{"id":"5cf38b07-3008-42b1-b19b-bc22f791f12a","arxiv_id":"2506.07057","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"A method-of-moments estimator using cross-moments of queue-length snapshots at Poisson times recovers the routing matrix, arrival rates, and service-time information of an infinite-server queueing network.","lead":"This paper develops a statistical method to recover the routing structure of a network of infinite-server queues from periodic snapshots of the number of customers at each node. If the method works, it lets researchers infer who influences whom in a queueing system without tracking individual customers.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3 defines Q = Omega_1^{-1} Omega_2 but never proves Omega_1 is nonsingular; Theorem 1's identifiability step and hence consistency rest on an unstated regularity condition.","rationale":"The paper's central deliverable is Theorem 1: consistency of the method-of-moments estimator for (lambda, Q) from Poisson-sampled population snapshots with known service-time distributions. The proof has three pillars: pointwise convergence of empirical moments (Proposition 4), uniform convergence via Lipschitz continuity (Lemma 3), and identifiability from the moment equations (Proposition 3). The linchpin is Proposition 3: it converts the moment equations into the linear system Omega_1 Q = Omega_2 and inverts Omega_1. Without nonsingularity of Omega_1, the moment map is not provably one-to-one, and the 'if and only if' in Proposition 4(iii) collapses; the Glivenko-Cantelli argument in Theorem 1 then no longer has a unique zero to converge to. The manuscript asserts invertibility implicitly by writing the inverse, but no proof or condition is given. I checked the n=1 exponential case, where Omega_1 reduces to a nonzero scalar, so the step is not vacuous, but this does not generalize. The paper's own Section 5.1 admits the analogous identifiability problem is unresolved in the parametric service-time extension, which strengthens the concern that the base-case inversion is nontrivial. I do not find grounds to reject: the moment derivations are careful, the numerical experiments in Section 7 support the estimator's behavior in tested instances, and the missing invertibility could plausibly be supplied as a lemma under Assumptions 1-2. Hence the appropriate disposition is still CONDITIONAL, unchanged from the reader's verdict. The reader identified exactly this gap as the weakest assumption, and I agree.","tokens_in":25685,"tokens_out":13059,"duration_ms":127873,"concrete_test":"Independently re-derive the algebra of Section 4.2 and test invertibility. For n=1, with a non-exponential service time (e.g., deterministic or Erlang-2) and p in [0,1-epsilon], compute Omega_1 as a scalar and check whether it vanishes for some p and beta > 0; if it does, Proposition 3 fails already in one station. For n=2 with exponential service times, sample a dense grid of valid (Q, lambda) satisfying Assumptions 1-2 (including self-loops q_ii > 0) and beta > 0, and compute the smallest singular value of Omega_1(alpha0, alpha1) at each point; a zero or near-zero singular value is a direct counterexample. As a complementary analytic check, attempt to prove a uniform lower bound on sigma_min(Omega_1) over Theta; failure to find such a bound would confirm that an explicit invertibility assumption must be added to Theorem 1.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The consistency proof for Theorem 1 (Section 4.3) reduces to Proposition 4(iii): the moment equations E[Psi(theta)] = 0 have unique solution theta = theta0. That uniqueness is obtained from Proposition 3 (Section 4.2), which asserts Q(alpha0, alpha1) = Omega_1^{-1} Omega_2. The paper defines Omega_1 and Omega_2 after rearranging the stationary moment equation, but it never shows that Omega_1 is nonsingular on the parameter space Theta. No eigenvalue bound, no condition on beta or the service-time LSTs, and no argument that the rearranged linear system has full rank is supplied. If Omega_1 is singular for some admissible (Q, lambda), then the moment-based mapping is not one-to-one at that point; the equations do not determine Q, so the claimed consistency cannot hold for those parameters. This is an internal proof gap, not a dispute with existing consensus. The issue is highlighted by the paper's own Section 5.1, where the analogous parametric service-time identifiability is left open with 'deriving a counterpart to Proposition 3 may prove to be more challenging'; the base-case invertibility is asserted even more briefly. The positive numerical evidence (Section 7, Experiment 1) shows that Omega_1 appears invertible in the tested topologies, but this does not establish the uniform invertibility the theorem requires.