{"id":"6228374b-ce32-4d01-8679-a87827b4e4b4","arxiv_id":"2412.04489","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A likelihood-based estimator recovers arrival rates, customers' perceived service value, and switching cost in a two-station queue from workload observations only, validated by simulation.","lead":"This paper builds a maximum-likelihood method that estimates how many customers would arrive at each of two service stations and how they value the service, using only observed waiting times. A manager could use these estimates to decide where to put servers or whether to change prices.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Likelihood derivation assumes one server per station (Eq. 6), contradicting the model's 'one or more servers'; multi-server workloads do not decrease at unit rate, so the estimator is not valid as stated.","rationale":"The paper's central claim is that the MLE recovers (λ1,λ2,θ,c) from workload observations. The derivation's key structural assumption is the deterministic linear decrease of the workload in Eq. (6). The model statement 'one or more servers' makes this assumption false for any station with more than one server, because the workload then decreases at the number of busy servers and the virtual waiting time is no longer equal to the workload. This is more fundamental than the observation-format concern raised by the reader: even with perfect continuous workload observations, the likelihood is misspecified for multi-server stations. The reader's weakest assumption (perfect observation of the workload process) is real but is a scope limitation; the multi-server issue is an internal inconsistency with the stated model. The simulations likely use single-server stations, but the paper does not say so, and the motivating server-allocation question requires multi-server capability. A conditional verdict remains appropriate: require an explicit single-server restriction or a re-derived likelihood for multi-server stations; hence the reader's CONDITIONAL verdict is unchanged.","tokens_in":12397,"tokens_out":22443,"duration_ms":201708,"concrete_test":"Analytically: for an M/M/2 station with two rate-1 servers and initial workload 2, the workload after t is 2−2t for t≤1, contradicting the 2−t decrease of Eq. (6). Then simulate data from a two-station system where one station has two servers and the other has one, using the Table 2 true parameters, and apply the paper's estimator; if estimates become biased or the fitted likelihood degrades substantially, the multi-server concern is confirmed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3 states 'Each station contains one or more servers,' and the introduction explicitly motivates the work with server-allocation decisions. However, the entire likelihood derivation rests on Eq. (6): Vs(Ãk−1+t) = max(0, Vs(Ãk−1) − t), i.e., the virtual waiting time decreases at unit rate until zero. This is only true for a single-server queue with unit service rate. In a station with m>1 servers, the total workload decreases at rate min(m, number in system) and the waiting time for a new arrival is not the workload nor a deterministic function of it; consequently, the effective-arrival rates in Eq. (5) and the likelihood expressions (15)–(17) are not the correct likelihood for the model as described. The simulation section does not state the number of servers; if it uses one server per station, the numerical validation does not extend to the multi-server case that the paper's motivating questions require. This is an internal inconsistency, not merely a generalization gap.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a maximum-likelihood estimator for a two-station queueing system in which customers, informed of both stations' waiting times, may join their local station, switch to the other station at a switching cost, or balk. Using observations of the workload processes at both stations, the authors derive a conditional likelihood for the effective arrival times and joined stations, distinguishing three cases based on the workload difference relative to the switching cost, and then estimate the potential arrival rates, the service-value distribution tail parameter, and the switching cost. The manuscript reports simulation experiments indicating that the estimator recovers the true parameters in most tested configurations, discusses identifiability issues near zero switching cost, and proposes a lower-bound statistic for the switching cost.","tokens_in":12601,"tokens_out":8719,"duration_ms":81595,"significance":"If the derivation is sound and the model consistent, the paper offers a practical way to estimate demand and customers' value perception from data that managers plausibly possess (workloads), without observing balking or the origin of joiners. This would be a useful contribution to queueing-system estimation, particularly for service operations in healthcare, restaurants, and similar settings. The simulation results, reproduced across a range of parameter values and service-time distributions, are encouraging, and the provision of code is a strength. However, the manuscript's central contribution is currently undermined by an inconsistency between the stated multi-server model and the single-server workload dynamics used throughout the likelihood derivation, and by garbled likelihood formulas in two of the three cases. The paper also lacks any formal consistency or asymptotic justification for the estimator and relies solely on synthetic data.","major_comments":[{"comment":"The model states that each station contains 'one or more servers' and the introduction motivates the work with server-allocation decisions, yet the likelihood derivation relies on Eq. (6), which asserts that the virtual waiting time decreases linearly at unit rate until zero. This is only true for