i^k(t)A_i^k(t) changes the objective unless only one service class is present. These are real flaws, not cosmetic.\n\nThere is also an unstated stationarity assumption. The drift-plus-penalty theorems require the primitives to be stationary/ergodic so that a stationary randomized policy with cost C* exists. Section 4.1 instead builds arrivals from the time-varying model of [48]. If that model is non-stationary, then C* in Theorems 2 and 3 is not well-defined. The self-citation itself is not the problem; the missing assumption is. And since the experiments generate arrivals from that same model, the simulations are partly a self-consistency check under the authors’ own process, not an independent validation. No code or data is provided, and the baselines are only cited, not specified.\n\nWho is this for? Someone working on Lyapunov offloading who wants to see a greedy matching variant in a multi-hop setting. It deserves a referee, not a desk rejection, because the model and empirical setup are competent and the theoretical mistakes are fixable in principle. But I would send it back with major revision, and if the proofs cannot be repaired, the O(ε) and “strong stability” claims need to be removed or downgraded to empirical observations.","headline":"Familiar Lyapunov offloading with a plausible greedy matching extension; the empirical story is reasonable, but the central O(ε) proof runs an inequality backwards and the stability theorem is nonstandard—major revision needed.","tokens_in":18077,"tokens_out":6753,"would_cite":false,"duration_ms":68936,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that LDSO, an online drift-plus-penalty offloading algorithm, solves the long-term cost minimization in Eq. (1) with an O(1/V) cost bound, strong queue stability, and roughly 10% lower cost and 18.75% more offloaded data th","keywords":["edge computing","service offloading","drift-plus-penalty optimization","queue stability","multi-hop collaboration","cost minimization","greedy matching"],"falsifier":"Run LDSO on a workload whose arrival rate grows over time (for instance, doubling halfway through the experiment) while keeping the control parameter fixed; if the average cost exceeds C*+B/V or queues grow without bound, the stationarity premise is violated. A second check is to compare LDSO against an exact offline optimal policy on a small M×K instance; if the cost gap is much larger than O(1/V), the greedy matching is losing the optimality that the theorem assumes.","tokens_in":16946,"feed_emoji":"📡","tokens_out":7604,"duration_ms":75034,"temperature":0.7,"pith_summary":"The paper tackles a horizontal edge-computing problem: many small base stations cooperate to serve user requests, but each server has a finite buffer, and the right split of work across servers is unclear when requests, channel rates, and processing rates change over time. It models the system as queues plus a per-slot cost that mixes energy and delay, then proposes LDSO, an online drift-plus-penalty offloading rule that needs no knowledge of future arrivals. The central claim is that LDSO drives the long-run average cost to within O(1/V) of the optimal value while keeping the queues strongly stable, with an O(1/ε²) convergence time when V=1/ε. Experiments on a real trace show about 10% lower cost and 18.75% more offloaded data than two recent baselines, so the paper intends to establish that stability-aware greedy matching is both provable and practical.","feed_headline":"Lyapunov offloading cuts edge cost 10%, offloads 19% more","feed_subtitle":"An online offloading policy keeps server queues bounded while driving average cost toward C* + B/V.","key_machinery":"The machinery is the Lyapunov function cluster: a quadratic Lyapunov function L(t)=1/2 Σ Q_i(t)², its one-slot drift ΔL(t), and the drift-plus-penalty ΔL(t)+V·Cost(t). Lemma 2 bounds this by B+V·Cost(t)+Σ Q_i(t) A_i(t), which turns the long-term stochastic problem into a per-slot deterministic minimization. The per-service cost C_i^k(t) is the object that carries the greedy matching: LDSO repeatedly selects the edge-server/service pair with minimal C_i^k(t), enforces Q_i(t)+A_i^k(t)≤Qmax_i, and updates queues via Q_i(t+1)=max{Q_i(t)−μ_i(t),0}+A_i(t).","core_discovery":"The paper's central claim is that the long-term offloading cost minimization in Eq. (1), subject to queue stability and capacity constraints, can be solved online by minimizing, at each slot, a drift-plus-penalty expression. This yields a per-service cost C_i^k(t) = V θ E_i^k(t) + V(1−θ) T_i^k(t) + Q_i^k(t) A_i^k(t), and LDSO greedily picks the smallest such cost while respecting queue limits. Theorem 2 states that with ε=1/V and T≥1/ε² the time-averaged cost is within O(ε) of the optimal C*, and the arrival rate is within O(ε) of the processing rate. Theorem 3 bounds the average cost by C*+B/V, and Theorem 4 gives a strong-stability bound on the squared queue lengths. Simulated comparisons","pith_inferences":["Editorial extension: the proofs rely on drift-plus-penalty theory's stationarity assumption; under a non-stationary workload with trends in arrival rate, the O(ε) cost guarantee would need re-derivation, and a sliding-window or regret-based variant would be a natural test.","Editorial inference: the same drift-plus-penalty core works for any per-slot separable objective—replacing energy/delay terms with latency percentiles, carbon cost, or revenue would only change the terms inside C_i^k(t).","Editorial