5.0 Article Within-Venue Monitoring of BTC/USDT Liquidity and Resiliency on Binance: A Queueing-Theoretic Framework Samir Varma Special Issue Market Microstructure and Digital Disruption Edited by Dr. Priyantha Mudalige and Dr. Damien Wallace https://doi.org/10.3390/jrfm19050372 Article Within-Venue Monitoring of BTC/USDT Liquidity and Resiliency on Binance: A Queueing-Theoretic Framework Samir Varma VS Asset Management, LLC, Cos Cob, CT 06807, USA; samir@vsasset.com Abstract This paper develops a queueing-organized framework for within-venue monitoring of BTC/USDT liquidity, signed-flow pressure, and resiliency on Binance. The model treats latent buy and sell pressure as occupancy processes and organizes three empirical diagnostics: the variance-per-BTC liquidity measure Rr , the mean-reversion rate θeff , and the companion proxy signed-flow proxy β eff . Using Binance trade data from 2020–2025, the empirical results show a pooled first-order variance–volume regularity away from the highest-volume tail, material rolling variation in liquidity and resiliency, and stronger next-day risk sorting from rolling Rr than from Amihud illiquidity or a Kyle-style minute-impact benchmark. In overlapping 30-day windows, θeff is positive by point estimate in roughly two-thirds of windows but clearly positive in only about two-fifths under a simple uncertainty buffer, implying that local recovery is often fragile or ambiguous. The one-hour symmetric benchmark is useful as a coarse center benchmark, but the heaviest tails require a Student-t VaR overlay. The framework is intended for risk managers, execution desks, market makers, and exchange surveillance teams that need indicators of thinning liquidity, weakening recovery, and one-sided signed flow. Queueing is useful because it turns those signals into one monitoring dashboard for market quality and short-horizon risk. Keywords: Bitcoin; market microstructure; order flow; queueing theory; market quality; liquidity risk; digital assets; market resiliency 1. Introduction Binance spot BTC/USDT is a useful venue for studying short-horizon liquidity and resiliency because it combines continuous high-turnover trading with auditable trade prints and aggressor-side information. Those features support venue-level diagnostics built directly from transaction data and make it possible to track how trading activity, signed flow, and recovery conditions evolve through time inside a single market. Academic Editors: Priyantha This paper develops a queueing-organized monitoring framework for three linked Mudalige and Damien Wallace objects: the variance-per-BTC liquidity diagnostic Rr , the effective mean-reversion rate θeff , proxy Received: 30 March 2026 and the companion signed-flow proxy β eff . The queueing layer models latent buy and Revised: 12 May 2026 sell pressure as occupancy processes. Customer-level queues are not directly observed, but Accepted: 12 May 2026 this state-space view places liquidity, signed-flow pressure, and recovery on one common Published: 21 May 2026 stock-flow accounting system instead of treating them as unrelated reduced-form measures. Copyright: © 2026 by the author. The practical question is simple: when do trade prints suggest that this venue is liquid and Licensee MDPI, Basel, Switzerland. resilient, and when do they instead suggest that it is becoming thin, one-sided, and fragile? This article is an open access article distributed under the terms and The paper sits at the intersection of four areas in the literature: queueing and inventoryconditions of the Creative Commons based market models (Cont & de Larrard, 2013; Garriott et al., 2025), order-flow and priceAttribution (CC BY) license. impact models (Amihud & Mendelson, 1980; Ho & Stoll, 1981; Kim & Stoll, 2014; Kyle, 1985; J. Risk Financial Manag. 2026, 19, 372 https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 2 of 41 Stoll, 1978), cryptocurrency market microstructure (Alexander et al., 2023; Anastasopoulos et al., 2026; Dimpfl, 2017), and short-horizon market-risk monitoring (Foucault et al., 2013). The contribution is a transparent single-venue monitoring framework for a continuously traded and fully auditable market, together with a direct comparison showing that its main liquidity diagnostic is related to, but not redundant with, feasible standard trade-based benchmarks. Who should care and why? The natural audience includes market makers adjusting inventory, execution desks assessing trading conditions, exchange risk and surveillance teams watching for one-sided markets, and researchers who want auditable market-quality measures. What matters to those users is whether the venue is becoming thin, whether directional pressure is unwinding quickly or only weakly, and whether stressed states are followed by worse near-term risk conditions. The aim is to show that a simple queueing state space can organize those questions using observables that are available in Binance trade data. The framework models latent buy-pressure and sell-pressure queues as two independent M/G/∞ systems. New directional pressure arrives stochastically, existing pressure unwinds after random holding times, and the resulting net imbalance is mapped into price changes through a linear imbalance-to-price mapping. Throughout the paper, “long” and “short” label latent directional pressure rather than directly observed leveraged financing positions, and the queues should be read as BTC-normalized latent occupancy units rather than as observed limit-order-book queues. These primitives yield a stationary Skellam distribution for net inventory, a variance–volume relation, and a local feedback extension that motivates the observable monitoring diagnostics used below. Empirically, three results largely underpin this paper. First, variance per traded BTC shows a pooled first-order regularity across volume bins, strongest away from the highestvolume tail, and the rolling Rr series shows that liquidity conditions move materially over time. Second, lagged hourly return autocorrelation yields a rolling effective mean-reversion estimate whose point estimate is positive in roughly two-thirds of 30-day windows but clearly positive in only about two-fifths under a simple uncertainty buffer. Third, the rolling diagnostics are not only contemporaneous summaries: high-Rr days are followed by worse next-day variance and tail outcomes than low-Rr days, non-positive resiliency windows are followed by worse next-day tail outcomes than clearly mean-reverting ones, and rolling Rr sorts those outcomes more sharply than rolling Amihud illiquidity or a closely matched Kyle-style minute-impact benchmark. The appendix recovery event study points in the same direction for large isolated hourly shocks. What is genuinely learned from queueing here? proxy Queueing earns its place here for three reasons. First, it makes Rr , θeff , and β eff measurements on one common stock-flow state space rather than three separate alarms. Second, it gives θeff a recovery-timescale interpretation in terms of latent pressure unwinding. Third, under the symmetric M/M/∞ special case it supplies one concrete turnover-versus-recovery translation. The empirical core remains observable: Rr and θeff proxy are directly estimated, and β eff is a companion proxy. The one-hour symmetry-based translations are kept for interpretation rather than presented as stand-alone structural estimates. For the intended audience, the payoff is that the same framework answers three operational questions at once: how much liquidity the venue is delivering per traded BTC, how quickly it appears to recover from directional pressure, and whether signed flow is becoming one-sided enough to treat current conditions as more fragile. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 3 of 41 Table 1 makes that added value concrete. The queueing layer does not replace reducedform evidence; it organizes that evidence and tells the reader how the main diagnostics fit together. Table 1. The table shows what the queueing layer adds beyond reduced-form monitoring. The table reports the observable objects used in the main paper and the queueing interpretation attached to them. Observable Object Reduced-Form Reading Queueing Reading Occupancy-based variance scale linking Rr Liquidity/variance ratio per traded BTC traded volume to latent imbalance Local short-horizon mean reversion from lag-1 Effective unwinding/resiliency timescale of θeff autocorrelation latent pressure proxy Arrival/exit pressure read in the same state (κ̃dir , bF , β eff ) Signed-flow pressure and feedback proxy space as occupancy decay Symmetry-based translations into E[ H ], λ± , and m± are reported only in Appendix B.3 as a joint full-sample illustration under the symmetric M/M/∞ bridge. They are not presented here as structural estimates. The layered assumption stack is summarized once in Table A1. The empirical discipline is simple: Rr and θeff are directly estimated from hourly variance, volume, and proxy lag-1 autocorrelation; β eff is a companion splice built from minute directional impact and one-hour signed BTC flow; and symmetry-based quantities such as E[ H ], λ± , and m± are reported in the appendix as contingent arithmetic. The empirical case for the paper therefore rests on three observable margins: a pooled variance-per-BTC regularity, a rolling resiliency classification built from θ̂eff , and a companion signed-flow measure that helps organize market phases. The symmetric one-hour increment benchmark is a coarse center check, and the heaviest tails require the Student-t mixture overlay reported in Appendix E. A full-sample contingent queueing illustration still helps interpretation: under the nearby symmetric one-hour splices reported in Appendix B.4, the observed liquidity and resiliency bundle maps to indicative mean holding times of about 5.20–7.63 h and total latent occupancy of about 8524–12,510 BTC-normalized units. Scope and limitations This is a focused single-venue paper. It is implemented on Binance spot BTC/USDT, uses a linear impact approximation, and does not run an exhaustive comparison with alternative microstructure models. Those choices are deliberate: they keep the framework transparent, auditable, and easy to implement on continuously updated trade data. The paper should therefore be read as a venue-level monitoring contribution. The central question proxy is whether the observable diagnostics—Rr , θeff , and β eff —remain jointly interpretable and practically useful inside one bookkeeping system. Table 2 summarizes the main outputs of the framework and their empirical implementation. Section 2 presents the queueing-theoretic framework, the feedback mechanism, the data, and the empirical implementation. Section 3 reports the main results, including variance–volume evidence, directional impact triangulation, distributional fit, rolling stability, and a brief pragmatic tail-risk overlay. Section 4 discusses the implications for short-horizon price formation, liquidity, and market monitoring. Section 5 concludes. Additional derivations, extensions, and robustness checks are reported in the appendix. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 4 of 41 Table 2. Baseline monitoring dashboard for the main paper (BTC/USDT, Binance, 2020–2025). The table emphasizes directly estimated quantities and one companion proxy used in the paper’s main claims. Quantity Construction Variance-per-BTC Rr Var[rt ]/E[νt ] = 1.44 × 10−8 with bootstrap s.e. 8.09 × 10−10 † Lag-1 return autocorrelation, θ̂eff = 0.0477 h−1 , implying Effective mean-reversion θeff descriptive half-life 14.54 h (roughly 10–25 h) Heuristic clear-positive 30-day windows One-standard-error rolling classification share, 39.2% proxy proxy Companion signed-flow proxy β eff (baseline splice) β eff = 21 κ̃dir bF = 7.24 × 10−2 h−1 † Estimates use a percentile bootstrap with 10,000 resamples of 24 h blocks. R r is the main liquidity diagnostic, and θeff comes from lag-1 autocorrelation under a local OU approximation, so the implied half-life is descriptive rather than literal. Under the baseline and doubled heuristic buffers, the clearly mean-reverting 30-day share ranges proxy from 17.2% to 39.2%, with ambiguity dominant under either rule. β eff is a companion signed-flow summary whose sign is stable across nearby-frequency alternatives, although its exact level is splice-sensitive. Symmetry translations are reported only in Appendix B.3. 2. Materials and Methods 2.1. Core Assumptions and Notation (M1) Independent Poisson arrivals of long (λ+ ) and short (λ− ) latent occupancy units, BTC-normalized by bookkeeping convention. (M2) I.i.d. holding times with finite means E[ H± ] (M/G/∞ queues). (M3) Linear imbalance-to-price mapping Pt = P0 + κXt with Xt = Bt − St . All subsequent sections build directly on these three postulates. In the present tradeprint dataset they are used as organizing modeling assumptions rather than as separately identified latent facts. Appendices B.5 and B.6 sketch nonlinear and finite-depth alternatives to (M3); the main text keeps the linear form because it is the clearest implementation of the monitoring system. 2.1.1. Notation and Scale Conventions Table 3 collects the core symbols and dimensions used throughout the paper. Throughout, Rr ≡ Var(rt )/E[ν1h ] denotes the observable returns-scale variance-per-BTC moment √ computed from one-hour return increments and one-hour traded BTC volume, and Rr its associated variance scale. The structural return-impact slope is κ̃ ≡ κ/P, with units BTC−1 ; once the one-hour holding-time correction c H is introduced below, the symmetrybased special-case translation is Rr = c H κ̃ 2 . Price-scale variance-per-BTC is R = P̄2 Rr . Numerical illustrations in USD set P∗ = $50,000; all calibrations and tests use returns units, so no reported estimate depends on the choice of P∗ . For empirical bookkeeping, queue occupancies are BTC-normalized latent occupancy units throughout: one latent unit corresponds by normalization to 1 BTC. This is a bookkeeping convention for the latent state, not a claim that literal customer orders are observed in exact 1-BTC blocks. A marked or compound-Poisson generalization with random chunk sizes would be the natural extension, but the present paper uses unit-normalized occupancies to keep the monitoring diagnostics tractable. Operationally, the latent state is tied to executed BTC turnover only at the first-moment level: the normalization is chosen so that expected openings and closures of latent occupancy units match expected executed BTC volume in the bookkeeping identity used below. A marked extension would preserve that first-moment bridge while changing higher moments and tail behavior. The stationary occupancy law is the general M/G/∞ result; the one-hour increment bridge used later for contingent structural translation is the narrower symmetric M/M/∞ special case. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 5 of 41 Table 3. Core symbols and dimensions. Symbol Meaning Units Bt , St Outstanding long/short latent occupancy units BTC-normalized latent occupancy units Xt = Bt − St Net inventory imbalance BTC-normalized latent occupancy units Pt , P0 Price at t/local reference level USD λ± Arrival rate of long/short latent occupancy units BTC-normalized latent occupancy units h−1 H± Holding-time random variable (long/short) h µ± = 1/E[ H± ] Exit (closure) rate h−1 m± = λ± E[ H± ] Steady-state latent occupancy BTC-normalized latent occupancy units κ Price-impact (price scale), ∆Pt ≈ κ ∆Xt USD · BTC−1 κ̃ Structural return-impact slope, κ̃ ≡ κ/P BTC−1 R Observable variance-per-BTC in returns units, Var(r1h )/ E[ν1h ] BTC−1 √r Rr Observable returns-scale variance coefficient BTC−1/2 P Sample-average price used to relate scales USD R Variance-per-BTC (price units; hourly unless otherwise stated) USD2 · BTC−1 β Latent momentum-feedback slope in the local state splice BTC h−1 per unit return Structural feedback-adjusted Ornstein–Uhlenbeck (OU) drift β eff = κ̃β h−1 under the latent-state splice proxy Reduced-form proxy feedback coefficient from lagged return β eff h−1 and signed BTC flow zt = κ̃Xt Latent local price-deviation state used in the feedback splice return θ = µ+ + µ− Aggregate exit intensity h−1 ϑ, ϑ± Gamma mixing scale (all/long/short) h ν1h Executed BTC volume in a one-hour bar BTC σP2 = √ R ν1h One-hour price-increment variance benchmark USD2 σr = Rr ν1h One-hour return standard deviation return 2.1.2. Queueing Mapping Queueing notation treats latent occupancy-unit openings as arrivals and holding times as service; for an M/G/∞ queue the steady-state occupancy is Poisson with mean λE[ H ], yielding a Skellam distribution for the long–short difference. A brief primer and the mapping from queueing language to trading pressure are provided in Appendix A. 2.2. Queueing-Theoretic Framework 2.2.1. Model Setup and Intuition The central modeling idea is to treat the market as a system of queues in which price pressure emerges from the imbalance between outstanding long and short latent occupancy units. Consider a Bitcoin market populated by two latent directional-pressure types: those contributing buy pressure (“longs”) and those contributing sell pressure (“shorts”). At any point in time, there are Bt outstanding long-side occupancy units and St outstanding short-side occupancy units. These are not directly observed financing books; they are state variables summarizing net pressure that remains active in the market. The key state variable is the net inventory imbalance: Xt = Bt − St (1) Over short horizons, local price deviations are determined by this inventory imbalance through a linear imbalance-to-price mapping: Pt = P0 + κXt (2) where P0 is a slowly moving local reference level and κ > 0 measures the local price effect per unit of net inventory. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 6 of 41 Impact-Kernel Interpretation Let latent occupancy-unit openings arrive at times ti with sign si ∈ {+1, −1} and holding time Hi . Then Xt = ∑ si 1{ti ≤ t < ti + Hi }, i and, taking expectations, the mean price follows E[ Pt ] = P0 + κ ∑ si G (t − ti ), G (u) = Pr( H > u). i This is a shot-noise/propagator representation of latent directional pressure with kernel G given by the holding-time survival function (Bouchaud et al., 2009). It is not a one-toone mapping from public trade prints to literal queue openings and exits; executed BTC enters the empirical implementation only through the first-moment bookkeeping bridge in Equation (12). In this interpretation, the price effect persists while latent pressure survives and decays as occupancy exits. The mean holding time H̄ is therefore an effective resiliency timescale of latent order-flow pressure; in the present implementation that timescale is inferred from lagged hourly return autocorrelation and then translated into E[ H ] under symmetry (Section 2.5). The dynamics of long and short occupancy units follow queueing processes. New long-side units arrive at rate λ+ and each unit is held for a random time H+ drawn from distribution G+ . Similarly, new short-side units arrive at rate λ− with holding times H− drawn from G− . The heterogeneity in trader horizons is captured by the entire distribution of holding times rather than a single parameter. The model is applied to Binance BTC/USDT spot transactions over the period 1 January 2020–9 July 2025. The empirical work does not directly estimate customer-level arrivals, exits, or holding times from minute buckets, nor does it observe order-book queue position. Instead, it estimates an observable variance scale, a minute directional-impact proxy, and an effective mean-reversion rate, then translates those objects into contingent structural quantities under symmetry for interpretation. Data conventions and empirical estimates are reported in Sections 2.4 and 3. These assumptions are most appropriate when trading is continuous, turnover is high, and short-horizon price adjustment is strongly influenced by order flow; BTC/USDT fits those conditions reasonably well over the sample horizon (see Appendix A). 2.2.2. Stationary Inventories and Skellam Law Outstanding long and short occupancy units each follow an M/G/∞ queueing system. New units arrive according to Poisson processes with rates λ+ and λ− , and holding times are drawn from G+ and G− with means E[ H+ ] and E[ H− ]. A standard queueing result is Palm’s theorem (Palm, 1943), which characterizes the steady-state distribution of an M/G/∞ system: Lemma 1 (Palm’s Theorem). In an M/G/∞ queueing system with arrival rate λ and service time distribution with mean E[ H ], the steady-state occupancy is Poisson distributed with parameter m = λE[ H ]. Applied to this setting, steady-state occupancies are B∞ ∼ Poisson(m+ ) (3) S∞ ∼ Poisson(m− ) (4) https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 7 of 41 with m+ = λ+ E[ H+ ] and m− = λ− E[ H− ]. Under the maintained baseline specification, the long and short queues are independent even though they operate in the same market. This independence is imposed through independent Poisson arrivals and independent holding periods across traders. The steady-state net inventory is therefore X∞ ∼ Skellam(m+ , m− ) =⇒ P∞ − P0 = κX∞ (5) This stationary result should be read as a local law for price deviations around the reference level P0 , not as a literal unconditional law for the full Bitcoin price path over 2020– 2025. The Skellam distribution—previously applied to tick-level price changes by Koopman et al. (2017)—emerges here as the stationary distribution of net inventory from explicit queueing primitives, linking arrival rates and holding times directly to local price-deviation moments. Illustrative symmetry-based translations for (m+ , m− ) and the associated Skellam parameters are reported only in Appendix B.3. 2.2.3. Price Distribution and Over-Dispersion The Skellam distribution has probability mass function:  k/2 m+ √ Pr{ X∞ = k } = e−(m+ +m− ) I|k| (2 m+ m− ) (6) m− where In is the modified Bessel function of the first kind of order n.1 From Equation (2), the local price-deviation distribution is:  k/2 −(m+ +m− ) m+ √ Pr{ P∞ = P0 + κk } = e I|k| (2 m+ m− ) (7) m− All moments of the local price-deviation distribution are available in closed form: E[ P∞ ] = P0 + κ (m+ − m− ) (8) 2 Var[ P∞ ] = κ (m+ + m− ) (9) m+ − m− Skew[ P∞ ] = (10) (m+ + m− )3/2 1 ExKurt[ P∞ ] = (11) m+ + m− These formulas reveal several important insights. (i) First, the expected local price deviation is proportional to the difference between  long and short pressure m+ − m− , formalizing the sign of the local price effect from net order imbalance. (ii) Second, local price-deviation volatility scales with the sum of these pressures  m+ + m− , so intense activity on either side widens the stationary distribution of price deviations. (iii) Third, under the baseline Poisson specification the excess kurtosis is 1 Kurt( Pt ) − 3 = , m+ + m− which vanishes as order-flow intensity grows, implying asymptotically normal tails. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 8 of 41 Empirically, hourly returns display extreme excess kurtosis (≈47.29), while the hourly buy-count and sell-count series are strongly over-dispersed relative to a Poisson benchmark. Fitting Gamma-mixed Poisson models to the hourly count series gives full-sample dispersion parameters α+ = 0.4922 and α− = 0.6423. These are trade-count burstiness proxies rather than BTC-commensurate structural calibrations of the latent queues, so the count-space extension below is retained only as a supplementary burstiness benchmark. 2.2.4. Negative–Binomial Over-Dispersion Observed hourly buy and sell trade counts are substantially more bursty than a pure Poisson benchmark. To document that without overloading the main paper, a Gammamixed Poisson extension is used only as a count-space burstiness proxy. Under a commonrate approximation for the buy and sell count mixtures, the hourly increment’s excess kurtosis can be decomposed into a tiny Poisson benchmark term plus a much larger overdispersion term. For Binance, the Poisson term is only 1.8 × 10−5 while the over-dispersion contribution is 5.2889, which is why the paper treats burstiness as a real feature of the count data. The exact common-rate derivation and parameterization are deferred to Appendix H; in the main paper this point is only supplementary and does not change the interpretation of the core liquidity and resiliency diagnostics. 2.2.5. Variance–Volume Relation At the first-moment level, the queueing framework implies a disciplined link between trading volume and the ex ante variance in price changes. Under the BTC-normalized latent-unit convention, observed executed BTC volume is bookkept by expected openings and expected closures of latent occupancy units. No event-level one-to-one mapping from public trades to literal latent queue openings or closures is being claimed. Because each side of the book is an M/G/∞ queue with mean inventory m± = λ± E[ H± ] and exit rate µ± = 1/E[ H± ], the BTC-normalized first-moment bookkeeping rate for executed volume is  ν = (λ+ + λ− ) + µ+ m+ + µ− m− = 2 λ+ + λ− , (12) where the first term captures arrivals and the second departures. The last equality follows by substituting m± . Under this normalization, each latent occupancy unit contributes one BTC when opened and one BTC when closed in the first-moment bookkeeping. Equation (12) is a bookkeeping identity under the BTC-normalized latent-unit convention, not direct observation of literal queue openings and closures. The model therefore does not independently predict venue volume. Instead, it uses a BTC-normalized first-moment convention to place observed turnover and latent occupancy on a common scale. The paper’s falsifiable content begins at the observable level: the variance-per-BTC moment Rr , the effective proxy mean-reversion estimate θeff , the proxy feedback diagnostic β eff , and the success or failure of the symmetric increment benchmarks. It does not begin at literal queue openings or at the contingent symmetry translations reported later only for interpretation. Sample Mean Executed Volume Given the hour-by-hour executed BTC volume νt , the long-run mean hourly executed BTC volume is estimated by 1 T T t∑ ν̂ = νt , (13) =1 where T is the number of hourly bars in the sample. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 9 of 41 Empirical Calibration √ Section 3.2 reports the empirical estimates of Rr , its square-root scale Rr , and the contingent structural translation κ̃ for Binance BTC/USDT. Table 2 summarizes the main outputs. Returns form (used in all calibrations and Q–Q plots): with rt := Pt /Pt−1 − 1 and κ̃ ≡ κ/P, Var[rt ] = Rr E[νt ] ( h = 1). 2.3. Feedback, Stability, and Resiliency Mechanism At intraday horizons, signed-flow intensities can respond to recent price movements. This extension adds a local state-dependent splice to motivate a proxy-based resiliency classifier within the queueing framework. In the empirical sections below, the object carried proxy forward is the local pair (θeff , β eff ), not a claim of direct structural estimation of the latent queue-stability boundary. 2.3.1. State-Dependent Arrivals The model is extended to allow arrival rates that depend on recent price changes, capturing local return-following or contrarian signed-flow behavior in a latent queueing splice. Standing Assumptions A1 (Queueing) Arrivals of long- and short-side latent occupancy units are independent Poisson with baseline rates λ0+ , λ0− ; holding times are independent and identically distributed (i.i.d.) with finite means E[ H± ]. A2 (Linear price impact) Price obeys Pt = P0 + κXt with constant κ > 0. A3 (Linear feedback splice) Latent arrival rates respond linearly to the local latent price-deviation state, Equations (14) and (15); the empirical sections below do not observe that state directly and therefore estimate a separate lagged-return proxy coefficient. Linear Feedback Specification Let θ = µ+ + µ− denote the aggregate exit intensity [h−1 ] and define the rescaled latent feedback coefficient   β eff := κ̃ β β eff is h−1 . To avoid confusion with observed close-to-close one-hour return increments, let zt := κ̃Xt denote the model’s local latent price-deviation state in returns units. The feedback equations in this subsection are written in terms of zt , whereas the empirical sections reserve rt for proxy observed one-hour close-to-close returns and estimate the separate proxy coefficient β eff from lagged returns and signed BTC flow. Let arrival rates respond linearly to the recent (left-limit) local price-deviation state zt− : λ+ (t) = λ0+ + β zt− (14) λ− (t) = λ0− − β zt− (15) where λ0± are baseline arrival rates and β ∈ R measures the strength of state dependence in the latent price-deviation state. When β > 0, positive recent deviations are associated with return-following latent pressure; when β < 0, positive deviations are associated with contrarian latent pressure. Because zt− is Ft− -measurable, this specification is predictable and introduces no simultaneity. Equations (14) and (15) are interpreted as a local linearisation, https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 10 of 41 with rates truncated at zero if needed; under the full-sample symmetry translation summarized in Appendix B.3, the implied perturbation βzt−1 is small relative to the baseline intensities λ0± , so truncation does not bind in practice. Linearized Dynamics For the local OU approximation in this subsection, symmetric exits µ+ = µ− =: µ = θ/2 are additionally imposed. Substituting zt− = κ̃Xt− into the net arrival rate gives feedback proportional to 2β eff Xt , so the drift of the expected net inventory is d E[ Xt ] = (λ0+ − λ0− ) − (θ/2 − 2β eff )E[ Xt ]. (16) dt Whenever the local effective mean-reversion rate θeff := θ/2 − 2β eff is positive, the corresponding local mean level is λ0+ − λ0− X̄ = . (17) θeff Defining the centered state Yt := Xt − X̄, the local linearized dynamics take the Ornstein– Uhlenbeck form √ dYt = −(θ/2 − 2β eff ) Yt dt + ν0 dWt , (18) where ν0 = 2(λ0+ + λ0− ) is the baseline event intensity and Wt is a standard Wiener process. The continuous-time stability condition is therefore θ > 4β eff . (19) Stability Analysis The centered local linearization is mean-reverting only when the drift coefficient is positive, i.e., when θeff = θ/2 − 2β eff > 0, equivalently θ > 4β eff under the symmetricexit splice. This inequality is best read as a latent heuristic for the sign logic behind the proxy observed pair (θeff , β eff ) rather than as a directly identified latent stability boundary. Baseline asymmetry λ0+ − λ0− shifts the local mean level X̄ but does not by itself change that local sign condition. Empirically, the paper uses this splice only to organize local regime classification through θ̂ and the companion proxy β ˆ ; the corresponding contingent proxy eff eff symmetry summary for ρ is reported only in Appendix C.10. Metrics The implied resiliency half-life—the time for the centered state Yt to decay halfway back toward its local mean after a shock—is t1/2 = ln 2/θeff , where θeff = θ/2 − 2β eff is the effective mean-reversion rate from (18). Because the structural θ is not directly observed, the main text focuses on the directly estimated pair (θeff , t1/2 ) and treats any further translation into θ, E[ H ], c H , or ρ as contingent on the symmetric proxy mapping summarized only in the appendix. For the calibrated parameters (Table 4), this yields the descriptive full-sample contingent illustration t1/2 ≈ 14.54 h. The empirical role of this subsection is therefore local and fragile by design: it supplies a queueing-organized interpretation for observed resiliency diagnostics, not a claim that the latent queue boundary is directly estimated from venue data. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 11 of 41 Table 4. Technical pooled-baseline input table for the main empirical implementation (BTC/USDT, Binance, 2020–2025). Table 2 is the baseline monitoring dashboard; this table records the directly estimated and proxy-based ingredients behind it. Symbol Description Value Status/Source Empirical inputs (Binance, 2020–2025) Var(r1h ) Variance in hourly close-to-close returns 4.722 × 10−5 Data moment E[r1h ] Mean hourly return p 8.034 × 10−5 Data moment σr Std. dev. of hourly returns, σr = Var(r1h ) 6.871 × 10−3 Derived E[ν1h ] Mean hourly traded volume (BTC) 3281 Data moment Nhourly Number of hourly bars 48,374 Sample size Ndaily Number of UTC days 2017 Sample size Directly estimated and proxy-based quantities √ p Rr Observable 1 h variance scale (BTC−1/2 ), Equation (20) 1.2 × 10−4 Var / E[ν] Descriptive minute signed-flow impact proxy (BTC−1 ), κ̃dir 2 × 10−5 Minute signed-flow regressions from the median of daily slopes in Equation (25) One-hour signed BTC-flow slope (BTC per unit return) bF 7250.26 Hourly flow on lagged return from the lagged-return flow regression ρ̂1 Lag-1 autocorrelation of simple hourly returns −0.0233 Direct sample autocorrelation θeff Effective mean-reversion rate (h−1 ), Equation (21) 0.0477 Lag-1 autocorrelation t1/2 Resiliency half-life (h), ln 2/θeff ≈14.54 Local OU summary proxy Companion signed-flow proxy (h−1 ), baseline β eff 7.24 × 10−2 1 2 κ̂dir b̂ F minute/hour splice  Heuristic clear-positive share of valid 30-day windows Pr θ̂eff > 0 clearly 39.2% Rolling autocorrelation under a one-standard-error autocorrelation buffer 95% uncertainty ranges are κ̃dir ∈ [1.96 × 10−5 , 2.05 × 10−5 ], bF ∈ [6145, 8355], θeff ∈ [0.0292, 0.0665], and proxy β eff ∈ [0.0613, 0.0837]. The rolling classification is heuristic: under the baseline and doubled buffers, the clearly mean-reverting share ranges from 17.2% to 39.2%, with ambiguity dominant under either rule. Nearby-frequency alternatives 0.0661 and 0.0417 show that the proxy sign is stable but the exact level is splice-sensitive. 2.4. Data The empirical implementation uses Binance BTC/USDT trade data from 1 January 2020 through 9 July 2025, a window that spans the 2021 bull market, the FTX collapse, and the 2024 exchange-traded fund (ETF) launch period. The working sample contains 48,374 hourly bars over 2017 Coordinated Universal Time (UTC) days. A complete UTC grid over that span would contain 48,408 h; the realized panel contains 48,374 because 34 h are missing on 15 UTC days. Those missing hours come from gaps in the source hourly summary files that feed the canonical hourly panel, not from discretionary filtering or from the later aggressor-side merge. The analysis retains those partial UTC days and computes daily variance and volume from the observed hourly bars. Dropping the 15 incomplete UTC days changes the headline variance-per-BTC estimate by less than 1% and leaves the lag-1 return autocorrelation essentially unchanged, so the main results are not driven by those omissions. For each UTC day, the workflow uses the raw tick file to construct minute-level signed BTC flow and returns, the workflow-generated hourly summary file for close-to-close returns and traded BTC volume, and the workflow-generated aggressor-side summary file to build an hourly panel of buyer-initiated BTC volume. All timestamps are converted to UTC. Binance spot BTC/USDT is used as a first venue because it combines deep continuous trading with directly auditable trade-print and aggressor-side records, which makes it a practical setting for a queueing-based monitoring exercise even though the broader Bitcoin market is multi-venue. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 12 of 41 All data processing, estimation, figure generation, and Monte Carlo validation were performed in Python 3.9.6 using NumPy 1.26.4, SciPy 1.13.1, pandas 2.2.2, and matplotlib 3.9.4. The full Python code that produces these results is provided as Supplementary Materials. The empirical pipeline therefore uses three aligned objects: an hourly return/volume panel, a day-level minute-regression panel, and an hourly signed-BTC-flow panel. The first supports the variance–volume and mean-reversion estimates, the second provides a directional-impact scale in BTC units, and the third provides the signed-flow input for the feedback and rolling-diagnostic results. Sign convention. Signed order flow uses the taker side: a trade has sign +1 if the aggressor is the buyer and −1 if the aggressor is the seller. Concretely, the convention takes isBuyerMaker=False as +1 and True as −1. Buyer-initiated and seller-initiated BTC volume are therefore observed directly at the trade level and then aggregated to minute or hourly frequency as needed. Cleaning and alignment. The construction is deliberately mechanical. The canonical hourly panel retains only rows with finite hourly close, traded BTC volume, and close-toclose return values, then sorts by UTC timestamp. In the minute regressions, raw trade records with non-finite price, quantity, or timestamp fields are dropped, timestamps are normalized to milliseconds, and signed BTC flow is aggregated within exact UTC minutes before minute returns are computed from the first and last trade in each occupied minute. The hourly BTC-flow panel is then matched hour-by-hour to the workflow-generated aggressor-side summary files; the workflow stops if any required day is missing or if the merge produces missing hourly buy quantities. No manual winsorization or discretionary outlier deletion is applied in these steps. 2.5. Empirical Implementation The framework is implemented at the one-hour horizon unless stated otherwise. The √ empirical procedure distinguishes four objects. First, Rr is an observable variance scale identified from the variance–volume moment condition. Second, θeff is an effective meanreversion rate identified from return autocorrelation. Third, κ̃dir is a minute directionalproxy impact slope in BTC units. Fourth, β eff is a proxy-based feedback rate obtained by combining κ̃dir with one-hour signed BTC flow. Structural quantities such as κ̃, θ, E[ H ], λ± , and m± are reported only as model-implied translations under the symmetry mapping. Throughout this section, all bar-level observables are one-hour totals: price-scale variance satisfies R = Var[∆P1h ]/ν1h and returns-scale variance satisfies Var[r1h ] = Rr E[ν1h ]. 2.5.1. Observable Variance Scale The variance–volume identity yields the one-hour variance scale s √ Var(r1h ) Rr : = . (20) E[ν1h ] This is an observable variance coefficient, not a directional-impact slope. Standard errors are computed with a block bootstrap using 10,000 resamples of 24 h blocks. 2.5.2. Effective Mean-Reversion The aggregate mean-reversion rate θeff is identified from the lag-1 autocorrelation of hourly returns. Under the OU linearisation (18), the lag-1 autocorrelation coefficient ρ̂1 satisfies ρ̂1 = (e−θeff − 1)/2. The estimate is  θ̂eff = − ln 1 + 2 ρ̂1 , (21) https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 13 of 41 which follows because the lag-1 autocorrelation of the OU increment rt = Xt − Xt−1 is (e−θeff − 1)/2, and is well-defined when 1 + 2 ρ̂1 > 0, i.e., ρ̂1 > −0.5. Applied to simple hourly returns, this is a local short-horizon approximation rather than an exact identity for the full return process. The resiliency half-life follows as t1/2 = ln 2/θeff , and the structural exit rate is implied under the proxy mapping as θ = 2θeff + 4β eff . Because θeff is identified from a single autocorrelation coefficient (SE ≈ 0.005), the half-life is imprecisely estimated; a two-standard-error band spans roughly 10–25 h. 2.5.3. Directional-Impact Scale To obtain a unit-consistent flow-feedback diagnostic, the implementation first estimates day-by-day minute regressions of one-minute returns on contemporaneous signed BTC flow: rd,m = ad + κd qd,m + ε d,m , (22) where qd,m is signed BTC volume (buyer-initiated minus seller-initiated) computed directly from trade records. The directional-impact scale used in the feedback mapping is κ̂dir := mediand |κ̂d |. This scale has units BTC−1 and is reported descriptively in Section 3.3. The day intercepts ad are estimated but not interpreted. 2.5.4. Feedback Hourly signed BTC flow Ft is then constructed from buyer-initiated minus sellerinitiated BTC volume, and the estimated regression is proxy 1 Ft = a F + bF rt−1 + ut , β eff := κ̃ bF , (23) 2 dir where Ft is measured in BTC within hour t, so bF has units BTC per unit return. A short discrete-time bridge makes the sign-and-units convention explicit. Let ∆ = 1 h and define the excess signed flow over hour t by Z t   Ft − F̄ ≈ (λ+ (s) − λ− (s)) − (λ0+ − λ0− ) ds ≈ 2β zt−1 ∆, t −1 where the second step is the local left-endpoint approximation under (14) and (15). Using the lagged one-hour return as the empirical proxy for the local deviation and κ̃dir as the minute-scale BTC-to-return slope proxy gives 2β∆ 2β∆ proxy κ̃dir bF Ft − F̄ ≈ r , bF ≈ , β eff ≈ κ̃dir β ≈ . κ̃dir t−1 κ̃dir 2∆ With ∆ = 1 h, this reduces to (23). This bridge is used only as a latent sign-and-units heuristic. Because rt−1 is an observed increment proxy for the unobserved latent state proxy rather than the state itself, the resulting object is reported as β eff , not as a direct estimate of the structural quantity β eff and not as a mapped latent feedback rate. The minute/hour splice is a deliberate technical compromise: minute regressions stabilize the directionalimpact slope, whereas hourly aggregation yields a less noisy signed-BTC-flow slope. That choice is useful for extracting a directional monitoring signal, but it leaves the absolute proxy level splice-sensitive; nearby-frequency robustness is reported in Appendix C.9. In the rolling 30-day results below, the signed-flow slope is re-estimated within each rolling window and combined with the same-window rolling median of daily directional-impact https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 14 of 41 slopes, with the observed missing UTC hours carried through the hourly grid rather than filled. When bF < 0 the flow is contrarian; when bF > 0 it is return-following. 2.5.5. Workflow and Units The proxy construction is intentionally mechanical. Step 1: estimate the minute directional scale κ̃dir in units BTC−1 . Step 2: estimate the hourly signed-flow slope bF in units BTC per unit return. Step 3: apply the one-hour discrete approximation above so that proxy β eff = 12 κ̃dir bF is an hourly signed-flow monitoring proxy in h−1 . This makes the splice proxy transparent, but it does not turn β eff into a directly identified arrival-rate parameter. 2.5.6. Model-Implied Structural Translation Under the symmetric M/M/∞ special-case translation used for interpretation, the structural holding time is E[ H ] = 2/θ. For one-hour return increments, Appendix B.2 shows that the holding-time correction factor is  c H = E[ H ] 1 − e−1/E[ H ] , √ so the structural return-impact slope follows as κ̃ = Rr /c H . This bridge is exact for the symmetric exponential-holding-time mapping; it is not a nonparametric M/G/∞ identification result. Additional occupancy translations, including λ± and m± under the same symmetry calibration, are reported only in Appendix B.3. All such quantities are reported as contingent model translations for interpretation, not as directly estimated facts. A related distinction matters for the empirical results below. The stationary Skellam law from Section 2.2.2 concerns the level of latent imbalance Xt , whereas the distributional diagnostics in Section 3.4 benchmark one-hour return increments generated from that same core. 3. Results 3.1. Main Parameter Estimates The empirical quantities are estimated as described in Section 2.5. See Appendix F for a Monte Carlo confirmation that the variance-scale estimator based on Var(r1h )/ E[ν1h ] is effectively unbiased in finite samples. √ The empirical mapping uses three observables: variance per traded BTC for Rr , lag-1 proxy hourly return autocorrelation for θeff , and the pair (κ̃dir , bF ) for β eff . These are the main empirical objects; any further translation into θ, E[ H ], c H , λ± , or m± is reported only in the appendix and remains contingent on the symmetric M/M/∞ bridge. Table 4 records the √ headline inputs. The full-sample values are Rr = 1.2 × 10−4 BTC−1/2 , θeff = 0.0477 h−1 , proxy implying a descriptive half-life of 14.54 h, and baseline-splice β eff = 7.24 × 10−2 h−1 . Nearby-frequency alternatives, 0.0661 and 0.0417, show that the proxy sign is stable while the exact level is splice-sensitive. Additional tail and supplementary diagnostics are reported in Appendix C. Public OHLCV aggregates are not used for calibration because the implementation relies on trade-print volume and aggressor-side information; brief reconciliation notes appear in Appendix D. 3.2. Variance–Volume Evidence Figure 1 is a visual summary of the empirical relationship between within-day variance and daily trading volume. The grouped scatter averages days within rounded daily volumepercentile bins labeled 0, 10, . . . , 100, so the two endpoint bins contain 101 days each and the interior bins contain about 201–202 days each. Within-day variance is computed from hourly close-to-close simple returns within each day; daily volume is total executed BTC volume that day. Log returns are not used anywhere in this test. The heavier evidentiary https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 15 of 41 work comes from the day-level ratio contrast below, the compact grouped-slope companion, and the within-bin dispersion evidence in Appendix C.1. Because that grouped-slope companion is built from daily observations rather than hourly bars, its uncertainty uses a 7-day moving-block bootstrap rather than the 24 h block bootstrap used for the hourly variance coefficient. Predicted slope Overall mean ratio 0.0030 OLS through origin 12 Daily variance-per-BTC ratio (×10 10) 0.0025 10 Within-day variance 0.0020 8 0.0015 6 0.0010 4 0.0005 Endpoint-bin median 2 (N=101, 14% of overall mean) 0.0000 0 