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies statistical inference for a network of infinite-server queues observed only through the population vector at Poisson sampling epochs. The main object is a method-of-moments estimator for the routing matrix, external arrival rates, and, in extensions, service-time parameters and observation probabilities. The authors derive explicit expressions for first moments and cross-moments at exponential and Erlang inter-observation times, define an estimator from matching these moments, and prove consistency of the estimator when service-time distributions are known (Theorem 1). They then sketch extensions to parametric service-time estimation, a model-free version, and noisy observations, and support the method with numerical experiments on several topologies.","tokens_in":25968,"tokens_out":4646,"duration_ms":50542,"significance":"If the central consistency theorem is fully established, the paper would provide a practically attractive way to infer directed network structure from population snapshots alone, going beyond the stationary distribution and exploiting asymmetric cross-moments. The moment derivations in Sections 3 and 4.5 are careful and nontrivial, and the numerical experiments in Section 7 give concrete evidence that the method works on the tested topologies. The proof strategy, using a majorizing M/G/infinity process and Glivenko-Cantelli arguments, is appropriate. However, the main theorem currently rests on an unproved invertibility assertion, and the extended settings in Sections 5 and 6 are not supported by the same level of proof; these gaps are load-bearing for the paper's broader claims.","major_comments":[{"comment":"The proposition defines Q(alpha^[0], alpha^[1]) = Omega_1(alpha^[0], alpha^[1])^{-1} Omega_2(alpha^[0], alpha^[1]) and states that there is a one-to-one relation between the moments and the parameters, but no proof is given that Omega_1 is nonsingular on the parameter space. This is not a minor gap: Proposition 4(iii) and hence Theorem 1 in §4.3 use exactly this uniqueness to conclude that the moment equations have only the true parameter as their solution. The text before Proposition 3 only says the equation is linear in Q and then asserts the inverse. The numerical experiments in Experiment 1 show invertibility for particular topologies, but they do not establish uniform invertibility over Theta, and the consistency theorem is stated for every admissible parameter. The authors should either prove invertibility under Assumptions 1–2, add an explicit assumption of nonsingularity of Omega_1 over Theta, or restrict Theorem 1 accordingly.","section":"§4.2, Proposition 3"},{"comment":"The claimed extensions to unknown service-time distributions are not proved. In §5.1 the paper explicitly states that \"deriving a counterpart to Proposition 3 may prove to be more challenging,\" which means the identifiability step, the essential ingredient of the consistency proof, is missing for the parametric variant. Section 5.2 similarly asserts that \"one again obtains consistency\" under Lipschitz assumptions, but it does not prove that the moment mapping is injective in the model-free parameter vector of dimension n^2+3n; having enough moment equations does not by itself imply a unique solution. Since the abstract and introduction present these variants as part of the contribution, the absence of identifiability proofs leaves the main claims for Sections 5.1 and 5.2 unsubstantiated.","section":"§5.1 and §5.2"},{"comment":"The section on unknown observation probabilities derives moment formulas in Lemma 4 and states that the estimation procedure \"works essentially as before,\" but no consistency theorem is stated or proved for this setting. Given that the introduction highlights the estimation of observation probabilities as part of the main contribution, the paper should either provide a formal consistency result with the required identifiability conditions or clearly label Section 6 as heuristic. As written, the claim that the procedure estimates all model parameters including the observation probabilities is not backed by the paper's theoretical results.","section":"§6"},{"comment":"The efficient procedure replaces the model-based first moment rho by the empirical first-moment vector psi^[0] inside the cross-moment equations. This is a plug-in estimating procedure, not exactly the estimator defined by equations (13)–(14) in §4.1. Since Theorem 1 is proved for the latter estimator, the numerical procedure in Section 7.2 is not directly covered by Theorem 1. The authors should clarify the relationship, or prove consistency for the plug-in version, which should follow from continuity of the relevant maps but is not stated.","section":"§7.2"}],"minor_comments":[{"comment":"Just