The Queue-Reactive Model

Early book simulators assumed order flow arrives at a fixed rate no matter what the book looks like — a constant $\lambda$ orders per second, always. Real books don't behave that way: a queue with 3,000 shares resting attracts and loses orders at very different rates than one with 30. The queue-reactive model builds that dependency in directly: every rate — new limit orders, cancellations, market orders — is a function $f(Q)$ of the current size of the queue it acts on, estimated straight from data rather than assumed.

How it's fit and used, worked through

Take historical data for the best-bid queue and bucket every observed state by its size $Q$. For each bucket, measure the empirical rate of each event type. Suppose the fitted rates come out roughly as:

Queue size $Q$	Arrival rate $\lambda(Q)$	Cancel rate $\mu(Q)$
100	6/s	0.5/s
1,000	9/s	3/s
5,000	10/s	9/s

Two things jump out. Arrivals rise modestly with queue size — a deep queue signals a popular, stable price, which itself attracts more orders. Cancellations rise much faster — more resting orders simply means more of them get pulled. At $Q=5{,}000$ the cancel rate has nearly caught up to the arrival rate, which is exactly the self-correcting behaviour real books show: very large queues don't grow without bound, because their own cancellation pressure pushes back.

To simulate forward from $Q = 1{,}000$: draw the next event from a mix weighted by $\lambda(1000)=9$, $\mu(1000)=3$, and the market-order execution rate at that state, then update $Q$ accordingly and re-look-up the rates for the new $Q$ before drawing the next event. The queue's own trajectory continuously changes the rates governing it — this feedback loop is the entire point of the model, and it's what's missing from a constant-rate simulator.

Where it's used. It's the workhorse for realistic order-book simulation: backtesting an execution or market-making strategy against a queue-reactive replay produces far more believable fill probabilities and impact than a Poisson-constant-rate model, because the simulated book actually reacts to being traded against, the same way a real one does. It's a direct input to Agent-Based Market Simulation and to fill-probability estimation for resting limit orders.