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REVIEW 3 major objections 4 minor 27 references

Estimating Potential Demand and Customers' Perception of Service Value in a Two-station Service System

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict Genuine two-station balking/switching likelihood, but the 'one or more servers' framing clashes with the single-server workload dynamics the estimator actually assumes. read the letter →

arxiv 2412.04489 v2 pith:IWSMAJM7 submitted 2024-11-21 eess.SY cs.SYmath.PR

Nishant Mangre , Jiesen Wang This is my paper · ORCID
classification eess.SYcs.SYmath.PR MSC 60K2562F1090B22
keywords queueingsystemsmaximumlikelihoodestimationbalkingbehaviorcustomerswitchingworkloadobservationservicevaluepotentialdemandtwo-stationsystem
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

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

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.

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 (3)
  1. [Section 3, Eq. (6)] 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.
  2. [Equations (16) and (17)] 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.
  3. [Section 4 (general)] 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.
minor comments (4)
  1. [Eq. (19)] 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.
  2. [End of Section 4] 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.
  3. [Eq. (20)] 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.
  4. [Literature review] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the MLE is derived from explicit model primitives and validated by simulation against independently generated data.

full rationale

The paper's central object is a likelihood function built directly from the stated primitives: Poisson potential arrivals with rates λ1 and λ2, i.i.d. service values with tail function H̃θ, observable workload processes, and a constant switching cost c. The estimator in Equation (18) is defined as the argmax of this likelihood, so the unknown parameters are not inserted into the derivation as inputs; they are the optimization variables. The simulation study generates workload traces from known parameters and then compares the argmax estimates with the true parameter values, which is a standard performance check rather than a prediction fitted to the data. No load-bearing self-citation appears: the closest methodological predecessor, reference [16] by Inoue, Ravner, and Mandjes, is written by different authors, and the likelihood derivation in Section 4 is self-contained given the model in Section 3. The lower-bound estimator c̃ in Equation (20) is an additional observable inequality and is not used to force the MLE. The reviewer concern about the assumption of one server per station, expressed through Equation (6), is a correctness and generality issue about whether the workload evolves as V_s(A_{k-1}+t)=max(0,V_s(A_{k-1})-t) when a station may have multiple servers; it is not a circularity, because Equation (6) is a stated modeling assumption rather than a re-use of the target parameter values or of the estimated quantities. Overall, the derivation chain does not reduce to its own inputs, and there are no fitted parameters renamed as predictions. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The model postulates four unknown scalars, two arrival rates, a value-distribution shape, and a switching cost, plus several structural assumptions about Poisson arrivals, iid Pareto service values, observable workload, and rational balking. The MLE estimates the scalars; the structural assumptions are inputs that are not tested against real data. No new physical or mathematical entities are invented.

free parameters (3)
  • λ1, λ2 (potential arrival rates) = e.g., 1.021 and 1.020 in the first row of Table 2
    Assumed unknown Poisson rates at the two stations; they are the primary targets of estimation, not ad hoc additions, but the model assumes they are constant over the observation window.
  • θ (service-value tail parameter) = e.g., 1.148 in the first row of Table 2
    Shape parameter of the Pareto tail H~θ(x)=1/(1+x)^θ used in the numerical study; this parametric assumption restricts the distribution of customers' perceived service value.
  • c (switching cost) = e.g., 0.489 in the first row of Table 2
    Assumed constant and identical across customers; estimated by MLE. Identifiability degrades when switching is rarely observed, as the paper itself notes.
assumptions (5)
  • domain assumption Customers arrive at each station as independent Poisson processes with rates λs.
    Section 3 states this as the arrival model; the likelihood derivation depends on the memoryless and independent-thinning properties of Poisson processes.
  • domain assumption Service values Rj are iid draws from a parametric distribution Hθ, with Hθ(x)=1 for x≤0.
    Section 3 defines Hθ and the condition Hθ(x)=1 for x≤0, ensuring all customers join when the cost is non-positive; the numerical section further assumes a Pareto tail.
  • domain assumption Customers observe exact current workloads and choose to join the cheaper station, switch, or balk according to Equation (1), with a constant switching cost c.
    This behavioral rule is the core of the model; if customers use expected waiting times or have heterogeneous switching costs, the likelihood is misspecified.
  • domain assumption The workload process is fully observable and evolves as V_s(t)=max(0,V_s(t-)-t) plus upward jumps of size Xk at effective arrivals.
    Equation (6) uses this deterministic linear decrease between arrivals; the method would not apply if only queue lengths or periodic workload snapshots are available.
  • standard math Effective arrivals form a time-inhomogeneous Poisson process by independent thinning of the potential arrival processes.
    Used in Section 4 to write the effective arrival rate as a sum of thinned Poisson rates; valid when each potential arrival independently decides to join based on current workloads.