before Proposition 2, the text says \"Applying (10) in combination with Proposition 8\"; this should refer to Proposition 1.","section":"§3.3"},{"comment":"In the displayed equation near the end of Section 7.2, the matrix diag{p} appears even though the subsection assumes p=1; this appears to be a leftover from the noisy-observation setting and should be removed or explained.","section":"§7.2"},{"comment":"The caption of Figure 1 labels topology (d) as \"2+3 Cliques,\" while the text in Experiment 1 calls it \"3 + 2 Cliques\"; the labels should be made consistent.","section":"Figure 1 and §7.3"},{"comment":"The proof of Theorem 1 assumes the existence of a solution to the moment equations and does not address the possibility of multiple solutions or the need to select one; Remark 2 mentions truncation/projection for finite samples, but the theoretical statement would be cleaner if the estimator were defined as any measurable solution or via a well-defined minimization over Theta.","section":"§4.3"},{"comment":"The notation G_i is used both for the service-time random variable and for its cumulative distribution function; the distinction is mostly clear from context, but a note would help avoid confusion in the derivations of Sections 3 and 4.","section":"§2"}],"recommendation":"major_revision","confidential_remarks":"The central obstruction is the unproved nonsingularity of Omega_1 in Proposition 3; this is a correctable but essential point. If the authors can supply a proof or an explicit assumption, and if they downgrade the extended claims in Sections 5 and 6 to match what is actually proved, the paper could become a solid contribution. The numerical study is useful but does not compensate for missing identifiability arguments in the theoretical statements."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nHere's the short version. The paper's real contribution is showing that directed routing structure in an infinite-server network can be recovered from Poisson-sampled population counts alone, using closed-form moment inversion. That is genuinely new. The cross-moment formulas (Propositions 1 and 2) and the explicit inversion to (λ, Q) (Proposition 3) are not in the earlier literature, which needed arrival and departure observations. The paper also demonstrates something useful: it can tell apart directed topologies that share the same stationary distribution.\n\nWhat the paper does well: the consistency proof for the base case (known service-time distributions, known observation probabilities) is credible. The majorizing M/G/∞ coupling and Glivenko-Cantelli argument are appropriate. The numerical work is substantial—1,000 runs per topology, large-scale networks, and a model-free variant that remains accurate under service-time misspecification.\n\nThe soft spot is the one the stress-test flags: Proposition 3 defines Q = Ω1^{-1} Ω2 and claims a one-to-one moment-to-parameter map, but never proves Ω1 is nonsingular on the parameter space. Theorem 1's identifiability step inherits that gap. The numerics suggest Ω1 is invertible for the topologies tested, but that is not a uniform condition. The paper's own Section 5.1 concedes that the analogous inversion for parametric service times 'may prove to be more challenging,' which makes the brief assertion in the base case more conspicuous. Sections 5 and 6 are otherwise sketches: the parametric and censored-observation variants are not given full consistency proofs. Minor points: no code is released, and in Experiment 3 the λ estimates deviate notably from truth even though effective arrival rates match.\n\nNet assessment: the core idea is sound and the main theorem is very likely true under an added regularity condition. The paper deserves a serious referee, but the referee should require the invertibility condition to be stated and proved, or Theorem 1 restricted accordingly. If you work on queueing network inference or network tomography, this is worth reading and citing for the moment formulas.\n\nI'd send it to review with that condition attached.","headline":"New moment-inversion result for topology recovery in infinite-server networks is real and mostly solid, but Theorem 1 hangs on an unproved invertibility condition that needs to be supplied.","tokens_in":26537,"tokens_out":2747,"would_cite":true,"duration_ms":27756,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K25","62F12","90B22"],"pacs":[],"model":"deepseek-v4-flash","headline":"Snapshot counts alone reveal the routing structure of a queueing network.","keywords":["infinite-server queues","routing matrix estimation","method of moments","Poisson sampling","network topology inference","identifiability","consistency","cross-moments"],"falsifier":"Find a two-station network and