a single-server station with unit service rate. In a station with m>1 servers, the workload decreases at rate min(m, number in system), and the waiting time of a new arrival is not simply the workload nor a deterministic function of it. Consequently, the effective-arrival rates in Eq. (5) and the densities in Eqs. (15)–(17) are not the likelihood of the model as described. The simulation section does not state the number of servers; if one server per station is used, the numerical validation does not cover the multi-server case that the motivating questions require. This internal inconsistency must be resolved, either by restricting the model to single-server stations or by deriving the correct workload dynamics for m servers.","section":"Section 3, Eq. (6)"},{"comment":"The likelihood formulas in cases (ii) and (iii) are garbled. In Eq. (16) the exponential term is split across lines so that the integration bounds and the arguments are misaligned, and Eq. (17) has the same problem. The notation 't < Ak - c' mixes the random variable Ak (an inter-arrival time) with the switching cost c, so it is dimensionally inconsistent. In addition, in case (ii) the text states that 'the probability of joining station i is 0' for t < Ak - c, but Eq. (10) immediately gives a non-zero expression for exactly that event. Since the likelihood is the core of the paper, these formulas must be rewritten with clear hazard rates and survival probabilities, and the time intervals over which each rate applies must be stated unambiguously.","section":"Equations (16) and (17)"},{"comment":"The paper provides no formal consistency or asymptotic-normality result for the MLE, and the simulation study is limited to parameter configurations chosen by the authors. The estimator is defined through a recursive conditional likelihood that depends on observed service times, so the standard likelihood theory for i.i.d. observations does not apply directly. A short consistency argument (or a specific reference establishing such a result for this type of conditional likelihood) is needed to support the claim that the estimates in Table 2 are close to the true parameters beyond the particular simulation settings; without it, the estimator's validity is an empirical conjecture.","section":"Section 4 (general)"}],"minor_comments":[{"comment":"The product is written as L_N with index n=1 to N, while elsewhere the number of effective arrivals is denoted K; moreover, the text in Section 4 uses both N and K inconsistently. Please unify the notation.","section":"Eq. (19)"},{"comment":"The sentence 'the likelihood does not include X, since X depends on the service time distribution, but not on balking or switching behavior' is misleading: the observed service times X do affect the workload evolution and hence the conditioning variables v(s)_{k-1} in the recursive likelihood. Please clarify that the likelihood is conditional on X and makes no assumption about the service-time distribution beyond the observed values.","section":"End of Section 4"},{"comment":"The lower-bound statistic for the switching cost is introduced but not connected to the estimation procedure. Either explain how it can inform the MLE or move it to a separate discussion; as written it appears as an isolated remark.","section":"Eq. (20)"},{"comment":"There are typos in the reference list: for instance, reference [12] appears to concatenate two distinct works ('Inferring balking behavior from transactional data' and 'Equilibrium strategies and the value of information in a two-line queueing system with threshold jockeying'). The full text also contains several spacing and punctuation errors that should be corrected.","section":"Literature review"}],"recommendation":"major_revision","confidential_remarks":"The paper is at an early stage, and the core idea is promising, but the multi-server inconsistency is a serious internal mismatch with the stated model, and the garbled likelihood formulas prevent the reader from verifying the central derivation. The authors should rewrite these sections carefully. The code availability is a positive asset, but the simulation study would be much stronger if it reported the number of servers and included a configuration with more than one server per station, or if the paper explicitly narrowed its scope to the single-server case."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core claim—that you can estimate potential arrival rates, the service-value parameter, and switching cost from workload-only observations in a two-station system—is plausible, and the likelihood construction is a genuine extension of the single-station balking estimator in Inoue et al. (2023). The three-case decomposition of the effective arrival rate, depending on whether the workload gap is within the switching cost, is a real piece of work. The simulation recovery is decent in most configurations, especially when switching rates are high, and the code is on GitHub, which helps. But there is a load-bearing mismatch between the model description and the likelihood. Section 3 says each station has 'one or more servers,' and the motivation is explicitly about server-allocation decisions. Yet Eq. (6) assumes the virtual waiting time decreases linearly at unit rate: V_s(t) = max(0, V_s(t0) - (t-t0)). That is only true for a single server per station with unit service rate. With m > 1 servers, the workload process does not evolve that way, and the effective arrival hazard in Eq. (5) is not the correct likelihood for the stated model. The simulation section never states the number of servers; if it uses one, the numerical validation only covers the single-server case, which is a subset of the stated model. This is not a minor generalization gap; it is an inconsistency between the problem statement and the math. The fix is either to restrict