check: because the greedy matching is a heuristic, the paper's theoretical bound assumes the per-slot minimization is exact; comparing LDSO with an exact small-instance solver would separate the drift-plus-penalty benefit from greedy loss.","Editorial caution: the 18.75% offload improvement is measured on one urban trace; re-running on multiple traces with different request distributions would indicate how general the gain is."],"forward_implications":["An operator can tune V to trade cost against buffer occupancy: raising V pushes average cost down toward C* while allowing more data to sit in queues, and Theorem 4 keeps the time-averaged squared backlog bounded.","No statistical knowledge of arrivals is needed at runtime; LDSO only observes current queues, current request counts, and channel state, so it can run in changing conditions without retraining.","The O(1/ε²) convergence time means that after roughly 1/ε² slots the algorithm is within O(ε) of the optimal cost, which is useful for deciding how long to warm up.","The per-slot greedy matching is cheap, making the approach applicable to systems with many services and many edge servers.","If the queue cap is removed, cost improves further, so the model cleanly separates the price of stability from the cost of storage limits."],"supporting_citations":[{"why":"Provides the drift-plus-penalty framework and the standard cost/stability bound used in Theorems 2 and 3.","marker":"[13]"},{"why":"Supplies the time-varying service request probability model that generates the arrival process A_i(t).","marker":"[48]"},{"why":"DSARA, the first comparison baseline whose cost and backlog performance LDSO is measured against.","marker":"[41]"},{"why":"MECNC, the second comparison baseline, giving the mobile-edge network control trade-off that LDSO extends.","marker":"[44]"},{"why":"Provides the convergence-time analysis of drift-plus-penalty that Theorem 2 uses to state the O(1/ε²) result.","marker":"[49]"},{"why":"Supplies the supporting hyperplane theorem used in the optimality-gap part of Theorem 2.","marker":"[50]"}],"fun_headline_variants":["Lyapunov-based offloading trims cost and stabilizes queues","Edge offloading with Lyapunov control lowers cost, keeps queues bounded","Queue-aware offloading via Lyapunov reduces edge cost","Lyapunov-driven offloading: lower cost, stable queues in edge"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The proofs assume the random workload seen by each server is statistically the same from slot to slot, so a fixed long-run optimal policy exists; the paper relies on this through the drift-plus-penalty theorem without stating it, and if it fails the convergence and cost bounds do not automatically hold.","fun_headline_variants_meta":{"raw":{"variants":["Lyapunov-based offloading trims cost and stabilizes queues","Edge offloading with Lyapunov control lowers cost, keeps queues bounded","Queue-aware offloading via Lyapunov reduces edge cost","Lyapunov-driven offloading: lower cost, stable queues in edge"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001137,"raw_usage":{"total_tokens":4533,"prompt_tokens":694,"completion_tokens":3839,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":438,"completion_tokens_details":{"reasoning_tokens":3764}},"tokens_in":438,"tokens_out":3839,"duration_ms":27982,"temperature":1.0,"reasoning_tokens":3764,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:32:17.592344+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run LDSO on a workload whose arrival rate grows over time (for instance, doubling halfway through the experiment) while keeping the control parameter fixed; if the average cost exceeds C*+B/V or queues grow without bound, the stationarity premise is violated. A second check is to compare LDSO against an exact offline optimal policy on a small M×K instance; if the cost gap is much larger than O(1/V), the greedy matching is losing the optimality that the theorem assumes.","supporting_citations":[{"cited_title":"Stochastic network optimization wit h application to communication and queueing systems","cited_arxiv_id":null,"evidence_quote":"Provides the drift-plus-penalty framework and the standard cost/stability bound used in Theorems 2 and 3."},{"cited_title":"Maximizing the Capacity of Edge- Caching Networks With User-Content Evolution Relationshi p","cited_arxiv_id":null,"evidence_quote":"Supplies the time-varying service request probability model that generates the arrival process A_i(t)."},{"cited_title":"Service characteristics-oriented joi nt optimization of radio and computing resource allocation in mobile-edge c omputing","cited_arxiv_id":null,"evidence_quote":"DSARA, the first comparison baseline whose cost and backlog performance LDSO is measured against."},{"cited_title":"Mobile edge computing network control: Tradeoff be tween de- lay and cost","cited_arxiv_id":null,"evidence_quote":"MECNC, the second comparison baseline, giving the mobile-edge network control trade-off that LDSO extends."},{"cited_title":"A Simple Convergence Time Analysis of Drift-Plus-Penalty for Stochastic Optimization and Convex Programs","cited_arxiv_id":"1412.0791","evidence_quote":"Provides the convergence-time analysis of drift-plus-penalty that Theorem 2 uses to state the O(1/ε²) result."},{"cited_title":"Convex analysis and optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the supporting hyperplane theorem used in the optimality-gap part of Theorem 2."}],"review_version":1}