parameter values satisfying Assumptions 1 and 2 for which the determinant of $\\Omega_1$ is zero while the moment system still has two distinct solutions $(\\boldsymbol{\\lambda}, Q)$, or compute $\\det \\Omega_1$ symbolically along a family of routing matrices and exhibit one that vanishes; either would give a concrete instance where the claimed identifiability fails.","tokens_in":25380,"feed_emoji":"🔀","tokens_out":6719,"duration_ms":74643,"temperature":0.7,"pith_summary":"This paper claims that the directed topology of a network of infinite-server queues can be recovered from nothing more than repeated snapshots of how many customers are at each station, taken at Poisson-distributed times. It proposes a method-of-moments estimator, built on explicit formulas for the mean population vector and for the cross-covariances between station populations one or two observation epochs apart, and proves that the estimator converges in probability to the true arrival rates and routing matrix as the number of snapshots grows, for any positive sampling rate. If true, this gives a statistical way to uncover who routes to whom, including link direction, without ever observing individual customers or their paths. The method also distinguishes networks that have the same stationary population distribution, because the lagged cross-covariances are asymmetric in the station indices.","feed_headline":"Snapshots alone reveal the routing structure of a queueing network","feed_subtitle":"A method-of-moments estimator recovers arrival rates and directed links from Poisson-time population snapshots.","key_machinery":"The machinery is the lag-one cross-moment matrix $\\mathbb{E}[\\boldsymbol{M}(0)\\boldsymbol{M}(T_\\beta)^\\top]$ for an independent exponential time $T_\\beta$ with rate $\\beta$. Its ingredients are $P(\\beta) = (I - \\mathrm{diag}\\{\\mathcal{G}(\\beta)\\}Q)^{-1}(I - \\mathrm{diag}\\{\\mathcal{G}(\\beta)\\})$, the probability that a fresh customer starting at one station is at another when the exponential clock rings, and its residual-service-time analogue $P^{(\\mathrm{res})}(\\beta)$; these enter through Proposition 1, and differentiating them in $\\beta$ gives the Erlang-2 cross-moments used when service-time parameters or observation probabilities must also be estimated. The routing matrix is then isolated as a solution of a linear matrix equation, $Q = \\Omega_1^{-1} \\Omega_2$, and the estimator plugs empirical moments into this expression, with projection back onto the parameter space when finite-sample estimates fall outside it.","core_discovery":"The central claim is that the moment equations $\\mathbb{E}[M_i(0)] = \\rho_i$ and $\\mathbb{E}[M_j(0) M_i(T_\\beta)] = [(\\boldsymbol{\\rho} \\boldsymbol{\\rho}^\\top + \\mathrm{diag}\\,\\boldsymbol{\\rho})\\, P^{(\\mathrm{res})}(\\beta) + \\beta^{-1} \\boldsymbol{\\rho} \\boldsymbol{\\lambda}^\\top P(\\beta)]_{ji}$ determine $(\\boldsymbol{\\lambda}, Q)$ uniquely, given the service-time Laplace-Stieltjes transforms. The paper derives these equations by conditioning on an independent exponential clock, defines the estimator as the solution of the empirical versions of these $n^2 + n$ equations, and proves consistency by combining pointwise almost sure convergence of the empirical moments, a Lipschitz and Glivenko-Cantelli argument on the compact parameter space, and identifiability through the linear relation $Q = \\Omega_1^{-1} \\Omega_2$. The consistency theorem is stated for every sampling rate $\\beta > 0$. A load-bearing step in the identifiability argument is the invertibility of $\\Omega_1$, which the paper uses to define $Q$; Section 4.2 does not supply a proof of non-singularity over the full parameter space.","pith_inferences":["Editorial extension: if the invertibility of $\\Omega_1$ is established for general topologies, the same moment-identification scheme should carry over to networks with batch arrivals or phase-type service times, since only Laplace transforms of service times enter the formulas.","Editorial extension: the asymmetry of the cross-moments suggests a general principle for causal discovery from aggregate counts: two snapshots at a known random lag can orient directed interactions even when the marginal distribution at each node is the same.","Editorial extension: for finite samples, the need to project estimates back into the parameter space could bias small-sample estimates, so resampling-based confidence intervals are a natural next step that the paper does not develop.","Editorial extension: the model-free variant, which treats $\\mathbb{E}[G_i]$, $\\mathcal{G}_i(\\beta)$, and $\\dot{\\mathcal{G}}_i(\\beta)$ as free parameters, points toward a nonparametric misspecification test: if estimates from a parametric fit and from the model-free fit diverge, the