the model to one server per station (and say so loudly) or to re-derive the likelihood with multi-server workload dynamics, which would be a much bigger project. Two other soft spots: the likelihood formulas in cases (ii) and (iii) are rendered in a way that is hard to verify, with missing parentheses and incomplete expressions in the arXiv text; a referee cannot check those integrals line by line as it stands. And the paper has no consistency proof and no real-data test, with visible biases in low-switching and low-arrival regimes. Those are acceptable at this stage for a methods paper, especially with a public repository, but they should be acknowledged more clearly. Who should read this: people working on inference in queueing systems, particularly those studying balking/switching with observable workloads. It deserves a serious referee, but the referee should push hard on the server-count issue. If that is fixed—in the model description, not just the simulations—the paper becomes a solid contribution to a niche but active literature. My advice: send it to review with the expectation of major revision.","headline":"Genuine two-station balking/switching likelihood, but the 'one or more servers' framing clashes with the single-server workload dynamics the estimator actually assumes.","tokens_in":729,"tokens_out":2727,"would_cite":false,"duration_ms":66752,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60K25","62F10","90B22"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proposes a maximum-likelihood estimator that recovers potential arrival rates, the service-value distribution parameter, and switching cost from continuous workload observations at two service stations.","keywords":["queueing systems","maximum likelihood estimation","balking behavior","customer switching","workload observation","service value estimation","potential demand","two-station service system"],"falsifier":"Simulate a two-station system with known parameters but record workloads only at discrete intervals (e.g., every 0.2 time units) and feed the same MLE to the discretized data; if the estimates of $c$ and $\\theta$ degrade markedly, that demonstrates the continuous-workload assumption is load-bearing. Independently, set $c=0$ and examine the profile likelihood over $\\lambda_1$: the paper's own logic predicts a flat surface along the sum $\\lambda_1+\\lambda_2$, which would confirm the identifiability boundary the method acknowledges.","tokens_in":12198,"feed_emoji":"📊","tokens_out":8004,"duration_ms":70186,"temperature":0.7,"pith_summary":"Two service stations show arriving customers both waiting times; customers then join their local queue, switch to the other queue for a switching cost, or leave. A manager sees only the workload processes — not who balked and not which station a joiner originally came from. This paper argues that, despite these hidden elements, a maximum-likelihood estimator built from the join times and chosen stations can recover the potential arrival rate at each station, the parameter of the customers' service-value distribution, and the switching cost. The virtue of the claim is operational: potential demand and perceived value are exactly the quantities a manager needs to decide server allocation, pricing, and service-rate upgrades, and they are normally private. Simulations with exponential and Pareto service times show averaged estimates close to the true parameters, with documented bias when switching is rare or arrival rates are low.","feed_headline":"Workload data alone reveals hidden demand and service value","feed_subtitle":"Managers can estimate potential arrivals and perceived service value without seeing who balks or switches.","key_machinery":"The machine of the paper is the conditional likelihood of an effective arrival. Given the workloads $(v^{(1)}_{k-1}, v^{(2)}_{k-1})$ left by the previous joiner, the next effective arrival occurs at random time $A_k$ and chooses station $I_k$, with a density that the paper evaluates by an integral over the inter-arrival interval. The key modelling fact is that between jumps each workload decreases linearly at unit rate until hitting zero, $V_s(\\tilde{A}_{k-1}+t)=\\max(0, v^{(s)}_{k-1}-t)$ (equation 6), so the time-dependent rate at which latent customers become joiners is known in closed form (equation 5). The derivation then splits into three cases according to how the workload difference compares with the switching cost $c$, producing the densities (15)–(17). Multiplying these conditional densities gives the full likelihood (19), and the service times $X_k$ drop out because the decision rule (1) involves only waiting times, the switching cost, and the private service value.","core_discovery":"The paper's central claim is that continuous observation of the two workload processes is sufficient to identify a four-parameter behavioral queueing model. The observations used are the effective-arrival inter-arrival times $A_k$ and the stations joined $I_k$; the latent ingredients are balking customers, the original station of each joiner, and the switching cost $c$. The likelihood is built case by case: depending on whether the workload difference is at most $c$, exceeds $c$, or is less than $-c$, the effective arrival process is a time-inhomogeneous Poisson process with a closed-form rate (equation 5), and the paper derives the conditional density of each $(A_k, I_k)$ given the previous workloads (equations 15–17). The product of these densities is the full likelihood (equation 19), and its maximizer (equation 18) ignores service-time observations $X$ because the join/switch/balk decision does not depend on them. The paper then reports simulation evidence that the averaged MLEs track the true $(\\lambda_1, \\lambda_2, \\theta, c)$ across configurations with exponential and Pareto service times, with the