Limit order books (LOB) are at the core of electronic financial markets. A LOB centralizes all orders of all market participants on a given exchange, matching buy and sell orders of all types.

In a first part, we observe that a LOB is a queueing system and that this analogy is fruitful to derive stationary properties of these structures. Using a basic Poisson model, we compute analytical formulas for the average shape of the LOB. Our model allows for non-unit size of limit orders, leading to new predictions on the granularity of financial markets that turn out to be empirically valid.

In a second part, we study the LOB during the call auction, a market design often used during the opening and closing phases of the trading day. We show that in a basic Poisson model of the call auction, the distributions for the traded volume and the range of clearing prices are analytically computable. In the case of a liquid market, we derive weak limits of these distributions and test them empirically.