Queue & Fill Probability

A limit order joins the back of its price level's queue. It fills only when everything in front of it has traded or been cancelled, or when price trades straight through the level. This view estimates that for every visible level at five horizons, and shows how far your own orders have moved up their queues. Open it with View → Queue & fill probability.

A public L2 feed has no order IDs, so queue position cannot be read directly. It is modelled from what the feed does show: each level's size on every update, and the trade tape.

1. 1
   Attribute prints

   A buy aggressor trades against the ask queue, a sell aggressor against the bid queue. Prints are folded onto the book's price buckets with the book's own rounding.
2. 2
   Split size changes

   Between updates a level changes by adds − cancels − trades. The tape gives the trades, so any shrink beyond them was cancelled. That gives a cancel hazard per distance rank: the share of the queue leaving per second.
3. 3
   Learn rates

   Per side: the passive trade rate λ and its burstiness (Σ size² per second). For the mid: its path, sampled every 250ms over 10 minutes. Rates use a 30-second half-life, so they adapt within a minute.
4. 4
   Move virtual orders

   Prints consume the queue from the front. Cancels are spread through the queue, so only the share in front of an order moves it forward. Size added later joins behind it.
5. 5
   Estimate

   Combine queue depletion with the chance price reaches or trades through the level (below).
6. 6
   Self-check

   Score every probe at 15 seconds against what actually happened.

With A₀ in front of you, a passive trade rate λ and a cancel hazard c, the queue ahead decays as dA/dt = −λ − c·A, so the expected volume gone from the front by τ is M(τ) = (A₀ + λ/c)(1 − e^(−cτ)). Trades arrive in bursts, and each burst's effect is itself eroded by later cancels, giving variance V(τ) = s²(1 − e^(−2cτ))/2c. The chance the queue in front of you is gone by τ is Φ(−A(τ)/√V).

const remaining = ahead - (ahead + lambda / c) * (1 - Math.exp(-c * tau));
const variance  = s2 * (1 - Math.exp(-2 * c * tau)) / (2 * c);
const floor     = 0.3 * ahead * Math.exp(-c * tau);          // model uncertainty
const pQueue    = normCdf(-remaining / Math.sqrt(variance + floor * floor));

// Touch: the queue clears, or price trades through the level.
pFill = 1 - (1 - pQueue) * (1 - reach(halfSpread + step / 2, tau));

// A level d from the touch: price must come to it, then a queue thinned by
// cancels must clear in the remaining time — or price trades through it.
pFill = 1 - (1 - reach(d, tau) * pQueue(ahead * Math.exp(-c * tau / 2), tau / 2))
          * (1 - reach(d + halfSpread + step / 2, tau));

Reach is the chance the mid moves a given distance toward the order within τ. It comes from the market's own path over the last 10 minutes: for every start point and horizon the largest move up and down is recorded, so trending and mean-reverting markets give asymmetric, realistic odds. Excursions missed between samples are corrected for discrete monitoring (Broadie–Glasserman). Until there is enough history, the Brownian reflection principle 2(1 − Φ(d/σ√τ)) stands in.

A public L2 feed has no order IDs, so queue position cannot be read directly. It is modelled from what the feed does show: each level's size on every update, and the trade tape.

1. 1
   Attribute prints

   A buy aggressor trades against the ask queue, a sell aggressor against the bid queue. Prints are folded onto the book's price buckets with the book's own rounding.
2. 2
   Split size changes

   Between updates a level changes by adds − cancels − trades. The tape gives the trades, so any shrink beyond them was cancelled. That gives a cancel hazard per distance rank: the share of the queue leaving per second.
3. 3
   Learn rates

   Per side: the passive trade rate λ and its burstiness (Σ size² per second). For the mid: its path, sampled every 250ms over 10 minutes. Rates use a 30-second half-life, so they adapt within a minute.
4. 4
   Move virtual orders

   Prints consume the queue from the front. Cancels are spread through the queue, so only the share in front of an order moves it forward. Size added later joins behind it.
5. 5
   Estimate

   Combine queue depletion with the chance price reaches or trades through the level (below).
6. 6
   Self-check

   Score every probe at 15 seconds against what actually happened.

With A₀ in front of you, a passive trade rate λ and a cancel hazard c, the queue ahead decays as dA/dt = −λ − c·A, so the expected volume gone from the front by τ is M(τ) = (A₀ + λ/c)(1 − e^(−cτ)). Trades arrive in bursts, and each burst's effect is itself eroded by later cancels, giving variance V(τ) = s²(1 − e^(−2cτ))/2c. The chance the queue in front of you is gone by τ is Φ(−A(τ)/√V).

const remaining = ahead - (ahead + lambda / c) * (1 - Math.exp(-c * tau));
const variance  = s2 * (1 - Math.exp(-2 * c * tau)) / (2 * c);
const floor     = 0.3 * ahead * Math.exp(-c * tau);          // model uncertainty
const pQueue    = normCdf(-remaining / Math.sqrt(variance + floor * floor));

// Touch: the queue clears, or price trades through the level.
pFill = 1 - (1 - pQueue) * (1 - reach(halfSpread + step / 2, tau));

// A level d from the touch: price must come to it, then a queue thinned by
// cancels must clear in the remaining time — or price trades through it.
pFill = 1 - (1 - reach(d, tau) * pQueue(ahead * Math.exp(-c * tau / 2), tau / 2))
          * (1 - reach(d + halfSpread + step / 2, tau));

Reach is the chance the mid moves a given distance toward the order within τ. Two estimates are blended:

• Empirical: the market's own mid path over the last 10 minutes. For every start point and horizon the largest move up and down is recorded, so trending and mean-reverting markets give asymmetric, realistic odds. Moves missed between 250ms samples are corrected for discrete monitoring (Broadie–Glasserman).
• Brownian: the reflection principle 2(1 − Φ(d/σ√τ)), with σ measured over 5-second windows so 100ms bid/ask bounce does not inflate it.
• Weighting: by how many independent τ-windows the history holds, n = history ÷ τ, as n ÷ (n + 8). A 15s estimate from 30s of history rests on two windows and leans on the Brownian one; after 10 minutes it is 40 windows and mostly empirical. Long horizons stay more Brownian for longer.

The scorecard shows the skill score: 1 − Brier ÷ (base rate × (1 − base rate)). Above zero the model knows more than the overall fill rate alone; at or below zero it does not. The raw Brier score is shown as well, but its meaning depends on the base rate.

• Without order IDs, queue position is an estimate. Hidden or iceberg size and the exchange's matching rules are not visible in L2.
• Binance depth arrives in 100ms batches; faster queue dynamics are invisible.
• Your real orders are assumed to be at the back of their level when first seen. If you placed one long before opening the view, it may be further forward than shown.
• Everything resets when the symbol or price grouping changes: rates and queue positions do not carry across buckets.
• The model warms up for 45 seconds. Probes start only after that, so early guesses are never scored.