0 200,000 400,000 600,000 0 100 200 300 Daily volume (BTC) Mean daily volume in bin (thousands of BTC) Figure 1. Variance–volume evidence for BTC/USDT, 2020–2025. The (left) panel shows within-day variance in simple hourly returns against same-day executed BTC volume; gray points are daily observations and blue points are grouped averages formed from rounded daily volume-percentile bins over the 2017 UTC trading days in the sample. The (right) panel reports the within-bin daily variance-per-BTC ratio with median and interquartile range by volume bin. The main takeaway is a pooled first-order rise of variance with volume together with a more stable middle-bin ratio pattern and visible attenuation only in the highest-volume endpoint bin. The visual deviation in Figure 1 is concentrated in the high-volume tail. In the grouped scatter, the highest-volume endpoint bin is visibly attenuated; in a separate day-level contrast, the mean daily ratio in the top 10% of days is only 0.569 times the corresponding ratio in the middle 60% of days. A compact quantitative companion (Table 5) gives the same message: excluding the highest-volume endpoint bin, the grouped n-weighted through-origin slope is 7.027 × 10−10 with 95% block-bootstrap interval [5.349 × 10−10 , 9.099 × 10−10 ], versus the theoretical slope 5.996 × 10−10 . The qualitative message is unchanged if all grouped bins are retained (4.581 × 10−10 ) or if the top two bins are excluded (7.950 × 10−10 ), which is why the endpoint exclusion is best read as a transparent display choice rather than an after-the-fact search for significance. The variance–volume evidence is therefore used as a descriptive first-order compatibility check rather than as a sharp invariance test. Appendix C.1 reports the within-bin dispersion of the daily variance-per-BTC ratio across volume bins, and Table 6 shows that the reduced-form Rr moment also shifts across broad sub-periods. Taken together, those checks support a pooled liquidity-monitoring regularity rather than a time-invariant liquidity law. Table 5. Compact quantitative companion to the variance–volume figure. The grouped slope excludes only the highest-volume endpoint bin and uses a 7-day moving-block bootstrap. Companion Statistic Estimate 95% Block-Bootstrap Interval Grouped slope (all bins) 4.581 × 10−10 — Interior grouped slope (bins 0–90) 7.027 × 10−10 [5.349 × 10−10 , 9.099 × 10−10 ] Grouped slope (bins 0–80) 7.950 × 10−10 — https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 16 of 41 Table 6. Broad sub-sample reduced-form checks. Including the intermediate 2023 phase shows directly that the reduced-form liquidity and resiliency moments move across major market phases, which is why the paper treats Rr as a pooled first-order diagnostic rather than as a constant law. Hours are reported so the pooled baseline benchmarks are not read as stable venue constants. Each sub-period estimate is computed from the same pooled formulas used for the full-sample benchmarks, restricted only to observations inside that sub-period. Period Hours Rr θeff (h−1 ) E[ν1h ] (BTC) 2020–2022 26,272 1.63 × 10−8 0.0471 3986 2023 8758 4.94 × 10−9 0.0850 4180 2024–2025 13,344 2.29 × 10−8 0.0336 1305 Figure 2 makes that liquidity movement visible at the same rolling horizon used later for the resiliency diagnostics. The rolling series is noisier than the pooled estimate, but it confirms that the liquidity moment itself is time-varying rather than merely a full-sample summary. The paper therefore treats Rr as a monitorable liquidity diagnostic whose pooled estimate remains useful for benchmarking, not as a constant law of the venue. In the daily variance–volume scatter, each day d has x-coordinate equal to total daily BTC volume νd and y-coordinate equal to the within-day sample variance in hourly returns. Since E[ d Vard (r1h )] ≈ Var(r1h ) while E[νd ] ≈ 24 E[ν1h ], the model’s variance–volume identity implies a predicted slope Var(r1h ) 1 Var(r1h ) Rr ≈ = . (24) E[νd ] 24 E[ν1h ] 24 With the full-sample moments in Table 4, this yields a predicted daily scatter slope of Rr /24 = 5.996 × 10−10 . Empirically, the day-by-day ratios Var d d (r1h )/νd have mean 6.65 × 10−10 , which is the primary estimate under the constant-ratio hypothesis. The day-level OLS slope through the origin is lower at 4.5 × 10−10 because it effectively volumeweights. High-volume days exhibit attenuated ratios when gross two-sided trading rises faster than the net directional pressure that drives prices. That gross-versus-net reading is plausible, but it should be treated as an interpretation of the attenuation rather than as a separately established result. This attenuation is especially pronounced in the highestvolume endpoint bin (mean ≈ 380,505 BTC/day, N = 101 days). In the median-based right panel of Figure 1, the endpoint-bin median is only about 14% of the overall mean ratio; at the day-level mean ratio, the same endpoint bin averages about 0.38 times the overall mean. One plausible reading is that gross two-sided activity rises faster than net directional pressure on those days, but the paper treats that only as an interpretation of the attenuation rather than as a separate empirical result. The fitted reference line in Figure 1 uses the theoretical slope Rr /24; the empirical OLS-through-origin slope is overlaid as a secondary comparison. 3.3. Directional Impact Cross-Check As an independent empirical cross-check on price impact, the analysis uses day-by-day minute signed-flow regressions only as descriptive triangulation and as a normalization device for the feedback proxy, not as a causal impact design or as structural identification. The day-by-day minute-level regressions relate one-minute returns to contemporaneous signed BTC order flow. For each day d, the regression is rd,m = ad + κd qd,m + ε d,m , (25) https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 17 of 41 where qd,m is signed BTC volume (buyer-initiated minus seller-initiated) computed directly from trade records as ∑(qty × side), with qty the Binance base-asset quantity in BTC and side ∈ {+1, −1} indicating trade direction; hence κd has units BTC−1 . The daily intercept ad is estimated but not used elsewhere in the paper. Across 2017 days, the median |κd | is 1.997 × 10−5 BTC−1 with interquartile range [1.401 × 10−5 , 2.619 × 10−5 ], and the median R2 is 0.2946. The absolute-median summary is used only to obtain a positive directionalimpact scale for the feedback proxy. In the present sample this choice is not driving the result mechanically: the median signed slope is also 1.997 × 10−5 BTC−1 , and negative daily slopes occur on fewer than 0.05% of days. These minute-level slopes are about 6.007× √ smaller than the observable variance scale Rr ≈ 1.2 × 10−4 , which is expected because √ Rr is a variance coefficient while κd is a directional slope in BTC units. The moderate median R2 of 0.2946 suggests that minute signed flow carries economically meaningful price information, but the regression is used only to supply the directional normalization in proxy β eff . The separate variance–volume moment condition provides the variance-consistent scale used in the diffusion calculations. 1 × 10−8 Rolling variance-per-BTC diagnostic 8 Full-sample Rr Rr (BTC−1) 4 ̂ 0 2020 2021 2022 2023 2024 2025 Figure 2. Rolling 30-day estimate of the one-hour variance-per-BTC moment R b r for BTC/USDT, sampled once per UTC day. The line is computed from hourly returns and hourly executed BTC volume inside each trailing 30-day window, and the dashed line marks the full-sample estimate. The main takeaway is that the liquidity diagnostic itself moves over time, so the paper’s monitoring interpretation is explicitly dynamic rather than purely pooled. 3.4. Variance-Matched One-Hour Skellam Benchmark and Falsification Check This subsection is intentionally a falsification check rather than a validation exercise. Figure 3 compares empirical hourly returns to a variance-matched one-hour Skellam benchmark. The object benchmarked here is not the exact increment law of the general M/G/∞ framework. Rather, it is the symmetric special-case reference distribution obtained after fixing the one-hour horizon, replacing realized volume by the sample mean one-hour BTC volume, and matching the observed one-hour variance scale Rr . This benchmark intentionally switches off feedback and regime variation so the symmetric core can be judged on its own. Specifically, the theoretical quantiles are generated from √ rtSk = r̄ + Rr Zt , Zt ∼ Skellam(ν̄/2, ν̄/2), where r̄ is the sample mean hourly return and ν̄ = E[ν1h ] is the sample mean hourly traded BTC volume. Figure 3 therefore benchmarks one-hour return increments under a symmetric variance-matched reference distribution; it does not fit the stationary price-level law from Section 2.2.2 directly, and it should be read as an unconditional benchmark rather than as a fully volume-conditioned fit. Sharper evidence appears in Table 7, which conditions √ on realized one-hour volume. After standardizing returns by Rr νt , the model-implied https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 18 of 41 central 80% and 50% bands cover 85.3% and 60.7% of standardized observations, with empirical/model width ratios 0.862 and 0.756. Ratios below one mean that the modelimplied central bands are wider than the empirical center. These results reject the symmetric benchmark as a full descriptive model for hourly spot BTC/USDT returns, but that does not make the framework useless: the practical value of the paper lies in monitoring liquidity per traded BTC, local recovery timescales, and signed-flow pressure, while the benchmark still serves as a transparent center-of-distribution reference that shows exactly where an additional tail overlay is needed. Additional burstiness and tail diagnostics are deferred to Appendices C.2 and C.4. Hourly BTC/USDT Returns: Q-Q vs Skellam Upper 1% Skellam 0.175 0.150 0.15 0.125 0.100 0.075 0.10 0.050 0.025 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.05 Empirical quantiles 0.00 0.05 Lower 1% 0.025 0.10 0.050 0.075 0.100 0.15 0.125 0.150 0.175 0.175 0.150 0.125 0.100 0.075 0.050 0.025 0.15 0.10 0.05 0.00 0.05 0.10 0.15 Theoretical quantiles Figure 3. Q–Q comparison of empirical hourly BTC simple returns with a variance-matched Skellam benchmark. Blue circles are empirical sample quantiles plotted against the variance-matched Skellam theoretical quantiles; the dashed line is the 45-degree reference indicating perfect agreement with the Skellam benchmark. The symmetric reference provides only a coarse description of the center and understates the most extreme moves. Table 7 contains the sharper evidence after conditioning on realized one-hour volume. Table 7. Volume-conditioned middle-quantile check for standardized one-hour returns. Conditioning on realized one-hour BTC volume improves the center-fit comparison relative to the unconditional Q–Q benchmark, but the symmetric core still remains too dispersed. Width ratios below one imply that the model-implied central bands are wider than the empirical center. Central Band Empirical Coverage/Model Width Diagnostic Value 80% band Empirical coverage of model-implied band 85.3% 80% band Empirical/model width ratio 0.862 50% band Empirical coverage of model-implied band 60.7% 50% band Empirical/model width ratio 0.756 https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 19 of 41 Even after conditioning on realized one-hour volume, the single-regime symmetric core remains too wide in the center and too thin in the far tails. Appendices C.2 and C.4 report the supplementary burstiness and tail diagnostics. 3.5. Rolling Regime Classification Under the Proxy Mapping The full-sample one-hour signed-BTC-flow slope is b̂F = 7250.26, and combining it with the directional-impact scale κ̂dir = 2 × 10−5 yields a positive companion signedflow proxy of baseline-splice β ˆ proxy eff = 7.24 × 10−2 h−1 with propagated 95% interval [0.0613, 0.0837]. The interval construction is simulation-based: 200,000 draws combine a HAC-normal approximation for bF with the bootstrap distribution of κ̃dir , so it should be read as sampling uncertainty conditional on the baseline splice. The effective meanreversion rate remains positive at θeff = 0.0477 h−1 , with interval [0.0292, 0.0665], implying a descriptive full-sample resiliency half-life of 14.54 h. A positive β effˆ by itself does not proxy imply instability: under the local OU splice, the key question is whether mean reversion remains strong enough for θ̂eff to stay positive. Nearby-frequency checks leave the sign intact but move the point estimate to 0.0661 and 0.0417 h−1 , so the broad practical range across the baseline and nearby alternatives is closer to [0.0417, 7.24 × 10−2 ] than to the conditional propagated interval alone. In that sense, the sign of β effˆ is more credible than proxy its exact level. The nearby-frequency checks are therefore used only to show that the splice is not sign-fragile; they do not convert β effˆ into a directly identified structural parameter. proxy Any ρ summary remains secondary and is left to the appendix. Rolling estimation sharpens that interpretation. The rolling analysis uses overlapping 30-day windows as a compromise between local stationarity and enough hourly observations to estimate the return autocorrelation and signed-flow slope without excessive noise. After the initial window-fill period, this yields 47,655 non-missing 30-day window estimates.2 In the baseline overlapping 30-day scheme, 66.2% of windows have θ̂eff > 0 by point estimate alone. That point-sign share is informative but too optimistic to stand on its own. Using a simple one-standard-error autocorrelation buffer of ±0.0373 around zero, 39.2% of overlapping 30-day windows are clearly mean-reverting, 49.4% are ambiguous, and 11.5% are non-positive. This buffer is a heuristic monitoring rule rather than a formal confidence interval under heteroskedastic, serially dependent returns. A more conservative doubled companion buffer leaves only 17.2% of overlapping windows clearly mean-reverting and 79.9% ambiguous. Across those two heuristic buffers, the clearly mean-reverting share therefore lies in a range of roughly 17.2–39.2%, with ambiguity dominant under either rule. The daily-sampled and non-overlapping companion splits are similar: (39.2%, 49.4%, 11.4%) and (38.8%, 49.3%, 11.9%). Appendix C.6 reports the full sensitivity tables, including shorter 15-day and longer 60-day windows and the conservative companion buffer. The rolling feedback proxy remains economically meaningful, but the central regime-classification result is the sign and magnitude of θ̂eff , not the behavior of any derived boundary ratio. Figure 4 reports the rolling feedback and mean-reversion diagnostics. The figure should be read first as a regime-classification device based on θ̂eff , with the rolling feedback proxy shown alongside it. Taken together, the full-sample and rolling results are consistent with return-following signed-flow pressure in BTC/USDT, but the mean-reverting component dominates only in a subset of windows once a near-zero heuristic buffer is imposed. The windows with ambiguous or non-positive θ̂eff are economically important precisely because they identify periods in which the baseline queueing interpretation becomes unreliable. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 20 of 41 (a) Proxy signed-flow diagnostic Full-sample proxy 0.15 (h−1) 0.10 proxy ̂ βeff 0.05 0.00 (b) Effective mean-reversion diagnostic ̂ =0 θeff 0.4 Full-sample mean reversion ̂ (h−1) 0.2 θeff 0.0 −0.2 2020 2021 2022 2023 2024 2025 Figure 4. Rolling feedback and mean-reversion diagnostics for BTC/USDT, Binance, 2020–2025. The underlying estimates are computed in overlapping 30-day windows updated each hour after the initial 30-day fill period, but the figure samples one end-of-day point per UTC day for display clarity. Panel (a) reports the rolling companion proxy β effˆ , and panel (b) reports the rolling effective proxy mean-reversion estimate θ̂eff . Under the baseline heuristic buffer, the hourly-updated 30-day split is 39.2% clear positive, 49.4% ambiguous, and 11.5% non-positive. Hatched amber shading marks ambiguous windows and cross-hatched gray shading marks non-positive windows, where the local OU interpretation is not used. 3.6. Simple Next-Day Monitoring-Value Check To check whether the rolling diagnostics do more than summarize the same-day sample, the analysis runs a deliberately simple day-ahead sort. For each UTC day d, the end-of-day 30-day rolling liquidity diagnostic R b r (d) is paired with the end-of-day rolling regime label from Figure 4, then compared with realized outcomes on day d + 1. The matched sample starts only once both predictors are fully filled 30-day end-of-day diagnostics. The outcomes are next-day realized variance in hourly returns, the next-day maximum absolute hourly return, and a tail-day indicator equal to one when the next-day maximum absolute hourly return falls in the top decile of the matched sample. This is not a forecast horse race and does not claim optimal monitoring thresholds; it is a descriptive check of whether the rolling outputs sort meaningfully different next-day risk conditions. Table 8 shows that the rolling liquidity diagnostic carries the clearest next-day signal. Moving from the low to the high rolling-Rr tercile raises next-day realized variance from 0.000026 to 0.000072, increases the next-day mean maximum absolute hourly return from 1.31% to 1.98%, and raises the next-day tail-day rate from 6.5% to 15.1%. The resiliency classification adds a weaker but directionally similar signal: non-positive windows are followed by larger next-day maximum hourly moves and higher tail-day rates than clearly mean-reverting windows (13.3% versus 8.9%). Under the doubled companion buffer from Appendix C.6, the clearly classified sample shrinks sharply (347 clear-positive days and 59 non-positive days), but the same tail-rate ordering remains directionally intact https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 21 of 41 (15.3% versus 6.9%). These patterns do not validate a trading rule or establish predictive superiority over a generic volatility-persistence monitor, but they do show that the rolling diagnostics carry useful next-day monitoring information rather than functioning only as contemporaneous summaries. To see whether rolling Rr is only a relabeling of familiar trade-based liquidity measures, Appendix C.8 applies the same 30-day day-ahead design to two feasible public-data benchmarks: rolling Amihud illiquidity and a rolling Kyle-style minute signed-flow impact slope. Because the Amihud series requires 30 non-missing daily return-volume pairs, that benchmark comparison begins one day later than Table 8. Rolling Rr is positively related to both, with correlations 0.304 and 0.630, but it sorts next-day tail outcomes more sharply. The high-minus-low tail-day spread is 8.6 percentage points for rolling Rr , versus 4.7 for rolling Amihud and 1.1 for the Kyle-style minute-impact slope. That does not make Rr a universal winner in every application, but it does show that the queueing-organized liquidity diagnostic is not redundant with these feasible trade-based benchmarks in the same day-ahead design. Table 8. Simple next-day monitoring-value check based on end-of-day 30-day rolling diagnostics. Predictors are measured at the end of UTC day d, and outcomes are computed from the observed hourly returns on day d + 1. The matched sample contains 1987 predictor–outcome day pairs, running from 30 January 2020 through 8 July 2025. The first 29 UTC days are omitted so that both end-of-day predictors are fully filled 30-day diagnostics, and the final sample day is omitted because it has no d + 1 outcome; 15 matched days contain fewer than 24 hourly returns because of the observed missing UTC hours. “Tail day” means that the next-day maximum absolute hourly return lies in the top decile of the matched sample, with threshold 2.89%. Because the predictors are persistent rolling states, this table is included only as a descriptive relevance check and does not establish incremental forecast evidence over generic volatility persistence. Predictor Bucket Days Next-Day Realized Variance Next-Day Max |r1h | Next-Day Tail-Day Rate End-of-day rolling Rr tercile Low 663 0.000026 1.31% 6.5% Mid 661 0.000044 1.56% 8.5% High 663 0.000072 1.98% 15.1% End-of-day rolling resiliency regime (baseline heuristic buffer) Clear positive 779 0.000048 1.53% 8.9% Ambiguous 982 0.000046 1.63% 10.2% Non-positive 226 0.000054 1.86% 13.3% 3.7. Secondary One-Hour Tail Overlay For the empirical risk application, the analysis stays in reduced-form observable notation and at the one-hour horizon only. Let ν̄ = E[ν1h ] denote average one-hour BTC volume, and let V b1h = Var(r1h ) = Rr ν̄ denote the observed one-hour return variance. The core one-hour benchmark is therefore the Normal approximation with variance V b1h . To monitor the most extreme tail events, the analysis adds a threshold-split Normal-plusStudent-t overlay re-fit every 24 h inside trailing 2160 h windows. Over 46,214 hourly forecasts, the rolling Gaussian benchmark breaches at 1.80% and the stress-state overlay at 1.13%. That overlay is included as a supporting parametric tail monitor; fuller benchmark comparisons and backtesting details remain in Appendices E and C.11. 4. Discussion 4.1. Short-Horizon Price Formation and Liquidity Interpretation The results support a market-microstructure interpretation in which inventory and trade imbalances shape short-horizon BTC/USDT price variation on Binance. The variance– volume relation shows that gross trading activity and short-horizon variance move together https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 22 of 41 in a disciplined way. At the same time, the BTC-flow diagnostics are consistent with positive returns being followed, on average, by additional buyer-initiated BTC flow at the hourly horizon. In practical terms, this means that high activity can raise volatility through both gross two-sided trading and directional signed-flow pressure, even though full-sample return autocorrelation remains negative. The queueing formulation is useful because, inside the model, arrival and holding-time assumptions organize occupancies, occupancies organize net imbalance, and net imbalance organizes the local price-deviation state. Empirically, the paper identifies monitoring diagnostics tied to that structure rather than the underlying queue primitives themselves. That decomposition explains what would be lost without queueing: the observable bundle would no longer live in one stock-flow state space, the recovery interpretation of θeff would lose its turnover counterpart, and the pooled occupancy translation would disappear. The model adds value by providing a disciplined accounting system that connects observable market states to short-horizon price formation, volatility, and resiliency. 4.2. Market Monitoring and Risk Application From a monitoring perspective, the strongest outputs are the rolling liquidity path in Figure 2, the rolling resiliency classification in Figure 4, and the next-day descriptive sort in Table 8. A rising rolling Rr indicates more variance per traded BTC, a longer descriptive half-life indicates slower local recovery, and ambiguous or non-positive θ̂eff flags periods in which resiliency is weaker or harder to classify. Appendix C.7 points in the same direction for isolated large hourly shocks: clear-positive windows reverse more over the next 24 h than ambiguous or non-positive windows. Appendix C.8 adds the benchmark point directly: rolling Rr is related to rolling Amihud illiquidity and a Kyle-style minute signed-flow impact slope, but it sorts next-day tail risk more sharply in that closely matched day-ahead benchmark sample. The rolling β ˆ series is best read as a companion signed-flow gauge rather than proxy eff as a stand-alone stability estimate. Its role is to sit beside θ̂eff and help distinguish momentum-heavy from more resilient market phases. Queueing matters here because it forces the rolling liquidity, resiliency, and signed-flow diagnostics to live on one occupancyaccounting scale rather than as separate alarms. Who Uses This Dashboard? The practical use is concrete. A market maker or execution desk can use Rr to see whether the venue is delivering less turnover-adjusted liquidity than usual, θeff to judge proxy whether directional pressure appears to unwind quickly or only weakly, and β eff to distinguish quiet deterioration from visibly one-sided flow. An exchange risk or surveillance team can use the same trio to separate ordinary busy trading from states in which liquidity is thin, resiliency is fragile, and next-day risk outcomes are worse. That is why the framework remains useful even though the heaviest unconditional tails call for an additional overlay. The tail lesson is separate and simple: the queueing core is useful for liquidity interpretation, resiliency monitoring, and a transparent benchmark for the center of one-hour returns, while the heaviest tails require an additional overlay. 4.3. Limitations The paper has several limitations. It is a single-venue Binance spot study even though Bitcoin trading is multi-venue; it uses a linear impact approximation rather than richer nonlinear or finite-depth dynamics; it assumes independent latent arrivals and i.i.d. holding times; and its feedback layer uses a minute-impact and hourly-flow splice as a proxy for latent directional pressure rather than directly observing customer-level queues. Those https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 23 of 41 latent-arrival and holding-time assumptions are imposed organizing approximations rather than separately tested features of the trade-print sample. On smaller venues with materially different maker/taker fee schedules, the signed-flow slope bF could also reflect fee-induced routing or inventory behavior differently, so any cross-venue comparison would require venue-specific fee controls rather than direct coefficient transfer. The stationary core is also designed for normal and moderately stressed periods rather than every crisis tail event, which is why the tail overlay remains separate. These choices keep the framework transparent and implementable, but they also bound the scope of the claims. Natural extensions include multi-venue implementations, nonlinear impact, and marked or compound-Poisson queueing generalizations. What Would Weaken the Framework? Three empirical patterns would materially weaken the interpretation advanced here. First, the framework would lose credibility if the variance-per-BTC relation became persistently unstable across volume bins or over rolling windows after using the same data construction. Second, it would be harder to maintain the resiliency interpretation if proxybased feedback remained persistently strong while θ̂eff repeatedly became small or nonpositive across rolling windows. Third, the queueing core would be too restrictive if it failed not only in the far tails but also in the middle quantiles of one-hour return increments once realized volume and horizon were fixed. Those are direct ways in which future evidence could narrow or overturn the usefulness of the present specification. 5. Conclusions This paper develops a queueing-organized framework for within-venue monitoring of BTC/USDT liquidity and resiliency on Binance. The practical outputs are three linked diagnostics: the variance-per-BTC liquidity measure Rr , the effective mean-reversion proxy rate θeff , and the companion signed-flow proxy β eff . Using Binance trade data from 2020–2025, the empirical results show a pooled first-order variance–volume regularity away from the highest-volume tail, rolling liquidity and resiliency measures that change materially across market phases, and a next-day descriptive sort in which high-Rr days and non-positive resiliency windows are followed by worse risk outcomes than quieter and clearly mean-reverting states. In a closely matched day-ahead benchmark sample, rolling Rr also sorts next-day tail risk more sharply than rolling Amihud illiquidity or a Kyle-style minute-impact benchmark. The paper’s contribution is to place those diagnostics inside one stock-flow bookkeeping system. Queueing is useful here because it ties liquidity, signed-flow pressure, and recovery timescales to the same occupancy accounting logic. That common state space is what turns three separate reduced-form monitors into one interpretable venue-level monitoring framework. The intended audience is any reader who must monitor venue conditions from trade prints: market makers, execution desks, exchange surveillance teams, and short-horizon risk managers. For that audience, the payoff is a compact dashboard with a coherent economic interpretation rather than three disconnected time series. The symmetric one-hour benchmark and the tail overlay are supporting checks; the core contribution is proxy the linked interpretation of Rr , θeff , and β eff . Scope and Falsifiability The paper remains deliberately focused. It is a single-venue study, it uses a linearimpact approximation, and it does not attempt an exhaustive comparison with alternative microstructure models. Within that scope, the framework should stand or fall on observable margins: a persistent breakdown of the variance-per-BTC relation, repeated failure of https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 24 of 41 effective mean reversion in rolling windows, or severe failure of the queueing core even as a center benchmark once realized one-hour volume is fixed would all count against the specification rather than as minor calibration noise. Supplementary Materials: The following supporting information can be downloaded at: https:// www.mdpi.com/article/10.3390/jrfm19050372/s1, a Python code package that reproduces all empirical outputs reported in this paper. Funding: This research received no external funding. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Raw Binance BTC/USDT trade data are publicly available through the Binance historical data portal https://data.binance.vision/?prefix=data/spot/daily/trades/ BTCUSDT/ (accessed on 28 July 2025). The empirical results in this paper are generated from those public trade data using a local workflow that constructs hourly and aggressor-side summaries and then reproduces the reported panels, figures, and macro files. Two public code archives are available. The broader project code base is archived at https://doi.org/10.5281/zenodo.18644286 (accessed on 28 July 2025). A code-only workflow archive from this project is archived at https://doi.org/ 10.5281/zenodo.19742219 (accessed on 28 July 2025); it contains workflow scripts and documentation from this project, but it does not include manuscript files, compiled PDFs, generated outputs, or raw Binance trade files. Acknowledgments: The author is grateful for computing resources provided by VS Asset Management, LLC. Conflicts of Interest: The author is a co-founder, Chief Investment Officer, and managing member of VS Asset Management, LLC. The author declares no conflicts of interest. Appendix A. Additional Discussion Appendix A.1. Queueing Theory Primer Queueing Theory Essentials Queueing theory studies systems where units arrive randomly, receive service, and depart. In finance, these elements are reinterpreted as follows: • Units → BTC-normalized latent occupancy units on the buy and sell sides • Arrivals → New latent occupancy units opened at rate λ • Service time → Holding period H until that latent occupancy exits • Queue length → Outstanding latent occupancy The M/G/∞ notation describes the system used here: • M = Markovian (Poisson) arrivals • G = General service time distribution • ∞ = Infinite servers (no capacity constraints) Key Result: In steady state, if latent occupancy units arrive at rate λ and are held for average time E[ H ], then outstanding occupancy follows a Poisson law with mean m = λE[ H ] (Palm’s occupancy result for M/G/∞ queues; Little’s Law gives the same mean). In the manuscript, occupancies are BTC-normalized, so one latent unit corresponds by normalization to 1 BTC. Poisson Arrivals See Time Averages (PASTA) ensures that sample-path observations of arrivals match time averages in M/G/∞ systems; see Wolff (1982). This framework https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 25 of 41 naturally captures how latent buy- and sell-side occupancy accumulates and dissipates in markets, providing the mathematical foundation for the pricing model. Table A1. Assumption ladder used in the paper. Each layer adds one extra assumption and delivers a narrower object than the layer above it. What It Does Not Layer Extra Assumption What It Delivers Establish Stationary occupancy law, Independent Poisson One-hour return law or Skellam imbalance, M/G/∞ stationary layer arrivals and i.i.d. holding literal queue openings variance–volume times from public trades bookkeeping Contingent one-hour Structural identification of Symmetric M/M/∞ Symmetry plus increment bridge, E[ H ], those quantities from one-hour bridge exponential holding times λ± , m± , and κ̃ illustration observables alone Local linearization plus Descriptive pair A directly identified latent proxy Local OU/proxy splice minute/hour splice for (θeff , β eff ) for rolling feedback boundary or signed-flow proxy regime classification exact queue stability law Appendix A.2. Applicability of Queueing Assumptions to Bitcoin The queueing framework is most useful when four conditions are approximately present: (1) limited direct intraday anchoring to externally observed fundamentals; (2) orderflow-dominated trading at intraday horizons; (3) a homogeneous traded asset; (4) continuous high-turnover trading. Bitcoin is a reasonable, though not unique, empirical setting for those conditions. Appendix B. Additional Model Extensions The extensions in this section are supplementary heuristic sketches and are not used in the empirical implementation reported in the paper. They indicate only how the baseline queueing framework could be generalized without changing the observable monitoring strategy used in the paper. Overview. This section first records the one-hour bridge, contingent symmetry translations, splice sensitivity, and reduced-form variance benchmark used elsewhere in the paper, and then adds three brief illustrative extensions: discrete time and price grids in Appendix B.1, together with nonlinear or finite-depth impact sketches in Appendices B.5 and B.6. All of these items are supplementary and do not add separate calibrated claims. Appendix B.1. Discrete Time and Price Grids Real-world trading is discrete in both time and price. The results derived in continuous time survive intact once (i) clock time is sliced into short intervals of length ∆ and (ii) prices are snapped to the nearest exchange tick. Appendix B.1.1. Discrete Time Fix a step of ∆ = 60 s (∆ = 1/60 h).3 At tick n, the numbers of newly opened longand short-side latent occupancy units, An,+ and An,− , are independent Poisson variates: An,+ ∼ Poisson(λ+ ∆) = Poisson(13.67), (A1) An,− ∼ Poisson(λ− ∆) = Poisson(13.67), (A2) where the BTC-normalized contingent symmetry translation in Appendix B.3 implies λ+ ≈ λ− ≈ 820 h−1 for the 2020–2025 Binance sample (Binance, 2025). https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 26 of 41 Because the mean holding time is E[ H ] ≈ 5.20 h, an existing occupancy unit survives one minute with probability ∆ 1/60 p+ = p− = 1 − = 1− ≈ 0.996792. (A3) E[ H ] 5.20 These dynamics define a GI/Geo/∞ queue whose steady-state occupancies equal those of the continuous-time model: B∞ ∼ Poisson(m+ ≈ 4262), S∞ ∼ Poisson(m− ≈ 4262). (A4) Running Example In the BTC-normalized contingent symmetry translation used throughout the appendix, discretisation leaves m± unchanged, so the net inventory remains X∞ ∼ Skellam(4262, 4262) and the same variance–volume relations continue to hold. Appendix B.1.2. Price Grid Let δP = 0.01 USD be the BTC/USDT tick size and q0 = 0.001 BTC the minimum trade lot. Grid-aligned impact is implemented as Pn = P0 + δP ⌊κXn /δP⌉, (A5) where ⌊·⌉ denotes rounding to the nearest integer and with the structural impact slope κ̃ = 1.26 × 10−4 BTC−1 (so κ = P̄ κ̃ ≈ $6.289 BTC−1 for P̄ = $50,000). At P0 = 50, 000 USD, one tick is a mere δP/P0 = 2 × 10−7 = 0.002 bp, so linearisation error is negligible: an inventory change of δP/κ ≈ 1.59 × 10−3 BTC (using P̄ = $50, 000) contracts moves the mid-price by exactly one tick. Summary Discretising time at one-minute intervals and snapping prices to the one-cent grid leaves the Poisson–Skellam backbone largely intact. In the present calibration, the continuous-time formulas remain adequate local approximations and any grid effects are second-order. Appendix B.2. One-Hour Increment Bridge The main text maps the observable variance-per-BTC moment into a structural returnimpact slope κ̃. This subsection gives the one-hour bridge explicitly under the symmetric M/M/∞ mapping used for that contingent translation. The general M/G/∞ occupancy result from Palm’s theorem still underlies the stationary inventory law; the extra step here is narrower because the increment covariance formula below uses exponential holding times. Let Bt and St denote stationary long and short occupancies with common mean holding time E[ H ] and common mean occupancy m = λE[ H ]. For an M/M/∞ queue, Cov( Bt , Bt+h ) = me−h/E[ H ] , Cov(St , St+h ) = me−h/E[ H ] . Therefore, for the net imbalance Xt = Bt − St ,  Var( Xt+h − Xt ) = 2 Var( Xt ) − 2 Cov( Xt+h , Xt ) = 4m 1 − e−h/E[ H ] . https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 27 of 41 Using m = λE[ H ] and the symmetric executed-volume identity E[νh ] = 4λh, the return increment rt,h ≈ κ̃ ( Xt+h − Xt ) satisfies −h/E[ H ]  Var(rt,h ) 2 E[ H ] 1 − e = κ̃ . (A6) E[νh ] h At the one-hour horizon used in the paper (h = 1), Var(r1h )  = κ̃ 2 c H , c H := E[ H ] 1 − e−1/E[ H ] . (A7) E[ν1h ] Equation (A7) is the bridge used for the symmetry-based structural translation in the main text and in the running examples in the appendix. The occupancy units in this bridge are latent and BTC-normalized. The bridge should therefore not be read as a one-to-one mapping from public trades to literal queue openings or exits; executed BTC enters only through the first-moment bookkeeping identity for E[νh ]. Appendix B.3. Additional Contingent Symmetry Translations Under the same symmetric M/M/∞ special-case translation used in the main text, mean executed one-hour volume implies baseline directional intensities E[ν1h ] λ+ = λ− = ≈ 820 BTC/h. Combining this with the translated holding time E[ H ] ≈ 5.20 h yields contingent occupancies m+ = m− = λ± E[ H ] ≈ 4262 BTC, where queue occupancies are BTC-normalized throughout, so one latent inventory unit corresponds by normalization to 1 BTC. The bridge to executed BTC volume is only a firstmoment bookkeeping convention: the normalization is chosen so that expected openings and closures of latent occupancy units match expected executed BTC turnover in the symmetric identity above. A marked or compound-Poisson extension would preserve that first-moment bridge while changing higher moments. These values are reported here only to make the symmetry arithmetic transparent. They are not separately identified empirical estimates. Appendix B.4. Splice Sensitivity of the Contingent Translation Because the contingent symmetry translation inherits the minute/hour proxy splice, the implied holding times and occupancies move when nearby-frequency alternatives are proxy substituted into β̂ eff . Table A2 reports the baseline splice together with the 5 min directional and 2 h flow alternatives. The point is only interpretive: the contingent translation remains useful for separating turnover from recovery, but the exact queue numbers should not be read as stable estimates when the proxy splice itself shifts. Across these nearby variants, the implied holding time ranges from about 5.20 to 7.63 h and the total latent occupancy from about 8524 to 12,510 BTC-normalized units. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 28 of 41 Table A2. Sensitivity of the contingent symmetry translation to nearby-frequency proxy-splice alternatives. These values are illustrative arithmetic under the symmetric bridge, not directly identified structural estimates. proxy Splice Variant β̂ eff (h−1 ) Contingent E[ H ] (h) Contingent m+ + m− Baseline splice 0.0724 5.20 8524 5 min directional alternative 0.0661 5.56 9116 2 h flow alternative 0.0417 7.63 12,510 Appendix B.5. Illustrative Nonlinear Price Impact The linear price impact assumption can be relaxed to accommodate realistic order book dynamics while maintaining substantial analytical tractability. Appendix B.5.1. Exponential Impact For multiplicative price effects, the illustrative extension uses: Pt = P0 exp(κ̃Xt ) (A8) Since X∞ ∼ Skellam(m+ , m− ), all moments of P∞ remain closed-form. The k-th moment is: h i k E[ P∞ ] = P0k exp −(m+ + m− ) + m+ ekκ̃ + m− e−kκ̃ (A9) The log-price Yt = ln Pt = ln P0 + κ̃Xt has variance κ̃ 2 (m+ + m− ), preserving the volume–volatility relationship for log-returns. Note: This section discusses log-prices only; all empirical returns used elsewhere are arithmetic. Appendix B.5.2. Saturating Impact (Hyperbolic Tangent) For bounded price impact that saturates at extreme inventory levels: Pt = P0 + κP0 tanh( Xt /L) (A10) where L > 0 controls the saturation scale. Moments exist as finite sums over the Skellam support: ∞ E[ P∞ ] = P0 + κP0 ∑ tanh(k/L) · Pr{X∞ = k} (A11) k =−∞ √ For L ≫ m+ + m− , a Taylor expansion yields: κP0 κP E[ P∞ ] ≈ P0 + (m+ − m− ) − 03 [(m+ − m− )3 + 3(m+ − m− )(m+ + m− )]. (A12) L 3L The cubic correction can introduce additional kurtosis relative to the linear model. Under the Binance full-sample calibration, these nonlinear transforms mainly matter in larger imbalance states and are therefore kept here only as illustrative alternatives rather than as separate empirical objects. Appendix B.6. Illustrative Finite Order Book Depth Real order books have finite depth, leading to state-dependent price impact. This extension models that feature through piecewise-linear impact functions that remain analytically tractable. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 29 of 41 Appendix B.6.1. Piecewise Linear Impact Schedule Define the impact function:   | X | < X1 κ1 ,  κ ( X ) = κ2 , X1 ≤ | X | < X2 (A13)    κ3 , | X | ≥ X2 with 0 < κ1 < κ2 < κ3 reflecting deeper impact at higher inventory levels. Appendix B.6.2. Price Distribution with Depth Tiers The price variance decomposes as: 3 Var[ P∞ ] = ∑ κr2 ∑ k2 Pr{ X∞ = k} − (E[ P∞ ] − P0 )2 (A14) r =1 k ∈Kr where K1 = {k : |k | < X1 }, K2 = {k : X1 ≤ |k | < X2 }, and K3 = {k : |k | ≥ X2 }. Each truncated sum uses the Skellam PMF. For efficient evaluation, use the stable ratio recursion r √  Pr{ X = k + 1} m+ Ik+1 2 m+ m− = √  , k ≥ 0, (A15) Pr{ X = k } m− Ik 2 m+ m− with the symmetric formula for negative k. Under the Binance full-sample calibration, this finite-depth illustration leaves the effective linear impact close to the top-of-book slope because most stationary mass remains in the first two depth tiers. The main implication is therefore qualitative: finite depth matters primarily in larger imbalance states and is not separately estimated in the paper. Appendix B.7. Reduced-Form One-Hour Variance Benchmark The main text now keeps the risk application at the one-hour horizon only. The relevant observable object is therefore b1h := Var(r1h ) = Rr ν̄, V ν̄ = E[ν1h ], which is the directly observed one-hour return variance used to normalize both the Gaussian benchmark and the threshold-split stress overlay. The local OU approximation and θeff remain useful as resiliency diagnostics, but the paper no longer treats them in the appendix as a separate multi-horizon return-law derivation. Using the Binance calibration, the one-hour benchmark standard deviation is σr ≈ 0.69% (about $344 at P∗ = $50,000), and the corresponding descriptive half-life from the local mean-reversion estimate remains t1/2 = ln 2/θeff = 14.54 h. This is the only variance object required by the one-hour VaR overlay used in the paper. Appendix C. Additional Empirical Results and Robustness Appendix C.1. Volume-Conditioned Dispersion of the Variance-per-BTC Ratio Figure A1 reports the within-bin dispersion of the daily variance-per-BTC ratio across the same volume bins used in the main paper’s scatter plot. The central bins cluster around the overall mean ratio, while the highest-volume bin is visibly attenuated. This is why the main text describes the variance–volume evidence as first-order compatibility rather than as a literal invariance result. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 30 of 41 1×10−9 Within-bin dispersion of the daily variance-per-BTC ratio Overall mean ratio 1.2 1.0 Daily variance-per-BTC ratio 0.8 0.6 0.4 0.2 Highest-volume bin 0.0 0 50 100 150 200 250 300 350 Mean daily BTC volume in bin (thousands of BTC) Figure A1. Within-bin dispersion of the daily variance-per-BTC ratio across volume bins for BTC/USDT, Binance, 2020–2025. Points show the within-bin median ratio and bars show the interquartile range. The dashed line is the overall mean ratio. The main takeaway is that the central bins cluster around a fairly stable ratio, while the highest-volume bin shows clear attenuation. Appendix C.2. Volume-Conditioned Variance-Matched Increment Benchmark The main text’s Q–Q plot fixes only the horizon and the sample-average hourly volume. Here the analysis conditions more tightly on realized one-hour volume. This subsection still studies a variance-matched benchmark rather than the exact increment law of the general M/G/∞ framework. For each hour t, define the standardized increment rt − r̄ zt := √ , Rr νt where rt is the simple one-hour return, r̄ is the sample mean one-hour return, Rr is the observable variance-per-BTC moment, and νt is realized one-hour BTC volume. Under the symmetric queueing benchmark, the corresponding conditional law is Zt z∗t = √ , Zt | νt ∼ Skellam(νt /2, νt /2). νt Using the observed νt sequence over 48,374 hourly bars and 8 simulated draws per hour, the empirical standardized middle quantiles are (q0.10 , q0.25 , q0.75 , q0.90 ) = (−1.097, −0.504, 0.514, 1.114), versus model-implied quantiles (−1.284, −0.675, 0.671, 1.281). The model-implied central 80% band therefore covers 85.3% of standardized observations, and the model-implied central 50% band covers 60.7%. The corresponding empirical/model width ratios are 0.862 for the 80% band and 0.756 for the 50% band. Ratios below one mean that the model-implied central bands are wider than the empirical center. Volume conditioning thus improves the middle-quantile comparison relative to the unconditional https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 31 of 41 benchmark in the main text, but the symmetric variance-matched core still overstates central dispersion somewhat and does not eliminate tail misspecification. Appendix C.3. Sub-Sample Reduced-Form Checks The main paper now reports 2020–2022, 2023, and 2024–2025 side by side so the reader can see directly that the reduced-form liquidity and resiliency moments move across major market phases. This appendix keeps the earlier endpoint contrast only as a compact descriptive comparison between the pre-transition window and the later ETF-era window; 2023 is treated in the main paper as the transition year rather than excluded from the evidentiary record. Interpretation The earlier sub-sample windows show somewhat heavier trading intensity (3986 BTC versus 1305 BTC per one-hour bar), while the later window shows a larger variance-perBTC estimate (1.4× the 2020–2022 value) together with slower effective mean reversion (0.0336 versus 0.0471). Those comparisons are descriptive rather than structural. Once the symmetry mapping is re-imposed on short sub-samples, contingent quantities such as E[ H ], λ± , and m± can move substantially even when the directly observed moments change only moderately. For that reason, the main text keeps the structural translations at the full-sample level and uses the sub-sample checks only to show that the reduced-form diagnostics remain regime-dependent. Appendix C.4. Addressing the Tail Risk Challenge Appendix C.4.1. Scope The Skellam (or NB–Skellam) core provides a useful baseline for normal trading periods and a coarse description of the center of the unconditional one-hour increment distribution. The far tails during stress periods necessarily require an additional component—such as a core–Normal/tail–Student-t mixture—to improve tail coverage relative to the Gaussian benchmark. The one-hour return series has empirical excess kurtosis 47.29. By comparison, the count-space Poisson term is only 1.8 × 10−5 and the count-space NB approximation rises only to 5.2889. Rather than invalidating the baseline interpretation, this gap delineates the domain over which the core specification is useful: What the Model Captures Well (the bulk of central observations): • Normal market conditions with continuous order flow. • Gradual order-flow shifts across descriptive market phases. • Routine volatility from imbalanced arrival rates. • Mean-reversion from latent occupancy unwinding. What Requires Extensions (stress-period tail events): • Exchange hacks or policy shocks. • Coordinated whale movements. • Cascading liquidations from leveraged positions. • Technological disruptions (forks, 51% attacks). Appendix C.4.2 shows that a two-component mixture improves unconditional tail coverage materially relative to a rolling Gaussian benchmark, but still falls short of exact 99% unconditional coverage and continues to show violation clustering during stress episodes. In that specification, the queueing-motivated core benchmark describes normal times while a heavy-tailed Student-t component captures crisis periods. The core model provides the baseline; practitioners must still add a stress-state layer for tail-risk work. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 32 of 41 This parallels established practice in equity markets, where models capture normal volatility but require separate treatment of crashes (Bates, 2000). The paper’s narrower contribution is to characterize normal order-flow dynamics before turning to extreme events. Appendix C.4.2. Heavy-Tail Regime Switch Pure Skellam dynamics capture normal order-flow fluctuations but miss the extreme tails seen in realised returns. The hourly return rt is therefore modeled as a two-component mixture described in Appendix E. This subsection uses a 1 h VaR overlay based on a core-Normal/tail-Student-t mixture with threshold at 2σemp , estimated by a threshold split rather than by full EM-MLE.  f r ( x ) = π0 N ( x; µ0 , σ02 ) +π1 tν x; µ1 , σ12 , π0 + π1 = 1, (A16) | {z } | {z } core stress where the core component is fit on the in-threshold observations and is only queueingmotivated rather than queueing-imposed, while the stress component adds a near-zeromean Student-t shock with scale σ1 and degrees of freedom ν.4 Parameter Estimation Equation (A16) is fit descriptively to the 2020–2025 Binance hourly close-to-close returns. The full-sample parameter estimates are π0 = 0.952, µ0 = 1.10 × 10−4 , σ0 = 0.00436, (A17) −4 π1 = 0.048, µ1 = −4.44 × 10 , σ1 = 0.02314, ν = 21.92. (A18) For risk management, however, the paper evaluates this specification in a rolling out-ofsample backtest rather than relying on the full-sample fit alone. The intent is pragmatic tail augmentation, not structural identification from the queueing model. Risk Metrics The one-hour 99% Value-at-Risk is the 1% quantile of the mixture. In the rolling backtest used in the main text, the average one-hour forecast is (1h) VaR0.99 = Fr−1(0.01) ≈ 0.0184 (1.84% on average). Back-Test Using trailing 2160 h estimation windows and re-fitting every 24 h, the mixture produces k = 520 VaR breaches over n = 46,214 out-of-sample hourly forecasts (1.13%). The Kupiec unconditional-coverage statistic is h   i LRUC = −2 (n − k) ln(1 − p) + k ln p − (n − k) ln 1 − nk − k ln nk ≈ 7.03, yielding a p-value of 0.008. Relative to the rolling Gaussian benchmark (breach rate 1.80%), the mixture improves unconditional coverage materially. It nevertheless still misses exact 99% unconditional coverage at conventional levels, whereas rolling historical simulation (1.07%) is slightly closer to nominal unconditional coverage in this sample. The mixture is retained because it is a compact parametric monitoring overlay tied to the same one-hour variance normalization as the core benchmark, not because it dominates nonparametric forecasting. Violation clustering remains visible in stress episodes even after the unconditional fit improves. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 33 of 41 Interpretation Only 4.8 % of the time does the market enter the stress state, but in that regime the scale of shocks (σ1 ≈ 0.02314) is roughly five times the core Skellam standard deviation (σ0 ≈ 0.00436), contributing substantially to the empirical excess kurtosis of 47.29 reported in the main paper’s empirical summary table. This appendix evidence sharpens the paper’s risk interpretation by the following: 1. Separating the descriptive full-sample fit from the rolling out-of-sample evaluation; 2. Showing that the stress-state layer improves on a Gaussian benchmark for unconditional 99% coverage without fully solving it; 3. Quantifying how much additional tail thickness is needed beyond the queueing core. Appendix C.5. Parameter Evolution and Stability The main paper reports rolling estimates of the feedback proxy β effˆ and the effective proxy mean-reversion rate θ̂eff across 30-day windows. The informative empirical result is the variation in those directly monitored quantities: the full-sample proxy is positive, while rolling mean reversion weakens materially and sometimes turns negative. Any accompanying symmetry-based quantities remain secondary descriptive translations rather than additional sources of identification. Appendix C.6. Rolling-Window-Length Sensitivity Because the 30-day rolling share of windows with θ̂eff > 0 is a visible headline number in the main paper, Table A3 checks the same descriptive statistic under shorter and longer overlapping windows and under more conservative sampling schemes for the 30-day window. The point of that table is modest: it does not create new identification, but it shows that the point-sign summary is not unique to a single exact window length or a single exact overlap convention. Table A4 then adds the uncertainty-aware 30-day classification used in the main text. The baseline classification uses a one-standard-error buffer in the √ √ lag-1 autocorrelation, ρ̂1 ∈ [−1/ 720, 1/ 720], which corresponds to the near-zero region ±0.0373 around zero in the autocorrelation scale. This is a heuristic monitoring rule rather than a formal confidence interval under heteroskedastic, serially dependent returns. As a more conservative companion heuristic, doubling the buffer to ±0.0745 leaves 17.2% of overlapping windows clear positive, 79.9% ambiguous, and 2.9% non-positive. The main qualitative message is unchanged: ambiguity dominates once one moves away from the raw point-sign share. Table A3. Sensitivity of the rolling sign summary for θ̂eff to window length. The shares remain descriptive because the windows overlap heavily.   Rolling Window Sampling Scheme Pr θ̂eff > 0 Pr θ̂eff ≤ 0 15-day Hourly-updated overlap 62.7% 37.3% 30-day Hourly-updated overlap 66.2% 33.8% 30-day Daily-sampled overlap 66.3% 33.7% 30-day Non-overlapping blocks 62.7% 37.3% 60-day Hourly-updated overlap 70.6% 29.4% https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 34 of 41 Table A4. Heuristic uncertainty-aware classification for the 30-day rolling summary. “Clear positive” √ √ means the implied lag-1 autocorrelation is below −1/ 720, “non-positive” means it is above 1/ 720, and the middle region is classified as ambiguous. The rule is a monitoring buffer, not a formal dependence-robust interval. 30-Day Scheme Clear Positive Ambiguous Non-Positive Hourly-updated overlap 39.2% 49.4% 11.5% Daily-sampled overlap 39.2% 49.4% 11.4% Non-overlapping blocks 38.8% 49.3% 11.9% In the day-ahead monitoring exercise reported in the main paper, this doubled companion buffer leaves 347 clear-positive predictor days and 59 non-positive predictor days, with corresponding next-day tail-day rates of 6.9% and 15.3%, respectively. The doubledbuffer comparison is therefore kept only as a directional companion check rather than as a primary classification result. Appendix C.7. Post-Shock Recovery by Rolling Resiliency Regime The main paper interprets θeff only as a local descriptive recovery-timescale diagnostic. To connect that language to realized paths, this appendix reports one narrow event study using the existing hourly returns and the existing rolling 30-day regime labels. Isolated large shocks are identified as the top 97.5% of absolute one-hour returns among hours with valid 30-day classifications, with only local maxima within ±6 h retained and candidate shocks de-overlapped within the following 24 h. This leaves 497 events. For an event at hour t with return rt and sign st = sign(rt ), define the aligned remaining displacement after h hours as st ∑hj=0 rt+ j Dt,h := . |r t | By construction, Dt,0 = 1. Lower values indicate more reversal of the initial shock, and Dt,h = 0 corresponds to a full reversal by horizon h. Table A5 and Figure A2 show a clear ordering for these rare isolated shocks. Mean initial shock magnitudes are similar across regimes (about 3% of price), but the subsequent paths differ. After 6 h, the mean aligned remaining displacement is 0.907 in clear-positive windows, versus 0.946 in non-positive windows and 1.025 in ambiguous windows. After 24 h, the same ordering remains: 0.795 in clear-positive windows, 0.890 in non-positive windows, and 1.017 in ambiguous windows. The corresponding mean 24 h reversal shares are 20.5%, 11.0%, and −1.7%. This is still only a threshold-specific descriptive check, but it gives a direct path-based complement to the next-day sorting exercise in the main text: the rolling resiliency classification is associated not only with next-day risk levels, but also with visibly different recovery paths after large isolated shocks. Table A5. Post-shock recovery by rolling 30-day resiliency regime for isolated large hourly shocks on Binance BTC/USDT, 2020–2025. Shocks are the top 97.5% of absolute one-hour returns among hours with valid rolling classifications, restricted to local maxima within ±6 h and de-overlapped within the following 24 h. Lower remaining displacement indicates faster reversal of the initial shock. Remaining Remaining Rolling 30-Day 24 h Reversal Events Mean Initial |r0 | Displacement at Displacement at Regime Share 6h 24 h Clear positive 161 3.11% 0.907 0.795 20.5% Ambiguous 256 3.01% 1.025 1.017 −1.7% Non-positive 80 3.07% 0.946 0.890 11.0% https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 35 of 41 Post-shock recovery by rolling 30-day resiliency regime 1.0 Aligned remaining displacement 0.8 0.6 Clear positive Ambiguous Non-positive 0.4 0.2 0.0 0 5 10 15 20 Hours after shock Figure A2. Aligned post-shock recovery paths by rolling 30-day resiliency regime for isolated large hourly shocks on Binance BTC/USDT, 2020–2025. Event hours are the top 97.5% of absolute hourly returns among observations with valid rolling classifications, restricted to local maxima within ±6 h and de-overlapped within the next 24 h. The vertical axis reports aligned remaining displacement, normalized so the shock hour starts at 1; lower paths indicate faster reversal, and the dotted zero line corresponds to full reversal of the initial shock. Clear-positive windows recover more over the next 24 h than ambiguous or non-positive windows in this narrow descriptive check; event counts by regime appear in Table A5. Appendix C.8. Comparison with Standard Trade-Based Liquidity Benchmarks To see whether the rolling liquidity monitor is just a renamed version of standard trade-based measures, this subsection compares rolling Rr with two benchmarks that can be constructed from the same public Binance trade-print data. Both benchmarks are constructed from the identical public Binance trade-print panel used for the main diagnostics. The first is a rolling 30-day Amihud illiquidity proxy, defined as the 30-day mean of daily |rd |/$Vd , where rd is the UTC-day close-to-close return and $Vd is daily dollar volume. The second is a rolling 30-day median of the absolute daily minute signed-flow slope |κd | from the day-by-day directional-impact regressions, used here as a trade-printfeasible Kyle-style minute signed-flow impact benchmark. The benchmark comparison follows the same 30-day day-ahead design as the monitoring exercise reported in the main paper, but it begins one day later because the rolling Amihud series needs 30 non-missing daily return-volume observations. The resulting matched benchmark sample contains 1986 predictor days from 31 January 2020 through 8 July 2025, with 15 matched days containing fewer than 24 hourly returns because of the observed missing UTC hours. Table A6 shows two things. First, rolling Rr is related to these familiar trade-based measures but not identical to them: its correlation is 0.304 with rolling Amihud illiquidity and 0.630 with the Kyle-style minute signed-flow impact benchmark. Second, rolling Rr sorts next-day stress more sharply in this matched benchmark sample. The high-minus-low next-day tail-day spread is 8.6 percentage points for rolling Rr , compared with 4.7 for rolling Amihud and 1.1 for the Kyle-style minute-impact benchmark. The corresponding high/low next-day realized-variance ratios are 2.76, 1.68, and 1.04. This still falls short of a formal forecast horse race, but it addresses the narrow benchmark question directly: the queueing-organized liquidity monitor is positively related to feasible standard trade-based measures while not being redundant with either of them in the same day-ahead design. Order-book depth measures are not added in this comparison because the public Binance data used here contain trade prints and aggressor-side information, not synchronized order-book snapshots. A depth comparison would require a different data source and a different reconstruction pipeline. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 36 of 41 Table A6. Matched benchmark-sample comparison of rolling Rr with standard trade-based liquidity benchmarks. All predictors are measured at the end of UTC day d using trailing 30-day windows and are compared with the realized outcomes on day d + 1. Correlations with rolling Rr are reported in the text; the table concentrates on the day-ahead stress-sorting comparisons. High/Low Low-Tercile Tail High-Tercile Tail High-Minus-Low Predictor Next-Day Rate Rate Tail Spread (pp) Variance Ratio Rolling Rr 6.5% 15.1% 8.6 2.76 Rolling Amihud illiquidity 8.3% 13.0% 4.7 1.68 Kyle-style minute impact |κd | 10.1% 11.2% 1.1 1.04 Appendix C.9. Nearby-Frequency Robustness of the Feedback Splice proxy The headline proxy β eff combines a directional-impact scale from signed-flow regressions with a lagged signed-BTC-flow slope. Because those ingredients are estimated at different frequencies, two nearby alternatives are reported here. The first replaces the baseline 1 min directional scale with a 5 min directional scale computed by the same day-by-day procedure on 5 min buckets. The second replaces the baseline 1 h flow regression with a non-overlapping 2 h block regression and rescales the resulting cumulative signed-flow slope back to a per-hour feedback coefficient. Table A7. Nearby-frequency robustness for the proxy feedback coefficient. All three full-sample variants remain positive and stay within the same order of magnitude as the baseline estimate. β effˆ proxy Specification Directional Scale Flow Slope Input (h−1 ) Baseline splice 1 min κ̂dir = 1.9971 × 10−5 1 h b̂F = 7250.26 0.0724 5 min directional alternative 5 min κ̂dir = 1.8246 × 10−5 1 h b̂F = 7250.26 0.0661 2 h flow alternative 1 min κ̂dir = 1.9971 × 10−5 2 h block slope b̂F = 8358.10 0.0417 The 5 min directional alternative is 0.914 times the baseline value, while the 2 h flow alternative is 0.576 times the baseline. These checks do not transform the splice into a directly identified structural estimate, but they do show that the positive sign and rough magnitude of the proxy are not tied to a single exact pair of sampling choices. Appendix C.10. Secondary ρ Summary The ratio 4 κ̃ β β ρ≡ = eff , θc := θ/4, θ θc remains available as a compact symmetry-based summary. In the full sample, the contingent structural translation gives ρ = 7.52 × 10−1 and δ = 1 − ρ = 0.247721. Those quantities are kept in the appendix rather than the main text because they do not add proxy independent identifying information beyond the directly monitored pair ( β eff , θeff ). Near the regime boundary, small movements in θ̂eff can make ρ move sharply, which is why the main paper treats it as secondary. Appendix C.11. Risk Management Performance This appendix backtests rolling one-hour 99% VaR forecasts rather than a fixed insample threshold. The pure queueing/Gaussian benchmark implies an average VaR of 1.51% and yields a breach rate of 1.80%, materially above the 1% target. The stressstate mixture raises the average forecast to 1.84% and lowers the breach rate to 1.13%, while rolling historical simulation gives 1.87% and 1.07%. The main text therefore uses https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 37 of 41 the mixture only as a pragmatic parametric tail overlay: it improves materially on the Gaussian benchmark, but its Kupiec p-value of 0.008 still rejects exact unconditional 99% coverage. Historical simulation remains slightly closer to nominal unconditional coverage in this sample. Appendix C.12. Robustness Tests The appendix robustness notes are kept short because the main paper already reports the core reduced-form evidence. Two points matter most. First, the rolling estimates reported in the main paper show local variation rather than a single stable resiliency regime: a non-trivial minority of windows have θeff ≤ 0 and are better interpreted as local momentum or stress windows than as mean-reverting calibrations. Second, the auxiliary robustness material here is meant only to show that reasonable specification changes leave the paper’s descriptive hierarchy intact: the main evidence remains the observable liquidity, signed-flow, and resiliency diagnostics reported in the main text. Exchange-specific data requirement: The present implementation relies on Binance trade-print files together with aggressor-side information recoverable from the Binance schema (including isBuyerMaker). Extending the workflow to other venues would require rebuilding venue-specific trade-print and aggressor-side panels, so multi-venue replication is left to future work rather than asserted as cross-venue equivalence here. Appendix C.13. Limitations and Future Data Work A dedicated cross-venue robustness study would require harmonised fill-level data and aggressor-side reconstruction across heterogeneous APIs. That exercise is therefore left to future work. Appendix D. Coverage and Reconciliation of Public OHLCV Feeds Public OHLCV aggregates (e.g., Binance “klines” and widely mirrored third-party files) do not provide the trade-print and aggressor-side fields required for the present calibration. The narrower point is that the present manuscript calibrates on full trade-print data rather than on public OHLCV aggregates, because the queueing, signed-flow, and aggressor-side diagnostics require fields that OHLCV files do not retain. Appendix E. Estimation Details for the Core/Stress Mixture The estimator uses a threshold split rather than full EM-MLE. The threshold is τ = 2σemp , and the mixture weights are π̂0 = |S0 |/T and π̂1 = 1 − π̂0 . The estimation procedure is as follows: 1. Compute the empirical hourly return standard deviation σemp ; set the stress threshold τ = 2σemp . 2. Core sample S0 = {rt : |rt | ≤ τ } → fit N (µ0 , σ02 ) via maximum-likelihood estimation. 3. Tail sample S1 = {rt : |rt | > τ } → fit Student-t(ν1 , µ1 , σ12 ), same optimisation call. 4. Mixture weights π̂0 = |S0 |/T, π̂1 = 1 − π̂0 . −1 5. 99 % one-period VaR is VaR0.99 = − Fmix (0.01). 6. Back-test: 520 breaches out of 46,214 one-hour forecasts ⇒ unconditional coverage 1.13 %. The identical numerical values are reported in Appendix C.4.2 (π̂0 = 0.952, µ̂1 = −4.44 × 10−4 , σ̂1 = 0.02314, ν̂1 = 21.92). Appendix F. Monte Carlo Validation of the Variance-Scale Estimator The Monte Carlo experiment simulates 1000 independent M/G/∞ paths with il- √ lustrative parameters λ± = 820 h−1 , H̄ = 2 h and ( Rr )true = 1.2 × 10−4 (chosen for https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 38 of 41 computational tractability; the bias test is scale-invariant). In each path, the variance-scale p coefficient is estimated from the same sample ratio Var(r )/ E[ν] used in the main paper’s empirical-implementation subsection. Table A8 summarizes bias and dispersion. √ Table A8. Finite-sample performance of the variance-scale estimator d Rr (1000 replications). T (h) Bias SD BiasSE RMSE 250 −2.66 × 10−7 5.18 × 10−6 1.64 × 10−7 5.19 × 10−6 500 −5.00 × 10−8 3.67 × 10−6 1.16 × 10−7 3.67 × 10−6 750 −8.74 × 10−8 3.11 × 10−6 9.83 × 10−8 3.11 × 10−6 1000 −2.79 × 10−8 2.74 × 10−6 8.68 × 10−8 2.74 × 10−6 Bias is statistically indistinguishable from zero and the RMSE follows the O( T −1/2 ) benchmark, confirming practical unbiasedness for sample windows of a few weeks or longer (see Figure A3). Convergence of Estimator: SD vs Sample Size MC simulation 12.1 OLS: slope = −0.46, R = 1.00 T=250 12.2 12.3 12.4 ln SD( ) 12.5 T=500 12.6 12.7 T=750 12.8 T=1000 12.9 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 T ln( ) Figure A3. Finite-sample precision of the variance-scale estimator. Natural-log plot of sample length √ d obtained from 1000 Monte Carlo replications (blue T (hours) versus the standard deviation  of√ Rr  dots). The red line is the OLS fit: ln SD ( dRr ) = −0.46 ln( T ) − 9.65, r2 = 1.00, confirming the expected SD ∝ T −1/2 rate. Appendix G. Occupancy of a Discrete-Time GI/Geo/∞ Queue Let ∆ > 0 be the time tick. In tick n ∈ Z≥0 the numbers of new long- and short-side latent occupancy units are independent An,+ ∼ Poisson(λ+ ∆), An,− ∼ Poisson(λ− ∆). An existing long-side occupancy unit survives to the next tick with probability p+ = 1 − ∆/E[ H+ ]; a short-side occupancy unit survives with p− = 1 − ∆/E[ H− ]. Let Bn and Sn denote the outstanding long- and short-side occupancies at the start of tick n and define Xn = Bn − Sn . Proposition A1. In steady state B∞ ∼ Poisson(m+ ), S∞ ∼ Poisson(m− ), m+ = λ+ E[ H+ ], m− = λ− E[ H− ]. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 39 of 41 Moreover, B∞ and S∞ are independent. Consequently, X∞ = B∞ − S∞ ∼ Skellam(m+ , m− ). Proof. Consider the long-side queue; the short-side queue follows identically. Given Bn , the number of long-side occupancy units that survive to the next tick is Binomial( Bn , p+ ). Thus B Bn+1 = Cn + An,+ , E[zCn | Bn ] = 1 − p+ + p+ z n .  With Gn (z) = E[z Bn ], independence of An,+ gives the recursion Gn+1 (z) = exp λ+ ∆ (z −   1) Gn 1 − p+ + p+ z . A stationary solution must satisfy G (z) = exp[λ+ ∆(z − 1)] G (1 − p+ + p+ z). The unique pgf with G (1) = 1 is G (z) = exp[m+ (z − 1)] provided m+ = λ+ ∆/(1 − p+ ) = λ+ E[ H+ ], establishing the Poisson law. Independence of B∞ and S∞ follows from independent arrivals and services together with the PASTA property of Poisson processes (Wolff, 1982). The Skellam result is the difference of two independent Poisson variates. Appendix H. Piecewise Linear Depth and Price Moments Partition the inventory space into three symmetric regions K 1 = { k ∈ Z : | k | < K1 } , K 2 = { k : K1 ≤ | k | < K2 } , K 3 = { k : | k | ≥ K2 } , and define the impact schedule   k ∈ K1 , κ1 ,  κ (k) = κ2 , k ∈ K2 , 0 < κ1 < κ2 < κ3 .    κ3 , k ∈ K3 , Let X∞ ∼ Skellam(m+ , m− ) and define P∞ = P0 + κ ( X∞ ) X∞ . Proposition A2. The first two unconditional moments of P∞ are 3 E[ P∞ ] = P0 + ∑ κr µr , µr = ∑ k Pr( X∞ = k ), (A19) r =1 k∈Kr 3  2 Var[ P∞ ] = ∑ κr2 σr2 − E[ P∞ ] − P0 , σr2 = ∑ k2 Pr( X∞ = k). (A20) r =1 k ∈Kr Each truncated sum involves a finite number of Skellam probabilities  k/2 √  m+ Pr( X∞ = k) = e−(m+ +m− ) m− I|k| 2 m+ m− . Proof. Write Y = κ ( X∞ ) X∞ = ∑3r=1 κr X∞ 1{ X∞ ∈Kr } . Linearity of expectation yields (A19).  2  For the second moment, E[Y 2 ] = ∑r κr2 E X∞ 1Kr because the regions are disjoint. Subtracting the square of the mean gives (A20). The explicit Skellam form follows by inserting the pmf. Derivation of the Count-Space Burstiness Approximation The main text’s count-space burstiness formula is an approximation for the excess kurtosis of the hourly increment ∆Xt = Nt+ − Nt− . This subsection first records the exact https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 40 of 41 common-rate Gamma–Poisson cumulant ratio and then shows the simpler proxy reported in the main text. Let Nt± | Λ± ∼ Poisson(Λ± t), Λ± ∼ Gamma(α± , ϑ ), where the buy and sell counts may have different shape parameters α± but share a common Gamma rate ϑ, matching the main text’s Negative-Binomial parameterization p± = ϑ± /(ϑ± + t). Write φ := t/ϑ, so n̄± = E[ Nt± ] = α± φ. Under this single shape–rate parameterization, the cumulant generating function of Nt± is  K± (u) = −α± log 1 − φ(eu − 1) . For the difference ∆Xt = Nt+ − Nt− , K∆X (u) = K+ (u) + K− (−u). Because odd cumulants change sign under u 7→ −u while even cumulants do not, the second and fourth cumulants of ∆Xt are just the sums of the side-specific even cumulants: κ2 (∆Xt ) = (α+ + α− ) φ(1 + φ), (A21) 2  κ4 (∆Xt ) = (α+ + α− ) φ(1 + φ) 1 + 6φ + 6φ . (A22) Writing n̄Σ := n̄+ + n̄− = (α+ + α− )φ, the exact common-rate excess kurtosis becomes κ4 (∆Xt ) 1 + 6φ + 6φ2 ExKurt(∆Xt ) = = (A23) κ2 (∆Xt )2 ( α + + α − ) φ (1 + φ ) 1 6 = + . (A24) n̄Σ (1 + φ) α+ + α− The main text uses the simpler proxy 1 6 ExKurt(∆Xt ) ≈ + , n̄Σ α+ + α− which preserves the familiar Poisson benchmark term 1/n̄Σ plus the over-dispersion term 6/(α+ + α− ). The approximation is therefore not an exact common-rate identity; it replaces the smaller exact remainder 1/[n̄Σ (1 + φ)] by the Poisson benchmark 1/n̄Σ to keep the decomposition transparent. In the paper, this is used only as a burstiness proxy for hourly buy/sell trade counts. It is not presented as a fully general return-law derivation. Appendix I. NB–Skellam Distribution When long- and short-side occupancy counts follow Negative-Binomial distributions (arising from Gamma-mixed Poisson processes), their difference follows a NB–Skellam distribution. Note on empirical relevance. With the empirical dispersion parameters α+ = 0.4922 and α− = 0.6423 (Gamma–Poisson mixing), the NB–Skellam does not collapse to the stan- √ dard Skellam. The arrival-rate coefficients of variation are 1/ α± ≈ 1.43 and 1.25, and the corresponding burstiness proxy contributes about 5.3 to the excess-kurtosis decomposition under the approximation above, both materially different from the Poisson–Skellam benchmark. In the Binance data, however, NB–Skellam still understates the extreme tails in hourly returns, which is why this count-space extension is retained only as a supplementary burstiness note alongside the separate stress overlay. https://doi.org/10.3390/jrfm19050372 J. Risk Financial Manag. 2026, 19, 372 41 of 41 Notes 1 The modified Bessel function of the first kind arises naturally in circular statistics and here characterizes the difference of Poisson variates. 2 “Non-missing” means windows for which the rolling return autocorrelation and signed-flow regression are both computable once the observed missing UTC hours are carried through the hourly grid. The count therefore differs slightly from the purely mechanical number of overlapping 30-day calendar windows. The sample contains 34 missing UTC hours overall, so a small number of windows lose one or more hourly observations even though the rolling clock continues to advance. 3 ∆ = 60 s keeps the Poisson means in the low teens, striking a good balance between numerical accuracy and computational speed. 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