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Cite this review

Pith. "Pith review of Estimating Potential Demand and Customers' Perception of Service Value in a Two-station Service System." pith.science (2026). https://pith.science/paper/IWSMAJM7

@misc{pith2026241204489,
  author       = {Pith},
  title        = {Pith review of: Estimating Potential Demand and Customers' Perception of Service Value in a Two-station Service System},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IWSMAJM7}},
  note         = {Machine review of arXiv:2412.04489}
}
read the original abstract

The potential demand in the market and customers' perception of service value are crucial factors in pricing strategies, resource allocation, and other operational decisions. However, this information is typically private and not readily accessible. In this paper, we analyze a service system operating across two stations, each with its own customer flow. Customers arriving at the system are informed of the waiting times at both stations and can choose to either join the local station, switch to the other station, or balk. Our objective is to estimate the arrival rates at each station and customers' perceived service value based on the observed workloads at both stations. A significant challenge arises from the inability to observe balking customers and the lack of distinction between local arrivals and customers switching from the other station, as the switching cost is unknown. To address this issue, we employ maximum likelihood estimation and validate the effectiveness of the estimator through a series of simulations.

Figures

Figures reproduced from arXiv: 2412.04489 by the authors.

Figure 1
Figure 1. An example of the queueing process, where the red ticks indicate the station each potential customer arrived at, and the dotted lines indicate the virtual waiting time they observed [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The mean throughput per unit time for different values of λ1, given that λ1+λ2 = 2, c = 2, H˜ θ(x) = 1 (x+1)4 . 8 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Virtual waiting time against time holds for all 0 < t < Ak. We can use this formula to deduce that H˜ (Vs(A˜ k−1 + t)) = H˜ (Vs(A˜ k−1) − t), since H˜ (x) = H˜ (0) = 1, for all x ≤ 0. Using (5), we find that the effective arrival rate at this time t equals λ1H˜ (v (1) k−1 − t) + λ2H˜ (v (2) k−1 − t), (7) and the probability P(Ak ∈ [a, a + ∆t), Ik = i | Ak > a, Vs(A˜− k−1 ) + Xk−11{Ik−1=s} = v (s) k−1 ) 0 < t < Ak =∆… view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Illustration of how the likelihood is derived. Let A = (a1, . . . , ak), I := (i1, . . . , ik), and X = (x1, . . . , xk) be realizations for A1, . . . , Ak, I1, . . . , Ik, and X1, . . . , Xk. We can now determine the full likelihood function LK(λ1, λ2, θ, c; A, I, X).…
Figure 5
Figure 5. Figure 5: Estimates from L = 1000 runs, given that the true value of (λ1, λ2, θ, c) = (5, 1, 3, 0.5), and the service time distribution for server 1 and server 2 has G1(x) = 1−e −x , and G2(x) = 1 − e −5x . The average joining and switching rates are 47.69% and 29.94%, respectiv…
Figure 6
Figure 6. Figure 6: Estimates from L = 1000 runs, given that the true value of (λ1, λ2, θ, c) = (1, 1, 3, 0.5), and the service time distribution for server 1 and server 2 has G1(x) = 1− 1 (x+1)2 , and G2(x) = 1 − 1 (x+1)6 . The average joining and switching rates are 78.93% and 6.71%, re…
Figure 7
Figure 7. Figure 7: Estimates from L = 1000 runs, given that the true value of (λ1, λ2, θ, c) = (1, 5, 3, 0.5), and the service time distribution for server 1 and server 2 has G1(x) = 1− 1 (x+1)2 , and G2(x) = 1 − 1 (x+1)6 . The average joining and switching rates are 67.07% and 4.06%, re…

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