assumed service-time family is wrong."],"forward_implications":["For known service-time distributions, $n^2 + n$ equations from means and one-step cross-moments exactly determine the $n^2$ routing probabilities and $n$ external arrival rates, with no individual customer tracking needed.","The estimator is consistent for any sampling rate $\\beta > 0$, so even sparse or slow Poisson snapshots suffice as the number of observations grows.","Because the cross-moments are not symmetric in the station indices, the method can orient edges and can tell apart networks that share the same stationary population distribution.","When service-time parameters or observation probabilities are unknown, adding Erlang-2 lag-two cross-moments supplies enough equations for consistency under the paper's regularity conditions.","In numerical experiments on line, circle, symmetric-circle, and clique topologies, the procedure recovers the routing structure, with empirical variances small relative to the estimated values."],"supporting_citations":[{"why":"Supplies the asymptotic-statistics backbone: identifiability definitions, Glivenko-Cantelli/uniform-convergence argument, and the theorem used to conclude consistency.","marker":"[16]"},{"why":"Provides the generic uniform convergence result invoked to lift pointwise convergence of the estimating equations to uniform convergence over the compact parameter space.","marker":"[1]"},{"why":"Gives the excess-life distribution used to compute residual service-time probabilities in the cross-moment formulas.","marker":"[3]"},{"why":"Foundational M/G/$\\infty$ estimation problem showing how inference from aggregate observations works in the single-node case, which this paper extends to networks.","marker":"[7]"},{"why":"Reference for infinite-server queueing models and the M/G/$\\infty$ majorant used in the coupling argument for the ergodic limits.","marker":"[17]"}],"fun_headline_variants":["Snapshots alone reveal queue network topology","Population snapshots unmask queueing routing","Moment estimator decodes network from Poisson samples","Queueing topology from population snapshot moments","Infer infinite-server network from snapshot data"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the matrix $\\Omega_1$ used to recover the routing matrix is invertible for every parameter vector in the allowed space; Section 4.2 defines $Q$ as $\\Omega_1^{-1} \\Omega_2$ but does not prove the inverse exists, and a singular $\\Omega_1$ would leave the moment equations unable to pin down the topology.","fun_headline_variants_meta":{"raw":{"variants":["Snapshots alone reveal queue network topology","Population snapshots unmask queueing routing","Moment estimator decodes network from Poisson samples","Queueing topology from population snapshot moments","Infer infinite-server network from snapshot data"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000165,"raw_usage":{"total_tokens":1235,"prompt_tokens":918,"completion_tokens":317,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":534,"completion_tokens_details":{"reasoning_tokens":251}},"tokens_in":534,"tokens_out":317,"duration_ms":3875,"temperature":1.0,"reasoning_tokens":251,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:43:18.032866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a two-station network and parameter values satisfying Assumptions 1 and 2 for which the determinant of $\\Omega_1$ is zero while the moment system still has two distinct solutions $(\\boldsymbol{\\lambda}, Q)$, or compute $\\det \\Omega_1$ symbolically along a family of routing matrices and exhibit one that vanishes; either would give a concrete instance where the claimed identifiability fails.","supporting_citations":[{"cited_title":"van der Vaart","cited_arxiv_id":null,"evidence_quote":"Supplies the asymptotic-statistics backbone: identifiability definitions, Glivenko-Cantelli/uniform-convergence argument, and the theorem used to conclude consistency."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generic uniform convergence result invoked to lift pointwise convergence of the estimating equations to uniform convergence over the compact parameter space."},{"cited_title":"Asmussen","cited_arxiv_id":null,"evidence_quote":"Gives the excess-life distribution used to compute residual service-time probabilities in the cross-moment formulas."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Foundational M/G/$\\infty$ estimation problem showing how inference from aggregate observations works in the single-node case, which this paper extends to networks."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reference for infinite-server queueing models and the M/G/$\\infty$ majorant used in the coupling argument for the ergodic limits."}],"review_version":1}