accuracy of $\\hat{c}$ tied to the number of observed switches.","pith_inferences":["A natural extension the paper leaves implicit is the multi-station network: the same latent-variable likelihood principle should carry over, but the number of case splits grows combinatorially and the switching-cost coupling becomes a graph, so identifiability of each pairwise cost would need a separate check.","The identifiability failure at $c=0$ suggests a practical safeguard: before interpreting $\\hat\\lambda_1$ versus $\\hat\\lambda_2$, one should verify that the lower-bound statistic $\\tilde c$ is not near zero; otherwise the two arrival rates are nearly confounded.","The reliance on continuous workload paths implies a concrete sensitivity test: in a simulator, subsample the workload observations at a sparse rate and re-run the MLE; the degradation in $\\hat\\theta$ and $\\hat c$ would quantify how much temporal resolution the method needs."],"forward_implications":["A manager can infer potential demand and service-value perception from workload records already available, avoiding surveys or transaction-level tracking of balkers.","The estimates directly inform capacity decisions: when $\\hat{\\lambda}_1$ is much larger than $\\hat{\\lambda}_2$, concentrating servers at station 1 can raise throughput, as the paper's own simulation illustrates.","The precision of the switching-cost estimate grows with the observed switch count, so effective use of the method requires observation windows long enough to see switches.","Because service-time observations $X$ drop out of the likelihood, the same estimation procedure applies without modification across different service-time distributions.","The switching-cost lower bound in equation (20) gives a quick pre-estimation diagnostic: if a switch ever happens from a longer to a shorter queue, $c$ is at least the maximal observed adverse waiting-time difference."],"supporting_citations":[{"why":"Supplies the conditional-likelihood approach for estimating queueing parameters from observations with unobserved balking, which this paper extends to two stations with switching.","marker":"[16]"},{"why":"Provides the modeling assumption that service durations are known at arrival so customers observe exact workloads, the informational basis of the decision rule in equation (1).","marker":"[25]"},{"why":"Introduces the reward-versus-expected-cost balking rule for joining queues that underlies the join/switch/balk choice in equation (1).","marker":"[24]"},{"why":"Survey of strategic queueing with observable workload that situates the information structure assumed here.","marker":"[14]"},{"why":"Survey of parameter and state estimation in queues that frames the estimation problem as distinct from analytic queueing models.","marker":"[2]"}],"fun_headline_variants":["Two-station workloads reveal hidden demand and value","MLE extracts demand and service value from workloads alone","Private service value inferred from workload timings only","Balking unobserved? Workloads still estimate demand and value","Pricing insights from workload data in two-station systems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The estimator assumes the manager sees the full continuous workload process at both stations and that workloads decay linearly at unit rate between arrivals, so all future waiting times are deterministically known from the last observation; if only queue counts or periodic snapshots are available, the likelihood no longer applies.","fun_headline_variants_meta":{"raw":{"variants":["Two-station workloads reveal hidden demand and value","MLE extracts demand and service value from workloads alone","Private service value inferred from workload timings only","Balking unobserved? Workloads still estimate demand and value","Pricing insights from workload data in two-station systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000188,"raw_usage":{"total_tokens":1332,"prompt_tokens":947,"completion_tokens":385,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":563,"completion_tokens_details":{"reasoning_tokens":307}},"tokens_in":563,"tokens_out":385,"duration_ms":4337,"temperature":1.0,"reasoning_tokens":307,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:21:24.104413+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Simulate a two-station system with known parameters but record workloads only at discrete intervals (e.g., every 0.2 time units) and feed the same MLE to the discretized data; if the estimates of $c$ and $\\theta$ degrade markedly, that demonstrates the continuous-workload assumption is load-bearing. Independently, set $c=0$ and examine the profile likelihood over $\\lambda_1$: the paper's own logic predicts a flat surface along the sum $\\lambda_1+\\lambda_2$, which would confirm the identifiability boundary the method acknowledges.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the conditional-likelihood approach for estimating queueing parameters from observations with unobserved balking, which this paper extends to two stations with switching."},{"cited_title":"K., and Kumar, S","cited_arxiv_id":null,"evidence_quote":"Provides the modeling assumption that service durations are known at arrival so customers observe exact workloads, the informational basis of the decision rule in equation (1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the reward-versus-expected-cost balking rule for joining queues that underlies the join/switch/balk choice in equation (1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Survey of strategic queueing with observable workload that situates the information structure assumed here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Survey of parameter and state estimation in queues that frames the estimation problem as distinct from analytic queueing models."}],"review_version":1}