  This paper focuses on computing the fill probabilities for limit orders
positioned at various price levels within the limit order book, which play a
crucial role in optimizing executions. We adopt a generic stochastic model to
capture the dynamics of the order book as a series of queueing systems. This
generic model is state-dependent and also incorporates stylized factors. We
subsequently derive semi-analytical expressions to compute the relevant
probabilities within the context of state-dependent stochastic order flows.
These probabilities cover various scenarios, including the probability of a
change in the mid-price, the fill probabilities of orders posted at the best
quotes, and those posted at a price level deeper than the best quotes in the
book, before the opposite best quote moves. These expressions can be further
generalized to accommodate orders posted even deeper in the order book,
although the associated probabilities are typically very small in such cases.
Lastly, we conduct extensive numerical experiments using real order book data
from the foreign exchange spot market. Our findings suggest that the model is
tractable and possesses the capability to effectively capture the dynamics of
the limit order book. Moreover, the derived formulas and numerical methods
demonstrate reasonably good accuracy in estimating the fill probabilities.
