[WITH CODE] Model: Queue reactive framework
Understand the dynamics of a limit order book by treating the arrivals and cancellations of orders
Table of contents:
Introduction.
What is the anatomy of the order book?
The queue reactive model.
Random arrivals and Poisson processes.
State-dependent intensities.
Incorporating queue size effects and neighboring levels.
Modeling the time until the next event.
Extention for the QR model.
Inter-queue interactions.
State-dependent intensities.
Feedback loops.
Introduction
Picture a digital bulletin board where buyers and sellers post orders. Buyers say: I’ll pay X and sellers say: I want Y. When X and Y match, boOM!—trade happens. This board is the limit order book or LOB. It is like Tinder for stocks. Swipe right (buy) or left (sell), but everyone sees your moves.
This analogy is a little bit weird, so let's be more technical. The LOB is the engine room of modern electronic markets. It’s where all the action happens: buyers and sellers place orders, and the market continuously updates the state of play as orders are added, canceled, or executed. At its most basic level, the LOB is a ledger that contains a list of buy orders—bids—and sell orders—asks. Here’s what you need to know:
Bid: The highest price a buyer is willing to pay for an asset.
Ask: The lowest price a seller is willing to accept.
Spread: The difference between the best bid and the best ask. This gap is often referred to as the “no man’s land” where prices hover until someone bridges it.
Queue: At each price level, orders are lined up in a “first-in, first-out” (FIFO) fashion. That’s why the timing of your order placement can be as important as the price you set.
If you want more information on this topic, be sure to visit this post:
What is the anatomy of the order book?
Think of the order book like a busy restaurant’s waiting list. When you call in a reservation—place a limit order—you get in line with everyone else. The sooner you call, the better your chances of being seated at the best table—getting executed at your desired price.
The LOB is the real-time ledger of buy and sell orders in an electronic market. It isn’t just a static list; it’s a living, breathing entity that reflects market sentiment and liquidity at every moment. Here are some key points:
Price levels and depth: Each price level has a queue of orders waiting to be executed. The depth refers to the volume available at each price. A deep book means plenty of orders—liquidity—while a shallow book can lead to larger price jumps when orders are executed.
Bid, ask, and spread:
Bid: The highest price buyers are willing to pay.
Ask: The lowest price sellers will accept.
Spread: The gap between the best bid and best ask. A narrow spread is usually a sign of high liquidity.
Order priority or FIFO:
Orders placed at the same price are typically executed on a first-in, first-out basis. The timing of your order placement matters—a bit like arriving early for a popular concert!
Okay, first-in, first-out… easy, right? Well, not so fast. It seems simple, but in simplicity lies complexity. There's actually a lot more to it, but for the sake of this article, let's focus on what we've been discussing for now and what's represented in this image:
Let's pause for a moment on this point because it will determine your trading approach. FIFO is the default method used by brokers for managing trades, particularly when selling shares or assets. Here's how it works and the role of share quantities:
Standard approach:
Brokers typically sell the oldest purchased lots first—earliest shares acquired are sold first—unless the trader explicitly specifies otherwise—e.g., selecting specific lots.
This applies regardless of the number of shares traded. For example, if you bought 100 shares in January and 100 shares in March, selling 50 shares would default to the January lot under FIFO.
Impact of share quantities :
The amount of shares bought (e.g., multiple lots at different times/prices) influences which lots are prioritized under FIFO, but the method itself remains unchanged.
Larger sell orders involve selling shares from multiple older lots sequentially until the total quantity is met.
Exceptions and alternatives :
Traders can override FIFO by specifying lots—e.g., selecting newer shares first for tax optimization—but brokers must default to FIFO unless explicit instructions are given.
LIFO or Last In, First Out is rarely used in stock trading but may apply to inventory accounting for businesse.
Now scale this to millions of traders. Orders arrive en masse, one after the other. They seem random, but there's a certain dependency that generates loops and more loops, as we'll see below. What do they consist of?
Queue reactive model
The QR model helps us understand the dynamics of a limit order book by treating the arrivals and cancellations of orders as state-dependent Poisson processes. In simpler terms, the model lets the rate at which orders arrive or get canceled depend on the current state of the queue (and even its neighbors), rather than assuming a constant rate.