Double-Sided Batch Queues with Abandonment: Modeling Crossing Networks

We study a double-sided queue with batch arrivals and abandonment. There are two types of customers, patient ones who queue but may later abandon, and impatient ones who depart immediately if their order is not filled. The system matches units from opposite sides of the queue based on a first-come first-served policy. The model is particularly applicable to a class of alternative trading systems called crossing networks that are increasingly important in the operation of modern financial markets. We characterize, in closed form, the steady-state queue length distribution and the system-level average system time and fill rate. These appear to be the first closed-form results for a double-sided queuing model with batch arrivals and abandonment. For a customer who arrives to the system in steady state, we derive formulae for the expected fill rate and system time as a function of her order size and deadline. We compare these system-and customer-level results for our model that captures abandonment in aggregate, to simulation results for a system in which customers abandon after some random deadline. We find close correspondence between the predicted performance based on our analytical results and the performance observed in the simulation. Our model is particularly accurate in approximating the performance in systems with low fill rates, which are representative of crossing networks.